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M.S. Ramaiah School of Advanced Studies, Bengaluru

Stress Strain Relations

(Constitutive Relations)

(Contd.)Session delivered by:

Mr. Nithin Venkataram.

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Session Objectives

At the end of session students would have understood:

The constitutive relations of isotropic and other types of

materials that establish the stress-strain relations.

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Session Topics

Introduction

Generalised Hooks Law

Constitutive Relation for Isotropic Material

Modulus of Rigidity

Bulk Modulus

Relations between the Elastic Constants

Displacement Equations of Equilibrium

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Generalised Hookes law

Introduction

In the problems arising in the practical application of the

Mechanics of Materials the stress and strain states are

frequently two- or three dimensional, so that the

generalization of the one-dimensional models to two or

three dimensions is necessary.

We will consider three dimensional constitutive laws in the

linear case (i.e., when the stresses and strains are related by

linear functions), considering isotropic, monotropic and

orthotropic materials.

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Isotropic Materials

Isotropic materials display symmetrical features in relation

to any plane. Therefore, the three planes define by the

reference axes in any rectangular Cartesian reference

system are symmetry planes in relation to the rheologicalbehaviour of the material.

Let us first consider the isolated action of the normal stress

x, as represented in figure below.

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Figure: Deformation caused by the isolated actuation of x:

(original configuration; deformed configuration)

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The relation between the normal stress (in this case x) andthe longitudinal strain in the same direction (in this case x)

is called the longitudinal modulus of elasticity or Youngs

modulus of the material.

The relation between the transversal and longitudinal

strains, multiplied by 1, is known as the Poissons

coefficient of the material (). The strains caused by the

stress xare then given by,

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If only infinitesimal deformations are considered, we can

accept that the parallelepipeds geometry remains

unchanged, when the effects of yand zare considered.

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Since the stress-strain relation is linear, it does not change

with

the actuation of x.

Thus, if the stresses yor zare applied after x, they cause

the same deformations that would occur under the isolated

action of each of them.

The total strains may therefore be computed by adding the

strains, which would be produced by the isolated action of

each stress.

This conclusion describes the so-called superposition

principle, which is valid for all solid bodies if the

deformations are small and if the material has a linearconstitutive law.

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The isolated actions of yand zwould cause the strains

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The superposition principle allows the computation of the

total strain by adding the strains caused by the isolated

actions of the stresses x, yand z, yielding

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These expressions were obtained from symmetry

considerations and relate the normal stresses with the

longitudinal strains.

The same symmetry considerations lead to the conclusion

that the shearing strains cause distortions only in their plane,

since the deformed parallelepiped must remain symmetrical

in relation to the plane containing the shearing stresses, as

represented in the figure below.

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Figure: Deformation caused by the shearing stress xy:

(original configuration; deformed configuration)

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The constant of proportionality between the shearing stress

and the shearing strain is known as the shear modulus of the

material (G), also called the transversal modulus of

elasticity.

The constitutive law of an isotropic material, defined in

terms of normal stresses and longitudinal strains by, is

completed by the relations between shearing stresses and

shearing strains.

As iven below14

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These expressions show that the shearing strain vanishes if

the shearing stresses have zero values.

Taking as reference system axes which are parallel to the

principal directions of the stress tensor, a strain tensor with

non-zero elements only in the diagonal is obtained, which

means that in an isotropic material the principal directions

of the stress and strain tensors coincide.

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Expressions to compute the stress for given strains may be

obtained by,

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Constitutive Relation for Isotropic Material

Based on observations from the previous section, toconstruct a general three-dimensional constitutive law for

linear elastic materials, we assume that each stress

component is linearly related to each strain component

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where the coefficients Cij are material parameters and the

factors of 2 arise because of the symmetry of the strain.

These relations can be cast into a matrix format as

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It can also be expressed in standard tensor notation by

writing

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Stress-Strain Relations for Isotropic

Materials We now make a further assumption that the ideal material

we are dealing with has the same properties in all

directions so far as the stress-strain relations are

concerned. This means that the material we are dealing

with is isotropic, i.e. it has no directional property.

