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# 22.4.1 HYPOELASTIC BEHAVIOR

Products: Abaqus/Standard Abaqus/CAE

# References

• “Material library: overview,” Section 21.1.1   
• “Elastic behavior: overview,” Section 22.1.1   
• \*HYPOELASTIC   
• “Creating a hypoelastic material model” in “Defining elasticity,” Section 12.9.1 of the Abaqus/CAE User’s Guide, in the HTML version of this guide

# Overview

The hypoelastic material model:

• is valid for small elastic strains—the stresses should not be large compared to the elastic modulus of the material;   
• is used when the load path is monotonic; and   
• must be defined by user subroutine UHYPEL if temperature dependence is to be included.

# Defining hypoelastic material behavior

In a hypoelastic material the rate of change of stress is defined as a tangent modulus matrix multiplying the rate of change of the elastic strain:

$$
d \pmb {\sigma} = \mathbf {D} ^ {e l}: d \pmb {\varepsilon} ^ {e l},
$$

where is the rate of change of the stress (the “true,” Cauchy, stress in finite-strain problems), $\mathbf { D } ^ { e l }$ is the tangent elasticity matrix, and $d \varepsilon ^ { e l }$ is the rate of change of the elastic strain (the log strain in finite-strain problems).

# Determining the hypoelastic material parameters

The entries in $\mathbf { D } ^ { e l }$ are provided by giving Young’s modulus, E, and Poisson’s ratio, , as functions of strain invariants. The strain invariants are defined for this purpose as

$$
I _ {1} = \mathrm{trace} \varepsilon^ {e l},
$$

$$
I _ {2} = \frac {1}{2} \left(\varepsilon^ {e l}: \varepsilon^ {e l} - I _ {1} ^ {2}\right),
$$

$$
I _ {3} = \det \left(\varepsilon^ {e l}\right).
$$

You can define the material parameters directly or by using a user subroutine.

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# Direct specification

You can define the variation of Young’s modulus and Poisson’s ratio directly by specifying $E , \nu , I _ { 1 } , I _ { 2 } ,$ , and .

Input File Usage: \*HYPOELASTIC

Abaqus/CAE Usage: Property module: material editor: Mechanical→Elasticity→Hypoelastic

# User subroutine

If specifying E and as functions of the strain invariants directly does not allow sufficient flexibility, you can define the hypoelastic material by user subroutine UHYPEL.

Input File Usage: \*HYPOELASTIC, USER

Abaqus/CAE Usage: Property module: material editor: Mechanical→Elasticity→Hypoelastic: Use user subroutine UHYPEL

# Plane or uniaxial stress

For plane stress and uniaxial stress states Abaqus/Standard does not compute the out-of-plane strain components. For the purpose of defining the above invariants, it is assumed that $I _ { 1 } = 0 ;$ ; that is, the material is assumed to be incompressible. For example, in a uniaxial stress case (such as a truss element) this assumption implies that

$$
I _ {1} = 0,
$$

$$
I _ {2} = \frac {3}{4} \left(\varepsilon_ {1 1} ^ {e l}\right) ^ {2},
$$

$$
I _ {3} = \frac {1}{4} \left(\varepsilon_ {1 1} ^ {e l}\right) ^ {3}.
$$

# Large-displacement analysis

For large-displacement analysis the strain measure in Abaqus is the integration of the rate of deformation. This strain measure corresponds to log strain if the principal directions do not rotate relative to the material. The strain invariant definitions should be interpreted in this way.

# Use with other material models

The hypoelastic material model can be used only by itself in the material definition. It cannot be combined with viscoelasticity or with any inelastic response model. See “Combining material behaviors,” Section 21.1.3, for more details.

# Elements

The hypoelastic material model can be used with any of the stress/displacement elements in Abaqus/Standard.

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# 22.5 Hyperelasticity

• “Hyperelastic behavior of rubberlike materials,” Section 22.5.1   
• “Hyperelastic behavior in elastomeric foams,” Section 22.5.2   
• “Anisotropic hyperelastic behavior,” Section 22.5.3

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# 22.5.1 HYPERELASTIC BEHAVIOR OF RUBBERLIKE MATERIALS

Products: Abaqus/Standard Abaqus/Explicit Abaqus/CAE

# References

• “Material library: overview,” Section 21.1.1   
• “Elastic behavior: overview,” Section 22.1.1   
• “Mullins effect,” Section 22.6.1   
• “Permanent set in rubberlike materials,” Section 23.7.1   
• \*HYPERELASTIC   
• \*UNIAXIAL TEST DATA   
• \*BIAXIAL TEST DATA   
• \*PLANAR TEST DATA   
• \*VOLUMETRIC TEST DATA   
• \*MULLINS EFFECT   
• “Creating an isotropic hyperelastic material model” in “Defining elasticity,” Section 12.9.1 of the Abaqus/CAE User’s Guide, in the HTML version of this guide

