Elastic materials

Isotropic linear elastic material - IsoLE

Linear isotropic material model. The model parameters are summarized in Table 1.

Table 1 Linear Isotropic Material - summary.

Description	Linear isotropic elastic material
Record Format	IsoLE num (in) d (rn) E (rn) n (rn) tAlpha (rn)
Parameters	num material model number d material density E Young modulus n Poisson ratio tAlpha thermal dilatation coefficient
Supported modes	3dMat, PlaneStress, PlaneStrain, 1dMat, 2dPlateLayer, 2dBeamLayer, 3dShellLayer, 2dPlate, 2dBeam, 3dShell, 3dBeam, PlaneStressRot
Features	Adaptivity support
Tests/Examples	benchmark/sm/tr2shell7.in, mpm/mpms01.in, partests/brazil_2d_nl2/brazil_2d_nl.oofem.in, sm/bondceb01.in, sm/linkslip02.in, sm/patch011.in, sm/patch140.in, sm/patch150.in, sm/spring05.in, sm/trshell02_test.in (and 199 more)

Orthotropic linear elastic material - OrthoLE

Orthotropic, linear elastic material model. The model parameters are summarized in Table 2.

Local coordinate system, which determines axes of material orthotropy can be specified using lcs array. This array contains six numbers, where the first three numbers represent directional vector of a local x-axis, and next three numbers represent directional vector of a local y-axis. The local z-axis is determined using the vector product. The right-hand coordinate system is assumed.

Local coordinate system can also be specified using scs parameter. Then local coordinate system is specified in so called “shell” coordinate system, which is defined locally on each particular element and its definition is as follows: principal z-axis is perpendicular to shell mid-section, x-axis is perpendicular to z-axis and normal to user specified vector (so x-axis is parallel to plane, with being normal to this plane) and y-axis is perpendicular both to x and z axes. This definition of coordinate system can be used only with plates and shells elements. When vector is parallel to z-axis an error occurs. The scs array contain three numbers defining direction vector . If no local coordinate system is specified, by default a global coordinate system is used.

For 3D case the material compliance matrix has the following form

\[\begin{split}\mbf{C}=\left[\begin{array}{cccccc} 1/E_X & -\nu_{xy}/E_x& -\nu_{xz}/E_x& 0& 0&0\\ -\nu_{yx}/E_y & 1/E_y& -\nu_{yz}/E_y& 0& 0&0\\ -\nu_{zx}/E_z & -\nu_{zy}/E_z& 1/E_z& 0& 0&0\\ 0 & 0 & 0 & 1/G_{yz} & 0 & 0\\ 0 & 0 & 0 & 0 & 1/G_{xz} & 0\\ 0 & 0 & 0 & 0 & 0 & 1/G_{xy} \end{array}\right]\end{split}\]

By inversion, the material stiffness matrix has the form

\[\begin{split}\mbf{D}=\left[\begin{array}{cccccc} d_{xx} & d_{xy} & d_{xz} & 0 & 0 & 0\\ & d_{yy} & d_{yz} & 0 & 0 & 0\\ \rm {sym} & & d_{zz} & 0 & 0 & 0\\ 0 & 0 & 0 & G_{yz} & 0 & 0\\ 0 & 0 & 0 & 0 & G_{xz} & 0\\ 0 & 0 & 0 & 0 & 0 & G_{xy} \end{array}\right]\end{split}\]

where \(\xi=1-(\nu_{xy}*\nu_{yx}+\nu_{yz}*\nu_{zy}+\nu_{zx}*\nu_{xz})-(\nu_{xy}*\nu_{yz}*\nu_{zx}+\nu_{yx}*\nu_{zy}*\nu_{xz})\) and

\[\begin{split}\begin{eqnarray} d_{xx}&=&E_X(1-\nu_{yz}*\nu_{zy})/\xi,\\ d_{xy}&=&E_y*(\nu_{xy}+\nu_{xz}*\nu_{zy})/\xi,\\ d_{xz}&=&E_z*(\nu_{xz}+\nu_{yz}*\nu_{xy})/\xi,\\ d_{yy}&=&E_y*(1-\nu_{xz}*\nu_{zx})/\xi,\\ d_{yz}&=&E_z*(\nu_{yz}+\nu_{yx}*\nu_{xz})/\xi,\\ d_{zz}&=&E_z*(1-\nu_{yx}*\nu_{xy})/\xi. \end{eqnarray}\end{split}\]

