Mae 640 Lec1

Continuum mechanics – MAE 640 Summer II – 2009

Dr. Konstantinos Sierros 263 ESB new add [email protected]

Important information

Class: 2:45pm – 3:50pm Everyday for the next 6 weeks (Jun 29 – Aug 06) Room: G84 – ESB Course blog: http://wvumechanicsonline.blogspot.com

Sources

Applied Mechanics of Solids Allan F. Bower

Course book An Introduction to Continuum Mechanics J. N. Reddy

Introduction CONTINUUM THEORY • Matter is formed of molecules which in turn consist of atoms and sub-atomic particles. Thus matter is not continuous. • In continuum mechanics we assume, for analysis purposes, that matter is continuous. • A hypothetical continuous matter is termed a continuum. Examples that can be described and analyzed using a continuum approach include:  The deflection of a structure under loads  The rate of discharge of water in a pipe under a pressure gradient  The drag force experienced by a body moving in the air

Continuum theory The theory that describes relationships between gross phenomena, neglecting the structure of material on a smaller scale, is known as continuum theory.

1. The continuum theory regards matter as indefinitely divisible. 2. Accept infinitesimal volume of materials referred to as a particle in the continuum. 3. In every neighborhood of a particle there are always neighboring particles. 4. Whether the continuum theory is justified in a given situation is a matter of experimental testing.

Example

∆m ρ = lim 3 ∆V →ε ∆V density: mass/unit volume Density of a material at a point is defined as the ratio of the mass Δm of the material to a small volume ΔV surrounding the point when ΔV tends to be ε3 • ε is small compared to the mean intermolecular distance • In fact, ε→0 A mathematical study of mechanics of such an idealized continuum is called continuum mechanics

Objectives • Study the conservation principles in mechanics of continua • Formulate related equations describing motion and mechanical behavior of materials • Present applications of these equations Examples - Fluid flow - Conduction of heat - Deformation of solid bodies

Solid mechanics Design of a diving board of a given length L (sufficient to allow the swimmer to gain enough momentum), fixed at one end and free at the other end. The board is initially straight and horizontal and has uniform cross section.

Process 2. Select the material (Young’s modulus E) 3. Select dimensions b and h such that the board caries the moving weight W of the athlete

Design criteria • Stresses developed must not exceed the allowable stress values • The deflection δ of the free end must not exceed the prespecified value A preliminary design can be done by using normal mechanics of materials equations

• Using equations of elementary beam theory we can relate δ, L, b, h, E and W

3

4WL δ= 3 Ebh

• Given the deflection δ and max W one can select L,b and h The final design in 3D must be done using more sophisticated equations such as 3D elasticity equations

Fluid mechanics Measure viscosity μ of a lubricating oil used in rotating machinery to prevent the damage of the parts that are in contact. (Viscosity is a material’s property used in calculation of shear stresses in fluid-solid interactions)

• A capillary tube is used in order to measure the viscosity

πd P1 − P2 µ= 128 L Q 4

Where; d is internal diameter, L is length of tube, P1 and P2 are pressures at two ends, Q is volume rate of flow

Heat transfer

Determine the heat loss through the wall of a furnace. The wall consists of a brick layer, cement mortar and cinder block. Each of these materials provides a different degree of thermal resistance.

dT q = −k dx • Using Fourier heat conduction law we can relate the heat flux q (heat flow/unit area) and the gradient of temperature T. Thermal conductivity is denoted by κ. • The negative sign in the equation indicates that heat flows from high temperature to low temperature region.

Governing equations • • • • • • •

Governing equations for the study of deformation and stress of a continuous material are analytical representation of the laws of conservation of mass, momenta and energy They are applicable to all materials that are treated as a continuum There are four categories supporting the study of motion and deformation of a continuum Kinematics (strain-displacement equations) Kinetics (conservation of momenta) Thermodynamics (1st and 2nd Law of thermodynamics) Constitutive equations (stress-strain relations) Kinematics – Study of geometric changes/deformation in a continuum without considering the forces that cause the deformation Kinetics – Study of static/dynamic equilibrium of forces and moments acting on a continuum Thermodynamics – Conservation of energy and relations between heat, mechanical work and thermodynamic properties of continuum Constitutive equations – Thermomechanical behaviour of the material of the continuum.

Summary • Concept of continuous medium discussed • Use of physical principles in order to derive the equations governing a continuous medium • Use these equations in the solution of specific problems • The four topics presented in the previous table form the chapters 3 through 6  Expressing mathematically the governing equations of a continuous medium requires the use of vectors and tensors  Vectors and tensors are covered in Chapter 2