Elasticity

The elastic material model, also referred to as a linear elastic material, is the simplest available form of elasticity. It defines the ability of a material to recover its original shape when applied forces are removed. The model requires the specification of the directional dependence of the elasticity and the elastic constants. It also supports an optional definition for a viscoelastic time scale and the ability to remove elastic stiffness in compression, tension, or both compression and tension.

This page discusses:

Isotropic Elasticity
Orthotropic Elasticity
Engineering Constraints
Lamina Elasticity
Anisotropic Elasticity
Transversely Isotropic Elasticity
Isotropic Shear Elasticity
Linear Elastic Response for Viscoelastic Materials
No Compression or No Tension

Linear elasticity is valid for small elastic strains, usually less than 5%. When you specify elasticity, you can select the following aspects of the elastic material model:

The elastic behavior can be isotropic, orthotropic, or fully anisotropic.
You can define properties that depend on temperature and other field variables; and
You can specify that compressive stress or tensile stress cannot be generated.

The total stress is defined from the total elastic strain as

σ = D^{e l} ε^{e l}

where

σ

is the total stress (“true,” or Cauchy stress in finite-strain problems),

D^{e l}

is the fourth-order elasticity tensor, and

ε^{e l}

is the total elastic strain (log strain in finite-strain problems). Do not use the linear elastic material definition when the elastic strains may become large; use a hyperelastic model instead (see Isotropic Hyperelasticity). Even in finite-strain problems the elastic strains should still be small (less than 5%).

Depending on the number of symmetry planes for the elastic properties, a material can be classified as either isotropic (an infinite number of symmetry planes passing through every point) or anisotropic (no symmetry planes). Some materials have a restricted number of symmetry planes passing through every point; for example, orthotropic materials have two orthogonal symmetry planes for the elastic properties. The number of independent components of the elasticity tensor $D^{e l}$ depends on such symmetry properties. You define the level of anisotropy and the method of defining the elastic properties, as described below. If the material is anisotropic, a local orientation must be used to define the direction of anisotropy.

Linear elastic materials must satisfy the conditions of material or Drucker stability. Stability requires that the tensor $D^{e l}$ be positive definite, which leads to certain restrictions on the values of the elastic constants. The stress-strain relations for several different classes of material symmetries are given below. The appropriate restrictions on the elastic constants stemming from the stability criterion are also given.

The following elasticity types are available:


Type	Description
Isotropic	A single elasticity definition for all material directions.
Orthotropic	A different elasticity definition for each primary material direction. The elasticity is defined using nine stiffness parameters, $D_{1111}$ , $D_{2222}$ , $D_{3333}$ , $D_{1122}$ , etc.
Engineering Constants	An alternate method to define orthotropic elasticity.
Lamina	Lamina elasticity is a special case of orthotropic elasticity valid only for two-dimensional structures.
Anisotropic	Provides a modeling capability for materials that exhibit highly anisotropic behavior, such as biomedical soft tissues and fiber-reinforced elastomers.
Transversely isotropic	A special subclass of orthotropy characterized by a plan of isotropy at every point in the material.
Shear	Describes the deviatoric response of materials whose volumetric response is governed by an equation of state.

Isotropic Elasticity

The simplest form of linear elasticity is the isotropic case, and the stress-strain relationship is given by

{\begin{matrix} ε_{11} \\ ε_{22} \\ ε_{33} \\ γ_{12} \\ γ_{13} \\ γ_{23} \end{matrix}} = [\begin{matrix} 1 / E & - ν / E & - ν / E & 0 & 0 & 0 \\ □ & 1 / E & - ν / E & 0 & 0 & 0 \\ □ & □ & 1 / E & 0 & 0 & 0 \\ □ & □ & □ & 1 / G & 0 & 0 \\ □ & s y m & □ & □ & 1 / G & 0 \\ □ & □ & □ & □ & □ & 1 / G \end{matrix}] {\begin{matrix} σ_{11} \\ σ_{22} \\ σ_{33} \\ σ_{12} \\ σ_{13} \\ σ_{23} \end{matrix}} = D^{e l} {\begin{matrix} ε_{11} \\ ε_{22} \\ ε_{33} \\ γ_{12} \\ γ_{13} \\ γ_{23} \end{matrix}} .

