Fea

What Is Fea?? Finite element analysis was first developed for use in the aerospace and nuclear industries where the safety of structures is critical. Today, the growth in usage of the method is directly attributable to the rapid advances in computer technology in recent years. As a result, commercial finite element packages exist that are capable of solving the most sophisticated problems, not just in structural analysis, but for a wide range of phenomena such as steady state and dynamic temperature distributions, fluid flow and manufacturing processes such as injection molding and metal forming. FEA consists of a computer model of a material or design that is loaded and analyzed for specific results. It is used in new product design, and existing product refinement. A company is able to verify that a proposed design will be able to perform to the client's specifications prior to manufacturing or construction. Modifying an existing product or structure is utilised to qualify the product or structure for a new service condition. In case of structural failure, FEA may be used to help determine the design modifications to meet the new condition. Mathematically, the structure to be analyzed is subdivided into a mesh of finite sized elements of simple shape. Within each element, the variation of displacement is assumed to be determined by simple polynomial shape functions and nodal displacements. Equations for the strains and stresses are developed in terms of the unknown nodal displacements. From this, the equations of equilibrium are assembled in a matrix form which can be easily be programmed and solved on a computer. After applying the appropriate boundary conditions, the nodal displacements are found by solving the matrix stiffness equation. Once the nodal displacements are known, element stresses and strains can be calculated Within each of these modeling schemes, the programmer can insert numerous algorithms (functions) which may make the system behave linearly or non-linearly. Linear systems are far less complex and generally ignore many subtleties of model loading & behaviour. Non-linear systems can account for more realistic behaviour such as plastic deformation, changing loads etc. and is capable of testing a component all the way to failure. Despite the proliferation and power of commercial software packages available, it is essential to have an understanding of the technique & physical processes involved in the analysis. Only then can an appropriate & accurate analysis model be selected, correctly defined and subsequently interpreted.

Stress & Displacement Analyses Introduction The most common application of FEA is the solution of stress related design problems. As a result, all commercial packages have an extensive range of stress analysis capabilities. What is Stress ?? Stress can be described as a measurement of intensity of force. As all engineers know, if this intensity increases beyond a limit known as yield, the component's material will undergo a permanent change in shape or may even be subjected a to dramatic failure. From a formal point of view, three conditions have to be met in any stress analysis, equilibrium of forces (or stresses), compatibility of displacements and satisfaction of the state of stress at continuum boundaries. These conditions, which are usually described mathematically in good undergraduate strength of material texts, are also applicable to non-linear analyses. How the Result is Achieved •

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It all starts off with the formulation of the components 'stiffness' matrix. This square matrix is formed from details of the material properties, the model geometry & any assumptions of the stress-strain field (plane stress or strain). Once the stiffness matrix is created, it may be used with the knowledge of the forces to evaluate the displacements of the structure (hence the term displacement analysis). On evaluation of the displacements, they are differentiated to give six strain distributions, 3 mutually perpendicular direct strains & 3 corresponding shear strains. Finally six stress distributions are determined via the stress/strain relationships of the material. Commercial packages usually go one further & calculate a range of more usuable stress fields from the six stress components such as the principal stresses & a host of failure prediction stressess as described by the most common yield criteria (Von Mises/Maxwell/Heckney, Guest/Tresca, Heubner/Thornton, etc.). The displacements can be used in conjunction with the element stiffnesses to determine the reaction forces & the forces internal to each element (otherwise known as the stress resultants).

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A point to note is that at least one of the displacements must be known before the rest can be determined (before the system of equations can be solved). These known displacements are referred to as boundary conditions and are oftentimes a zero value. Without these boundary conditions, we would get the familiar singularity or zero-pivot error message from the solver, indicating that no unique solution was obtainable.

An alternative solution An alternative solution may be obtained via the force matrix method (otherwise known as the flexibility method). In the previous description, the displacements were the unkown, and solution is said to be obtained via the stiffness method. In the force method, the forces are the nodal unknowns, while the displacements are known. The solution is obtained for the unknown forces via the flexibility matrix & the known displacements. The stiffness method is more powerful & applicable than the flexibility approach. Non-Linear Analyses In order to explain non-linearity in stress analyses, lets examine the nature of linear solutions. Many assumptions are made in linear analyses, the two primary ones being the stress/strain relationship & the deformation behaviour. The stress is assumed to be directly proportional to strain and the structure deformations are proportional to the loads. The second assumption is oftentimes mistaken to derive from the first, a fishing rod is an example of a non-linear structure made of linear material. A stress analysis problem is linear only if all conditions of proportionality hold. If any one of them is violated, then we have a Non-Linear problem. Most real life structures, especially plastics, are non-linear, perhaps both in structure and in material. Most plastic materials have a non-linear stress strain relationship. The non-linearity arising from the nature of material is called 'Material Non-linearity'. Furthermore, thin walled plastic structures exhibit a nonlinear load-deflection relationship, which could arise even if the material were linear (fishing rod). This kind is called geometric non-linearity. All non-linearities are solved by applying the load slowly (dividing it into a number of small loads increments). The model is assumed to behave linearly for each load increment, and the change in model shape is calculated at each increment. Stresses are updated from increment to increment, until the full applied load is reached. In a nonlinear analysis, initial conditions at the start of each increment is the state

of the model at the end of the previous one. This dependency provides a convenient method for following complex loading histories, such as a manufacturing process. At each increment, the solver iterates for equilibrium using a numerical technique such as the Newton-Raphson method. Due to the iterative nature of the calculations, non-linear FEA is computationally expensive, but reflects the real life conditions more accurately than linear analyses. The big challenge is to provide a convergent solution at minimum cost (the minimum number of increments). See the 'non-linear' section of solution types for more details of such analyses & how solutions are achieved. For details on applying loads & boundary conditions, see the 'improving results' and 'faster analyses' sections on the menu.

Vibration: Resonance & Mode Shapes What is Vibration ?? Vibration usually becomes a concern when it's amplitudes grow large enough to cause either excessive stress, or if it disturbs the people in, on or near the vibrating object(s). As far as most structures are concerned, vibration will disturb the people around the structure long before stress becomes an issue. There are many items of equipment (balances, microscopes, cameras, transmission equipment etc.) that are very sensitive to vibration. Modal analyses are important in machines where there is likely to be cyclic out of balance forces, such as in rotating machinery (engines, electric & penumatic motors, generators, industrial equipment, etc.) and fluid flow applications (due to alternating vortex shedding). The chief aim of any vibration analysis is to ensure that the system is not subject to a dangerous resonant condition during the range of operation. A point to note is that although the response of the system is time dependant, any excitation will be harmonic, and the solution may be obtained using the eigenvalue approach. It is important to note that many applications fall in a category beyond this range, and full dynamic analyses are required. A More Formal Approach If a system is given some initial disturbance, then it will vibrate at some frequency known as it's natural frequency. The natural frequency of a system is defined as the frequency at which the system oscillates if the forcing function is identically zero. If harmonic loading is applied, the solution becomes transient in nature, but modal analyses can still be carried out for systems.

You may recall from elementary vibrations lectures that the square of the natural frequency is referred to as an eigenvalue. For a single mass-spring system, there is one eigenvalue, for distributed mass systems (all practical applications), an infinite number of eigenvalues exist. The lowest natural frequency, usually referred to as the fundamental frequency, has the lowest potential or strain energy, and hence the reason why it is often regarded as the 'lazy mode'. The fundamental frequency is usually the one of most interest to design engineers, as most systems are designed to operate below it. Oftentimes, an operating frequency is higher than the fundamental, hence as the equipment speeds up or slows down, it experiences a momentary 'shudder' period as it passes through the resonance zone. There is a corresponding mode shape which describes the displacement of the system due to the vibration. Eignevalues are otherwise known as latent roots and characteristic values, the square root of the eigenvalue is known as a natural frequency or resonant frequency. There is also a number of terms used to describe mode shapes, they are also known as eigenvectors, normal modes, characteristic vectors or latent vectors. The first five modes of vibration for an aerofoil are given below:

mode 1

mode 2

mode 3

mode 4

mode 5

Design Applications of Frequency Analysis In many practical problems the natural frequencies and mode shapes are all that are required. Designers use modal analyses to determine if there are any natural frequencies within the range of operation. Alternatively, measured mode shapes and natural frequencies of a structure can be compared with those predicted by FEA in a condition monitoring program to verify structural integrity. There are also situations where the response of the structure to a particular forcing excitation is required. This is usually found using a technique known as modal superposition. The overall response is described in terms of a sum of modal responses, with the contribution of a particular mode given by the proximity of the forcing frequency to the natural frequency and the amount of damping present in the system. The response is dominated by modes close to the excitation frequency and therefore the modal series is often truncated to reduce computation. Modal superposition methods can only be applied in applications with a harmonic excitation, otherwise the response becomes non-linear & cannot be solved using the eigenvalue extraction approach. The results from a forced harmonic analysis can be used to determine whether the displacement of a particular structure is within acceptable limits. By calculating

the stress induced by the vibration it is also possible to predict the fatigue life of a particular component. How Eigenvalues are Extracted Having discretised the component (continuum) into elements and described the variation of the displacement within each element, the kinetic and potential energy of the structure are the calculated to find the natural frequencies and mode shapes in terms of various nodal values. Numerically this equates to solving an eigenvalue problem expressed in terms of ‘mass’ and ‘stiffness’ matrices. You will remember from elementary engineering mathematics that once the eigenvalues are known, the eigenvectors may be evaluated. There are a large number of ways to determine eigenvalues & eigenvectors, the best choice depends on the form of the equations being solved. The main methods are the power, subspace, LR, QR, Givens, Householder & Lanczos methods. Each method is usually well documented in advanced Engineering Math texts, except for Lanczos, which is relatively new. There is a very useful check to evaluate if the eigenvalues have been extracted successfully. This is known as the Sturm sequence check. This is a method where the number of eigenvalues below a certain value can be evaluated, and is useful for finding if there are a large number of low frequency secondary components modes that would result in a long analysis time. It is also useful for indicating blunders with units (mass, length, etc), and can be used with subspace iteration & Lanczos methods as a cut off point (i.e. only extract the first five natural frequencies). When Modal Analysis is Appropriate A structure deforms as load is applied. If the load is cyclic, but with a much longer period than that of the fundamental frequency, then it is unlikely that the input will excite a resonance condition. A good rule of thumb is to implement a modal analysis if the forcing frequency is more than one-third the structure's fundamental frequency. However, if the excitation is random or applied suddenly, the problem becomes non-linear & eigenvalue based analyses will not provide correct results, hence a full dynamic analysis is required. Natural Frequency & Modal-Dynamic Differences Frequency based analyses perform eigenvalue extraction to calculate the natural frequencies and corresponding mode shapes of a 'free system' (i.e. with no time dependant loads applied). Modal-dynamic analyses are transient in nature. They give the response for the

model as a function of time where a cyclic (sinusoidal) load is applied to the structure. Modal-dynamic analyses is also referred to as forced harmonic response analysis. Complex displacements and phase angles are evaluated and deflections & stresses may be calculated at specific times. This analysis type is formulated on the principle of modal superposition, and so a natural frequency analysis must be carried out first. The modal amplitudes are integrated through time & the response is subsequently evaluated. This analysis solution must be linear in nature (in time domain), as superposition & eigenvalue extraction techniques cannot be applied to non-linear time domain applications. Boundary Conditions Theoretically, no boundary conditions are required. However, I do advise to apply at least one to a model, as it will help the solver overcome any potential problems due to rigid body motion (otherwise known as ill-conditioning). Just fix any point on the model with a zero displacement, and the model will always solve. If you intend applying boundary conditions, note that the results of eigenvalue analyses are very sensitive to the way in which BC's are applied. Therefore, always try to replicate the physical boundary conditions as closely as possible. If you cannot achieve this for some reason, carry out a sensitivity analysis by modifying the BCs slightly & comparing the results with your previous output. This will give you an indication of the influence the BCs have on the results. Avoid Applying Symmetric Boundary Conditions In all eigenvalue problems (buckling or vibration), symmetry of geometry, material properties, loading & boundary conditions does not guarantee symmetry of displacements. Therefore, it is advisable to try to implement a full 3D analysis where possible & only use symmetrically idealised models with great care. Since eigenvalue problems are less sensitive to mesh density that other analysis types, coarse meshes can be used for the 3D model, so long as it is graded & refined towards load & BC fixing points. You shouldn't use overly coarse meshes though, as it will result in a stiffer structure with resulting higher modes of vibration than is actually the case.