# Overview

The hyperelastic material model:

• is isotropic and nonlinear;   
• is valid for materials that exhibit instantaneous elastic response up to large strains (such as rubber, solid propellant, or other elastomeric materials); and   
• requires that geometric nonlinearity be accounted for during the analysis step (“General and linear perturbation procedures,” Section 6.1.3), since it is intended for finite-strain applications.

# Compressibility

Most elastomers (solid, rubberlike materials) have very little compressibility compared to their shear flexibility. This behavior does not warrant special attention for plane stress, shell, membrane, beam, truss, or rebar elements, but the numerical solution can be quite sensitive to the degree of compressibility for three-dimensional solid, plane strain, and axisymmetric analysis elements. In cases where the material is highly confined (such as an O-ring used as a seal), the compressibility must be modeled correctly to obtain accurate results. In applications where the material is not highly confined, the degree of compressibility is typically not crucial; for example, it would be quite satisfactory in Abaqus/Standard to assume that the material is fully incompressible: the volume of the material cannot change except for thermal expansion.

Another class of rubberlike materials is elastomeric foam, which is elastic but very compressible. Elastomeric foams are discussed in “Hyperelastic behavior in elastomeric foams,” Section 22.5.2.

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We can assess the relative compressibility of a material by the ratio of its initial bulk modulus, $K _ { 0 }$ , to its initial shear modulus, $\mu _ { 0 }$ . This ratio can also be expressed in terms of Poisson’s ratio, $\nu ,$ since

$$
\nu = \frac {3 K _ {0} / \mu_ {0} - 2}{6 K _ {0} / \mu_ {0} + 2}.
$$

The table below provides some representative values.
<table><tr><td> $K_0/\mu_0$ </td><td>Poisson&#x27;s ratio</td></tr><tr><td>10</td><td>0.452</td></tr><tr><td>20</td><td>0.475</td></tr><tr><td>50</td><td>0.490</td></tr><tr><td>100</td><td>0.495</td></tr><tr><td>1000</td><td>0.4995</td></tr><tr><td>10,000</td><td>0.49995</td></tr></table>

# Compressibility in Abaqus/Standard

In Abaqus/Standard it is recommended that you use solid continuum hybrid elements for almost incompressible hyperelastic materials with initial Poisson’s ratio greater than 0.495 (i.e., the ratio of $K _ { 0 } / \mu _ { 0 }$ greater than 100) to avoid potential convergence problems. Otherwise, the analysis preprocessor will issue an error. Except for fully incompressible hyperelastic materials, you can use the “nonhybrid incompressible” diagnostics control to downgrade this error to a warning message.

In plane stress, shell, and membrane elements the material is free to deform in the thickness direction. Similarly, in one-dimensional elements (such as beams, trusses, and rebars) the material is free to deform in the lateral directions. In these cases special treatment of the volumetric behavior is not necessary; the use of regular stress/displacement elements is satisfactory.

Input File Usage: Use the following option to downgrade an error message to a warning message:

\*DIAGNOSTICS, NONHYBRID INCOMPRESSIBLE=WARNING

# Compressibility in Abaqus/Explicit

Except for plane stress and uniaxial cases, it is not possible to assume that the material is fully incompressible in Abaqus/Explicit because the program has no mechanism for imposing such a constraint at each material calculation point. Instead, we must provide some compressibility. The difficulty is that, in many cases, the actual material behavior provides too little compressibility for the algorithms to work efficiently. Thus, except for plane stress and uniaxial cases, you must provide enough compressibility for the code to work, knowing that this makes the bulk behavior of the model softer than that of the actual material. Some judgment is, therefore, required to decide whether or not the solution is sufficiently accurate, or whether the problem can be modeled at all with Abaqus/Explicit because of this numerical limitation.

