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Module 3: 3D Constitutive Equations
Lecture 10: Constitutive Relations: Generally Anisotropy to Orthotropy
The Lecture Contains:
Stress Symmetry
Strain Symmetry
Strain Energy Density Function
Material Symmetry
Symmetry with respect to a Plane
Symmetry with respect to two Orthogonal Planes
Homework
References
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Module 3: 3D Constitutive Equations
Lecture 10: Constitutive Relations: Generally Anisotropy to Orthotropy
Introduction
In this lecture, we are going to develop the 3D constitutive equations. We will start with the the
generalized Hookes law for a material, that is, material is generally anisotropic in nature. Finally, we
will derive the constitutive equation for isotropic material, with which the readers are very familiar.
The journey for constitutive equation from anisotropic to isotropic material is very interesting and will
use most of the concepts that we have learnt in earlier Lecture 9.
The generalized Hookes law for a material is given as
(3.1)
where, ij is a second order tensor known as stress tensor and its individual elements are the
stress components.
is another second order tensor known as strain tensor and its individual
elements are the strain components. C ijkl is a fourth order tensor known as stiffness tensor. In the
remaining section we will call it as stiffness matrix, as popularly known. The individual elements of
this tensor are the stiffness coefficients for this linear stress-strain relationship. Thus, stress and
) 9 components each and the stiffness tensor has (=) 81 independent
strain tensor has (
elements. The individual elements =) 81 are referred by various names as elastic constants,
moduli and stiffness coefficients. The reduction in the number of these elastic constants can be
sought with the following symmetries.
Stress Symmetry:
The stress components are symmetric under this symmetry condition, that is,
. Thus, there
are six independent stress components. Hence, from Equation. (3.1) we write
(3.2)
Subtracting Equation (3.2) from Equation (3.1) leads to the following equation
(3.3)
There are six independent ways to express i and j taken together and still nine independent ways to
express k and l taken together. Thus, with stress symmetry the number of independent elastic
) 54 from 81.
constants reduce to (
Strain Symmetry:
The strain components are symmetric under this symmetry condition, that is,
. Hence, from
Equation (3.1) we write
Subtracting Equation (3.3) from Equation (3.2) we get the following equation
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(3.4)
It can be seen from Equation (3.4) that there are six independent ways of expressing i and j taken
together when k and l are fixed. Similarly, there are six independent ways of expressing k and l taken
independent constants
together when i and j are fixed in Equation (3.4). Thus, there are
for this linear elastic material with stress and strain symmetry.
With this reduced stress and strain components and reduced number of stiffness coefficients, we can
write Hookes law in a contracted form as
(3.5)
where
(3.6)
Note: The shear strains are the engineering shear strains.
For Equation (3.5) to be solvable for strains in terms of stresses, the determinant of the stiffness
matrix must be nonzero, that is
The number of independent elastic constants can be reduced further, if there exists strain energy
density function W, given as below.
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Module 3: 3D Constitutive Equations
Lecture 10: Constitutive Relations: Generally Anisotropy to Orthotropy
Strain Energy Density Function (W):
The strain energy density function W is given as
with the property that
(3.7)
(3.8)
It is seen that W is a quadratic function of strain. A material with the existence of W with property in
Equation (3.8) is called as Hyperelastic Material.
The W can also be written as
(3.9)
Subtracting Equation (3.9) from Equation (3.7) we get
(3.10)
which leads to the identity
. Thus, the stiffness matrix is symmetric. This symmetric matrix
has 21 independent elastic constants. The stiffness matrix is given as follows:
(3.11)
The existence of the function W is based upon the first and second law of thermodynamics. Further,
it should be noted that this function is positive definite. Also, the function W is an invariant (An
invariant is a quantity which is independent of change of reference).
The material with 21 independent elastic constants is called Anisotropic or Aelotropic Material.
Further reduction in the number of independent elastic constants can be obtained with the use of
planes of material symmetry as follows.
