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Continuum mechanics MAE 640
Summer II 2009
Dr. Konstantinos Sierros
263 ESB new add
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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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Cauchy Stress Tensor and Cauchys Formula
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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Cauchy Stress Tensor and Cauchys Formula
stress vector
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Cauchy Stress Tensor and Cauchys Formula
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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Cauchy Stress Tensor and Cauchys Formula
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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Cauchy Stress Tensor and Cauchys Formula
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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Cauchy Stress Tensor and Cauchys Formula
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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Cauchy Stress Tensor and Cauchys Formula
and using these expressions;
And dividing by we have;
when the tetrahedron shrinks to a point (i.e.h 0);
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Cauchy Stress Tensor and Cauchys Formula
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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Cauchy Stress Tensor and Cauchys Formula
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
Cauchy Stress Tensor and Cauchys 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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Cauchy Stress Tensor and Cauchys Formula
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.