Time Dependant Dynamic Analyses: Modelling Impulse Problems Introduction

If the excitation applied to a structure is impulsive rather than harmonic, many modes contribute to the response and it becomes more appropriate to use direct integration methods rather than modal analysis. There are a large number of applications where transient analyses are necessary. Many structures are subject to time varying loads such as impluse, blast, impact & seismic loadings. Transient dynamic analysis determines the time-response history of a structure subjected to a forced displacement function. The structure may behave linearly, or in some cases, friction, plasticity, large deflections or gaps may produce nonlinear behavior. Once the time response history is known, complete deflection and stress information can be obtained for specific times. The first step in any dynamic analysis should be the determination of the frequencies and shapes of the natural vibration modes. In a 3-D structure there are three dynamic degrees of freedom (DDOF) for every unrestrained node with nonzero mass and there is potentially a natural vibration mode for each DDOF. Thus, there are usually many potential vibration modes in a typical structure, but usually only a small number of vibration modes with the lowest frequencies that are of interest. In a multi-storey building, for example, it might be only a few in each of two horizontal directions, plus one or two torsional modes that have to be considered. Frequency & Transient Analysis Differences While frequency analyses take place in the frequency domain, transient analyses are studies in the time domain. It is always possible to go from the time domain to the frequency domain via a fourier transform. Correspondingly, a change from the frequency domain to the time domain may be achieved by implementing an inverse fourier transform. Due to mathematical difficulties, solutions in the frequency domain can only be linear in nature. Therefore if the application requires a solution that is a non-linear function of time, then a time domain analysis must be carried out. The solution can subsequently be projected to the frequency domain if required. Frequency analyses can be solved using no boundary conditions, while transient analyses must be fully constrained. Transient Solutions: Modal & Direct There are usually two approaches one can take when carrying out a transient analysis, modal solutions or purely direct solutions. The modal approach involves evaluating the relevant natural frequencies of a structure first. Once this is carried out, the response is converted to the time domain and is included in the evaluation of transient response of the structure. Modal analyses are usually used where there many natural frequencies within the operation range. It is important to evaluate the natural frequencies above and

below that of the analysis range. This is due to the fact that, in practice, there is never just one distinct mode of vibration due to an excitation, but one dominant mode with a range of additional harmonics from the adjacent upper and lower modes. Direct solutions are used to evaluate the response of a structure within a very narrow frequency range of interest, and are usually used for models that subject to high frequency impulses. The solution is purely transient, no frequency extraction is carried out first. The Solution Approach As a transient load is applied, the solution must follow the response of the structure. To achieve this, the overall time period being studied is divided into a number of linear time pieces, each one being referred to as a time step. The successful implementation of any time domain analysis is dependant on a suitable number of time steps being selected. If the time step is too large, portions of the response (such as spikes) could be missed or truncated. On the other hand, if the time step is too small, the analysis will become excessively long or even prohibitive. The progression of the solution from one time step to the next is achieved by implementing time integration techniques. Despite many packages providing automatic time stepping estimates, the full response of the structure may not be captured, and manual intervention will be required. If there is a discontinuity in your automatically time stepped results, chances are there is a spike in the response that is not being fully captured. Stepping Schemes: Time Integration Many time stepping algorithms have been developed, each having their advantage over others under certain circumstances. However, three main types of solution dominate, backward difference (Implicit), central difference (Crank-Nicolson) & forward difference (Explicit). The explicit & implicit techniques are often referred to as Euler's rule & the backwards Euler's rule respectively. Explicit schemes, which are conditionally stable (stability of solution not guaranteed), find the response at the end of the time step in terms of the conditions at the start of the time step. In other words, the calculation of the solution at time (t+∆t) is obtained by considering the situation at time t. The advantage of this approach is that the underlying system of equations that comprise the model (stiffness matrix, capacitance matrix, flexibility matrix) does not have to be solved at each time step. Furthermore, the material & time matrices can be diagonalised to become uncoupled, and so the solution can be calculated explicitly. Very fast calculations of individual time steps can be achieved as no matrix factorisation is required. However, the technique is much less stable than

the implicit method, so very small time steps must be used to ensure an appropriate solution. Implicit schemes, which are unconditionally stable, find the response at the end of the time step in terms of the conditions at the end of the time step. In other words, the calculation of the solution at time (t+∆t) is found by considering the response at time (t+∆t). An important point to note is that the solution at each time step involves matrix factorisation (evaluating the system of equations that comprise the model), which is a computationally intensive process. Despite this disadvantage, implicit schemes are often used, as the solution is inherently reliable & robust. Implicit analyses allow much larger time steps than the others, and so the solution can be obtained with fewer calculation increments. As implicit schemes are always stable, the time step length is governed by considerations of accuracy alone. The Crank-Nicolson approach evaluates the next step of the solution by using the prediction at the centre of the time step. As with the backward difference scheme, this is an implicit solution which is conditionally stable (results in an oscillatory solution if the critical time step for stability is exceeded). The central difference method is more accurate than both the purely implicit or explicit techniques since neither favours the response at the start or end of the time step. Response Spectrum Analysis Response spectrum analysis (RSA) is a procedure for computing the statistical maximum response of a structure to a ground bourne excitation. Each vibration mode considered may be assumed to respond independently as a single-degree-offreedom system. Design guideline codes specify response spectra that determine the base acceleration applied to each mode according to its period (the number of seconds required for a cycle of vibration). The design response spectrum is then usually obtained by multiplying the basic acceleration coefficient by a factor based on required structural performance, risk & location. Having determined the response of each vibration mode to the excitation, it is necessary to obtain the response of the structure by combining the effects of each vibration mode. Because the maximum response of each mode will not necessarily occur at the same instant, the statistical maximum response, where damping is zero, is taken as the square root of the sum of the squares of the individual responses. Response spectrum analysis produces a set of results for each excitation load case which is in the form of an envelope. All results are absolute values, each value represents the maximum absolute value of displacement, moment, shear, etc. that is likely to occur during the event which corresponds to the input response spectrum.

Concepts associated with Dynamic Strucural Analyses SHAKEDOWN ANALYSIS: If load intensities on a structure remain sufficiently low, the response of the body is purely elastic (with the exception of stress singularities). If the load intensities become sufficiently high, the instantaneous load-carrying capacity of the structure becomes exhausted (unconstrained plastic flow and damage evolution occurs) & collapses. If the plastic strain increments in each load cycle are of the same sign then, after a sufficient number of cycles, the total strains (and therefore displacements) become so large that the structure departs from its original form and becomes unserviceable. This phenomenon is called incremental collapse or ratchetting. If the strain increments change sign in every cycle, they tend to cancel each other and total deformation remains small leading to alternating plasticity. In this case, however, the material at the most stressed points begins to fails due to low-cycle fatigue. If, after some time plastic flow and damage evolution cease to develop further and the accumulated dissipated energy in the whole structure remains bounded such that the structure responds purely elastically to the applied variable loads, one says that the structure shakes down FLUTTER is a dynamic instability that involves coupling of aerodynamic forces and elastic and inertial forces of the structure. In a flow, an oscillating structure generates unsteady aerodynamic forces. These unsteady aerodynamic forces introduce coupling into the structure and cause phase shifts between the motions of the structure (degrees of freedom). The speed of the flow affects the amplitude ratios and phase shifts between the various degrees of freedom in such a way that energy is extracted from the airstream. At the critical airspeed, the energy dissipated is exactly equal to the available structural damping. At speeds greater than the critical speed, the extracted energy dissipated is less than available structural damping and the motion is divergent.

Analysis of Temperature & Heat Flow Introduction Thermal analysis is used to determine the temperature distribution, heat accumulation or dissipation, and other related thermal quantities in an object. The nodal degrees of freedom (primary unknown data) are the temperatures. The

primary heat transfer mechanisms are conduction, convection and radiation. In addition, less dominant phenomena such as change of phase (melting or freezing) & internal heat generation can occur. Conduction Conduction is governed by Fourier's law, which is a differential equation describing the rate of heat transfer as a function of temperature gradient, material thermal capacitance & the rate of internal heat generation. This law describes the temperature within the solid body, but does not account for how heat will flow to & from the component. In order to carry out analyses using a conduction model alone, temperatures must be described as part of the boundary condition description. Heat flows (otherwise known as heat flux) are oftentimes specified along boundaries in addition to temperature BC's. Radiation Radiation type boundary conditions are applied if there is a significant temperature difference between bodies in an enclosed space, or if there is a far field heat source/sink (such as the sun or a very cold enviornment). This heat transfer mechanism occurs exclusively at the surface and is a function of the fourth power of the absolute temperatures (Kelvin), the emissivity of the bodies & a value known as the Stefan-Boltzman constant. The emissivity is dependent on surface properties such as the colour & finish. Radiation type boundary conditions are highly non-linear due to the difference between fourth order absolute temperatures. A further complication is due to incidents where the surfaces of two adjacent radiating bodies are not flat and parallel to each other. This case is overcome by introducing a shape factor (otherwise known as a view, angle or interception factor) to the solution. Convection The convection heat transfer mechanism is due to the temperature gradient between a fluid and a solid. This mechanism is complex as a boundary layer usually exists within the fluid adjacent to the solid boundary. The heat flux is a function of the temperature difference, ∆T, and a heat transfer coefficient, h. The heat transfer coefficient is dependent on many factors such as fluid pressure, velocity, density, specific heat (ratio of specific heats if the fluid is compressible), viscosity & conductivity. It is also dependent on surface properties such as roughness & geometry. Due to the extreme non-linear nature of convection type phenomena, solutions are usually based on imperical relations such as log laws.