Material Symmetry:
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It should be recalled that both the stress and strain tensor follow the transformation rule and so does
the stiffness tensor. The transformation rule for these quantities (as given in Equation (3.1)) is known
as follows
(3.12)
where
are the direction cosines from i to j coordinate system. The prime indicates the quantity in
new coordinate system.
When the function W given in Equation (3.9) is expanded using the contracted notations for strains
and elastic constants given in Equation (3.11) W has the following form:
(3.13)
Thus, from Equation (3.13) it can be said that the function W has the following form in terms of strain
components:
(3.14)
With these concepts we proceed to consider the planes of material symmetry. The planes of the
material, also called elastic symmetry are due to the symmetry of the structure of anisotropic body.
In the following, we consider some special cases of material symmetry.
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Module 3: 3D Constitutive Equations
Lecture 10: Constitutive Relations: Generally Anisotropy to Orthotropy
(A) Symmetry with respect to a Plane:
Let us assume that the anisotropic material has only one plane of material symmetry. A material with
one plane of material symmetry is called Monoclinic Material.
Let us consider the x 1 - x 2 ( x 3 = 0) plane as the plane of material symmetry. This is shown in Figure
3.1. This symmetry can be formulated with the change of axes as follows
(3.15)
With this change of axes,
(3.16)
This gives us along with the use of the second of Equation (3.12)
(3.17)
Figure 3.1: Material symmetry about x1-x2 plane
First Approach: Invariance Approach
Now, the function W can be expressed in terms of the strain components
. If W is to be invariant,
then it must be of the form
(3.18)
Comparing this with Equation (3.13) it is easy to conclude that
(3.19)
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Thus, for the monoclinic materials the number of independent constants are 13. With this reduction
of number of independent elastic constants the stiffness matrix is given as
(3.20)
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Module 3: 3D Constitutive Equations
Lecture 10: Constitutive Relations: Generally Anisotropy to Orthotropy
Second Approach: Stress Strain Equivalence Approach
The same reduction of number of elastic constants can be derived from the stress strain equivalence
approach. From Equation (3.12) and Equation (3.16) we have
(3.21)
The same can be seen from the stresses on a cube inside such a body with the coordinate systems
shown in Figure 3.1. Figure 3.2 (a) shows the stresses on a cube with the coordinate system x 1 ,
.
x 2 ,x 3 and Figure 3.2 (b) shows stresses on the same cube with the coordinate system
Comparing the stresses we get the relation as in Equation (3.21).
Now using the stiffness matrix as given in Equation (3.11), strain term relations as given in Equation
(3.17) and comparing the stress terms in Equation (3.21) as follows:
Using the relations from Equation (3.17), the above equations reduce to
Noting that
, this holds true only when
Similarly,
This gives us the
matrix as in Equation (3.20).
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Figure 3.2: State of stress (a) in x 1 , x 2 , x 3 system
(b) with x 1 -x 3 plane of symmetry
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Module 3: 3D Constitutive Equations
Lecture 10: Constitutive Relations: Generally Anisotropy to Orthotropy
Symmetry with respect to two Orthogonal Planes:
Let us assume that the material under consideration has one more plane, say x 2 -x 3 is plane of
material symmetry along with x 1 -x 2 as in (A). These two planes are orthogonal to each other. This
transformation is shown in Figure 3.3.
This can be mathematically formulated by the change of axes as
(3.22)
And
(3.23)
This gives us the required strain relations as (from Equation (3.12)).
or using contracted notations, we can write,
(3.24)
Figure 3.3: Material symmetry about x 1 -x 2 andx 2 -x 3
planes
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Module 3: 3D Constitutive Equations
Lecture 10: Constitutive Relations: Generally Anisotropy to Orthotropy
First Approach: Invariance Approach
and using contracted notations
We can get the function W simply by substituting in place of
for the strains in Equation (3.18). Noting that W is invariant, its form in Equation (3.18) must now be
restricted to functional form
(3.25)
From this it is easy to see that
Thus, the number of independent constants reduces to 9. The resulting stiffness matrix is given as
(3.26)
When a material has (any) two orthogonal planes as planes of material symmetry then that material
is known as Orthotropic Material. It is easy to see that when two orthogonal planes are planes of
material symmetry, the third mutually orthogonal plane is also plane of material symmetry and
Equation (3.26) holds true for this case also.