\(E_i\) is Young’s modulus in the \(i\)-th direction, \(G_{ij}\) is the shear modulus in \(ij\) plane, \(\nu_{ij}\) is the major Poisson ratio, and \(\nu_{ji}\) is the minor Poisson ratio. Assuming that \(E_x>E_y>E_z\), \(\nu_{xy} > \nu_{yx}\) etc., then \(\nu_{xy}\) is referred to as the major Poisson ratio, while \(\nu_{yx}\) is referred as the minor Poisson ratio.

Note that there are only nine independent material parameters, because of symmetry conditions. The symmetry condition yields

\[\nu_{xy}E_y=\nu_{yx}E_x,\ \ \nu_{yz}E_z=\nu_{zy}E_y,\ \ \nu_{zx}E_x=\nu_{xz}E_z\]

The model description and parameters are summarized in Table 2.

Table 2 Orthotropic, linear elastic material – summary.

Description	Orthotropic, linear elastic material
Record Format	OrthoLE num (in) d (rn) Ex (rn) Ey (rn) Ez (rn) NYyz (rn) NYxz (rn) NYxy (rn) Gyz (rn) Gxz (rn) Gxy (rn) tAlphax (rn) tAlphay (rn) tAlphaz (rn) [lcs (ra)] [scs (ra)]
Parameters	num material model number d material density Ex, Ey, Ez Young moduli for x,y, and z directions NYyz, NYxz, NYxy major Poisson’s ratio coefficients Gyz, Gxz, Gxy shear moduli tAlphax, tAlphay, tAlphaz thermal dilatation coefficients in x,y,z directions lcs Array defining local material x and y axes of orthotrophy scs Array defining a normal vector n. The local x axis is parallel to plane with n being plane normal. The material local z-axis is perpendicular to shell mid-section.
Supported modes	3dMat, PlaneStress, PlaneStrain, 1dMat, 2dPlateLayer, 2dBeamLayer, 3dShellLayer, 2dPlate, 2dBeam, 3dShell, 3dBeam, PlaneStressRot
Tests/Examples	benchmark/Tr2shell7XFEM_adaptiveDelam.in, benchmark/Tr2shell7XFEM_stressRecovery.in, sm/layered_cube.in, sm/layered_cube_lcs.in, sm/materOrient01.in, sm/plate_a1000.0_n10_h100.0.in

General anisotropic linear elastic material - AnisoLE

Linear elastic material model with completely general material stiffness (21 independent elastic constants). The model parameters are summarized in Table 3. The material stiffness matrix is a completely arbitrary symmetric \(6 \times 6\) matrix. The input line must contain an array with the upper triangle of this matrix. This array has length 21 and the stiffness coefficients are listed in each row from the diagonal to the last column.

Table 3 Anisotropic, linear elastic material – summary.

Description	Anisotropic, linear elastic material
Record Format	OrthoLE num (in) d (rn) stiff (ra) tAlphax (ra)
Parameters	num material model number d material density stiff real array of length 21 with stiffness coefficients \(D_{11}\), \(D_{12}\), \(D_{13}\ldots D_{66}\) tAlpha real array of length 0 or 3 with thermal dilatation coefficients in x,y,z directions
Supported modes	3dMat, PlaneStress, PlaneStrain, 1dMat
Tests/Examples	benchmark/Tr2shell7XFEM_adaptiveDelam.in, benchmark/Tr2shell7XFEM_stressRecovery.in, sm/layered_cube.in, sm/layered_cube_lcs.in, sm/materOrient01.in, sm/plate_a1000.0_n10_h100.0.in

1D linear elastic material with different tension and compression stiffness - isoAxysymm1D

Bi-linear material model for 1D elasticity, with different elastic moduli in tension and compression. The model parameters are summarized in Table 4.

Table 4 Bi-Linear Elastic Material - summary.

