Difference between revisions of "Isotropic, Hyperelastic-Plastic Material"
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<math> \mathbf{F} = \mathbf{F}_{e}. \mathbf{F}_{p} </math> | <math> \mathbf{F} = \mathbf{F}_{e}. \mathbf{F}_{p} </math> | ||
Where '''F'''e and '''F'''p are the elastic and plastic deformation gradient tensors respectively, with det '''F'''<sub>p</sub>, that supposes the plastic flow to be isochoric. The Neo-Hookean elastic stored energy, represented by its uncoupled volumetric-deviatoric internal energy form, is consistent with the fundamental idea that the elastic-plastic deviatoric response is assumed to be uncoupled from the elastic volumetric response | Where '''F'''e and '''F'''p are the elastic and plastic deformation gradient tensors respectively, with det ('''F''')<sub>p</sub>, that supposes the plastic flow to be isochoric. The Neo-Hookean elastic stored energy, represented by its uncoupled volumetric-deviatoric internal energy form, is consistent with the fundamental idea that the elastic-plastic deviatoric response is assumed to be uncoupled from the elastic volumetric response | ||
In finite strain plasticity, the stored energy is based on the additive decomposition of the stored energy into elastic We and plastic Wp internal energies. The elastic stored energy is related to the intermediate configuration, and the plastic stored energy is expressed in term of plastic state variables α. | In finite strain plasticity, the stored energy is based on the additive decomposition of the stored energy into elastic We and plastic Wp internal energies. The elastic stored energy is related to the intermediate configuration, and the plastic stored energy is expressed in term of plastic state variables α. | ||
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<math>W_{e} ={\kappa\over 2 }({1\over 2 }(J_{e}^2-1)-ln J_{e}) + {\mu \over 2 } (\bar I_{e1}-3)</math> | <math>W_{e} ={\kappa\over 2 }({1\over 2 }(J_{e}^2-1)-ln J_{e}) + {\mu \over 2 } (\bar I_{e1}-3)</math> | ||
Where J<sub>e</sub>=det ('''F''')<sub>e</sub> and <math>I_{1e} = Trace(\mathbf{\bar B}_{e} </math> is the deviatoric part of the left Cauchy-Green strain tensor and is the deviatoric part of the elastic deformation gradient. κ and μ are interpreted in small strains as bulk and shear modulus respectively. | |||
The elastic stress-strain constitutive law, which derives from the elastic storage energy, is given here in term of Kirchhoff stress tensor by: | |||
The | Where | ||
\mathbf{\sigma} = p \mathbf{I} + \bar \mathbf{\sigma} </math> given by: | |||
<math> \mathbf{\sigma} =K(J-1)\mathbf{I} + {G_{1} \over J^{5/3} } (\mathbf{B}-{I_{1} \over3}\mathbf{I}) + {G_{2} \over J^{7/3}} (I_{1} \mathbf{B}-\mathbf{B^2}-{2I_{2} \over3}\mathbf{I})</math> | <math> \mathbf{\sigma} =K(J-1)\mathbf{I} + {G_{1} \over J^{5/3} } (\mathbf{B}-{I_{1} \over3}\mathbf{I}) + {G_{2} \over J^{7/3}} (I_{1} \mathbf{B}-\mathbf{B^2}-{2I_{2} \over3}\mathbf{I})</math> | ||
Revision as of 12:42, 23 September 2013
Constitutive Law
The HEIsotropic material HEIsotropic is an isotropic, elastic-plastic material in large strains using the hyperelastic formulation, using a Neo-Hookean strain energy function Mooney-Rivlin.
The formulation of finite strain plasticity is based on the notion of a stress free intermediate configuration and uses a multiplicative decomposition of the deformation gradient F given by:
[math]\displaystyle{ \mathbf{F} = \mathbf{F}_{e}. \mathbf{F}_{p} }[/math]
Where Fe and Fp are the elastic and plastic deformation gradient tensors respectively, with det (F)p, that supposes the plastic flow to be isochoric. The Neo-Hookean elastic stored energy, represented by its uncoupled volumetric-deviatoric internal energy form, is consistent with the fundamental idea that the elastic-plastic deviatoric response is assumed to be uncoupled from the elastic volumetric response
In finite strain plasticity, the stored energy is based on the additive decomposition of the stored energy into elastic We and plastic Wp internal energies. The elastic stored energy is related to the intermediate configuration, and the plastic stored energy is expressed in term of plastic state variables α.
[math]\displaystyle{ W =W_{e} (\mathbf{B}_{e}) + W_{p} (\alpha) }[/math]
The particular Neo-Hookean stored energy considered here is recalled: [math]\displaystyle{ W_{e} ={\kappa\over 2 }({1\over 2 }(J_{e}^2-1)-ln J_{e}) + {\mu \over 2 } (\bar I_{e1}-3) }[/math]
Where Je=det (F)e and [math]\displaystyle{ I_{1e} = Trace(\mathbf{\bar B}_{e} }[/math] is the deviatoric part of the left Cauchy-Green strain tensor and is the deviatoric part of the elastic deformation gradient. κ and μ are interpreted in small strains as bulk and shear modulus respectively.
The elastic stress-strain constitutive law, which derives from the elastic storage energy, is given here in term of Kirchhoff stress tensor by:
Where
\mathbf{\sigma} = p \mathbf{I} + \bar \mathbf{\sigma} </math> given by:
[math]\displaystyle{ \mathbf{\sigma} =K(J-1)\mathbf{I} + {G_{1} \over J^{5/3} } (\mathbf{B}-{I_{1} \over3}\mathbf{I}) + {G_{2} \over J^{7/3}} (I_{1} \mathbf{B}-\mathbf{B^2}-{2I_{2} \over3}\mathbf{I}) }[/math]
where [math]\displaystyle{ I_{1} = J^{2/3} \bar I_{1} \qquad {\rm and} \qquad I_{2} = J^{4/3} \bar I_{2} }[/math] .
Material Properties
The constants involved in the strain energy function, are equivalent in small strains to properties of isotropic elastic material with Poisson's ratio [math]\displaystyle{ {\nu} }[/math] as well as shear G = G1 + G2 and bulk modulus K given by
[math]\displaystyle{ G = {E \over 2({1+\nu })} \qquad {\rm and} \qquad K = {E \over 3({1-2\nu })} }[/math].
| Property | Description | Units | Default |
|---|---|---|---|
| E | Elastic modulus | MPa | none |
| G1, G2 | Shear modulus | MPa | none |
| alpha | Thermal expansion coefficient | ppm/M | 40 |
History Variables
None
Examples
These commands model polymer as an isotropic, hyperelastic material with G1=G2 =G/2 (using scripted or XML commands):
Material "polymer","polymer","Mooney" E 2500 nu .4 alpha 60 rho 1.2 Done
<Material Type="8" Name="polymer"> <rho>1.2</rho> <G1>35.714285714</G1> <G2>35.714285714</G2> <K>166.66666666</K> <alpha>60</alpha> </Material>