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Continuum mechanics MAE 640

Summer II 2009

Dr. Konstantinos Sierros263 ESB new add

kostas.sierros@mail.wvu.edu

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Material Symmetry

Further reduction in the number of independent stiffness (or compliance) parameterscomes from the so-called material symmetry.

Suppose that (x1, x2, x3) denote the coordinate system with respect to which Eqs.(6.2.5)(6.2.16) are defined.

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Material Symmetry

We shall call the (x1, x2, x3) , a material coordinate system (we will use mostly this).

The coordinate system (x, y, z) used to write the equations of motion and strain-displacement equations will be called theproblem coordinates

The phrase material coordinates used in connection with the material descriptionshould not be confused with the present term.

Both are fixed in the body, and the two systems are oriented with respect to each other.

When elastic material parameters at a point have the same values for every pair ofcoordinate systems that are mirror images of each other in a certain plane, that plane iscalled a material plane of symmetry

Examples include: symmetry of internal structure due to crystallographic form,regulararrangement of fibers or molecules

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Material Symmetry

We note that the symmetry under discussion is a directional property and not apositional property.

Thus, a material may have certain elasticsymmetry at every point

of a material body and the properties may

vary from point to point.

Positional dependence of material properties is what we called the inhomogeneity of thematerial.

We need to discuss various planes of symmetry and forms ofassociated stressstrain relations.

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The use of the tensor components of stress and strain is necessary because thetransformations are valid only for the tensor components.

Material Symmetry

The second order tensor components i jand i jand the fourth-order tensor

components Ci jkltransform according to the formulae;

Transformation

direction cosines associated with thecoordinate systems ( x1,x2,x3)

and (x1, x2, x3)

components of the fourth-order tensorC in the barred and unbarred coordinates

systems

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Material Symmetry

A trivial symmetry transformation is one in which the barred coordinate system isobtained from the unbarred coordinate system by simply reversing their directions:

x1 = x1, x2 = x2, and x3 = x3.This transformation is satisfied by all materials, and they are called triclinicmaterials.

The associated transformation matrix is given by;

For this transformation, it can be shownthat Eq. (6.2.17) gives the trivial result

Ci jkl= Ci jkl.

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Symmetry transformations

Monoclinic Materials

When the elastic coefficients at a point have the same value for every pair of coordinate

systems which are the mirror images of each other with respect to a plane, thematerial is called a monoclinic material.

For example, let (x1, x2, x3) and ( x1,x2,x3) be two coordinates systems, with thex1, x2-plane parallel to the plane of symmetry.

Choosing the x3-axis such that x3 = x3 (never mind about the left-handed coordinate system as it does not affect the discussion)

Then one system is the mirror

image of the other.

This symmetry transformation can beexpressed by the transformation

matrix ( x1 =x1, x2 =x2, x3 = x3).

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The requirement that Ci jklbe the same as Ci jklunder the transformation (6.2.19)yields (because 23 = 23, 31 = 31, 23 = 23, 31 = 31 under the sametransformation):

Symmetry transformations

Monoclinic Materials

Thus, all Cs with an odd number of index 3 are zero.

For the two systems;

But also;

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Symmetry transformations

The elastic parameters Ci j are the same for the two coordinate systemsbecause they are the mirror images in the plane of symmetry.

..and

Using theAbove

EquationsWe have

Similar discussion with the two alternative expressions of the remainingstress components yield C24 = 0 and C25 = 0; C34 = 0 and C35 = 0; and

C46 = 0 and C56 = 0.

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Symmetry transformations

Therefore,

From 21 material parameters, wehave only

21 8 = 13 independent parameters,as indicated below;

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Symmetry transformations

When three mutually orthogonal planes of material symmetry exist, the

number of elastic coefficients is reduced to 9.

Such materials are called orthotropic.

ransformation matrices associated with the 3 planes of symmetry are shown b

The stressstrain relationsfor an orthotropic material

Orthotropic Materials

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Symmetry transformations

Orthotropic Materials

Most often, the material properties are determined in a laboratory in termsof the engineering constants such as Youngs modulus, shear modulus etc.

These constants are measured using simple tests like uniaxial tension testor pure shear test. Because of their direct and obvious physical meaning,engineering constants are used in place of the more abstract stiffness

coefficients Ci j

Next we can discuss how to obtain the strainstress relations (6.2.23) andrelate Si j (compliance coefficient inverse of stiffness coefficient) into theengineering constants.

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Symmetry transformations

Orthotropic Materials

One of the consequences of linearity (both kinematic and material

linearizations) is that the principle of superposition applies.

Principle of superposition

If the applied loads and geometric constraints are independent ofdeformation, the sum of the displacements (and hence strains) produced by

two sets of loads is equal to the displacements (and strains) produced by thesum of the two sets of loads.i.e. The strains of the same typeproduced by the application of

individual stress components can besuperposed.

In physics and systems theory, the superposition principle, also knownas superposition property, states that, for all linear systems,

The net response at a given place and time caused by two or more stimuli is the sum

of the responses which would have been caused by each stimulus individually.

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Symmetry transformations

Orthotropic Materials

Example

The extensional strain (1)11 in the material coordinate directionx1 due tothe stress 11 in the same direction is 11/E1, where E1 denotes Youngsmodulus of the material in thex1 direction, as shown in the figure below;

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The extensional strain (2)11 due to the stress 22 applied in thex2-direction

is (a result of the Poisson effect) 21 (22/E2), where 21 is Poissons ratio(note that the first subscript in i j , i j , corresponds to the load directionand the second subscript refers to the directions of the strain) E2 is Youngsmodulus of the material in thex2-direction.

Orthotropic Materials

Symmetry transformations

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Symmetry transformations

Orthotropic Materials

Similarly, 33 produces a strain (3)11 equal to 31(33/E3).

Therefore we have;

total strain 11 due to thesimultaneous

application of all three normal stresscomponents

the direction of loading is denoted by the superscr

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Symmetry transformations

Orthotropic Materials

Similarly, we have;

he shear tests with an orthotropic material give the following results;

i j (i j ) denotes the corresponding

shear stress in thexi -xj plane, and Gij (i =j) are the shear moduli in

thexi -xj plane.

Note that

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Symmetry transformations

Orthotropic Materials

In matrix form we have;

E1, E2, E3 are Youngs moduli in 1, 2, and 3 material directions

i j is Poissons ratio, defined as the ratio of transverse strain in thejth

direction to the axial strain in the ith direction when stressed in the i-direction,

G23,G13,G12

are shear moduli in the 2-3, 1-3, and1-2 planes

The nine independent material coefficients for an orthotropic material areE1 E2 E3 G23 G13 G12 12 13 23