Mae 640 Lec14

Continuum mechanics – MAE 640 Summer II – 2009

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

Conservation of Momenta Principle of Conservation of Linear Momentum •The principle of conservation of linear momentum, or Newton’s second law of motion, applied to a set of particles (or rigid body) can be stated as; “The time rate of change of (linear) momentum of a collection of particles equals the net force exerted on the collection”

total mass velocity

resultant force on the collection of particles

•For constant mass;

•Newton’s second law for a control volume Ω can be expressed as;

resultant force

vector representing an area element of the outflow

Example 5.3.1 Suppose that a jet of fluid with area of cross-section A and mass density ρ issues from a nozzle with a velocity v and impinges against a smooth inclined flat plate, as shown in Figure 5.3.1. Assuming that there is no frictional resistance between the jet and plate, determine the distribution of the flow and the force required to keep the plate in position.

Example 5.3.1 Since there is no change in pressure or elevation before and after impact, the velocity of the fluid remains the same before and after impact. Let the amounts of flow to the left be QL and to the right be QR. Then the total flow Q = vA of the jet is equal to the sum (by continuity equation);

Next, we use the principle of conservation of linear momentum to relate QL and QR;

Example 5.3.1

Solving the two equations for QL and QR, we obtain;

•Thus, the total flow Q is divided into the left flow of QL and right flow of QR as given above.

Principle of Conservation of Angular Momentum

e principle of conservation of angular momentum states that;

the time rate of change of the total moment of momentum for a continuum is equal to vector sum of the moments of external forces acting on the continuum.

control volume Ω

control surface Γ

Problem 5.8 (p.174) A jet of air (ρ = 1.206 kg/m3) impinges on a smooth vane with a velocity v = 50 m/s at the rate of Q = 0.4 m3/s. Determine the force required to hold the plate in position for the two different vane configurations shown in Fig. P5.8. Assume that the vane splits the jet into two equal streams, and neglect any energy loss in the streams.

Ch. 6: Constitutive Equations Introduction •Constitutive equations are those relations that connect the primary field variables (e.g., ρ, T, x, and u or v) to the secondary field variables (e.g., e, q, and σ). •Constitutive equations are not derived from any physical principles, although they are subject to obeying certain rules •Constitutive equations are mathematical models of the behavior of materials that are validated against experimental results. •The differences between theoretical predictions and experimental findings are often attributed to inaccurate representation of the constitutive behavior. First, we review certain terminologies that were already introduced in beginning courses on mechanics of materials.

Ch. 6: Constitutive Equations Introduction •A material body is said to be homogeneous if the material properties are the same throughout the body (i.e., independent of position).

a heterogeneous body, the material properties are a function of position. •An anisotropic body is one that has different values of a material property in different directions at a point, i.e., material properties are direction dependent. •An isotropic material is one for which every material property is the same in all directions at a point. •An isotropic or anisotropic material can be nonhomogeneous or homogeneous.

Ch. 6: Constitutive Equations Introduction •Materials for which the constitutive behavior is only a function of the current state of deformation are known as elastic. •If the constitutive behavior is only a function of the current state of rate of deformation, such materials are termed viscous.

In this study, we are concerned with; (a) Elastic materials for which the stresses are functions of the current deformation and temperature (b) Viscous fluids for which the stresses are functions of density, temperature, and rate of deformation. Special cases of these materials are the Hookean solids and Newtonian fluids.

Ch. 6: Constitutive Equations •The approach typically involves assuming the form of the constitutive equation and then restricting the form to a specific one by appealing to certain physical requirements, including invariance of the equations and material frameisindifference •This chapter primarily focused on Hookean solids and Newtonian fluids. The constitutive equations presented in Section 6.2 for elastic solids are based on small strain assumption. •Thus, we make no distinction between the material coordinates X and spatial coordinates x and between the Cauchy stress tensor σ and second Piola–Kirchhoff stress tensor S.

Elastic Solids Introduction •A material is said to be (ideally or simple) elastic or Cauchy elastic when, under isothermal conditions, the body recovers its original form completely upon removal of the forces causing deformation For Cauchy elastic materials, the Cauchy stress σ;

F is the deformation gradient tensor •A material is said to be hyperelastic or Green elastic if there exists a strain energy density function U0(ε) such that

Elastic Solids Introduction

For an incompressible elastic material (i.e., material for which the volume is preserved and hence J = 1 or div u = 0), the above relation is written as;

hydrostatic pressure

Generalized Hooke’s Law •The linear constitutive model for infinitesimal deformations is referred to as the generalized Hooke’s law. •To derive the stress–strain relations for a linear elastic solid, begin with the quadratic form of U0;

Using…

Generalized Hooke’s Law

•Cmn have the same units as σmn, and they represent the residual stress components of a solid.

Generalized Hooke’s Law •We assume, that the body is free of stress prior to the load application so that we may write;

The coefficients Cijkl are called elastic stiffness coefficients. The components Cijkl satisfy the following symmetry conditions

Generalized Hooke’s Law

us, the number of independent coefficients in Cijmn is reduced to 21;

We express Eq. (6.2.8) in an alternate form using single subscript notation for stresses and strains and two subscript notation for the material stiffness coefficients:

Generalized Hooke’s Law

•The single subscript notation for stresses and strains is called the engineering notation or the Voigt-Kelvin notation. Therefore; In matrix form…

Generalized Hooke’s Law

We assume that the stress–strain relations are invertible. Thus, the components of strain are related to the components of stress by;

Si j are the material compliance coefficients with [S] = [C]−1 (i.e., the Compliance tensor is the inverse of the stiffness tensor: S = C−1).

Generalized Hooke’s Law

In matrix form…