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Continuum mechanics MAE 640

Summer II 2009

Dr. Konstantinos Sierros

263 ESB new add

kostas.sierros@mail.wvu.edu

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Isotropic Materials

Isotropic materials are those for which the material properties are independent of the

direction, and we have;

The stressstrain relations take the form;

Inverse relation

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Isotropic Materials

Conversely, the application of a shearing stress to an anisotropic material causes

shearing strain as well as normal strains.

Normal stress applied to an orthotropic material at an angle to its principal material

directions causes it to behave like an anisotropic material.

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Transformation of Stress and Strain Components

The constitutive relations and for an orthotropic material are written in terms of the

stress and strain components that are referred to the material coordinate system.

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Transformation of Stress and Strain Components

We can use the above transformation equations of a second-order tensor to write the

stress and strain components (ij, ij) referred to the material coordinate system in terms

of those referred to the problem coordinates.

Let (x, y, z) denote the coordinate system used to write the governing equations of a

problem, and let (x1, x2, x3) be the principal material coordinates such thatx3-axis isparallel to the z-axis

i.e., thex1x2-plane and thexy-plane are parallel

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Thex1-axis is oriented at an angle of +counterclockwise (when looking down) from

thex-axis, as shown in the figure below.

Transformation of Stress and Strain Components

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The coordinates of a material point in the two coordinate systems are related as follows

(z=x3):

Transformation of Stress and Strain Components

Inverse

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Transformation of Stress Components

Let denote the stress tensor which has components

11, 12, . . . , 33 in the material

(m) coordinates (x1, x2, x3)

xx, xy, . . . , zzin the problem

(p) coordinates (x, y, z)

Since stress tensor is a second-order tensor, it transforms according to the formula;

components of the stress tensor in the

material coordinates (x1, x2, x3)

components of the same stress tensor in

the problem coordinates (x, y, z)

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Transformation of Stress Components

In matrix form

rearranging the equations in terms of the single-subscript stress components in (x, y,

z) and (x1, x2, x3) coordinate systems, we have

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Transformation of Stress Components

The inverse relationship between {}m and {}p is given by;

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Transformation of Strain Components

Transformation equations derived for stresses are also valid fortensorcomponents of

strains;

Inverse relation

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Nonlinear Elastic Constitutive Relations

Most materials exhibit nonlinear elastic behavior for certain strain threshold. Beyond

that threshold Hookes law is not valid.

Past certain nonlinear elastic range, permanent deformation ensues, and the material issaid be inelastic or plastic, as shown in the figure below;

We review constitutive relations for two well-known nonlinear elastic materials, the

MooneyRivlin and neo-Hookean materials.

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Nonlinear Elastic Constitutive Relations

For a hyperelastic material, there exists a free energy function = (F) such that;

F is deformation

gradient tensor

Some materials (e.g., rubber-like materials) undergo large deformations without

appreciable change in volume (i.e., J 1).

Such materials are called incompressible materials.

For incompressible elastic materials, the stress tensor is not completely determined

by deformation.

The hydrostatic pressure also affects the stress.

pressure

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Nonlinear Elastic Constitutive Relations

For a hyperelastic elastic material we can also have;

B is the left CauchyGreen tensor

B = F FT

The free energy function takes different forms for different materials.

It is often expressed as a linear combination of unknown parameters and principalinvariants of Green strain tensorE, deformation gradient tensorF, or left CauchyGreen

strain tensorB.

The parameters characterize the material and they are determined through suitable

experiments.

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Nonlinear Elastic Constitutive Relations

For incompressible materials, the free energy function is taken as a linear function of

the principal invariants ofB

constants principal invariants ofB

(there is also a third invariant whichIs equal to unity in this case)

Materials for which the strain energy

functional is given by the above equation

are known as the MooneyRivlin materials.

The stress tensor in this

case has the form;

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Nonlinear Elastic Constitutive Relations

The MooneyRivlin incompressible material model presented previously is most

commonly used to represent the stress-strain behavior of rubber-like solid materials.

Now

If the free energy function is of the form = C1(IB 3);

The constitutive equation

takes the form;

Materials whose constitutive behavior is described by the above equation are called the

neo-Hookean materials.

The neo-Hookean model provides a reasonable predictionof the constitutive behavior of

natural rubber for moderate strains.