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TRANSCRIPT
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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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