Elastic materials


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

Table 1: Linear Isotropic Material - summary.

1.1.2 Orthotropic linear elastic material - OrthoLE

Orthotropic, linear elastic material model. The model parameters are summarized in Tab. 2. Local coordinate system, which determines axes of material orthotrophy can by 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.

where ξ = 1 - (ν_xy * ν_yx + ν_yz * ν_zy + ν_zx * ν_xz) - (ν_xy * ν_yz * ν_zx + ν_yx * ν_zy * ν_xz) and

E_i is Young’s modulus in the i-th direction, G_ij is the shear modulus in ij plane, ν_ij is the major Poisson ratio, and ν_ji is the minor Poisson ratio. Assuming that E_x > E_y > E_z, ν_xy > ν_yx etc., then ν_xy is referred to as the major Poisson ratio, while ν_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


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

Table 2: Orthotropic, linear elastic material – summary.

1.1.3 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 Tab. 3. The material stiffness matrix is a completely arbitrary symmetric 6 × 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.


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 D₁₁, D₁₂, D₁₃…D₆₆
	- tAlpha real array of length 0 or 3 with thermal dilatation coefficients in x,y,z directions
Supported modes	3dMat, PlaneStress, PlaneStrain, 1dMat

Table 3: Anisotropic, linear elastic material – summary.

1.1.4 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 Tab. 4.


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

Table 4: Bi-Linear Elastic Material - summary.

1.1.5 Hyperelastic material - Simo-Pister Material

This material model can describe elastic behavior at large strains. A hyperelastic model postulates the existence of free energy potential. Existence of the potential implies reversibility of deformations and no energy dissipation during loading process. Here we use the free energy function introduced in [21]

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 E is the Green-Lagrange strain. Then stress-strain law can be derived from (9) as

where S is the second Piola-Kirchhoff stress, E is the Green-Lagrange strain and C is the right Cauchy-Green tensor. The model description and parameters are summarized in Tab. 5.


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

Table 5: Hyperelastic material - summary.

1.1.6 Hyperelastic material - Compressible Mooney-Rivlin

where C₁ and C₂ 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), I₁ = J^-I₁, I₂ = J^-I₂, where I₁ and I₂ are the first and the second principal invariants of the right Cauchy-Green deformation tensor C. Compressible neo-Hookean material model is obtained by setting C₂ = 0. Then stress-strain law can be derived from (11) as


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

Table 6: Compressible Mooney-Rivlin - summary.

1.1.7 Hyperelastic material - Compressible Ogden

where μ_I and α_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), λ_i = J^-λ_i are the deviatoric stretches.


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

Table 7: Compressible Ogden material - summary.

1.1.8 Hyperelastic material - Blatz-Ko

where μ is initial shear modulus, I₂ and I₃ are the second and third invariants of the Cauchy-Green tensor C.


Description	Blatz-Ko material

Record Format	blatzkomat (in) # d(rn) # mu(rn) #
Parameters	- material number
	- d material density
	- mu shear modulus
ν is fixed to 0.25

Table 8: Blatz-Ko material - summary.

1.1 Elastic materials