## Linear Elastic Material Models for Stress Analysis

Linear elastic materials models are supported by all stress elements (rod/cable, beam, 2D, shell, and 3D elements). Isotropic, orthotropic, and anisotropic material properties can be specified.

## 1. Linear Elastic Isotropic Material

Linear elastic isotropic materials are supported by all stress elements (rod/cable, beam, 2D, shell, and 3D elements).

material id type isotropic e v   [nu v]   [alpha v]   [failure ... end]   [density v] end

The following parameters are specified:

e v

Specifies the modulus of elasticity.

nu v

Specifies Poisson's ratio. Default is 0.

alpha v

Specifies the thermal expansion coefficient. Default is 0.

failure ... end

Specifies a failure criterion (optional).

density v

Specifies the material density. Default is 0.

## 2. Linear Elastic Orthotropic Material

Linear elastic orthotropic material models are supported by all 2D, shell, and 3D stress elements). Depending on the element, not all material constants need to be specified.

material id type orthotropic   [planestrain]   e1|e2|e3 v   nu12|nu13|nu23 v   g12|g13|g23 v   [alpha11|alpha22|alpha33|alpha12|alpha13|alpha23 v]   [failure ... end]   [density v] end

The following parameters are specified:

planestrain

Specifies plane-strain elasticity. For 2D elements, default is plane-stress. Is ignored by non-2D elements (shell elements are always plane-stress, and 3D elements are neither plane-strain nor plane-stress).

e1|e2|e3 v

Specifies the modulus of elasticity in the material directions.

nu12|nu13|nu23 v

Specifies Poisson's ratio relating the different material directions. Shell elements and 2D elements make use of nu13 and nu23 for the plane-stress and plane-strain (2D elements only) conditions, respectively.

g12|g13|g23 v

Specifies the shear modulus. 2D elements will ignore g13 and g23.

alpha11|alpha22|alpha33|alpha12|alpha13|alpha23 v

Specifies the thermal expansion coefficients. Default is 0. 2D elements will ignore alpha33|alpha13|alpha23.

failure ... end

Specifies a failure criterion (optional).

density v

Specifies the material density. Default is 0.

Example:

material 1 type orthotropic
e1 124000.0
e2 9000.0
nu12 0.3
g12 5100.0
g13 5100.0
g23 5000.0
density 1.53e3
end

## 3. Linear Elastic Anisotropic Material

Linear elastic anisotropic material models are supported by all shell and 3D stress elements.

material id type anisotropic   [planestrain]   c1111|c1122|c1133|c1112|c1113|c1123 v         c2222|c2233|c2212|c2213|c2223 v               c3333|c3312|c3313|c3323 v                     c1212|c1213|c1223 v                           c1313|c1323 v                                 c2323 v   [alpha11|alpha22|alpha33|alpha12|alpha13|alpha23 v]   [failure ... end]   [density v] end

The following parameters are specified:

planestrain

Specifies plane-strain elasticity. For 2D elements, default is plane-stress. Is ignored by non-2D elements (shell elements are always plane-stress, and 3D elements are neither plane-strain nor plane-stress).

cijkl v

Specifies an entry in the consitutive matrix. Default is 0 for all entries.

alpha11|alpha22|alpha33|alpha12|alpha13|alpha23 v

Specifies the thermal expansion coefficients. Default is 0. 2D elements will ignore alpha33|alpha13|alpha23.

failure ... end

Specifies a failure criterion (optional).

density v

Specifies the material density. Default is 0.

## 4. Laminates

Laminates are supported by shell elements and by hexahedral and prismatic 3D elements.

material id type laminate   t alpha id   t alpha id   ... end

The plies of the laminate are arranged in the element thickness direction from the bottom surface to the top surface, with the first ply located at the bottom and the last ply located at the top. A ply is defined by the following parameters:

t

Specifies the ply thickness.

alpha

Specifies angle of rotation (in degrees) around the element surface normal and according to the right-hand rule.

id

Specifies the identifier of the material for this layer. The same material model must be used for all plies. Materials for stress analysis and heat analysis are supported.

## 5. ABD Material

The ABD material model is a linear equivalent-stiffness-layer material model for shell analysis with the plane stress assumption. It allows to directly specify the A,B,D matrices from classical laminate theory. It is supported by all shell elements.

material id type abd   [membrane_matrix  "[" C1111 C1122 C1112 C2222 C2212 C1212 "]"]   [bending_matrix  "[" C1111 C1122 C1112 C2222 C2212 C1212 "]"]   [coupling_matrix  "[" C1111 C1122 C1112 C2222 C2212 C1212 "]"]   [transverse_shear_matrix  "[" C1313 C2313 C2323 "]"]   [membrane_alpha "[" A11 A22 A12  "]"]   [bending_alpha "[" A11 A22 A12  "]"]   [mass v] end

membrane_matrix "[" C1111 C1122 C1112 C2222 C2212 C1212 "]"

Specifies the upper part of the symmetric 3x3 matrix of the membrane shell section stiffness. It corresponds to the A-matrix of classical laminate theory.