In order to implement convective heat transfer in FEA, boundary conditions for specific cases have been developed. Examples of which are Vertical Plate in horizontal flows, flow over isothermal inclined flat plates, flow through horizontal cylinders, flow over an inclined surface, vertical enclosed space flow, flow in horizontal tubes & ducts, generic convection as a function of temperature difference or grashof & prandtl numbers, flow along a rotating disk, etc.. Each boundary condition may have automatic implementation for each of the three flow types, laminar, transition & turbulent. Information defining all parameters of each BC type must be input, this can usually be carried out manually or via tables of data. An important point to note is that very few packages have the capability to apply convective BC's to the level described here. MSc/Thermal and SC03, the Rolls-Royce proprietary code, both have extensive capabilities for applying convective type BC's. At least two other FEA software vendors are currently considering implementing such capability into their codes. Non-Linear & Transient Analyses If temperatures are much higher or lower than the average temperature in certain locations of the model, there is a good chance that the heat transfer coefficients & conductivity will themselves become temperature dependent. The problem becomes non-linear, as the heat transfer rate is not directly proportional to temperature. The approach to a solution is similar to that of a non-linear displacement analysis, the load is divided into a number of smaller ones that are applied incrementally. The solution becomes an iterative procedure rather than one of matrix factorisation alone. If the thermal load is impulsive in nature (time dependent), then a solution through time is required. This is carried out by dividing the overall time range into a number of smaller time steps & applying time integration techniques to handle the evaluation of the solution from one time step to the next. As with structural analyses, there are three main types of time integration techniques, Implicit, explicit & central difference (Crank-Nicholson). Implicit analyses are stable but computationally expensive, explicit integration is fast but unstable, and Crank-Nicholson is a mix of the two, but is also unstable. Thermal Stress Analysis Often an object will fail because of stresses induced by uneven heating, rapid temperature change or differences in thermal properties. A coupled analysis, which models both thermal and stress variations, can be effective in predicting the overall structural response. It facilitates effective

prediction of incidents where thermal expansion is an important consideration, such as in reciprocating & gas turbine engine design. The usual procedure is to carry out a thermal analysis which evaluates the temperature distribution. These temperatures can then be used to prime the displacement analysis, and hence thermal deflections, strains & stresses can be evaluated. It is also possible to have fully coupled analyses where the temperatures & displacements are a function of each other. This is most evident in analyses that involve fluid flows, such as in a gas turbine or rocket. The heat transfer rates are dependent on massflow rates, but massflow rates are a function of valve & seal clearances (labrynth seals in gas turbines, nozzles in rockets). Therefore, we end up with a scenario where clearances are a function of temperature and temperature is a function of clearances. This type of problem can only be solved via a nonlinear and fully coupled solution. Iterative Analyses & Submodelling In analysis of large & complex systems (such as reciprocating & turbine engines), it is usual to carry out isolated design work on specific components of the overall system. For example, we may want to refine the design of a piston, and would like a realistic temperature distribution, but don't want to incorporate the rest of the engine into the analysis model. In order to do so effectively, information is required about the state of temperature at the sub-system extremities. One way of providing these temperatures is to carry out a coarse analysis on the overall model and use the temperatures at relevant points to 'prime' the subsystem model (otherwise known as a submodel). FEA Packages with such facilities have the capability of writing out the temperature-time values from the coarse model to a file. This file can be subsequently used in the submodel, and so an accurate representation of the thermal enviornment can be provided while studying the finer details contained within the sub-system model. This approach is extremely rich as regards saving on analysis times. Despite the high level of idealisation being implemented, it provides very accurate & realistic modelling of the physical conditions being investigated.

Buckling Analyses: Sudden Collapse

Introduction Buckling is a critical state of stress and deformation, at which a slight disturbance causes a gross additional deformation, or perhaps a total structural failure of the part. Structural behaviour of the part near or beyond 'buckling' is not evident from the normal arguments of statics. Buckling failures do not depend on the strength of the material, but are a function of the component dimensions & modulus of elasticity. Therefore, materials with a high strength will buckle just as quickly as low strength ones. If a structure has one or more dimensions that are small relative to the others (slender or thin-walled), and is subject to compressive loads, then a buckling analysis may be necessary. From an FE analysis point of view, a buckling analysis is used to find the lowest multiplication factor for the load that will make a structure buckle. The result of such an analysis is a number of buckling load factors (BLF). The first BLF (the lowest factor) is always the one of interest. If it is less than unity, then buckling will occur due to the load being applied to the structure. The analysis is also used to find the shape of the buckled structure. Evaluating Linear Instabilities From a formal point of view, buckling is an eigenvalue problem that is a function of the material & geometric stiffness matrices. Consequently, there will be a number of buckling modes and corresponding mode shapes. As with a frequency analysis, eigenvalue extraction may be carried out using a number of available methods, the best choice depends on the form of the equations being solved. The main methods are the power, subspace, LR, QR, Givens, Householder & Lanczos methods. An important note is that the eigenvalue method does not take into account of any initial imperfections in the structure and so the results rarely correspond with practical tests. Eigenvalue solutions usually over estimate the buckling load and give no information about the post-buckling state of the structure. Sudden buckling simply does not occur in the real world. So how should we know if a linear buckling analysis is sufficient ?? Carry out both a linear static analysis and a linear (eigenvalue) buckling analysis. If the max stress is significantly less than yield, and the buckling load factor is greater than 1.0, then buckling will probably not occur. If however the BLF is less than 1.0, then the buckling analysis will be linear provided that the max stress is far below yield. In all other cases, a non-linear buckling analysis should be carried out. If the component is critical to the safe operation of a system, full displacememnt analyses should be carried out.

Non-Linear Buckling A more practical approach is to carry out a large displacement analysis, where buckling can be detected by the change of displacement in the model. A large displacement problem is non-linear in nature. Geometric non-linearity arises when deformations are large enough to significantly alter the way load is applied, or load is resisted by the structure. The approach to a non-linear buckling solution is achieved by applying the load slowly (dividing it into a number of small loads increments). The model is assumed to behave linearly for each load increment, and the change in model shape is calculated at each increment. Stresses are updated from increment to increment, until the full applied load is reached. The solution becomes an iterative procedure rather than one of matrix factorisation alone, and consequently is computationally expensive. An interesting variation arises in the case of automotive applications. In the case of front end collision, the hood is expected to crumple (buckle) in order to absorb the energy of collision, as well as to save the passenger compartment. In such cases, we are not designing against, but for buckling. Avoiding Instabilities Any structure is most efficient when subjected to evenly distributed tensile or compressive stress, such as occurring in cables, strings etc. Evidently, such modes of loading makes the best use of the material, and its strength. On the other hand bending (flexing) is the least efficient way of loading a structure. A high flexural stiffness of the structure means high resistance to buckling. This is true even if the load is entirely in-plane, since when buckling is imminent, the only stiffness that counts is flexural. Eccentricity of loading promotes buckling. Eccentricity means that the resultant load does not pass through the centroid of the load bearing cross section. It is safe to assume that in 100% of practical applications, loads are eccentric. When buckling occurs, symmetry of the part does not apply. There is no symmetry of the buckled shape, although both the part, and the loading may be symmetric. Correspondingly, when carring out an FE buckling investigations, it is advisable to implement a full 3D analysis of the structure under inspection. The non-linear stress strain behavior of the material reduces the stiffness at higher stress (load) levels, and hence elastic formulas from the handbooks tend to be highly unconservative.

If a component is structurally slender, and is made of plastic, then the component faces buckling from three directions; from the low material stiffness, the large deflections producing eccentricity during deformation, and from the non-linearity of the material itself. By and large it is true that buckling usually occurs when compressive stress is present. But what is not evident that compressive stress can prevail in un-expected places. Shallow domes under internal pressure can develop local compressive stress regions, and make it vulnerable to instabilities. Bifurcation & Snap Through Buckling In many systems a smooth change in a control parameter (the load) can lead to an abrupt change in the behaviour of the system. A simple example is the buckling of a rod. If a straight rod is compressed by a small load, it shrinks to some extent, but remains straight. For larger loads, however, it starts to buckle. Mathematically, the solution corresponding to a straight rod still exists, but it is unstable for the large load applied and very small transverse perturbations make the rod buckle. The transition from the unbuckled to the buckled state occurs via a bifurcation, that is, at the onset of the instability a new solution corresponding to the buckled rod comes into existence. In bifurcation buckling, there are two equilibrium solutions at the bifurcation point, the ordinary static strength of materials solution and the instable (buckling) solution. Snap through buckling occurs when a structure is subject to an increasing load that at some point causes the structure to undergo a gross deformation. Subseqent to this deformation, the structure regains sufficient stability to carry load, usually in a configuration that changes the structural load from being initially compressive to tensile. An example of this is a shallow dome in compression. If the load becomes too great, it buckles and snaps through so that the load is supported in tension.

Electromagnetics & Related Analyses Introduction Many kinds of electromagnetic phenomenon can be modeled from the propagation of microwaves to the torque in an electric motor. Analysis of electrostatic and magnetic fields passing through and around a structure provides insight into the response, and hence a means for regulating these fields to attain specific responses.

FEA can be used to analyse the linear electric or magnetic behaviour of devices. Analyses typically involve the evaluation of magnetic, electric and thermal fields. Further applications include the analysis of shape-memory materials & piezoelectric effects. An analysis can be static, harmonic or transient state in nature. Due to the complexity of the practical applications of the technique, it is not unusual to have magnetic, dielectric and thermal couplings in a single model. Such complex analyses generally make realistic modelling an ardouous task. Application Areas The application areas include, but is not limited to the design of: •

• • • •

Rotating machines (DC motors, synchronous machines, induction motors, stepper motors, coupling devices, brushless motors, switched reluctance motors, PM motors, generators) Energy transfer and conversion modules (transformers, cables, high voltage devices, insulators, connectors & fuses). Electrical actuators (linear motors, electromagnetic brakes, contactors, magnetic bearings, fuel injectors, electromagnetic launchers). Sensors (capacitive and inductive, speed, eddy currents non destructive testing, magnetoscopy, resolvers, electric meters). Field generators (mass spectrometers, magnetic recording, polarisation fields, magnetisation devices).

Analysis Types ELECTROSTATIC analyses involve the computation of electric potential and fields in the absence of current. It can also involve the calculation of capacitance between conductors, and dielectric strength of insulators. VOLUMIC CONDUCTION type simulations involve the determination of current distribution and electric potential in complex-shaped conductors. Calculation of electric resistance and distribution of Ohmic losses can also be carried out. LINEAR MAGNETOSTATICS analyses attempt to evaluate magnetic fields in linear materials due to DC currents. Applications include electromagnets, contactors, motors, electron guns. AC MAGNETICS invloves the calculation of magnetic fields in solid conducting materials due to high-frequency AC currents. Typical applications are non destructive testing, surface hardening, and superconductivity.

PIEZOELECTRIC analyses study crystals that develop an electrical charge when exposed to mechanical stress. Conversely, the application of an electric field to a piezoelectric crystal leads to a physical deformation of the crystal, and so may be of interest to the designer. SHAPE MEMORY analyses study the phenomenon where material deformed in the room temperature can be returned to its original shape by heating. The shape memory effect results from the fact that a Shape Memory Alloy has two states, austenitic and martensitic, between which its metallurgical structure can be transformed. MAGNETORESISTIVE analyses study the resistance of a semiconductor as it changes in the presence of magnetic field. The magnitude of the resistance change depends on the shape of the sample and is best seen in samples being short & wide. The applications are usually in the design of magnetic sensors. ELECTROSTRICTIVE analyses examine the deformation of material in an electric field. Similar to the piezoelectric effect, electrostriction occurs in ferroelectric materials but in contrast to the piezoelectric effect, it does not require asymmetric crystal structure. Electrostriction applications are usually in the area of actuation.

