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Continuum mechanics MAE 640

Summer II 2009

Dr. Konstantinos Sierros

263 ESB new add

[email protected]

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Conservation of Momenta

Principle of Conservation of Linear Momentum

The principle of conservation of linear momentum, or Newtons 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 forceon the

collection of

particles

For constant mass;

Newtons second law for a control volume can be expressed as;

resultant force

vector representing

an area element of

the outflow

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Example 5.3.1

Suppose that a jet of fluid with area of cross-sectionA and mass density issues from a

nozzle with a velocity vand 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.

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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 QLand 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;

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Example 5.3.1

Solving the two equations forQL and QR, we obtain;

Thus, the total flow Q is divided into the left flow ofQL and right flow ofQRas givenabove.

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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 isequal to vector sum

of the moments of external forces acting on the continuum.

control volume control surface

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A jet of air ( = 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 jetinto two equal streams, and neglect any energy loss in the streams.

Problem 5.8 (p.174)

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Ch. 6: Constitutive Equations

Introduction

Constitutive equations are those relations that connect theprimaryfield variables (e.g.,

,

T, x, and u orv) to the secondaryfield variables (e.g., e, q, and ).

Constitutive equations are notderived 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.

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Introduction

A material body is said to be homogeneous if the material properties are the

same throughout the body (i.e., independent of position).

heterogeneous body, the material properties are a function of position.

An anisotropic body is one that has different values of a material property indifferent directions at a point, i.e., material properties are direction

dependent.An isotropic material is one for which every material property is the same inall directions at a point.

An isotropic or anisotropic material can be nonhomogeneous orhomogeneous.

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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 ofdeformation, such materials are termed viscous.

n this study, we are concerned with;

(a) Elastic materials for which the stresses are functions of the currentdeformation 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.

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The approach typically involves assuming the form of the constitutiveequation and then restricting the form to a specific one by appealing to

certain physical requirements, including invariance of the equations andmaterial frame indifferenceThis chapter is primarily focused on Hookean solids and Newtonian fluids.The constitutive equations presented in Section 6.2 for elastic solids arebased on small strain assumption.

Thus, we make no distinction between the material coordinates X andspatial coordinates x and between the Cauchy stress tensor and second

PiolaKirchhoff stress tensor S.

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Elastic Solids

Introduction

A material is said to be (ideally or simple) elasticorCauchy elasticwhen, under

isothermal conditions, the body recovers its original form completely upon removal of the

forces causing deformation

For Cauchy elastic materials, the Cauchy stress

;

F is the deformation gradient tensor

A material is said to be hyperelasticorGreen elasticif there exists a strain energy

densityfunction U0() such that

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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

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Generalized Hookes Law

The linear constitutive model for infinitesimal deformations is referred to as the

generalized Hookes law.

To derive the stressstrain relations for a linear elastic solid, begin with the quadratic

form ofU0;

Using

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Cmn have the same units as mn, and they represent the residual stresscomponents of a solid.

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We assume, that the body is free of stress prior to the load application sothat we may write;

The coefficients Cijkl are called elastic stiffness coefficients.

The components Cijkl satisfy thefollowing symmetry conditions

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s, 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 forstresses and strains and two subscript notation for the material stiffnesscoefficients:

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The single subscript notation for stresses and strains is called theengineering notation or the Voigt-Kelvin notation.


Therefore;

In matrix form

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

Si jare the material compliance coefficients

with [S] = [C]1 (i.e., the

Compliance tensor is the inverse of the

stiffness tensor: S = C1).

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In matrix form

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