bending_matrix "[" C1111 C1122 C1112 C2222 C2212 C1212 "]"

Specifies the upper part of the symmetric 3x3 matrix of the bending shell section stiffness. It corresponds to the D-matrix of classical laminate theory.

coupling_matrix "[" C1111 C1122 C1112 C2222 C2212 C1212 "]"

Specifies the upper part of the symmetric 3x3 membrane-bending coupling matrix. It corresponds to the B-matrix of classical laminate theory.

transverse_shear_matrix "[" C1313 C2313 C2323 "]"

Specifies the upper part of the symmetric 2x2 matrix containing the transverse-shear shell section stiffnesses. It is required by shear-deformable shell elements such as the MITC elements. This matrix must incorporate the shear correction factor. For example, with isotropic materials, the two diagonal elements would have a value of 5 6 E t 2( 1+ ν) where t is the shell thickness, E the modulus of elasticity, and ν Poisson's ratio.

membrane_alpha "[" A11 A22 A12 "]"

Specifies the thermal expansion coefficients for the membrane components of the strain tensor.

bending_alpha "[" A11 A22 A12 "]"

Specifies the thermal expansion coefficients for the bending components of the strain tensor.

mass v

Specifies the mass per unit area.

All shell elements are assumed to have a thickness of 1 and an eccentricity of 0, regardless of what is specified otherwise. Hence, the matrices are not scaled by a thickness, instead, they are utilized as is specified.

The material law is as follows:

[ σ m σ b σ s ] = [ A B 0 B T D 0 0 0 S ] ( [ ε m ε b ε s ] - ΔT [ α m α b 0 ] )

where ΔT is the difference between the interpolated temperature to the reference temperature (see the temperatures command). It is evaluated at the shell mid-surface (it is the average of the values at the bottom and top shell surface).

The in-layer strains and stresses are not calculated and are not written to the database. Instead, the shell section strains, expressed in the material reference frame, are stored in the STRAIN_SHELL_MATERIAL dataset/baspl++ Field, with the components aligned with the ABD material matrix. The shell section stresses, also expressed in the material reference frame, are stored in the STRESS_SHELL_MATERIAL dataset/baspl++ Field.

## 6. PSHELL Material

The PSHELL material model is a linear equivalent-stiffness-layer material model for shell analysis with the plane stress assumption. It is supported by all shell elements. Its purpose is to provide compatibility with FE models that were created for Nastran.

material id type pshell   [membrane_mid mid]   [bending_mid mid]   [tranverse_shear_mid mid]   [coupling_mid mid]   [bending_scale_factor f]   [shear_correction_factor f]   [thickness t]   [non_structural_mass m] end

The MITC elements are Reissner-Mindlin shell elements and as such require a nonzero but finite transverse-shear stiffness. The B2000++ implementation of PSHELL ensures this: If no material id for the transverse shear is given, transverse-shear stiffness will be derived from the membrane material. If no membrane material is specified, transverse-shear stiffness will be derived from the bending material.

According to Nastran conventions, the transverse shear stiffness is extracted as follows according to the type of material: For isotropic materials, g is used. For orthotropic materials, g13 and g23 are used. For anisotropic materials, the sub-matrix [[C1111,C1122],[C1122,C2222]] is used.

If no thickness is specified for the PSHELL material, a thickness for the shell element must be specified. The thickness is used to scale the membrane matrix and the transverse shear stiffnesses by t and the bending matrix by t 3 12 . The coupling matrix is not scaled.

To account for non-homogeneous materials, the bending_scale_factor (default 1) can be modified.

The shear_correction_factor (default 5/6) should not be set to zero.

## 7. Additional Notes

The following figure displays the spatial orientation components of the 2D and 3D stress tensors and of the heat flux directions.

The strain-stress (compliance) relation of a linear elastic orthotropic 3D material is defined as follows:

Equation 1. Compliance relation of a linear elastic orthotropic 3D material

{ ε 11 ε 22 ε 33 γ 12 γ 23 γ 13 } = ( 1 E 1 ν 12 E 1 ν 13 E 1 0 0 0 1 E 2 ν 23 E 2 0 0 0 1 E 3 0 0 0 1 G 12 0 0 1 G 23 0 symm 1 G 13 ) { σ 11 σ 22 σ 33 τ 12 τ 23 τ 13 }

The strain-stress (compliance) relation of a linear elastic orthotropic 2D material is defined as follows:

Equation 2. Compliance relation of a linear elastic orthotropic plane-stress material (left) and plane-strain material (right)

{ ε 11 ε 22 γ 12 } = 1 E 1 ν 12 E 1 0 1 E 2 0 symm 1 G 12 { σ 11 σ 22 τ 12 } , { ε 11 ε 22 γ 12 } = 1 ν 12 2 E 1 ν 12 ( ν 12 1 ) E 1 0 1 ν 12 2 E 2 0 symm 1 G 12 { σ 11 σ 22 τ 12 }