FE analysis of Fluid Flow Problems Introduction Fluid flow problems arise in almost all industrial sectors: food processing, water treatment, marine engineering, automotive, aerodynamics, and gas turbine design. FEA facilitates the prediction of fluid flow, heat & mass transfer, and chemical reactions (explosions) and related phenomena. By solving the fundamental equations governing fluid flow processes, FE analyses provide information on important flow characteristics such as pressure loss, flow distribution, and mixing rates. This results in better designs, lower risk, and faster time to the marketplace for product or processes. Models can be developed for physical phenomena such as turbulence, multiphase flow, chemical reactions, and radiative heat transfer. Solution Approach

The foundation of fluid dynamics is based on the Navier-Stokes equations, the set of partial differential equations that describe fluid flow. In FEA, this equation is rewritten as algebraic equations that relate the velocity, temperature, pressure, and other variables, such as species concentrations. The resulting equations are then solved numerically, yielding a complete picture of the flow. The equations are solved iteratively using the method of weighted residuals. The main method of solution is achieved via the Galerkin method, but others exist. One such variant is the Petrov-Galerkin method, which is used to solve instances of viscous, high Reynolds number flows. During the solution, unsymmetric solution matrices may exist. Therefore, it is often necessary to use many relatively small & higher order elements in order to obtain convergence. Due to this problem, it is not uncommon to implement reduced integration type elements in an analysis. For transient problems, a special time integration technique known as the semiexplicit scheme is used in large analyses, as it is more economical than other methods available. Coupled Analyses Due to the complex nature of the physical processes being modelled, it is not unusal to conduct coupled analyses as part of a design program. Fluid-structural, fluid-thermal & fluidacoustic analyses are not uncommon. Fluid-structural interaction is important in the design of offshore structures & their pipelines (also known as risers). Strong tidal currents & wave bombardment have a critical influence on the life of such components. The structural design of dams usually requires a model which involves hydrostatic loading & seepage flows. Fluid-thermal analyses are important in many applications such as viscous flows through restrictors & valves (both process & medical applications). Fluidacoustic evaluation is required in order to ensure that noise emissions from systems such as gas turbines are within tolerable limits. Metal Forming Techniques Solidification modeling of complex castings can significantly reduce casting porosity and improve casting yields. Large strain analysis of forging operations can provide residual strains/stresses generated in products and provide estimates of product "spring back". Despite the difficulty one may have in visualising a metal behaving as a fluid, FEA uses slow non-newtonian flow techniques for simulating metal forming processes. This type of analysis is

divided into two primary sections, steady state problems such as extrusion & rolling processes, and transient problems such as forging & stamping. Both steady-state & transient solutions are usually highly non-linear in nature. Therefore they are computationally expensive to implement. CFD: An Alternative Technique Despite the fact that most fluid-flow type problems can be implemented successfully using FEA, it is not the paramount technology. Due to the nature of the fluid formulations for solution via FEM, long solution times & poor convergence can be experienced. As a result, a more convenient solution is obtained by using a method known as CFD. This method is based around finitedifference & finite volume solution techniques. The solution-adaptive grid capability of CFD is particularly useful for accurately predicting flow fields in regions with large gradients, such as free shear layers and boundary layers. In comparison to solutions on structured or block structured grids, this feature significantly reduces the time required to generate a "good" grid and arrive at a suitable solution.

Linear Steady State Solutions Introduction Despite the fact that all physical phenomena are non-linear and time dependant to some degree, linear static analyses remain the most useful and prolific form of FE analyses carried out today. The reason for its widespread use is that linear analyses are fast, oftentimes sufficiently representative of the physical phenomena and very easy to carry out. The meaning of Linear Static Linear analyses deal with problems in which the structural response is linear. Therefore, if the applied forces are doubled, then the displacements and internal stresses also double. Problems that fall outside this domain are usually classified as non-linear.

Static or steady state analyses are those where the solution is independent of time. Inertial forces are either ignored or neglected and so there is no requirement to calculate actual time derivatives. Problems that require inertial terms to be evaluated are usually classified as dynamic and/or transient analyses. Linear static analyses are usually sufficient for situations where loads are known and the instance at which peak stress occurs is obvious. When performing a linear static stress analysis, the analyst applies static loads (forces, pressures or prescribed displacements) to the model. Assumptions of Linear Static Analyses As with all types of analyses, linear static ones are based on a set of assumptions. The main assumptions are listed here: o

All deformations and strains are small.

o

Structural deformations are proportional to the loads applied. This infers that the loading pattern does not changed due to the deformed shape and no geometric stiffening occurs due to the application of the load.

o

All materials behave in a linear elastic fashion. Therfore, the material deforms along the straight line portion of the stress-strain curve (no plasticity or failures occur). Highly localised stress concentrations are usually permitted as long as gross yielding does not take place.

o

Loads are all static. This means that the loads are applied to the structure in a slow or steady fashion and in a way that makes them time independent (are assumed to be constant for an infinite period of time).

o

No boundary condition varies with time or application of load.

Limitations of Linear Static Analyses There is a point when linear static analyses are not sufficient to represent the real behaviour of the system being modelled. As mentioned previously, all phycical phenomena are non-linear and dynamic to some degree, some are negligibly nonlinear/transient and some are grossly non-linear and time dependent. In between these two extremes is a grey area. The decision to go NL and/or Transient should be based on careful consideration of the physics of the problem at hand. Some suggested considerations are: o

If any of the above linear static assumptions are clearly violated.

o

If there is a very low factor of safety applied to the components being designed.

o

When non-linear behaviour cannot be estimated from the linear results.

o

If system behaviour is unclear (e.g. in buckling analyses).

Nonlinear Analyses: The Real World

Introduction Every physical phenomena in the real world is nonlinear to some extent. Engineers and scientists observe linear behaviour to be suitably representative for many applications. However, there are many cases where the nonlinearity of the problem cannot be ignored and so methods of obtaining nonlinear solutions have to be adopted. Due to the recent advances in computer hardware, more and more engineering firms are turning to nonlinear finite element analysis to understand the behaviour of their products. nonlinear analyses can be either transient or steady-state (static) in nature, this document will outline the main points of nonlinear static analyses. View the documents on dynamic or transient type analyses for more details on time-dependent problems. Causes of Nonlinear Behaviour There are generally four ways in which structural nonlinear behaviour can occur. They are: o

Material nonlinearity This is where the material stressstrain relationship is actively nonlinear. In this case, material behavior depends on current deformation state and possibly past history of the deformation. Material nonlinearity can be observed in structures undergoing nonlinear elasticity, plasticity, viscoelasticity, creep or other inelastic effects.

o

Geometric nonlinearity This is where there is a nonlinear straindisplacement relationship. In this case, the change in geometry (as the structure deforms) is taken into account when forming the strain-

displacement and hence the equilibrium equations. Geometric nonlinearity maybe due to large strains (membrane analyses or metal forming) or small strains but with large displacements and/or rotations (cables, leaf-springs, arches, fishing rods, snap-through buckling). o

Application of nonlinear forces This is where the magnitude or direction of the applied forces change with application to the structure (nonlinear force-deflection relationship). This could be due to pressure loadings, gyroscopic forces or follower forces. Stress stiffening is a major contributor to nonlinear force application. It is observed with an increase or decrease in structural stiffness due to the stress state. This occurs due to the coupling between the in-plane and transverse deflections within a structure (which often occurs in structures that are weak in bending e.g. pressurised membranes, turbine blades rotating at a high speed).

o

Displacement boundary condition nonlinearities This is where the displacement boundary conditions depend on the deformation of the structure. (nonlinear displacement-deformation relationship). The most important and obvious application is in contact problems, the displacement is highly dependant on the relationship between two contact surfaces (normal force and friction present). It is important to note that the bodies in contact could be in a state of linear elastic stress; the nonlinearities all come from the contact definition. Nonlinear contact occurs in impact (crash), assembly of mechanical components or sliding frictional interfaces. Accurate resolution of this type of problem lies in the identification and application of an appropriate physical representation. The two main friction types used today are coloumb and shear friction. Most software packages allow user-defined friction relationships to be defined.

Identifying Nonlinear Behaviour Sometimes nonlinear behaviour is difficult to indentify in an analysis, there are a number of signs that indicate that the problem has left the linear zone. Some of these are: o

Stresses that exceed that of the limit of proportionality of the material.

o

Major changes in geometry

o

Changes in geometry that remain after the process is finished.

o

Processes involves buckling, crushing, wrinkling or plastic flows.

o

Temperatures exceeding the melting temperature of the material.

o

Large strains, finite strains can occur in hyperelastic materials.

o

Nonlinear stress-strain laws, some materials have diffferent compressive than tensile strengths.

o

Boundary conditions change due to the application of load (contact).

o

The direction of load application changes with deformation (follower forces such as pressures).

Irreversibility and Superposition Nonlinear analyses are usually irreversible, that is when the load is removed, the model will not return to it's original configuration. As a result residual strains and spring-back are common although oftentimes ignored. As the nonlinear solution is usually dependent on a combination of loads and restraint interactions, they cannot be equally solved by superposition (which states that the resultant deflection, stress, or strain in a system due to several forces is the algebraic sum of their effects when separately applied). Tangent Stiffness The equilibrium path of a linear analysis is a straight line; when the load is increased, the deflection increases correspondingly. With a non-linear analysis however, the equilibrium path is non-linear, so the relationship between load and displacements is not linear. An equilibrium path for a non-linear analysis is shown. If we get the tangent to this curve (the slope), we have a property of the system known as the tangent stiffness. Based on this, we observe that the tangent stiffness is constant in a linear analysis, while it is variable in a nonlinear analysis. The tangent to an equilibrium path may be loosely viewed as the ratio of force increment over displacement increment. This is by definition a stiffness or, more precisely, the tangent stiffness associated with the force-displacement. The reciprocal ratio is called the flexibility or compliance of the system. The sign of the tangent stiffness is closely associated with stability of an equilibrium state. A negative stiffness is always associated with unstable equilibrium while a positive stiffness is necessary but on its own is not sufficient for stability.

Solution Techniques In a nonlinear analysis, the structural stiffness matrix and load vector may depend on the solution, they are both unknown. To solve the problem, an iterative procedure has to be used. This involves splitting up the equilibrium path into a set of linear analyses. During each analysis step (which is usually referred to as an increment), equilibrium is sought using some sort of numerical scheme. In order that the selected scheme is controlled, there are a number of different numerical procedures that have to be incorporated into the overall solution of nonlinear problems. As the complete procedure is difficult to describe absolutely without equations, it will be explained using a step by step example for a structural analysis: o o

Setting up the analysis Using our infinite wisdom on the problem being solved, we decide that the load should be divided into 2 increments in order that the equilibrium path is followed using a linear piecewise solution (breaking the equilibrium curve into into 2 linear bits as shown). The first increment will apply half the total load, the second will increment this up to its final value.

o o

Increment 1, part 1 The load applied is incremented (increased) to half the total load (always strive for the first increment to be within the linear range of the system being modelled, if there is a linear portion to the equilibrium curve). The stiffness matrix is formed using the initial conditions. This increment is indicated by the yellow portion of the equilibrium curve.

o

Increment 1, part 2 The linear stiffness matrix is referred to as the Tangent stiffness matrix in a non-linear analysis (this was discussed in the previous section). The tangent stiffness is used in the second increment.

o

Increment 2, part 1 The external load is incremented by a set amount, (to its final value in this case) and the Newton-Raphson expression is formed.

o

Increment 2, part 2 Using the external load vector and the stiffness matrix (tangent stiffness) from the pevious increment, the current iteration displacement is estimated by extrapolation via the Newton-Raphson method.

o

Increment 2, part 3 Once the first-guess displacement is determined, the internal forces resisting the load are calculated. This value is subtracted from the external load vector to determine a residual force.

o

Increment 2, part 4 This residual is compared with a set tolerance, if it is within the convergence tolerance, then the solution has been determined. If not, the stiffness matrix is reevaluated using the updated displacement and internal force vectors. This is achieved using LU decomposition or any other appropriate method, the updated tangent stiffness results.

o

Increment 2, part 5 The internal load vector and the tangent stiffness matrix from the pevious iteration are used to estimate a new displacement. This is done by extrapolation via the Newton-Raphson method.

o

Increment 2, part 6 Using the second-guess displacement vector, the internal forces are again determined and compared with the applied load. A residual exists, there is still no convergence.

o

Increment 2, part 7 Using the second-guess displacement and updated internal force vectors, the the tangent stiffness matrix is updated, again via LU decomposition of the sytem equations.

o o

Increment 2, part 8 This procedure of guessing the displacement and updating the internal force vector is continued until zero residual force is obtained (within a tolerance). When this is achieved, it indicates that the solution has converged and that the current load and displacement vectors are final for this increment.

o o

Increment 2, part 7 The results are written to memory as those for the increment and to file if required. When this is achieved the current load increment has been completed. The final equilibrium path for this increment is shown here. As can be seen, three iterations were required to obtain convergence.

o

Further Load Increments The solution for this example has been found, and the analysis now terminates. Had it required further increments of load, the procedure for

each increment would be exactly the same as for increment 2 in our example.