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If no value is given for the material compressibility in the hyperelastic model, by default Abaqus/Explicit assumes $K _ { 0 } / \mu _ { 0 } = 2 0$ , corresponding to Poisson’s ratio of 0.475. Since typical unfilled elastomers have $K _ { 0 } / \mu _ { 0 }$ ratios in the range of 1,000 to 10,000 ( 0.4995 to $\nu = 0 . 4 9 9 9 5 )$ and filled elastomers have $K _ { 0 } / \mu _ { 0 }$ ratios in the range of 50 to 200 $( \nu = 0 . 4 9 0$ to $\nu = 0 . 4 9 7 )$ , this default provides much more compressibility than is available in most elastomers. However, if the elastomer is relatively unconfined, this softer modeling of the material’s bulk behavior usually provides quite accurate results. Unfortunately, in cases where the material is highly confined—such as when it is in contact with stiff, metal parts and has a very small amount of free surface, especially when the loading is highly compressive—it may not be feasible to obtain accurate results with Abaqus/Explicit.

If you are defining the compressibility rather than accepting the default value, an upper limit of 100 is suggested for the ratio of $K _ { 0 } / \mu _ { 0 }$ . Larger ratios introduce high frequency noise into the dynamic solution and require the use of excessively small time increments.

# Isotropy assumption

In Abaqus all hyperelastic models are based on the assumption of isotropic behavior throughout the deformation history. Hence, the strain energy potential can be formulated as a function of the strain invariants.

# Strain energy potentials

Hyperelastic materials are described in terms of a “strain energy potential,” $U ( \varepsilon )$ , which defines the strain energy stored in the material per unit of reference volume (volume in the initial configuration) as a function of the strain at that point in the material. There are several forms of strain energy potentials available in Abaqus to model approximately incompressible isotropic elastomers: the Arruda-Boyce form, the Marlow form, the Mooney-Rivlin form, the neo-Hookean form, the Ogden form, the polynomial form, the reduced polynomial form, the Yeoh form, and the Van der Waals form. As will be pointed out below, the reduced polynomial and Mooney-Rivlin models can be viewed as particular cases of the polynomial model; the Yeoh and neo-Hookean potentials, in turn, can be viewed as special cases of the reduced polynomial model. Thus, we will occasionally refer collectively to these models as “polynomial models.”

Generally, when data from multiple experimental tests are available (typically, this requires at least uniaxial and equibiaxial test data), the Ogden and Van der Waals forms are more accurate in fitting experimental results. If limited test data are available for calibration, the Arruda-Boyce, Van der Waals, Yeoh, or reduced polynomial forms provide reasonable behavior. When only one set of test data (uniaxial, equibiaxial, or planar test data) is available, the Marlow form is recommended. In this case a strain energy potential is constructed that will reproduce the test data exactly and that will have reasonable behavior in other deformation modes.

# Evaluating hyperelastic materials

Abaqus/CAE allows you to evaluate hyperelastic material behavior by automatically creating response curves using selected strain energy potentials. In addition, you can provide experimental test data for a material without specifying a particular strain energy potential and have Abaqus/CAE evaluate the

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material to determine the optimal strain energy potential. See “Evaluating hyperelastic and viscoelastic material behavior,” Section 12.4.7 of the Abaqus/CAE User’s Guide, for details. Alternatively, you can use single-element test cases to evaluate the strain energy potential.

# Arruda-Boyce form

The form of the Arruda-Boyce strain energy potential is

$$
\begin{array}{l} U = \mu \left\{\frac {1}{2} \left(\bar {I} _ {1} - 3\right) + \frac {1}{2 0 \lambda_ {m} ^ {2}} \left(\bar {I} _ {1} ^ {2} - 9\right) + \frac {1 1}{1 0 5 0 \lambda_ {m} ^ {4}} \left(\bar {I} _ {1} ^ {3} - 2 7\right) \right. \\ \left. + \frac {1 9}{7 0 0 0 \lambda_ {m} ^ {6}} (\overline {{{I}}} _ {1} ^ {4} - 8 1) + \frac {5 1 9}{6 7 3 7 5 0 \lambda_ {m} ^ {8}} (\overline {{{I}}} _ {1} ^ {5} - 2 4 3) \right\} + \frac {1}{D} \left(\frac {J _ {e \ell} ^ {2} - 1}{2} - \ln J _ {e \ell}\right), \\ \end{array}
$$

where $U$ is the strain energy per unit of reference volume; $\mu , \lambda _ { m }$ , and D are temperature-dependent material parameters; $\overline { { I } } _ { 1 }$ is the first deviatoric strain invariant defined as

$$
\overline {{I}} _ {1} = \overline {{\lambda}} _ {1} ^ {2} + \overline {{\lambda}} _ {2} ^ {2} + \overline {{\lambda}} _ {3} ^ {2},
$$

