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Concept
Strain energy density function
Summary
A strain energy density function or stored energy density function is a scalar-valued function that relates the strain energy density of a material to the deformation gradient.
Equivalently,
where is the (two-point) deformation gradient tensor, is the right Cauchy–Green deformation tensor, is the left Cauchy–Green deformation tensor,
and is the rotation tensor from the polar decomposition of .
For an anisotropic material, the strain energy density function depends implicitly on reference vectors or tensors (such as the initial orientation of fibers in a composite) that characterize internal material texture. The spatial representation, must further depend explicitly on the polar rotation tensor to provide sufficient information to convect the reference texture vectors or tensors into the spatial configuration.
For an isotropic material, consideration of the principle of material frame indifference leads to the conclusion that the strain energy density function depends only on the invariants of (or, equivalently, the invariants of since both have the same eigenvalues). In other words, the strain energy density function can be expressed uniquely in terms of the principal stretches or in terms of the invariants of the left Cauchy–Green deformation tensor or right Cauchy–Green deformation tensor and we have:
For isotropic materials,
with
For linear isotropic materials undergoing small strains, the strain energy density function specializes to
A strain energy density function is used to define a hyperelastic material by postulating that the stress in the material can be obtained by taking the derivative of with respect to the strain. For an isotropic hyperelastic material, the function relates the energy stored in an elastic material, and thus the stress–strain relationship, only to the three strain (elongation) components, thus disregarding the deformation history, heat dissipation, stress relaxation etc.
For isothermal elastic processes, the strain energy density function relates to the specific Helmholtz free ener
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Related concepts (2)
Deformation (physics)
In physics and continuum mechanics, deformation is the transformation of a body from a reference configuration to a current configuration. A configuration is a set containing the positions of all particles of the body. A deformation can occur because of external loads, intrinsic activity (e.g. muscle contraction), body forces (such as gravity or electromagnetic forces), or changes in temperature, moisture content, or chemical reactions, etc. Strain is related to deformation in terms of relative displacement of particles in the body that excludes rigid-body motions.
Elasticity (physics)
In physics and materials science, elasticity is the ability of a body to resist a distorting influence and to return to its original size and shape when that influence or force is removed. Solid objects will deform when adequate loads are applied to them; if the material is elastic, the object will return to its initial shape and size after removal. This is in contrast to plasticity, in which the object fails to do so and instead remains in its deformed state. The physical reasons for elastic behavior can be quite different for different materials.