The next few sections will refer to this example to describe the full richness of contemporary commercial finite element solvers. The explanations will not encapsulate all the issues that commercial solvers deal with, it will merely give an overview. Solution Control Techniques In the example above, the force was increased by a set amount during each increment and the displacement was used as a solution parameter. This is known as the force control method. An alternative to this would be to drive the solution by increasing the displacement by a set amount and using the force as a solution parameter. Therfore this alternative is known as the displacement control method. Both force and displacement control methods are based on incrementing one variable and extrapolating the other based on the current state of the tangent stiffness matrix. Therefore the increments are calculated by a prediction step, and so are known as predictor or extrapolator type methods. Both force control and displacement control could cause problems when the equilibrium path comprises of locally curved pockets such as that shown here. This equilibrium curve exhibits snap-through behaviour (for a small load applied at the first maximum of the curve, the component will undergo a relatively large displacement). In order that the snap-through is captured a third option known as the arc-length control method can be used. The applied load is incremented to achieve equilibrium under the control of a specified arc-length of the equilibrium path. This arc-length is automatically calculated by the solver and helps avoid drift errors. Arc-length control is known as a predictor-corrector or incremental-iterative type method. Non-Linear Solution Algorithm. In the example, we used the Newton-Raphson method to obtain convergence during the second increment. There are many ways of obtaining the solution for this non-linear set of equations. Most for finite element analyses are based around the Newton-Raphson solution method. In this example, we calculated the tangent stiffness at each iteration step. This can be trecharous and computationally expensive for problems where there may be multiple roots. Therfore, the method can be modified so that the first tangent

stiffness equation is used at every iterative step. This speeds up the solution process also, as the system equations do not require solving at each and every step (even though more steps required to achieve convergence). This method is known as the modified Newton-Raphson method. In addition, to the two methods outlined above, there is the strain correction method, the secant method, the direct substitution method and quasi-Newton Methods (such as BFGS and Inverse Broyden). Stiffness Updating Algorithm In the example, we used the LU decomposition method to evaluate the updated stiffness matrix at each increment. However, there are many solution techniques to choose. There are two basic types of solvers for linear algebraic systems of equations, linear and iterative. The LU decomposition method is a linear (direct) solution method. Others include matrix inversion methods such as the method of co-factors or Gauss-Jordan, or the famous elimination procedure known as Gauss-elimination. There are also other variants of the LU decomposition method such as Choleski decomposition method. Among iterative methods are Jacobi, Gauss-Seidel, relaxation techniques and preconditioned conjugate gradient methods. The choice of algorithm used depends on the problem being solved. Most vendors use flavoured versions of these standard methods and they usually have many solution methods in their software libraries. Stopping the Iteration Process - Termination Control Convergence should be evaluated at the end of each iteration, if the control parameter is within a specified and realistic tolerance, then the procedure should be terminated. Very loose tolerances will lead to inaccurate results, while very tight tolerances will needlessly increase computational cost. In our example, we used the displacement as the termination criterion (when two successive displacement vectors equalled each other, within a tolerance, iterations were stopped). This is not the only way of terminating the iterations. A force control method could have equally been used as the iteration control parameter. Since the residual between the load vector applied and the current load vector will differ, ther residual (difference between the two) measures the departure from equilibrium. This is known as the residual convergence test. Another criterion that can be used is the internal energy during each iteration. This is is the work done by the residual forces through the incremental displacements. Since this convergence criterion involves both the displacements

and forces, it is an attractive termination scheme. On rare occasions, Newton iteration schemes will neither converge nor diverge, but just oscillate in the iterative ether. To avoid excessive and unnecessary computations, it is worth imposing a limit on the maximum number of iterations per increment. Kinematics of Geometrically Nonlinear Analyses. In our example, the displacements were small relative to the linear displacements, thus it was considered a small strain, small displacement analysis and so we could use a static coordinate sytstem. However, there are many analyses where coordinate system updating needs to be considered. If a body we are modelling moves through space, we need to track its motion in some way. The key difference between geometrically linear and nonlinear structural analysis lies in its kinematics. Equilibrium conditions must be posed in the deformed geometry, this introduces a historical ingredient. In dynamic analyses, the configuration of the structure must be tracked as the loads change. This tracking process must involve a kinematic description with respect to a reference state. There are two main ways of implementing a constantly changing reference system, via Lagrangian or Eulerian kinematic expressions. The total Lagrangian formulation is used when the equations are to be written with respect to the original reference state only. The reference frame does not change throughout the analysis. It is usually applicable to large deflection type problems such as the one we were solving in our example. The updated Lagrangian method is a variant of the total Lagrangian formulation, the only difference being that the reference state is periodically updated to a new reference state (the mesh coordinates are updated) after each increment. It is applied to problems featuring large inelastic strains such as metal forming. The Eulerian method is used where the mesh is fixed in space and the material flows through it. Therefore, Eulerian finite elements undergo no distortion due to material motion. Despite this, the treatment of constitutive equations and updates is complicated due to the convection of material through the elements. This type of reference frame is most suitable for steady-state problems such as extrusion and fluid mechanics problems. A Final Word Approach all nonlinear analyses with care, black-box codes are easily used and frequently mis-used. Even codes that can solve difficult nonlinear problems will

not produce accurate results if modeled improperly (the solver can only answer the questions posed, it is not magic). It is necessary to rigorously study each computed solution, try to you understand what each solution tells you before running another analysis.

Introduction to Finite Element Analysis A Brief History Finite Element Analysis (FEA) was first developed in 1943 by R. Courant, who utilized the Ritz method of numerical analysis and minimization of variational calculus to obtain approximate solutions to vibration systems. Shortly thereafter, a paper published in 1956 by M. J. Turner, R. W. Clough, H. C. Martin, and L. J. Topp established a broader definition of numerical analysis. The paper centered on the "stiffness and deflection of complex structures". By the early 70's, FEA was limited to expensive mainframe computers generally owned by the aeronautics, automotive, defense, and nuclear industries. Since the rapid decline in the cost of computers and the phenomenal increase in computing power, FEA has been developed to an incredible precision. Present day supercomputers are now able to produce accurate results for all kinds of parameters. What is Finite Element Analysis? FEA consists of a computer model of a material or design that is stressed and analyzed for specific results. It is used in new product design, and existing product refinement. A company is able to verify a proposed

design will be able to perform to the client's specifications prior to manufacturing or construction. Modifying an existing product or structure is utilized to qualify the product or structure for a new service condition. In case of structural failure, FEA may be used to help determine the design modifications to meet the new condition. There are generally two types of analysis that are used in industry: 2-D modeling, and 3-D modeling. While 2-D modeling conserves simplicity and allows the analysis to be run on a relatively normal computer, it tends to yield less accurate results. 3-D modeling, however, produces more accurate results while sacrificing the ability to run on all but the fastest computers effectively. Within each of these modeling schemes, the programmer can insert numerous algorithms (functions) which may make the system behave linearly or non-linearly. Linear systems are far less complex and generally do not take into account plastic deformation. Non-linear systems do account for plastic deformation, and many also are capable of testing a material all the way to fracture. How Does Finite Element Analysis Work? FEA uses a complex system of points called nodes which make a grid called a mesh (Figure 2). This mesh is programmed to contain the material and structural properties which define how the structure will react to certain loading conditions. Nodes are assigned at a certain density throughout the material depending on the anticipated stress levels of a particular area. Regions which will receive large amounts of stress usually have a higher node density than those which experience little or no stress. Points of interest may consist of: fracture point of previously tested material, fillets, corners, complex detail, and high stress areas. The mesh acts like a spider web in that from each node, there extends a mesh element to each of the adjacent nodes. This web of vectors is what carries the material properties to the object, creating many elements. (Theory) A wide range of objective functions (variables within the system) are available for minimization or maximization: • • • •

Mass, volume, temperature Strain energy, stress strain Force, displacement, velocity, acceleration Synthetic (User defined)

There are multiple loading conditions which may be applied to a system. Next to Figure 3, some examples are shown: • • • • •

Point, pressure (Figure 3), thermal, gravity, and centrifugal static loads Thermal loads from solution of heat transfer analysis Enforced displacements Heat flux and convection Point, pressure and gravity dynamic loads

Each FEA program may come with an element library, or one is constructed over time. Some sample elements are: • • • • • • • • •

Rod elements Beam elements Plate/Shell/Composite elements Shear panel Solid elements Spring elements Mass elements Rigid elements Viscous damping elements

Many FEA programs also are equipped with the capability to use multiple materials within the structure such as: • • •

Isotropic, identical throughout Orthotropic, identical at 90 degrees General anisotropic, different throughout

Types of Engineering Analysis Structural analysis consists of linear and non-linear models. Linear models use simple parameters and assume that the material is not plastically deformed. Non-linear models consist of stressing the material past its elastic capabilities. The stresses in the material then vary with the amount of deformation as in Figure 4. Vibrational analysis is used to test a material against random vibrations, shock, and impact. Each of these incidences may act on the natural vibrational frequency of the material which, in turn, may cause resonance and subsequent failure.

Fatigue analysis helps designers to predict of a material or structure by showing the of cyclic loading on the specimen. Such can show the areas where crack propagation likely to occur. Failure due to fatigue may show the damage tolerance of the material 5). Heat Transfer analysis models the conductivity or thermal fluid dynamics of the or structure (Figure 1). This may consist of a

the life effects analysis is most also (Figure

material steady-

state or transient transfer. Steady-state transfer refers to constant thermoproperties in the material that yield linear heat diffusion. Results of Finite Element Analysis FEA has become a solution to the task of predicting failure due to unknown stresses by showing problem areas in a material and allowing designers to see all of the theoretical stresses within. This method of product design and testing is far superior to the manufacturing costs which would accrue if each sample was actually built and tested.

The History and Brief Introduction of Fracture by: Sean Grealis for MSE2094 Introduction The history of analytical approaches to studying fracture has been relatively short lived in comparison with other methods. Only beginning around the turn of this century, there has not been much time in which to do research and theoretical analysis of fracture, yet the advances made have been significant, if not the backbone of much of our advancement in this century. The work of such men as A. A. Griffith and G. R. Irwin are just a few of the people who have helped to make these advances possible. Griffith's theory of brittle fracture helps us to understand why brittle fracture occurs in a material. Likewise, as we will later see, Irwin took Griffith's work and applied it to ductile materials, which is also beneficial. Also work in the areas of fatigue and stress concentration have enabled us to make more advances as far as the uses of specific materials are concerned.

Work of Griffith A. A. Griffith started his work in around the 1920s. At this time, it was accepted that the theoretical strength of a material was taken to be E/10, where E is Young's Modulus for the particular material. He was only considering elastic, brittle materials, in which no plastic deformation took place. However, it was observed that the true values of critical strength was as much as 1000 times less than this predicted value, and Griffith wished to investigate this discrepancy. He discovered that there were many microscopic cracks in every material which were present at all times. He hypothesized that these small cracks actually lowered the overall strength of the material because as a load is applied to these cracks, stress concentration is experienced. This stress concentration magnifies the stresses at the crack tip, and these cracks will grow much more quickly, thus causing the material to fracture long before it ever reaches its theoretical strength. It should be noted that Griffith believed that, at the crack tips, the value of stress actually reached the theoretical maximum, but the overall average of the stress was lowered. It should also be

noted that this phenomenon of stress concentration is not only relegated to microscopic cracks in a material. Any void in the material (holes that have been machined or drilled out), corners, or hollow areas in the internal area of the material also cause stress concentration to occur, and most times, fracture will begin in one of these areas simply because of this phenomenon. (fig 22.3 Reed-Hill) From this work with stress concentration and working with elastic, brittle materials, Griffith formulated his own theory of brittle fracture, using elastic strain energy concepts. His theory described the behavior of crack propagation of an elliptical nature by considering energy methods. The equation basically states that when a crack is able to propagate enough to fracture a material, that the gain in the surface energy is equal to the loss of strain energy, and is considered to be the primary equation to describe brittle fracture.