where the deviatoric stretches $\overline { { { \lambda } } } _ { i } = J ^ { - \frac { 1 } { 3 } } \lambda _ { i } ; J$ is the total volume ratio; $J ^ { e \ell }$ is the elastic volume ratio as defined below in “Thermal expansion”; and $\lambda _ { i }$ are the principal stretches. The initial shear modulus, $\mu _ { 0 }$ , is related to $\mu$ with the expression

$$
\mu_ {0} = \mu (1 + \frac {3}{5 \lambda_ {m} ^ {2}} + \frac {9 9}{1 7 5 \lambda_ {m} ^ {4}} + \frac {5 1 3}{8 7 5 \lambda_ {m} ^ {6}} + \frac {4 2 0 3 9}{6 7 3 7 5 \lambda_ {m} ^ {8}}).
$$

A typical value of $\lambda _ { m }$ is 7, for which $\mu _ { 0 } = 1 . 0 1 2 5 \mu$ . Both the initial shear modulus, $\mu _ { 0 }$ , and the parameter $\mu$ are printed in the data (.dat) file if you request a printout of the model data from the analysis input file processor. The initial bulk modulus is related to D with the expression

$$
K _ {0} = \frac {2}{D}.
$$

# Marlow form

The form of the Marlow strain energy potential is

$$
U = U _ {d e v} (\overline {{I}} _ {1}) + U _ {v o l} (J _ {e \ell}),
$$

where U is the strain energy per unit of reference volume, with $U _ { d e v }$ as its deviatoric part and $U _ { v o l }$ as its volumetric part; $\overline { { I } } _ { 1 }$ is the first deviatoric strain invariant defined as

$$
\overline {{I}} _ {1} = \overline {{\lambda}} _ {1} ^ {2} + \overline {{\lambda}} _ {2} ^ {2} + \overline {{\lambda}} _ {3} ^ {2},
$$

where the deviatoric stretches ${ \overline { { \lambda } } } _ { i } = J ^ { - \frac { 1 } { 3 } } \lambda _ { i } ;$ J is the total volume ratio; $J _ { e \ell }$ is the elastic volume ratio as defined below in “Thermal expansion”; and $\lambda _ { i }$ are the principal stretches. The deviatoric part of the potential is defined by providing either uniaxial, equibiaxial, or planar test data; while the volumetric

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part is defined by providing the volumetric test data, defining the Poisson’s ratio, or specifying the lateral strains together with the uniaxial, equibiaxial, or planar test data.

# Mooney-Rivlin form

The form of the Mooney-Rivlin strain energy potential is

$$
U = C _ {1 0} (\overline {{I}} _ {1} - 3) + C _ {0 1} (\overline {{I}} _ {2} - 3) + \frac {1}{D _ {1}} (J ^ {e \ell} - 1) ^ {2},
$$

where U is the strain energy per unit of reference volume; $C _ { 1 0 } , C _ { 0 1 }$ , and $D _ { 1 }$ are temperature-dependent material parameters; $\overline { { I } } _ { 1 }$ and $\overline { { I } } _ { 2 }$ are the first and second deviatoric strain invariants defined as

$$
\overline {{I}} _ {1} = \overline {{\lambda}} _ {1} ^ {2} + \overline {{\lambda}} _ {2} ^ {2} + \overline {{\lambda}} _ {3} ^ {2} \qquad \mathrm{and} \qquad \overline {{I}} _ {2} = \overline {{\lambda}} _ {1} ^ {(- 2)} + \overline {{\lambda}} _ {2} ^ {(- 2)} + \overline {{\lambda}} _ {3} ^ {(- 2)},
$$

where the deviatoric stretches ${ \overline { { \lambda } } } _ { i } = J ^ { - { \frac { 1 } { 3 } } } \lambda _ { i } ;$ ; J is the total volume ratio; $J ^ { e \ell }$ is the elastic volume ratio as defined below in “Thermal expansion”; and $\lambda _ { i }$ are the principal stretches. The initial shear modulus and bulk modulus are given by

$$
\mu_ {0} = 2 (C _ {1 0} + C _ {0 1}), \qquad K _ {0} = \frac {2}{D _ {1}}.
$$

# Neo-Hookean form

The form of the neo-Hookean strain energy potential is

$$
U = C _ {1 0} (\overline {{I}} _ {1} - 3) + \frac {1}{D _ {1}} (J ^ {e \ell} - 1) ^ {2},
$$

where U is the strain energy per unit of reference volume; $C _ { 1 0 }$ and $D _ { 1 }$ are temperature-dependent material parameters; $\overline { { I } } _ { 1 }$ is the first deviatoric strain invariant defined as