Description	Bi-Linear elastic material
Record Format	IsoAxysymm1D num (in) d (rn) Et (rn) Ec (rn) tAlpha (rn) [m (rn)]
Parameters	num material model number d material density Et Modulus of elasticity in tension Ec Modulus of elasticity in compression tAlpha thermal dilatation coefficient m optional regularization coefficient, default value set to 15, higher value makes response close to trully bilinear
Supported modes	1dMat
Features	Adaptivity support

Hyperelastic material - Simo-Pister Material

This material model can describe elastic behavior at large strains. A hyperelastic model postulates the existence of a free energy potential. The existence of the potential implies reversibility of deformations and no energy dissipation during the loading process. Here, we use the free energy function introduced in [SimoHughes].

(1)\[\rho_0 \psi = \frac{1}{4} \left(K - \frac{2}{3}G \right) \left( J^2 - 2 \ln J - 1 \right) + G \left(\mathbf{E} : \mathbf{I} - \ln J \right)\]

where \(K\) is the bulk modulus, \(G\) is the shear modulus, \(J\) is the Jacobian (determinant of the deformation gradient, corresponding to the ratio of the current and initial volume), and \(\mathbf{E}\) is the Green-Lagrange strain. Then, the stress-strain law can be derived from ((1)) as

\[\mathbf{S} = \rho_0 \frac{\partial \psi}{\partial \mathbf{E}} = \frac{1}{2}\left(K - \frac{2}{3}G \right) \left(J^2 - 1 \right)\mathbf{C}^{-1} + G \left(\mathbf{I} -\mathbf{C}^{-1} \right)\]

where \(\mathbf{S}\) is the second Piola-Kirchhoff stress, \(\mathbf{E}\) is the Green-Lagrange strain, and \(\mathbf{C}\) is the right Cauchy-Green tensor.

The model description and parameters are summarized in Table 5.

Table 5 Hyperelastic material - summary.

Description	Hyperelastic material
Record Format	SimoPisterMat in d (rn) K (rn) G (rn)
Parameters	material number d material density K bulk modulus G shear modulus
Supported modes	3dMat
Tests/Examples	benchmark/sm/contact/friction_wip/visual/contact3d_two_bricks_press_rotate_visual.in

Hyperelastic material - Compressible Mooney-Rivlin

The Mooney-Rivlin strain energy function is expressed by

(2)\[\rho_0 \psi = C_1(\bar{I}_1-3) + C_2(\bar{I}_2-3) + \frac{1}{2}K (\ln J)^2\]

where \(C_1\) and \(C_2\) are material constants, \(K\) is the bulk modulus, \(J\) is the Jacobian (determinant of the deformation gradient, corresponding to the ratio of the current and initial volume), \(\bar{I}_1 = J^{-\frac{2}{3}} I_1\), \(\bar{I}_2 = J^{-\frac{2}{3}} I_2\), where \(I_1\) and \(I_2\) are the first and the second principal invariants of the right Cauchy-Green deformation tensor \(\mathbf{C}\). Compressible neo-Hookean material model is obtained by setting \(C_2 = 0\). Then stress-strain law can be derived from ((2)) as

\[\mathbf{P} = \rho_0 \frac{\partial \psi}{\partial \mathbf{F}} = C_1 \frac{\partial\bar{I}_1}{\partial\mathbf{F}} + C_2\frac{\partial\bar{I}_2}{\partial\mathbf{F}} + K \ln J \mathbf{F}^{-T}\]

where \(\mathbf{P}\) is the first Piola-Kirchhoff stress,

\[\frac{\partial \bar{I}_1}{\partial\mathbf{F}} = \frac{2}{J^{\frac{2}{3}}}\mathbf{F} - \frac{2}{3}\bar{I}_1\mathbf{F}^{-T}\]

and

\[\frac{\partial \bar{I}_2}{\partial\mathbf{F}} = 2\bar{I}_1\mathbf{F}-\frac{4}{3}\bar{I}_2\mathbf{F}^{-T}-\frac{2}{J^\frac{4}{3}}\mathbf{F}\cdot \mathbf{C}\]