Work of Irwin Griffith's work was significant, however it did not include ductile materials in its consideration. Another man, G. R. Irwin, in the 1950s, began to see how the theory would apply to ductile materials. He determined that there was also a certain energy from plastic deformation that had to be added to the strain energy originally considered by Griffith in order for the theory to work for ductile materials as well, creating what is known as the strain energy release rate.

Stress Intensity The term stress intensity is not to be confused with stress concentration work done by Griffith. The stress concentration is how the stress is amplified at a crack tip, whereas the stress intensity is used to describe the distribution of stress around a particular flaw. This term is used when investigating modes of fracture (link to part about K1c), in particular, mode I fracture, which is the most common. This term is used when computing the plane stresses and strains which exist in front of a moving crack. This value is dependent upon many things and is different for each material. Among the things which it depends on is the applied stress, the size and placement of the crack, as well as the geometry of the specimen.

Fatigue Fatigue is a special kind of failure in which fracture occurs not because of an instantaneous load that is a applied, causing a crack to grow. Rather, it is because a stress is applied for some period of time in which the cracks gradually grow until they finally reach a critical level. This concept is especially important when dealing with metals because it is the single most common cause of failure in metallic structures. There has been much study done on the concept, and much has been learned since the beginning of

the study of fatigue. Since much has been learned about fatigue, much has been done in the way of learning how to prevent it. For instance it is almost universally accepted that the cracks in fatigue always start on the surface of a material, so therefore, in order to prevent fatigue from occurring, one should strengthen the surface of the material, making it more difficult to fatigue.

Energy Concepts for Fracture by Jireh J. Yue Introduction In selecting materials for a given application one must have an idea of the final geometry and the dimension of the part. Under certain environmental conditions and given loadings the part must be able to function properly. One way to make this decision is by comparing the failure criterion to a critical load factor. Fracture is a very complex process that involves the nucleation and growth of micro and macro voids or cracks, mechanisms of dislocations, flip bands, and propagation of microcracks, and the geometry of the material. There has been no one set theory "set in stone" to handle all of these factors in fracture. However there are many proposed theories used to understand the complex nature of fracture in the material. One such class of theories involved energy concepts. In order to understand the complex nature of fracture in materials, one must understand the nature and character of initial cracking. This is only possible if we know the distribution of internal stressed in the body, but also the stress needed to initiate fracture and the length of the crack as shown in Figure 1.

Fig. 1 A plate with a crack growing with an applied stress. (From Parton V.Z., Fracture Mechanics from Theory to Practice, Pg. 69, Figure 48, Gordon and Breach Science Publishers.) Griffith Theory of Brittle Fracture

One such introductory model was developed by a young English scientist called AA Griffith. He recognized the macroscopic potential energy of the system consisting of the internal stored elastic energy and the external potential energy of the applied loads, varied with the size of the crack. Therefore fracture is associated with the consumption of energy. U : the total potential energy of the system U0: the elastic energy of the uncracked plate. Ua: the decrease in the elastic energy caused by introducing the crack in the plate. Uγ: the increase in the elastic-surface energy caused by the formation of the crack surfaces. Once a crack is propagated throughout a material as illustrated in Figure 1. , the extension of the crack resulted in the creation of new crack surface. New free surfaces are created at the faces of a crack, which increases the surface energy of the system. Such new surfaces can be seen in Figure 2 and Figure 3.

Fig. 2 Fractograph of ductile cast iron showing a transgranular fracture surface. (From Callister, W.D. Jr. , Materials Science and Engineering : An Introduction, Pg. 187, Figure 8.6, John Wiley and Sons, Inc.)

Fig. 3 Fractograph of an intergranular fracture surface. (From Callister W.D. Jr., Materials Science and Engineering : An Introduction , Pg. 187 Figure 8.6, John Wiley and Sons, Inc.) One such model used to demonstrate the propagation of a crack in a brittle material is called the elastic strain energy model.

E: modulus of elasticity γs = specific surface energy a = one half the length of an internal crack In today's material world many materials also experience some plastic deformation during fracture during fracture. Therefore the crack extension involves more than just an increase in surface energy. γp represents a plastic deformation energy associated with crack extension. γs + γp can be substituted into the above equation to model materials that undergo some plastic deformation.

G: the strain energy release rate. γp: plastic deformation energy associated with crack extension. γs: the specific surface energy Please note that crack propagation can only occurs when it exceeds the critical value of G. Example Problem If the specific surface energy for polmethyl acrylate is 36.5 ergs/ cm2 and its corresponding modulus of elasticity is 2.38 GPa. Compute the critical stress required for propagation of an internal crack length is 0.03m. Solution (36.5 ergs / cm2)*(1 Joule / 1.0 * 107 ergs)*(100 cm / m)2 = .0365 J / m2 2.38 GPa = 2.38 *109 Pa

σc= (2(2.38*109 Pa)(0.0365 J / m2 ) / ( * (0.03m)/2) ) ^ (1/2) σc= 60,719 Pa Griffith - Orowan - Irwin Failure Criteria The process of fracture consists of crack initiation and crack propagation. The condition necessary for crack initiation is if the crack-like cut is able to propagate. If there was a perfectly elastic body with a slit already present. If the slit is to propagate thereby increasing its surface, the slit will need a certain amount of energy. This energy is called the energy of fracture. With the formation of a new surface the strain in the corresponding area will be reduced which results in the release of corresponding elastic energy from the body. δτ: the energy of fracture necessary for the formation of a new fracture surface area. G :the energy released into the crack tip per unit area of the crack (rate of elastic strain energy release). δ: the crack growth increment.

Energy Release Rate The energy release rate often denoted by G is the amount of energy, per unit length along the crack edge, that is supplied by the elastic energy in the body and by the loading system in creating the new fracture surface area. In terms of the stress intensity factor there is relationship called the Irwin relationship. Note that there our two models for the stress intensity factor one for plane stress and plane strain.

(Plane Strain) G: the energy release rate. ν: Poisson's Ratio. K: the stress intensity factor. E: the modulus of elasticity. G=K2/E (Plane Stress) G: the energy release rate. K: the stress intensity factor. E: the modulus of elasticity. The total energy release rate in combined mode cracking can be obtained by adding the energies from the different modes (Figure 4.),

Fig. 4 Three modes of crack surface displacements Mode I (opening or tensile mode), Mode II (sliding mode), and Mode III (tearing mode). (From Parton V.Z. Fracture Mechanics from Theory to Practice Pg. 66 Figure 47, Gordon and Breach Science Publishers.)

These models by Irwin started the foundation of linear elastic fracture mechanics (LEFM). This discipline of fracture mechanics characterizes the state of material loading over a volume of sufficient size that the fracture strength of many engineering materials can be given in terms of the critical (maximum) stress intensity factor, KIC.

Fig. 5 A cracked body with a force (F) and (a) is the crack length. (From Portela A., Dual Boundary Element Analysis of Crack Growth, Pg. 26 Figure 2.4, Computational Mechanics Publications.) Shown in the figure is a cracked body with a force being exerted on it and the propagation of a crack (Figure 5.). P: the potential energy of the external forces. F: the generalized force per unit thickness. ∆: the corresponding load-point displacement. According to Clapyron's Theorem, the strain energy is:

U: the strain energy. F: the generalized force per unit thickness. ∆: the corresponding load-point displacement. Crack Speed and Kinetic Energy

In the previous sections, we assumed that the crack growth was slow. Fracture instability occurs when the energy release rate G remains larger than the crack resistance. The surplus of energy is converted into kinetic energy which governs the speed at which the crack will propagate through the material. The total amount of energy that is converted into kinetic energy after a crack growth a.

Ekin: the kinetic energy. G: the energy release rate. R: the force of crack resistance. The assumptions with the following model are: 1. crack propagation takes place under constant stress 2. the elastic energy release rate does not depend upon crack speed 3. the crack growth resistance R is constant The crack resistance is a function of the plastic behavior of the material at the crack tip and of its fracture characteristics. This particular property is dependent upon strain rate. At the tip of a crack moving at high velocity the strain rates are very high, and it must be expected that the material behaves in a more brittle manner the higher the crack speed. Crack Growth using Energy Theorems So far we have assumed that the crack resistive force is independent of crack length. This is true only for crack under plain strain. For plane stress, the crack resistance varies with amount of crack growth. When a particular specimen is loaded, the crack starts propagating, a further increase of the stress is required to maintain crack growth, although the crack is longer it can withstand a higher stress. During stable crack growth the energy release rate is equal to the crack resistive force. As can be seen in Figure 6. the growth rate of the crack increases with an increase in crack size. For example in a ductile material the energy required for crack growth is the same as the amount of work for formation of a new plastic zone at the tip of the advancing crack, plus the work required for initiation, growth and coalescence of microvoids.

Fig. 6 A graph of the increase of growth rate with crack size. (From Broek D., Elementary Engineering Fracture Mechanics, Pg. 145 Figure 6.2, Kluwer Academic Publishers Group.) Fracture Energy of Specimen of Different Sizes In a plastic, cylindrical test specimen deforms in a way that the degrees of deformation is nearly constant along the whole cross section. Research has shown that deformation of geometrically similar specimens of different diameters made from similar material is the same in any phase of the tensile test and not only the reductions of area but the contour lines in the vicinity of necking. The absolute value of the stress is independent of the size is the strain rate is constant. A proportion law of L. Gillemot states that to the same deformation of two geometrically similar specimens from similar material, similar specific energy is necessary if the cross head speed is proportional with the diameters of the specimen.

U1 and U2: cross head speeds applied during the tensile test. d1 and d2: diameters of the specimen. Impact Energy Testing Methods In order to learn more about the complex nature of fracture in materials impact testing conditions were established. The conditions that were judged the most relative to the potential for fracture are:

1. deformation at low temperatures 2. a high strain rate (rate of deformation) 3. a triaxial stress state Two tests called the Charpy and Izod tests are used to measure the impact energy (also known as notch toughness). These tests are important, because one can obtain information to model the behavior of actual structures so that the laboratory test results can be used to predict service performance under different environments. With the Charpy V-notch (CVN) technique, the specimen is in the shape of a bar of square cross section with a V notch. The load is applied as an impact below from a weighted pendulum hammer that is released from a position h. The pendulum with a knife edge strikes and fractures the specimen at the notch. The pendulum continues its swing, rising to a maximum height h', which is lower than h. The energy necessary to fracture the test piece is directly calculated from the difference in initial and final heights of the swinging pendulum (Figure 7.). The impact energy (toughness) from the Charpy test is related to the area under the total stress-strain curve. The difference in the Charpy and the Izod techniques is in the way that the specimens are supported in the apparatus machine. One can expect that materials with large values of strength and ductility to have large impact fracture energies. One has to also note that the impact data are very sensitive to test conditions such as temperature, specimen size, and notch configuration as can be seen in Figure 8.

Fig. 7 Illustration of Charpy and Izod Impact Tests. (From Callister W.D. Jr., Materials Science and Engineering : An Introduction , Pg. 198 Figure 8.13, John Wiley and Sons, Inc.)

Fig. 8 A graph of the temperature dependence on the Charpy V-notch impact energy (curve A) and percent shear fracture (curve B). (From Callister W.D. Jr., Materials Science and Engineering : An Introduction , Pg. 199 Figure 8.14, John Wiley and Sons, Inc.) References 1. Callister, William, " Materials Science and Engineering : An Introduction", John Wiley and Sons, New York, New York 1994. 2. Portela A., "Dual Element Analysis of Crack Growth", Computational Mechanics Publications, 1993. 3. Parton V.Z., " Fracture Mechanics : From Theory to Practice", Gordon and Breach Science Publishers, 1992. 4. Rolfe, Stanley, Barson John. "Fracture and Fatigue Control in Structures", Prentice Hall, Inc., 1977. 5. Broek, D. "Elementary Engineering Fracture Mechanics", Kluwer Academic Publishers Group, 1982. 6. Shackelford, J. " Introduction to Materials Science for Engineers", Macmillan Publishing Company, 1985.