Assuming that the material is isotropic, one can show that

only two independent constants are involved in the

generalized statement of Hooke's law.

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It was shown that at any point there are three faces on which

the resultant stresses are purely normal. The stresses on

these faces were termed as principal stresses.

Also it was shown, that at a point, a small rectangular block,

the faces of which remain rectangular after strain, can be

found. The normals to these faces were termed the principal strain

axes.

If the material is isotropic, then there is no reason why a

symmetrical system of purely normal stresses should

produce asymmetrical distortion.

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Hence, it is evident that in material which has no directional

property, the directions of the principal stresses and of the

principal strains must coincide.

Therefore, in the most general statement of Hooke's law for

isotropic materials, we have to relate the three principal

stresses 1, 2and 3 with the three principal strains 1, 2,and 3.

For 1we should have,

where a,band c are constants.

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But we observe that b and cshould be equal since the effect

of 1 in the directions of 2and 3, which are both at right

angles to 1 must be the same for an isotropic material.

Hence, for ithe equation becomes,

But (1+ 2+ 3) is the first invariant of strain or the cubical

dilatation.

Denoting

bby and

(a-b)by 2 , the equation for 1

becomes

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Similarly, for 2and 3we get,

and are called Lame's coefficients.

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Modulus of Rigidity

Let the coordinate axes ox, oy and oz coincide with the

principal

Stress axes.

Consequently, for an isotropic body, the principal strain axes

will

also be along ox, oy and oz.

Consider another frame of reference ox', oy', oz', such that the

direction cosines of ox', are nx1, ny1and nz1and those of Oy'

are nx2, ny2, and nz2

Since Ox' and Oy' are perpendicular to each other,

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The normal stress x'and the shear stress x'v'are obtained from

Cauchy's formula as

Similarly, if 1,2and 3are the principal strains

Substituting for 1, 2and 3

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From the previous equations,

The above equation relates the shear stress with its associated

shear strain. Comparing this with the relation used in

elementary strength of materials, one observes that is the

modulus of rigidity, usually denoted by the letter G.

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Bulk Modulus

Using the expression,

and substituting for 1,2and 3

Similarly,

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Adding,

since is an invariant, and is also an invariant.

From elementary strength of materials, when

we have

where K is the bulk modulus.

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Young's Modulus and Poisson's Ratio

We have,

From elementary strength of materials

whereE is Young's modulus, and is Poisson's ratio.

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Comparing the above two equations,

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Relations between the Elastic Constants

In elementary strength of materials, we are familiar with Young'sModulusE, Poisson's ratio, shear modulus or modulus of

rigidity G, and bulk modulus K. Among these, only two are

independent andE and are generally taken as the independent

constants. The other two, namely, G and K, are expressed as

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For an isotropic material, the 36 elastic constants involved in the

Generalized Hooke's Law, can be reduced to two independent

elastic constants. These two elastic constants are Lame'scoefficients and . The second coefficient . is the same as

the rigidity modulus G. In terms of these, the other elastic

constants can be expressed as

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It should be observed that for the bulk modulus to be positive, the

value of Poisson's ratio cannot exceed 1/2. This is the upper

limit for . For =

A material having Poisson's ratio equal to 1/2 is known as an

incompressible material, since the volumetric strain for such a

material is zero.

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Displacement Equations of Equilibrium

It was shown that if a solid body is in equilibrium, the six

rectangular stress components have to satisfy the three

equations of equilibrium.

It is possible relate the stress components to the strain

components using the stress-strain relations. Hence, stress equations of equilibrium can be converted to

strain equations of equilibrium.

The strain components are related to the displacement

components.

Therefore, the strain equations of equilibrium can be converted

into displacement equations of equilibrium.

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Using the notation,

the displacement equations of equilibrium are

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These are known as Lame's equations.

They involve a synthesis of the analysis of stress, analysis of

strain and the relations between stresses and strains.

These equations represent the mechanical, geometrical and

physical characteristics of an elastic solid.

Consequently, Lame's equations play a very prominent role in

the solutions of problems.

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M S R i h S h l f Ad d St di B l

Summary

The constitutive relations of isotropic and other types of

materials which establish the stress-strain relations are

explained.

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