Inelastic Strain Contributions
Many of the material models in COMSOL Multiphysics will compute a stress based on an elastic strain. The elastic strain tensor is obtained after removing any inelastic deformation contribution from the total deformation from the displacements. There are several possible inelastic strain contributions:
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Initial strain, ε0
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External strain, εex
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Thermal strain, εth
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Hygroscopic strain, εhs
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Plastic strain, εpl
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Creep strain, εcr
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Viscoplastic strain, εvp
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Viscoelastic strain, εve
Additive Decomposition
In a geometrically linear analysis, the elastic strain is computed by a straightforward subtraction of the inelastic strain:
where
Additive decomposition of strains can also be used in a geometrically nonlinear analysis. In this case, it can be justified as long as the strains are small, or when using Hencky (logarithmic) strains. In the case of large deformations, the different strain contributions may not even be commutative.
Choose to use additive decomposition also for geometric nonlinearity in the Geometric Nonlinearity section in the settings for Linear Elastic Material or Nonlinear Elastic Material.
Multiplicative Decomposition
In the finite deformation case, the inelastic strain can be removed by using a multiplicative decomposition of the deformation gradient tensor. The elastic deformation gradient tensor is the basis for all strain energy formulations in hyperelastic materials, and also for the elastic strain in linear and nonlinear elasticity. It is derived by removing the inelastic deformation from the total deformation gradient tensor.
The total deformation gradient tensor is defined as the result of two successive operations, an inelastic deformation followed by an elastic deformation:
(3-4)
The order is important here, multiplication from the left makes the elastic deformation act on the inelastically deformed state.
Since a deformation gradient tensor describes a mapping from one frame to another, there are actually three frames involved in this operation. The F tensor is defined by the displacements as usual and describes the mapping from the material frame to the spatial frame. The Finel tensor, however, describes a mapping from the material frame to an intermediate frame, and the Fel tensor describes a mapping from the intermediate frame to the spatial frame.
When the inelastic deformation gradient tensor is known, the elastic deformation gradient tensor is computed as
(3-5)
so the inelastic deformations are removed from the total deformation gradient tensor. The inelastic deformation tensor Finel is derived from inelastic processes, such as thermal expansion or plasticity. When there are several inelastic contributions, they are applied sequentially to obtain the total inelastic deformation tensor Finel:
where Fi is the inelastic strain contribution from subnode i under a Linear Elastic Material, Nonlinear Elastic Material, or Hyperelastic Material.
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The order is important when deformations are finite. The contributions are applied in the same order as the subnodes appear in the model tree. If a Thermal Expansion node appears before a Plasticity node, then the physical process can be viewed as a thermal expansion followed by a plastic deformation.
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When a certain inelastic strain contribution is small, the order is not significant.
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If the inelastic strain is a pure isotropic volume change, as is often the case for thermal expansion and hygroscopic swelling, the order is not significant.
The right Cauchy–Green deformation tensor C is computed from the deformation gradient F, as well as the Green–Lagrange strain tensor ε
and
The elastic right Cauchy–Green deformation tensor is then computed from Fel
and the elastic Green–Lagrange strain tensor is computed as
The internal variables for the elastic right Cauchy–Green deformation tensor in the local coordinate system are named solid.Cel11, solid.Cel12, and so on; and for the elastic Green–Lagrange tensor in local coordinates solid.eel11, solid.eel12, and so on.
The elastic, inelastic, and total volume ratios are related as
or
The internal variables for the elastic, inelastic, and total volume ratio are named solid.Jel, solid.Ji, and solid.J.
After Equation 3-5, the elastic Green–Lagrange strain depends on the inelastic deformation as
(3-6)
where the inelastic Green–Lagrange strain reads
When using multiplicative decomposition of deformation gradients, Hooke’s law for a Linear Elastic Material produces a second Piola–Kirchhoff stress tensor which is linearly related to the elastic Green–Lagrange strain
This is however, a stress tensor defined in the intermediate configuration. Introducing a pull-back operation of this tensor, Hooke’s law for the multiplicative decomposition of elastic and inelastic strains reads
Logarithmic Decomposition
For finite deformations, the inelastic strain can be removed by using an additive decomposition of logarithmic stretches. The elastic Hencky tensor is the basis for the strain energy formulation. It is computed by removing the inelastic strains from the total Hencky strain tensor:
(3-7)
here, the inelastic logarithmic strain, εinel, is computed as the accumulation of all possible inelastic effects
The right Cauchy–Green deformation tensor C is computed from the deformation gradient F, as well as the total Hencky strain tensor εH
and
The Padé approximation (Ref. 51) of the matrix logarithm reads
which can be faster than the analytical formula for moderate stretches.
When using an additive decomposition of logarithmic strains, Hooke’s law for a Linear Elastic Material produces a Mandel stress tensor which is linearly related to the elastic Hencky strain