$$
\overline {{I}} _ {1} = \overline {{\lambda}} _ {1} ^ {2} + \overline {{\lambda}} _ {2} ^ {2} + \overline {{\lambda}} _ {3} ^ {2},
$$

where the deviatoric stretches ${ \overline { { \lambda } } } _ { i } = J ^ { - { \frac { 1 } { 3 } } } \lambda _ { i } ;$ J is the total volume ratio; $J ^ { e \ell }$ is the elastic volume ratio as defined below in “Thermal expansion”; and $\lambda _ { i }$ are the principal stretches. The initial shear modulus and bulk modulus are given by

$$
\mu_ {0} = 2 C _ {1 0}, \qquad K _ {0} = \frac {2}{D _ {1}}.
$$

# Ogden form

The form of the Ogden strain energy potential is

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$$
U = \sum_ {i = 1} ^ {N} \frac {2 \mu_ {i}}{\alpha_ {i} ^ {2}} (\overline {{\lambda}} _ {1} ^ {\alpha_ {i}} + \overline {{\lambda}} _ {2} ^ {\alpha_ {i}} + \overline {{\lambda}} _ {3} ^ {\alpha_ {i}} - 3) + \sum_ {i = 1} ^ {N} \frac {1}{D _ {i}} (J ^ {e \ell} - 1) ^ {2 i},
$$

where $\overline { { \lambda } } _ { i }$ are the deviatoric principal stretches $\overline { { { \lambda } } } _ { i } = J ^ { - \frac { 1 } { 3 } } \lambda _ { i } ; \lambda _ { i }$ are the principal stretches; N is a material parameter; and $\mu _ { i } , \alpha _ { i }$ , and $D _ { i }$ are temperature-dependent material parameters. The initial shear modulus and bulk modulus for the Ogden form are given by

$$
\mu_ {0} = \sum_ {i = 1} ^ {N} \mu_ {i}, \quad K _ {0} = \frac {2}{D _ {1}}.
$$

The particular material models described above—the Mooney-Rivlin and neo-Hookean forms—can also be obtained from the general Ogden strain energy potential for special choices of $\mu _ { i }$ and $\alpha _ { i }$ .

# Polynomial form

The form of the polynomial strain energy potential is

$$
U = \sum_ {i + j = 1} ^ {N} C _ {i j} (\overline {{I}} _ {1} - 3) ^ {i} (\overline {{I}} _ {2} - 3) ^ {j} + \sum_ {i = 1} ^ {N} \frac {1}{D _ {i}} (J ^ {e \ell} - 1) ^ {2 i},
$$

where $U$ is the strain energy per unit of reference volume; N is a material parameter; $C _ { i j }$ and $D _ { i }$ are temperature-dependent material parameters; $\overline { { I } } _ { 1 }$ and $\overline { { I } } _ { 2 }$ are the first and second deviatoric strain invariants defined as

$$
\overline {{I}} _ {1} = \overline {{\lambda}} _ {1} ^ {2} + \overline {{\lambda}} _ {2} ^ {2} + \overline {{\lambda}} _ {3} ^ {2} \qquad \mathrm{and} \qquad \overline {{I}} _ {2} = \overline {{\lambda}} _ {1} ^ {(- 2)} + \overline {{\lambda}} _ {2} ^ {(- 2)} + \overline {{\lambda}} _ {3} ^ {(- 2)},
$$

where the deviatoric stretches ${ \overline { { \lambda } } } _ { i } = J ^ { - \frac { 1 } { 3 } } \lambda _ { i } ;$ ; J is the total volume ratio; $J ^ { e \ell }$ is the elastic volume ratio as defined below in “Thermal expansion”; and $\lambda _ { i }$ are the principal stretches. The initial shear modulus and bulk modulus are given by

$$
\mu_ {0} = 2 (C _ {1 0} + C _ {0 1}), \qquad K _ {0} = \frac {2}{D _ {1}}.
$$

For cases where the nominal strains are small or only moderately large $( < 1 0 0 \% )$ , the first terms in the polynomial series usually provide a sufficiently accurate model. Some particular material models—the Mooney-Rivlin, neo-Hookean, and Yeoh forms—are obtained for special choices of $C _ { i j }$ .

# Reduced polynomial form

The form of the reduced polynomial strain energy potential is

$$
U = \sum_ {i = 1} ^ {N} C _ {i 0} (\overline {{I}} _ {1} - 3) ^ {i} + \sum_ {i = 1} ^ {N} \frac {1}{D _ {i}} (J ^ {e \ell} - 1) ^ {2 i},
$$