The first elasticity tensor is derived as

\[A_{ijkl} = \frac{\partial P_{ij}}{\partial F_{kl}} = C_1 A^1_{ijkl} + C_2 A^2_{ijkl} + K(F_{ji}^{-1}F_{lk}^{-1} - \ln J F_{jk}^{-1}F_{li}^{-1})\]

where \(A^1_{ijkl} = \frac{2}{3} J^{-\frac{2}{3}}\left[3\delta_{ik}\delta_{jl} + I_{1}F_{jk}^{-1}F_{li}^{-1} - 2F_{lk}^{-1}F_{ij}+\frac{2}{3}I_1 F_{ji}^{-1}F_{lk}^{-1}-2F_{ji}^{-1}F_{kl} \right]\)

and

(3)\[\begin{split}A^2_{ijkl} = 2 J^{-\frac{4}{3}}\left[ I_1 \delta_{ik} \delta_{jl} + 2 F_{ij}F_{kl}-\frac{4}{3}I_1 F_{ij} F_{lk}^{-1} -\frac{8}{9}I_2 F_{ji}^{-1}F_{lk}^{-1}-\frac{4}{3}I_1 F_{ji}^{-1}F_{kl} \nonumber\right.\\ \left. + \frac{4}{3} F_{kn}C_{nl}F_{ji}^{-1} +\frac{2}{3}I_2 F_{li}^{-1}F_{jk}^{-1} + \frac{4}{3}F_{kl}^{-1}F_{im}C_{mj} - \delta_{ik}C_{lj} - F_{il}F_{kj} + F_{km}F_{im}\delta_{jl}\right]\end{split}\]

The model description and parameters are summarized in Table 6.

Table 6 Compressible Mooney-Rivlin - summary.

Description	Mooney-Rivlin
Record Format	MooneyRivlinCompressibleMat in d (rn) K (rn) C1 (rn) C2 (rn)
Parameters	material number d material density K bulk modulus C1 material constant C2 material constant
Supported modes	3dMat, PlaneStrain
Tests/Examples	benchmark/sm/contact/friction_wip/visual/contact2d_inclined_plane_slip.in, sm/contact2d.in, sm/contact2d_linesearch.in, sm/contact2d_projection_boundary.in, sm/contact2d_projection_outside.in, sm/contact2d_sliding.in, sm/contact3d_generalized_edge_direct.in, sm/friction_wip/contact2d_friction.in, sm/mooneyrivlin1.in, sm/mooneyrivlin2.in (and 16 more)

Hyperelastic material - Compressible Ogden

The Ogden strain energy function is expressed by

(4)\[\rho_0 \psi = \sum_{I=1}^N \frac{\mu_I}{\alpha_I} \left( \bar{\lambda}_1^{\alpha_I}+\bar{\lambda}_2^{\alpha_I}+\bar{\lambda}_3^{\alpha_I}-3\right)+ U(J)\]

where \(\mu_I\) and \(\alpha_I\) are material constants, \(U(J)\) is the volumetric part of energy, \(J\) is the Jacobian (determinant of the deformation gradient, corresponding to the ratio of the current and initial volume), \(\bar{\lambda}_i = J^{-\frac{1}{3}} {\lambda}_i\) are the deviatoric stretches.

The model description and parameters are summarized in Table 7.

Table 7 Compressible Ogden material - summary.

Description	Ogden material
Record Format	OgdenCompressibleMat in d (rn) K (rn) alpha (ra) mu (ra)
Parameters	material number d material density K bulk modulus alpha array of material constants mu array of material constants
alpha and mu have to have the same size
Tests/Examples	sm/ogden1.in, sm/ogden2.in, sm/timestepreduction.in

Hyperelastic material - Blatz-Ko

The Mooney-Rivlin strain energy function is expressed by

(5)\[\rho_0 \psi = \frac{\mu}{2}\left(\frac{I_2}{I_3} + 2\sqrt{I_3}-5 \right)\]

where \(\mu\) is initial shear modulus, \(I_2\) and \(I_3\) are the second and third invariants of the Cauchy-Green tensor \(\mbf{C}\).

The model description and parameters are summarized in Table 8.

Table 8 Blatz-Ko material - summary.

Description	Blatz-Ko material
Record Format	blatzkomat in d (rn) mu (rn)
Parameters	material number d material density mu shear modulus
\(\nu\) is fixed to 0.25
Tests/Examples	sm/blatzko1.in