The elastic properties are completely defined by giving the Young's modulus,

E

, and the Poisson's ratio,

ν

. The shear modulus,

G

, can be expressed in terms of

E

and

E

G = E / 2 (1 + ν)

. These parameters can be given as functions of temperature and of other predefined fields, if necessary.

The stability criterion requires that $E > 0$ and $- 1 < ν < 0.5$ . Values of Poisson's ratio approaching 0.5 result in nearly incompressible behavior. With the exception of plane stress cases (including membranes and shells) or beams and trusses, such values require the use of “hybrid” elements in implicit analyses and generate high frequency noise and result in excessively small stable time increments in explicit simulations.

Parameters


Input Data	Description
Young's Modulus	$E > 0$ .
Poisson's Ratio	$- 1 < ν < 0.5$ .
Use temperature-dependent data	Specifies material parameters that depend on temperature. A Temperature field appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.
Number of field variables	Specifies material parameters that depend on field variables. Field columns appear in the data table for each field variable you add. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.

Orthotropic Elasticity

Linear elasticity in an orthotropic material can also be defined by giving the nine independent elastic stiffness parameters, as functions of temperature and other predefined fields, if necessary. In this case the stress-strain relations are of the form

{\begin{matrix} σ_{11} \\ σ_{22} \\ σ_{33} \\ σ_{12} \\ σ_{13} \\ σ_{23} \end{matrix}} = [\begin{matrix} D_{1111} & D_{1122} & D_{1133} & 0 & 0 & 0 \\ □ & D_{2222} & D_{2233} & 0 & 0 & 0 \\ □ & □ & D_{3333} & 0 & 0 & 0 \\ □ & □ & □ & D_{1212} & 0 & 0 \\ □ & s y m & □ & □ & D_{1313} & 0 \\ □ & □ & □ & □ & □ & D_{2323} \end{matrix}] {\begin{matrix} ε_{11} \\ ε_{22} \\ ε_{33} \\ γ_{12} \\ γ_{13} \\ γ_{23} \end{matrix}} = D^{e l} {\begin{matrix} ε_{11} \\ ε_{22} \\ ε_{33} \\ γ_{12} \\ γ_{13} \\ γ_{23} \end{matrix}} .

The restrictions on the elastic constants because of material stability are

D_{1111}, D_{2222}, D_{3333}, D_{1212}, D_{1313}, D_{2323} > 0

| D_{1122} | < {(D_{1111} D_{2222})}^{1 / 2}

| D_{1133} | < {(D_{1111} D_{3333})}^{1 / 2}

| D_{2233} | < {(D_{2222} D_{3333})}^{1 / 2}

\det (D^{e l}) = D_{1111} D_{2222} D_{3333} + 2 D_{1122} D_{1133} D_{2233} - D_{2222} D_{1133}^{2} - D_{1111} D_{2233}^{2} - D_{3333} D_{1122}^{2} > 0.

These restrictions in terms of the elastic stiffness parameters are equivalent to the restrictions in terms of the "engineering constants."

Parameters


Input Data	Description
Dnnnn	Nine independent elastic stiffness parameters $D_{i j k l}$ .
Use temperature-dependent data	Specifies material parameters that depend on temperature. A Temperature field appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.
Number of field variables	Specifies material parameters that depend on field variables. Field columns appear in the data table for each field variable you add. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.

Engineering Constraints

Orthotropic elasticity can be defined by providing the engineering constants: the Young's moduli, Poisson's ratios, and shear moduli associated with the three principal material directions.

{\begin{matrix} ε_{11} \\ ε_{22} \\ ε_{33} \\ γ_{12} \\ γ_{13} \\ γ_{23} \end{matrix}} = [\begin{matrix} 1 / E_{1} & - ν_{21} / E_{2} & - ν_{31} / E_{3} & 0 & 0 & 0 \\ - ν_{12} / E_{1} & 1 / E_{2} & - ν_{32} / E_{3} & 0 & 0 & 0 \\ - ν_{13} / E_{1} & - ν_{23} / E_{2} & 1 / E_{3} & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 / G_{12} & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 / G_{13} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 / G_{23} \end{matrix}] {\begin{matrix} σ_{11} \\ σ_{22} \\ σ_{33} \\ σ_{12} \\ σ_{13} \\ σ_{23} \end{matrix}} .