Fatigue by: Shawn M. Kelly Overview

• • • • • • •

Cyclic Stresses S-N Curve Crack Initiation and Propagation Propagation Rate Factors That Affect Fatigue Life Example Problems References

The concept of fatigue is very simple, when a motion is repeated, the object that is doing the work becomes weak. For example, when you run, your leg and other muscles of your body become weak, not always to the point where you can't move them anymore, but there is a noticeable decrease in quality output. This same principle is seen in materials. Fatigue occurs when a material is subject to alternating stresses, over a long period of time. Examples of where Fatigue may occur are: springs, turbine blades, airplane wings, bridges and bones. This page will cover the topics included in Materials Science and Engineering, and Introduction by Callister, as well as other information that may be helpful to the student in an introductory materials science class.

Cyclic Stresses There are three common ways in which stresses may be applied: axial, torsional, and flexural. Examples of these are seen in Fig. 1.

Figure 1 Visual examples of axial stress, torsional stress, and flexural stress. There are also three stress cycles with which loads may be applied to the sample. The simplest being the reversed stress cycle . This is merely a sine wave where the maximum stress and minimum stress differ by a negative sign. An example of this type of stress cycle would be in an axle, where every half turn or half period as in the case of the sine wave, the stress on a point would be reversed. The most common type of cycle found in

engineering applications is where the maximum stress (σmax)and minimum stress (σmin) are asymmetric (the curve is a sine wave) not equal and opposite. This type of stress cycle is called repeated stress cycle. A final type of cycle mode is where stress and frequency vary randomly. An example of this would be automobile shocks, where the frequency magnitude of imperfections in the road will produce varying minimum and maximum stresses.

The S-N Curve A very useful way to visualize time to failure for a specific material is with the S-N curve. The "S-N" means stress verse cycles to failure, which when plotted uses the stress amplitude, σa plotted on the vertical axis and the logarithm of the number of cycles to failure. An important characteristic to this plot as seen in Fig. 2 is the fatigue limit.

Figure 2 A S-N Plot for an aluminum alloy The significance of the fatigue limit is that if the material is loaded below this stress, then it will not fail, regardless of the number of times it is loaded. Material such as aluminum, copper and magnesium do not show a fatigue limit, therefor they will fail at any stress and number of cycles. Other important terms are fatigue strength and fatigue life. The stress at which failure occurs for a given number of cycles is the fatigue strength. The number of cycles required for a material to fail at a certain stress in fatigue life.

Crack Initiation and Propagation Failure of a material due to fatigue may be viewed on a microscopic level in three steps: 1. Crack Initiation: The initial crack occurs in this stage. The crack may be caused by surface scratches caused by handling, or tooling of the material; threads ( as in a screw or bolt); slip bands or dislocations intersecting the surface as a result of previous cyclic loading or work hardening. 2. Crack Propagation: The crack continues to grow during this stage as a result of continuously applied stresses

3. Failure: Failure occurs when the material that has not been affected by the crack cannot withstand the applied stress. This stage happens very quickly.

Figure 3 A diagram showing location of the three steps in a fatigue fracture under axial stress One can determine that a material failed by fatigue by examining the fracture sight. A fatigue fracture will have two distinct regions; One being smooth or burnished as a result of the rubbing of the bottom and top of the crack( steps 1 & 2 ); The second is granular, due to the rapid failure of the material. These visual clues may be seen in Fig. 4:

Figure 4 A diagram showing the surface of a fatigue fracture. Notice that the rough surface indicates brittle failure, while the smooth surface represents crack propagation Other features of a fatigue fracture are Beachmarks and Striations. Beachmarks, or clamshell marks, may be seen in fatigue failures of materials that are used for a period of time, allowed to rest for an equivalent time period and the loaded again as in factory usage. Striations are thought to be steps in crack propagation, were the distance depends on the stress range. Beachmarks may contain thousands of striations. Visual Examples of Beachmarks and Striations are seen below in Fig. 5 and 6:

Figure 5 An example of beachmarks or "clamshell pattern" associated with stress cycles that vary in magnitude and time as in factory machinery

Figure 6 An example of the striations found in fatigue fracture. Each striation is thought to be the advancement of the crack. There may be thousands of striations in a beachmark Demonstration of Crack Propagation Due to Fatigue

The figure above illustrates the various ways in which cracks are initiated and the stages that occur after they start. This is extremely important since these cracks will ultimately lead to failure of the material if not detected and recognized. The material shown is pulled in tension with a cyclic stress in the y ,or horizontal, direction. Cracks can be initiated by several different causes, the three that will be discussed here are nucleating slip planes, notches. and internal flaws. This figure is an image map so all the crack types and stages are clickable. For more information on clickable maps and how to do them see the clickable map tutorial. Other Useful Links • • •

Fatigue Fractography (Halahan, Mutter), Excellent pictures of fatigue fracture Energy Methods and Crack Initiation and Propagation (Yue) Fatigue: Experimental (Meyer), Additional diagrams of crack propagation

Propagation Rate

The rate at which a crack grows has considerable importance in determining the life of a material. The propagation of a crack occurs during the second step of fatigue failure. As a crack begins to propagate, the size of the crack also begins to grow. The rate at which the crack continues to grow depends on the stress level applied. The rate at which a crack grows can be seen mathematically in equation 8.16 in Callister by:

Eq. 1 The variables A and m are properties of the material, da is the change in crack length, and dN is the change in the number of cycles. K is the change in the stress intensity factor or by equation 8.17(a & b):

Eq. 2 Rearrangement and integration of Eq. 1 gives us the relation of the number of cycles of failure, Nf, to the size of the initial flaw length, ao, and the critical crack length, ac, and Eq. 2:

Note: Nf is an estimate of the number of cycles to failure Eq. 3

Factors That Affect Fatigue Life and Solutions The Mean Stress, discussed in Callister, 8.8, is defined as:

Eq. 4 The Mean stress has the affect that as the mean stress is increased, fatigue life decreases. This occurs because the stress applies is greater. I mentioned previously that scratches and other imperfections on the surface will cause a decrease in the life of a material. Therefore making an effort to reduce these imperfections by reducing sharp corners, eliminating unnecessary drilling and stamping, shot peening, and most of all careful fabrication and handling of the material. Another Surface treatment is called case hardening, which increases surface hardness and fatigue life. This is achieved by exposing the component to a carbon-rich atmosphere at

high temperatures. Carbon diffuses into the material filling interstisties and other vacancies in the material, up to 1 mm in depth.

Figure 7 A case hardened steel gear. Notice the effect of diffusion of Carbon into the material produces a "case" around the gear. Exposing a material to high temperatures is another cause of fatigue in materials. Thermal expansion, and contraction will weaken bonds in a material as well as bonds between two different materials. For example, in space shuttle heat shield tiles, the outer covering of silicon tetraboride (SiB4) has a different coefficient of thermal expansion than the Carbon-Carbon Composite. Upon re-entry into the earth's atmosphere, this thermal mismatch will cause the protective covering to weaken, and eventually fail with repeated cycles. Another environmental affect on a material is chemical attack, or corrosion. Small pits may form on the surface of the material, similar to the effect etching has when trying to find dislocations.

Figure 8 Example of pits formed by corrosion on the surface of LiF. The "chemical" attacks weak spots on the surface of the material, especially where dislocations intersect the surface. This chemical attack on a material can be seen in unprotected surface of an automobile, whether it be by road salt in the winter time or exhaust fumes. This problem can be solved by adding protective coatings to the material to resist chemical attack. Other Useful Links Dealing With Design Examples • •

Fatigue, Experimental (Meyer) Fracture Design Examples (Gordon)

Example problems 1. Consider a flat plate of some metal alloy that is to be exposed to repeated tensilecompressive cycling in which the mean stress is 25 MPa. If: ao = 0.25 mm, ac = 5.0 mm, m = 4.0, A = 5 * 10-15, Y = 2.0, and Nf =3.2 * 105 cycles . Find: Estimate the maximum tensile stress to yield the fatigue life prescribed Solution: Use Equations 3 above to solve for ∆σ.

Eq. 3

Comments or Questions? Email Shawn Kelly.

References 1

Beer, Ferdinand P, and E. Russell Johnston, Jr. Mechanics of Materials. 2nd ed. New York: McGraw-Hill, Inc. 1992. Images: Fig. 2.54(b), Fig. 3.8(b), Fig. 4.19 Reed-Hill, Robert E, and Reza Abbaschian. Physical Metallurgy Principles. 3rd ed. Boston: PWS Publishing Company, 1994. The following figures appear in Reed-Hill : 2

Fig. 21.34, page 752

3

Fig. 21.43, page 761

4

Fig. 21.30, page 749

5

Fig. 21.31, page 749

Callister, William D Jr. Materials Science and Engineering, an Introduction. 3rd ed. New York: John Wiley & Sons, Inc., 1994. The following figures appear in Callister: 6

Figure 8.24, page 209

7

Figure 5.0 , page 89

8

Reed-Hill: Fig. 5.3, page 127

Special thanks to those who provided links: Chris Meyer, Jireh Yue, Jared Mutter, Ron Halahan, and Matt Gordon Thanks to Brian Seal for his HTML skillz.

STRESS CONCENTRATION Author: Anita Noble Stress Concentration The fracture of a material is dependent upon the forces that exist between the atoms. Because of the forces that exist between the atoms, there is a theoretical strength that is typically estimated to be one-tenth of the elastic modulus of the material. However, the experimentally measured fracture strengths of materials are found to be 10 to 1000 times below this theoretical value. The discrepancy is explained to exist because of the presence of small flaws or cracks found either on the surface or within the material. These flaws cause the stress surrounding the flaw to be amplified where the magnification is dependent upon the orientation and geometry of the flaw. Looking at fig. 1, one can see a stress profile across a cross section containing an internal, ellipticallyshaped crack. One can see that the stress is at a maximum at the crack tip and decreased to the nominal applied stress with increasing distance away from the crack. The stress is concentrated around the crack tip or flaw developing the concept of stress concentration. Stress raisers are defined as the flaws having the ability to amplify an applied stress in the locale.

Fig. 1: (a) The geometry of surface and internal cracks. (b) Schematic stress profile along the line X-X' in (a), demonstrating stress amplification at crack tip positions. Determination of the Maximum Stress at the Crack Tip

If the crack is assumed to have an elliptical shape and is oriented with its long axis perpendicular to the applied stress, the maximum stress, σ m can be approximated at the crack tip by Equation 1.

Eqn. 1: Determination of the maximum stress surrounding a crack tip. The magnitude of the nominal applied tensile stress is σ o; the radius of the curvature of the crack tip is ρ; and a represents the length of a surface crack, or half the length of an internal crack.

Determination of Stress Concentration Factor The ratio of the maximum stress and the nominal applied tensile stress is denoted as the stress concentration factor, Kt, where Kt can be calculated by Equation 2. The stress concentration factor is a simple measure of the degree to which an external stress is amplified at the tip of a small crack.

Eqn. 2: Determination of the stress concentration factor. Stress Concentration Considerations It is important to remember that stress amplification not only occurs on a microscopic level (e.g. small flaws or cracks,) but can also occur on the macroscopic level in the case of sharp corners, holes, fillets, and notches. Fig. 2 depicts the theoretical stress concentration factor curves for several simple and common material geometries.

