Example:
An Incompressible Neo-Hookean Material undergoing Homogeneous Pure Stretch
Motion
Consider the deformation of a cubic block of material, with sides of length 1, initially occupying the region . The subsequent motion of the material is given by
This represents a stretching of the material in the directions, and a consequent thinning of the material in the direction, as illustrated.
The spatial description of the motion is obtained by inverting the motion equations,
Deformation
The deformation gradient is
The Jacobian determinant is
so thematerial is incompressible.
The right Cauchy-Green tensor is
and the left Cauchy-Green tensor is and equals C is this example.
The deformation can be decomposed into stretch tensors through or , where U and v are, respectively, the right (material) and left (spatial) stretch tensors. Since Cand F are diagonal, , and , so this is a pure stretch with no rotation.
Rates of Deformation
The velocity is
The velocity in the spatial description can be obtained by substituting into the above expression the motion , to get
Note that , as expected for an incompressible material. The acceleration is
In terms of the spatial coordinates, this is
The acceleration can also be obtained directly from the spatial velocity using the material time derivative,
giving the same result.
Note that one can take the material time derivative of , which yields . One finds that this expression is zero, since is always zero.
The spatial velocity gradient is
which is symmetric and so equal to the rate of deformation d, with the spin (as expected, since there are clearly no rotations of material particles).
Stress
Suppose now that the Cauchy stress is
The Equations of Motion
The equations of motion in the spatial description are
It can be seen that the Cauchy stress is uniform throughout the material at any time instant, , so body forces must act in the direction to equilibrate the acceleration:
The PK1 stress is
The equations of motion in the material description are . Again, and a body force acts to equilibrate the accelerations:
Note that since the material is incompressible, the spatial density is equal to the density of the material in the reference configuration, , and .
Angular Momentum
The principle of conservation of angular momentum requires that the Cauchy stress be symmetric, which is clearly the case, or, in the material description, that , which can be seen to hold.
Balance of Mechanical Energy
The stress power (in the spatial description) is now
In the material description, first note that the PK2 stress is
Also,
and then the stress power is, as expected, (note that here). The same result can be obtained by using the alternative stress power expression, .
Since the material is incompressible, the volume does not change, and is equal to unity for the initial unit cube configuration, so the total stress power is
Let us look now at the termin the mechanical energy balance equation, where t is the (applied) traction acting over the surface of the material (in the current configuration), inducing the deformation, and v is the velocity over the surface. The velocity is zero over three of the sides (the sides with unit normals ), and is non-zero over the three sides with unit normals . The traction is given by Cauchy’s law, , and is shown in the figure.
Also, the surface area over which acts is
and the velocity here is . Similarly for the other surface, giving
It can be seen that this power of the surface tractions equals the stress power, so the mechanical energy is balanced:
(the power of the body forces equals the rate of change of kinetic energy, since the body force is equal to the accerelation).
Strain Energy and Conservative Force System
The above Cauchy and PK2 stresses were actually derived from the strain energy function
This is the strain energy of the Neo-Hookean material. It is a hyperelastic model (since the stress is obtained from a strain energy function) of an isotropic material (since the Cauchy stress is a function of the left Cauchy-Green strain b only). Differentiating W gives
Here we assume that the faces with normals are free of normal stress, so that, in which case one finds that the hydrostatic pressure is
and the Cauchy stress is as given above. Note that the constitutive equation of the material can be written in the form
showing the one-to-one relationship between the stress and the state of deformation, showing the independence of the stress on the history of deformation, before time t.
The PK2 stress can also be obtained directly through
leading to the PK2 stress given earlier.
Explicitly, from , the strain energy is
It can be seen that this time derivative of the strain energy is precisely the stress power, as expected for a hyperelastic material:
Note that the complete strain energy function, including the hydrostatic term, is . The time derivative of the hydrostatic term is zero (since for an incompressible material).
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