The quantity

ν_{i j}

has the physical interpretation of the Poisson's ratio that characterizes the transverse strain in the j-direction, when the material is stressed in the i-direction. In general,

ν_{i j}

is not equal to

ν_{j i}

: they are related by

ν_{i j} / E_{i} = ν_{j i} / E_{j}

. The engineering constants can also be given as functions of temperature and other predefined fields, if necessary.

The restrictions on the elastic constants due to material stability are

E_{1}, E_{2}, E_{3}, G_{12}, G_{13}, G_{23} > 0

| ν_{12} | < {(E_{1} {/ E}_{2})}^{1 / 2}

| ν_{13} | < {(E_{1} {/ E}_{3})}^{1 / 2}

| ν_{23} | < {(E_{2} {/ E}_{3})}^{1 / 2}

\det (D^{e l}) = {1 - ν}_{12} ν_{21} {- ν}_{23} ν_{32} {- ν}_{31} ν_{13} - 2 ν_{21} ν_{32} ν_{13} > 0.

Using the relations

ν_{i j} / E_{i} = ν_{j i} / E_{j}

the second, third, and fourth restrictions in the above set can also be expressed as

| ν_{21} | < {(E_{2} {/ E}_{1})}^{1 / 2}

| ν_{31} | < {(E_{3} {/ E}_{1})}^{1 / 2}

| ν_{32} | < {(E_{3} {/ E}_{2})}^{1 / 2} .

Parameters


Input Data	Description
E1	Young's modulus in the first local direction, $E_{1}$ .
E2	Young's modulus in the second local direction, $E_{2}$ .
E3	Young's modulus in the third local direction, $E_{3}$ .
Nu12	Poisson's ratio in the plane defined by the first and second local directions, $ν_{12}$ .
Nu13	Poisson's ratio in the plane defined by the first and third local directions, $ν_{13}$ .
Nu23	Poisson's ratio in the plane defined by the second and third local directions, $ν_{23}$ .
G12	Shear modulus in the plane defined by the first and second local directions, $G_{12}$ .
G13	Shear modulus in the plane defined by the first and third local directions, $G_{13}$ .
G23	Shear modulus in the plane defined by the second and third local directions, $G_{23}$ .
Use temperature-dependent data	Specifies material parameters that depend on temperature. A Temperature field appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.
Number of field variables	Specifies material parameters that depend on field variables. Field columns appear in the data table for each field variable you add. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.

Lamina Elasticity

Under plane stress conditions, such as in a shell element, only the values of $E_{1}$ , $E_{2}$ , $ν_{12}$ , $G_{12}$ , $G_{13}$ , and $G_{23}$ are required to define an orthotropic material. (In all of the plane stress elements the $(1, 2)$ surface is the surface of plane stress, so that the plane stress condition is $σ_{33} = 0$ .) The shear moduli $G_{13}$ and $G_{23}$ are included because they may be required for modeling transverse shear deformation in a shell. The Poisson's ratio $ν_{12}$ is implicitly given as $ν_{12} = (E_{2} / E_{1}) ν_{21}$ . In this case the stress-strain relations for the in-plane components of the stress and strain are of the form

{\begin{matrix} ε_{1} \\ \begin{matrix} \begin{matrix} ε_{2} \end{matrix} \\ γ_{12} \end{matrix} \end{matrix}} = [\begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} 1 / E_{1} \end{matrix} \\ \begin{matrix} \begin{matrix} - ν_{12} / E_{1} \end{matrix} \\ 0 \end{matrix} \end{matrix} & \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} - ν_{12} / E_{1} \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} 1 / E_{2} \end{matrix} \\ 0 \end{matrix} \end{matrix} & \begin{matrix} \begin{matrix} 0 \end{matrix} \\ \begin{matrix} \begin{matrix} 0 \end{matrix} \\ 1 / G_{12} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix}] {\begin{matrix} σ_{11} \\ \begin{matrix} \begin{matrix} σ_{22} \end{matrix} \\ τ_{12} \end{matrix} \end{matrix}}

The restrictions on the elastic constants because of material stability are

E_{1}, E_{2}, G_{12}, G_{13}, G_{23} > 0

| ν_{12} | < {(E_{1} {/ E}_{2})}^{1 / 2} .

