mae 640 lec16
TRANSCRIPT
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Continuum mechanics MAE 640
Summer II 2009
Dr. Konstantinos Sierros
263 ESB new add
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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.