Mechanical Behavior of Materials Lecture Note (8) 2/4/2011

It denotes complete information of the stress state at a point, with the coordinate system. Stress is a tensor.

The in-plane stress-strain relations for plane stress (σzz = σxz = σyz = 0) and plane strain (ezz = exz = eyz = 0) as given in any textbook on elasticity, are

Plane Stress:

Plane Strain:

Remember that:

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2

1 0

1 01

1 20 0

2

xx xx

yy yy

xy xy

eE

e

e

1 0

1 01 1 2

1 2 20 0

2

xx xx

yy yy

xy xy

eE

e

e

Mechanical Behavior of Materials Lecture Note (8) 2/4/2011

γxy is the engineering shear strain exy is the shear strain (tensorial)

2

2 1

xy xy

xy xy

G

e

EG

A state of plane stress usually occurs in structural elements where one dimension is very small compared to the other two, i.e. the element is flat or thin. In this case, the stresses are negligible with respect to the smaller dimension as they are not able to develop within the material and are small compared to the in-plane stresses.

Therefore, the face of the element is not acted by loads and the structural element can be analyzed as two-dimensional, e.g. thin-walled structures such as plates subject to in-plane loading or thin cylinders subject to pressure loading. The other three non-zero components remain constant over the thickness of the plate. The stress tensor can then be approximated by:

11 12

21 22

0 0

0 0

0 0 0 0 0 0

xx xy

ij yx yy

Tensor is a simple geometrical representation which is useful, for example, in deriving the magnitude of a material property in a

given direction

The corresponding strain tensor is:

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Mechanical Behavior of Materials Lecture Note (8) 2/4/2011

11 12

21 22

33

020 0

0 0 02

0 0 0 0 0 0

xyxx

xx xyyx

ij yx yy yy

xxxx

ee e e e

e e e e e e

e e e

In which the non-zero exx term arises from the Poisson's effect. This strain term can be temporarily removed from the stress analysis to leave only the in-plane terms, effectively reducing the analysis to two dimensions.

Representing stress as a tensor

Consider a block laying in 3-D space, for a given load, the faces of this block experience tensile stresses normal to the faces and shear stresses tangent to the faces. See Figure below.

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Mechanical Behavior of Materials Lecture Note (8) 2/4/2011

The nine stresses shown in the above figure may be arranged in an orderly array (matrix) called the stress tensor, which represents the general state of stress at a point. The first horizontal row shows the stresses on the x face, the second row those on the y face, and so on.

In order for a stress not to move the material, the stress tensor must be symmetric; it has mirror symmetry about the diagonal

The general stress tensor has six independent components and could require us to do a lot of calculations. To make things easier it can be rotated into the principal stress tensor by a suitable change of axes. (Later)

Hydrostatic and deviatoric components of stress

The tensor can be broken down into two components: stresses; that act on the block to change the volume and stresses; that act to distort the volume [act to change the shape only]. The former stresses are referred to as hydrostatic [mean] stresses and the latter as deviatoric [differential] stresses.

Since the hydrostatic stresses maintain the original proportions of the volume (change volume), it makes sense that they would be equal. Thus the hydrostatic stresses are simply the mean of the principal stresses.

1 2 3

3m

The deviatoric stresses consist simply of the hydrostatic stresses subtracted from the original stress tensor. The resulting matrix includes tensile stresses that elongate the volume as well as shear stresses that cause angular distortion—in other words, stresses that cause the volume to deviate from its original proportions (change shape).

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Mechanical Behavior of Materials Lecture Note (8) 2/4/2011

Hydrostatic component Deviatoric component

An easy way to think about the mean and deviatoric stress components is to consider that

the mean stress acts to change volume the deviatoric stress acts to change shape

Example

For a simple example, consider the cube in Figure below, subject to an evenly distributed compressive load of 210 lb in the y-direction [the loading is uniaxial]. Assume each face has an area of 1 in2.

It is apparent that the cube will compress in the y-direction. It is also apparent that the cube will expand in the x- and z-directions due to Poisson’s effect.

Block subject to 210-lb compressive load.

The hydrostatic stresses on each face may be calculated as:

Thus the hydrostatic stresses here are represented as 70 psi stresses on each face that act to compress the volume.

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Mechanical Behavior of Materials Lecture Note (8) 2/4/2011

Deviatoric stresses are simply

Thus the contraction/expansion of the volume is represented as a 140 psi compressive stress in the y-direction and 70 psi tensile stress in the x and z-directions.

In crystalline metals plastic, deformation occurs by slip, a volume-conserving process that changes the shape of a material through the action of shear stresses. On this basis, it might therefore be expected that the yield stress of a crystalline metal does not depend on the magnitude of the hydrostatic stress; this is in fact exactly what is observed experimentally.

In amorphous metals, a very slight dependence of the yield stress on the hydrostatic stress is found experimentally

Dilatational and deviatoric components of strain

Dilatation

11 12 13

11 12 13

21 22 23 21 22 23

31 32 33

31 32 33

0 03 3

0 03 3

0 03 3

e e ee e e

e e e e e e

e e ee e e

Dilatational Deviatoriccomponent componentVolume change Shape change

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11 22 33

Ve e e

V

Mechanical Behavior of Materials Lecture Note (8) 2/4/2011

Principal stresses

The magnitudes of the components of the stress tensor depend on how we have defined the orthogonal 1, 2 and 3 axes.

For every stress state, we can rotate the axes, so that the only non-zero components of the stress tensor are the ones along the diagonal:

That is, there are no shear stress components, only normal stress components.

This is an example of a principal stress tensor of all the tensors we could use to express the stress state that exists. The elements σ1, σ2, σ3 are the principal stresses. The positions of the axes now are the principal axes. While it may be that σ1 > σ2 > σ3, it only matters that the 1, 2 and 3 axes define the directions of the principal stresses.

The largest principal stress is bigger than any of the components found from any other orientation of the axes. Therefore, if we need to find the largest stress component that the body is under, we simply need to diagonalize the stress tensor.

Remember – we have not changed the stress state, and we have not moved or changed the material – we have simply rotated the axes we are using and are looking at the stress state seen with respect to these new axes

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