Parameters


Input Data	Description
E1	Young's modulus in the first in-plane local direction, $E_{1}$ .
E2	Young's modulus in the second in-plane local direction, $E_{2}$ .
Nu12	Poisson's ratio in the plane defined by the first and second local directions, $ν_{12}$ .
G12	In-plane shear modulus, $G_{12}$ .
G13	Shear modulus in the plane defined by the first and third local directions, $G_{13}$ .
G23	Shear modulus in the plane defined by the second and third local directions, $G_{23}$ .
Use temperature-dependent data	Specifies material parameters that depend on temperature. A Temperature field appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.
Number of field variables	Specifies material parameters that depend on field variables. Field columns appear in the data table for each field variable you add. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.

Anisotropic Elasticity

The anisotropic model provides a modeling capability for materials that exhibit highly anisotropic behavior, such as biomedical soft tissues and fiber-reinforced elastomers.

For fully anisotropic elasticity 21 independent elastic stiffness parameters are needed. The stress-strain relations are as follows:

{\begin{matrix} σ_{11} \\ σ_{22} \\ σ_{33} \\ σ_{12} \\ σ_{13} \\ σ_{23} \end{matrix}} = [\begin{matrix} D_{1111} & D_{1122} & D_{1133} & D_{1112} & D_{1113} & D_{1123} \\ □ & D_{2222} & D_{2233} & D_{2212} & D_{2213} & D_{2223} \\ □ & □ & D_{3333} & D_{3312} & D_{3313} & D_{3323} \\ □ & □ & □ & D_{1212} & D_{1213} & D_{1223} \\ □ & s y m & □ & □ & D_{1313} & D_{1323} \\ □ & □ & □ & □ & □ & D_{2323} \end{matrix}] {\begin{matrix} ε_{11} \\ ε_{22} \\ ε_{33} \\ γ_{12} \\ γ_{13} \\ γ_{23} \end{matrix}} = D^{e l} {\begin{matrix} ε_{11} \\ ε_{22} \\ ε_{33} \\ γ_{12} \\ γ_{13} \\ γ_{23} \end{matrix}} .

When the material stiffness parameters (the $D_{i j k l}$ ) are given directly, the constraint $σ_{33} = 0$ is imposed for the plane stress case to reduce the material's stiffness matrix as required.

The restrictions imposed on the elastic constants by stability requirements are too complex to express in terms of simple equations. However, the requirement that $D^{e l}$ is positive definite requires that all its eigenvalues be positive.

Parameters


Input Data	Description
Dnnnn	Independent elastic stiffness parameter $D_{i j k l}$ .
Use temperature-dependent data	Specifies material parameters that depend on temperature. A Temperature field appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.
Number of field variables	Specifies material parameters that depend on field variables. Field columns appear in the data table for each field variable you add. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.

Transversely Isotropic Elasticity

A special subclass of orthotropy is transverse isotropy, which is characterized by a plane of isotropy at every point in the material. Assuming the 1–2 plane to be the plane of isotropy at every point, transverse isotropy requires that $E_{1} = E_{2} = E_{p}$ , $ν_{31} = ν_{32} = ν_{t p}$ , $ν_{13} = ν_{23} = ν_{p t}$ , and $G_{13} = G_{23} = G_{t}$ , where $p$ and $t$ stand for “in-plane” and “transverse,” respectively. Thus, while $ν_{t p}$ has the physical interpretation of the Poisson's ratio that characterizes the strain in the plane of isotropy resulting from stress normal to it, $ν_{p t}$ characterizes the transverse strain in the direction normal to the plane of isotropy resulting from stress in the plane of isotropy. In general, the quantities $ν_{t p}$ and $ν_{p t}$ are not equal and are related by $ν_{t p} / E_{t} = ν_{p t} / E_{p}$ . The stress-strain laws reduce to