Fig. 2: Stress concentration factor plots for three different macroscopic flaw situations. Stress raisers are typically more destructive in brittle materials. Ductile materials have the ability to plastically deform in the region surrounding the stress raisers which in turn evenly distributes the stress load around the flaw. The maximum stress concentration factor results in a value less than that found for the theoretical value. Since brittle materials cannot plastically deform, the stress raisers will create the theoretical stress concentration situation.

Reference: Callister, William D. Materials Science and Engineering: An Introduction - 3rd Edition. John Wiley & Sons, Inc.: New York, 1994

Stress Intensity

The Liberty Bell (Philadelphia, PA) Jefferson Kim, MSE 2094, Term Project (Edited R.D. Kriz 3-5-00) Stress Intensity Factor, K, is used in fracture mechanics to more accurately predict the stress state ("stress intensity") near the tip of a crack caused by a remote load or residual stresses. When this stress state becomes critical a small crack grows ("extends") and the material fails. The load at which this failure occurs is referred to as the fracture strength. The experimental fracture strength of solid materials is 10 to 1000 times below the theoretical strength values, where tiny internal and external surface cracks create higher stresses near these cracks, hence lowering the theoretical value of strength. The large crack seen in the picture of the Liberty Bell was the result of small cracks and internal residual stresses not known at the time. The original, "as fabricated" cracks were very small and hard to see with naked eyes, and according to Hertzberg, during the war against the British, the bell was polished whenever they saw a crack on the surface. Hardly a solution based on what we understand today. Unlike "stress concentration", Stress Intentsity, K, as the name implies, is a parameter that amplifies the magnitude of the applied stress that includes the geometrical parameter Y (load type). These load types are categorized as Mode-I, -II, or -III. The Mode-I stress intensity factor, KIc is the most often used engineering design parameter in fracture mechanics and hence must be understood if we are to design fracture tolerant materials used in bridges, buildings, aircraft, or even bells. Polishing just won't do if we detect a crack. Typically for most materials if a crack

can be seen it is very close to the critical stress state predicted by the "Stress Intensity Factor".

Stress Analysis of Cracks

Generally there are three modes to describe different crack surface displacement in Fig.8.3 (Hertzberg, p321). Mode I is opening or tensile mode where the crack surfaces move directly apart. Mode II is sliding or in-plane shear mode where the crack surfaces slide over one another in a direction perpendicular to the leading edge of the crack. Mode III is tearing and antiplane shear mode where the crack surfaces move relative to one another and parallel to the leading edge of the crack. Mode I is the most common load type encountered in engineering design and will be explained here in more detail. The value of the stress intensity factor, K, is a function of the applied stress, the size and the position of the crack as well as the geometry of the solid piece where the cracks are detected, Fig.8.5 (Hertxberg, p323). The tensile stress in X and Y directions, and the shear stress in the X-Y plane can calculated in terms of K and position can be written as: Mode-I

Mode-I

Fracture Toughness, KIc Engineers are mostly worried about the brittle fracture because the brittle fractures bring most devastating accidents and happen rapidly, and usually the brittle fractures take place when the applied stress increases such that the stress state at the crack tip reaches a critical value. The fracture toughness can be defined in terms of the stress intensity factor, K, but at a critical stress state. as:

where Y is a dimensionless parameter that depends on both the specimen and crack geometry in Fig8.11(Callister, p193), and the greek symbol "Sigma" is an applied stress and "a" is crack length. Generally, for the elliptical shaped crack,the equation is modified to include the geometry of the crack with three different Y's.

However, Y factor is 1.0 for the plate of infinite width and 1.1 for a plate of semi-infinite width. When the thickness of specimen is very large with respect to the crack length, the stress intensity factor for Mode I is often called the plane strain "Fracture Toughness". This modification of stress intensity into a plane strain fracture toughness parameter can be approximated by a relationship that includes specimen geometry, and yield strength.

Hence the specimen thickness is shown to be the most significant parameter that controls the transition of fracture toughness from "plane stress" to "plane strain", see Fig. 8.12 (Callister, p194).

The plain strain fracture toughness for Mode I, KIc is also a function of many other factors such as temperature, strain-rate and microstructure. Hence KIc is unique for a particular material and is a fundamental material property so it is a very important consideration for material selection and design.

Designing and Preventing Fracture with KIc KIc, stress, and Y factor are important variables for engineers to design and to determine the safety of machinery, and often the size of the cracks is a very important factor to make decisions such that the maximum allowable size of the crack can be written as

Also because KIc is unique for a particular material, engineers can use this variable for selecting appropriate materials for a range of different applications. From the table in Appendix B engineers can also decide how much load and stress can be allowed for a particular specimen geometry. This critical information helps engineers to optimize the

design and the safety on the operations and to prevent or minimize possible accidents. For example, in aircraft components, there are a lot of rivet holes and small cracks which bring Y calibration factor high up to the critical stress. What engineers do is measure the length of cracks to calculate the maximum cracks length and to compare with safety measurement. They can also make a hole at the tip of cracks, which brings down Y calibration factor and the also the the stress concentration. Additionally, engineers clean the fracture surfaces to prevent further damages. Not only does cleaning lower the Y calibration, but it also helps to protect the surface from undesireable chemical reactions. Various cleaning methods are described in the table below.

Table A.1 (Hertzberg, p752)

Conclusion Stress intensity and fracture toughness are critically important fracture mechanics parameters used by materials engineers and designers. We saw that there are a lot of factors that determine fracture of a material. KIc is an unique material property, that is used by engineers to design and manufacture products for durability and safe operation. Appendix B (Hertzberg, p757) K Calibrations for Typical Test Specimen Geometries

References Callister, W. Materials Science and Engieering. John Wiley and Sons, New York1994. Hertzberg, R. Deformation and Fracture Mechanics of Engineering Materials. John Wiley and Sons, New York 1996.

Theory of Finite Element Analysis Finite Element Analysis (FEA) is a numerical method which provides solutions to problems that would otherwise be difficult to obtain. In terms of fracture, FEA most often involves the determination of stress intensity factors. FEA, however, has applications in a much broader range of areas; for example, fluid flow and heat transfer. While this range is growing, one thing will remain the same: the theory of how the method works. The most efficient method of learning is by example. Therefore, I would like to present to you a simple FEA problem: the case of a three-member truss. The method of solution to this problem should demonstrate the basic concepts of FEA which are present in any analysis. Before introducing specific quantities for our example, let's first take a look at our structure:

The overall objective of our analysis will be to determine the displacements of the truss members given the load P. The first thing we must do is choose our elements. For our situation this is easy: each truss member should be one element. Further division would accomplish nothing, since each truss member can only support axial loads. Let us now examine a single truss member:

(more info. on nodes)

Nodes are located at each end of the bar, each of which can have displacements in the x and y directions. The displacements are denoted u1, u2, u3, and u4. Corresponding forces due to these displacements are F1, F2, F3, and F4. The bar has a uniform cross-sectional area A and Young's Modulus E. The general relationship between force and displacement is Fi = kij*uj, where Fi is the force in direction i, uj is the displacement in direction j, and kij is the "stiffness" coefficient relating Fi to uj. In our particular example of a horizontal truss element, we have the following system of equations: F1 = k11u1 + k12u2 + k13u3 + k14u4 F2 = k21u2 + k22u2 + k23u3 + k24u4 F3 = k31u1 + k32u2 + k33u3 + k34u4 F4 = k41u1 + k42u2 + k43u3 + k44u4 Alternatively, in matrix form:

The matrix kij is called the " stiffness matrix." It is the matrix which defines the geometric and material properties of the bar. Stiffness matrices are a fundamental part of FEA. These matrices always define inherent properties of the system being studied. For the system at hand, we need to determine the stiffness matrix. The way we will go about doing this may seem a little strange at first, but try to follow the reasoning as it does make sense. Let's begin by assuming u1 = 1 and u2 = u3 = u4 = 0. Then our matrix takes the form:

Each force Fi is equal to kj1. Now, recall from mechanics of materials that the displacement of a rod is given by u = FL/AE. With displacement u1 = 1, force 1 is F1 = AE/L. To maintain equilibrium, we must also have a force F3 = -AE/L:

Since our Fi's equal our ki1's, we have:

It important to remember that our element can support only axial loads. Therefore, displacements u2 and u4 can not give rise to stresses in the bar since these displacements are perpendicular to the axis of the bar. Thus, the stiffness coefficients of these displacements must be zero: ki2 = ki4 = 0. Finally, a displacement u3 = 1 will result in forces just opposite to those from u1 = 1, so ki3 = -ki1. Our stiffness matrix is:

It must be emphasized that the stiffness matrix just derived is only valid for bars parallel to the x-axis. Through a similar derivation it can be shown that the stiffness matrix for any bar oriented at an angle "theta" to the x-axis is:

where c = cos"theta" and s = sin"theta". Note that when "theta" = 0, this stiffness matrix reduces to the one we derived for a horizontal bar. Now knowing the stiffness matrix for any axially loaded bar, we can apply it to a real situation with specific quantities. Consider the following truss:

The displacements and external forces are:

Note the symbols we are using: R is an external force on the truss; F is an internal force resulting from the stresses imposed on the structure during a displacement. Knowing the orientations of each element, we can set up matrices for them. Using "theta" = 90 degrees for element 1, "theta" = 135 degrees for element 2, and "theta" = 0 degrees for element 3 we obtain the following matrices:

Element 1:

Element 2:

Element 3: We can now generate a set of equilibrium equations for each node. Consider the following figure:

The nodal forces (resulting from element displacements) must be equal and opposite to the externally applied forces. Note that we have all forces drawn in positive x and y directions. Thus, for equilibrium at node 1: x - direction: R2 - F2(element3) - F2(element2) = 0 y - direction: R1 - F1(element3) - F1(element2) = 0 We want to solve for R1 and R2. Obtaining the nodal forces F2(element3), F2(element2), F1(element3), and F1(element2) from our previously determined matrices we get: R1 = AE/L ( 3u1/2 - u2/2 - u3 - u5/2 + u6/2 ) R2 = AE/L ( -u1/2 + u2/2 + u5/2 - u6/2 ). Similarly, from equilibrium of nodes 2 and 3 we obtain: R3 = AE/L ( -u1 + u3 ) R4 = AE/L ( u4 - u6 ) R5 = AE/L ( -u1/2 + u2/2 + u5/2 - u6/2 ) R6 = AE/L ( u1/2 - u2/2 - u4 - u5/2 +3u6/2 )

We can now combine all of our external forces into one matrix:

Now recall what we are trying to do here: given a load P, we want to solve for the displacements at each node. Observing that node 2 is pinned and that node 3 is on a roller, the displacements u3, u4, and u5 must equal 0. These values are quite important because without them we wouldn't be able to solve the problem. As a matter of fact, values such as these are always needed in finite element analyses; they are known as " boundary conditions." Next, we must state the reactions which are known from our particular loading. We can see from the truss that R1 = 0, R2 = -P, and R6 = 0. Entering the known displacements and reactions into our matrix we get:

This matrix reduces to:

We can now finish our problem by solving this matrix for u1, u2, and u6: u1 = - PL/AE u2 = -4PL/AE u6 = -PL/AE This application of FEA to a simple three-member truss shows in general how the method works. Most applications to engineering problems, however, are much more complex. Such analyses require large numbers of elements and nodes in order to accurately represent the physical system being studied. These analyses inevitably require the application of a computer.

For Virginia Tech engineering students interested in FEA, one undergraduate course is available: ESM 4734 - An Introduction to the Finite Element Method. In this course, students study the theory and application of FEA to problems in various fields of engineering and applied sciences. The pre-requisite for the course is: ESM 2074 Computational Methods. References Finite Element Analysis on Microcomputers, Nicholas M. Baran, McGraw-Hill Book Company, 1988. Finite Element Primer, Bruce Irons and Nigel Shrive, John Wiley & Sons, Inc., 1983.