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Continuum mechanics MAE 640

Summer II 2009

Dr. Konstantinos Sierros

263 ESB new add

[email protected]

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Chapter 4: Stress Measures

Introduction

All materials have certain threshold to withstand forces, beyond which they fail to

perform their intended function.The force per unit area, called stress, is a measure of the capacity of the material to

carry loads.

It is necessary to determine the state of stress in a material.

In this chapter we study the concept of stress and its various measures.

For example, stress at a point in a three-dimensional continuum can be measured in

terms of nine quantities, three per plane, on three mutually perpendicular planes at the

point.

These nine quantities may be viewed as the components of a

second-order tensor, called stress tensor.

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Cauchy Stress Tensor and Cauchys Formula

First we introduce the true stress which is the stress in the deformed configuration that

is measured per unit area of the deformed configuration .

The surface force acting on a small element of area in a continuous medium depends

not only on the magnitude of the area but also upon the orientation of the area.

It is common to denote the direction of a plane area by means of a unit vector drawn

normal to that plane, as discussed in Section 2.2.3.

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Let the unit normal vector be denoted by n. Then the area is expressed as A =An.

If we denote by df( n) the force on a small area nda located at the position x, the

stress vectorcan be defined.

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stress vector

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The stress vector is a point function of the unit normal n which denotes the orientation

of the surface a.

The component oft that is in the direction of n is called the normal stress.

The component oft that is normal to n is called the shearstress.

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cause of Newtons third law for action and reaction, we have;

At a fixed point x for each given unit vector n, there is a stress vectort( n) acting on

the plane normal to n.

Note that t( n) is, in general, not in the direction of n. It is good to establish a

relationship between t and n.

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To establish the relationship between t and n, we now set up an infinitesimal

tetrahedron in Cartesian coordinates, as shown in the figure below;

t1,t2,t3, and t denote the stress vectors in the outward directions on the faces

Areas of the infinitesimal tetrahedron

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Using Newtons second law for the mass inside the tetrahedron,

volume of the tetrahedr

densitybody force per unit

mass

acceleration

Since the total vector area of a closed surface is zero (using the gradient theorem), we

have;

The volume of the element vcan be expressed as;

perpendicular

distance from

the origin to theslant face.

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and using these expressions;

And dividing by we have;

when the tetrahedron shrinks to a point (i.e.h 0);

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using the summation

convention

stress dyadic

orstresstensor

The stress tensor is a property of the medium that is independent of the n.

Therefore we have;

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he stress vector t represents the vectorial stress on a plane whose normal is

Cauchy stress formula

Cauchy

stress

tensor.

the current force per unit deformed area,

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In Cartesian component form, the Cauchy formula


ti= njji

Can be written

as;

and in matrix form

stress (force per unit area) on a plane perpendicular to thexicoordinate and in thexj

coordinate direction

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The component i jrepresents the stress (force per unit area) on a plane perpendicular

to thexicoordinate and in thexjcoordinate direction

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Transformation of Stress Components

The Cauchy stress is a second-order tensor therefore we can define;

Its invariants,

Transformation law

Eigenvalues and eigenvectors.

Invariants

Transformation law

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Principal stresses and principal planes

The determination of maximum normal stresses and shear stresses at a point is of

considerable interest in the design of structures because failures occur when the the

magnitudes of stresses exceed the allowable (normal or shear) stress values, called

strengths, of the material.It is of interest to determine the values and the planes on which the stresses are the

maximumTherefore, we must determine the eigenvalues and eigenvectors associated with the

stress tensor

Eigenvalues and eigenvectors

Yielding a cubic equation for, called the characteristic equation, the solution of which

yields three values of.

The eigenvalues of are called theprincipal stresses and the associated

eigenvectors are called theprincipal planes.

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