{\begin{matrix} ε_{11} \\ ε_{22} \\ ε_{33} \\ γ_{12} \\ γ_{13} \\ γ_{23} \end{matrix}} = [\begin{matrix} 1 / E_{p} & - ν_{p} / E_{p} & - ν_{t p} / E_{t} & 0 & 0 & 0 \\ - ν_{p} / E_{p} & 1 / E_{p} & - ν_{t p} / E_{t} & 0 & 0 & 0 \\ - ν_{p t} / E_{p} & - ν_{p t} / E_{p} & 1 / E_{t} & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 / G_{p} & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 / G_{t} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 / G_{t} \end{matrix}] {\begin{matrix} σ_{11} \\ σ_{22} \\ σ_{33} \\ σ_{12} \\ σ_{13} \\ σ_{23} \end{matrix}} = D^{e l} {\begin{matrix} ε_{11} \\ ε_{22} \\ ε_{33} \\ γ_{12} \\ γ_{13} \\ γ_{23} \end{matrix}} .

where

G_{p} = E_{p} / 2 (1 + ν_{p})

and the total number of independent constants is only five.

The restrictions on the elastic constants due to material stability are

E_{p}, E_{t}, G_{p}, G_{t} > 0

| ν_{p} | < 1

| ν_{p t} | < {(E_{p} {/ E}_{t})}^{1 / 2}

| ν_{t p} | < {(E_{t} {/ E}_{p})}^{1 / 2}

\det (D^{e l}) = 1 - ν_{p}^{2} - 2 ν_{t p} ν_{p t} - 2 ν_{p} ν_{t p} ν_{p t} > 0.

Parameters


Input Data	Description
Parallel Young's modulus	Young's modulus in the parallel direction, $E_{p}$ .
Normal Young's modulus	Young's modulus in the normal direction, $E_{t}$ .
Parallel Poisson's ratio	Poisson's ratio in the parallel direction, $ν_{p}$ .
Normal Poisson's ratio	Poisson's ratio in the normal direction, $ν_{t p}$ .
Parallel shear modulus	Shear modulus in the parallel direction, $G_{p}$ .

Isotropic Shear Elasticity

In an explicit analysis you can define isotropic shear elasticity to describe the deviatoric response of materials whose volumetric response is governed by an equation of state (see EOS (Equation of State)). In this case the deviatoric stress-strain relationship is given by

S = 2 μ e^{e l}

where

S

is the deviatoric stress and

e^{e l}

the deviatoric elastic strain. You must provide the elastic shear modulus, μ, when you define the elastic deviatoric behavior.

The stability criterion requires that $μ > 0$ .

Parameters


Input Data	Description
Shear Modulus	Shear modulus, $μ > 0$ .
Use temperature-dependent data	Specifies material parameters that depend on temperature. A Temperature field appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.
Number of field variables	Specifies material parameters that depend on field variables. Field columns appear in the data table for each field variable you add. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.

Linear Elastic Response for Viscoelastic Materials

In an implicit analysis the elastic response of a viscoelastic material can be specified by defining either the instantaneous response or the long-term response of the material. To define the instantaneous response, experiments to determine the elastic constants have to be performed within time spans much shorter than the characteristic relaxation time of the material. Alternatively, if the long-term elastic response is used, data from experiments have to be collected after time spans much longer than the characteristic relaxation time of the viscoelastic material.

Parameters


Input Data	Description
Moduli time scale (for viscoelasticity)	Select Long Term or Instantaneous.

No Compression or No Tension

The modified elastic behavior is obtained by first solving for the principal stresses assuming linear elasticity and then setting the appropriate principal stress values to zero. The associated stiffness matrix components will also be set to zero. These models are not history dependent: the directions in which the principal stresses are set to zero are recalculated at every iteration. No compression and no tension definitions modify only the elastic response of the material.

Parameters


Input Data	Description
Tension and compression	Specifies whether the elasticity model includes a tensile response only, a compressive response only, or full tensile and compressive response.