Note: Unidirectional fibrous composites are an example of orthotropic materials.
Second Approach: Stress Strain Equivalence Approach
The same reduction of number of elastic constants can be derived from the stress strain equivalence
approach. From the first of Equation (3.12) and Equation (3.23) we have
(3.27)
The same can be seen from the stresses on a cube inside such a body with the coordinate systems
shown in Figure 3.3. Figure 3.4 (a) shows the stresses on a cube with the coordinate system x 1 , x 2 ,
x 3 and Figure 3.4 (b) shows stresses on the same cube with the coordinate system
Comparing the stresses we get the relation as in Equation (3.27).
Now using the stiffness matrix given in Equation (3.20) and comparing the stress equivalence of
Equation (3.27) we get the following:
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This holds true when
This gives us the
. Similarly,
matrix as in Equation (3.26).
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Module 3: 3D Constitutive Equations
Lecture 10: Constitutive Relations: Generally Anisotropy to Orthotropy
Figure 3.4: State of stress (a) in x 1 , x 2 , x 3 system
(b) with x 1 -x 2 andx 2 -x 3 planes of symmetry
Alternately, if we consider x 1 -x 3 as the second plane of material symmetry along with x 1 -x 2 as
shown in Figure 3.5, then
(3.28)
And
(3.29)
This gives us the required strain relations as (from Equation (3.12))
or in contracted notations, we write
Substituting these in Equation (3.18) the function W reduces again to the form given in Equation
(3.25) for W to be invariant. Finally, we get the reduced stiffness matrix as given in Equation (3.26).
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Figure 3.5: Material symmetry about x 1 -x 2 andx 1 -x 3
planes
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Module 3: 3D Constitutive Equations
Lecture 10: Constitutive Relations: Generally Anisotropy to Orthotropy
The stress transformations for this coordinate transformations are (from the first of Equation (3.12)
and Equation (3.29))
The same can be seen from the stresses shown on the same cube in x 1 , x 2 , x 3 and
coordinate systems in Figure 3.6 (a) and (b), respectively. The comparison of the stress terms leads
to the stiffness matrix as given in Equation (3.26).
Note: It is clear that if any two orthogonal planes are planes of material symmetry the third mutually
orthogonal plane has to be plane to material symmetry. We have got the same stiffness matrix when
we considered two sets of orthogonal planes. Further, if we proceed in this way considering three
mutually orthogonal planes of symmetry then it is not difficult to see that the stiffness matrix remains
the same as in Equation (3.26).
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Module 3: 3D Constitutive Equations
Lecture 10: Constitutive Relations: Generally Anisotropy to Orthotropy
Homework:
1. Starting with hyperelastic material, first take x 2 -x 3 plane as plane of material symmetry and
obtain the stiffness matrix. Is this matrix the same as in Equation (3.20) ? Justify your answer.
2. Starting with the stiffness matrix obtained in the above problem, take x 1 -x 3 as an additional
plane of symmetry and obtain the stiffness matrix. Is this matrix the same as in Equation
(3.26)? Justify your answer.
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Module 3: 3D Constitutive Equations
Lecture 10: Constitutive Relations: Generally Anisotropy to Orthotropy
References:
AE Green, W Zerna. Theoretical Elasticity. Oxford, Clarendon Press, 1963.
SG Lekhnitskii. Theory of Elasticity of an Anisotropic Body. Mir Publishers, Moscow,
1981.
IS Sokolnikoff. Mathematical Theory of Elasticity, First Edition, McGraw Hill
Publications, New York.
LE Malvern. Introduction to Mechanics of a Continuous Medium, Prentice-Hall, Inc. New
Jersey, 1969.
CT Herakovich. Mechanics of Fibrous Composites, John Wiley & Sons, Inc. New York,
1998.
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