\*BIAXIAL TEST DATA \*PLANAR TEST DATA Multiple unloading-reloading curves from different strain levels for any given test type can be entered by repeated specification of the appropriate test data option. # Abaqus/CAE Usage: Property module: material editor: Mechanical→Damage for Elastomers→Mullins Effect: Definition: Test Data Input: enter the values for up to two of the values r, m, and beta. In addition, enter data for at least one of the following Suboptions→Biaxial Test, Planar Test, or Uniaxial Test # User subroutine specification An alternative method for specifying energy dissipation involves defining the damage variable in user subroutine UMULLINS in Abaqus/Standard and VUMULLINS in Abaqus/Explicit. Optionally, you can specify the number of property values needed as data in the user subroutine. You must provide the damage variable, , and its derivative, $\begin{array} { r } { \left( \frac { d \eta } { d \tilde { U } _ { - } } \right. } \end{array}$ . The latter contributes to the Jacobian of the overall system of equations and is necessary to ensure good convergence characteristics in Abaqus/Standard. If needed, you can specify the number of solution-dependent variables (“User subroutines: overview,” Section 18.1.1). These solution-dependent variables can be updated in the user subroutine. The damage dissipation energy and the recoverable part of the energy can also be defined for output purposes. # Input File Usage: \*MULLINS EFFECT, USER, PROPERTIES=constants # Abaqus/CAE Usage: Property module: material editor: Mechanical→Damage for Elastomers→Mullins Effect: Definition: User Defined # Elements The model can be used with all element types that support the use of the elastomeric foam material model. # Procedures The model can be used in all procedure types that support the use of the elastomeric foam material model. In linear perturbation steps in Abaqus/Standard the current material tangent stiffness is used to determine the response. Specifically, when a linear perturbation is carried out about a base state that is on the primary curve, the unloading tangent stiffness will be used. In Abaqus/Explicit the unloading tangent stiffness is always used to compute the stable time increment. As a result, the inclusion of stress-softening effects may lead to more increments in the analysis, even when no unloading actually takes place. # Output In addition to the standard output identifiers available in Abaqus (“Abaqus/Standard output variable identifiers,” Section 4.2.1, and “Abaqus/Explicit output variable identifiers,” Section 4.2.2), the following variables have special meaning when energy dissipation is present in the model:

DMENER	Energy dissipated per unit volume by damage.
ELDMD	Total energy dissipated in element by damage.
ALLDMD	Energy dissipated in whole (or partial) model by damage. The contribution from ALLDMD is included in the total strain energy ALLIE.
EDMDDEN	Energy dissipated per unit volume in the element by damage.
SENER	The recoverable part of the energy per unit volume.
ELSE	The recoverable part of the energy in the element.
ALLSE	The recoverable part of the energy in the whole (partial) model.
ESEDEN	The recoverable part of the energy per unit volume in the element.

The damage energy dissipation, represented by the shaded area in Figure 22.6.2–1 for deformation until $c ^ { ' }$ , is computed as follows. When the damaged material is in a fully unloaded state, the augmented energy function has the residual value $U ( \mathbf { I } , \eta _ { m } ) = \phi ( \eta _ { m } )$ . The residual value of the energy function upon complete unloading represents the energy dissipated due to damage in the material. The recoverable part of the energy is obtained by subtracting the dissipated energy from the augmented energy as $\eta \tilde { U } ( \hat { \lambda } _ { i } ) +$ $\phi ( \eta ) - \phi ( \eta _ { m } )$ . The damage energy accumulates with progressive deformation along the primary curve and remains constant during unloading. During unloading, the recoverable part of the strain energy is released. The latter becomes zero when the material point is unloaded completely. Upon further reloading from a completely unloaded state, the recoverable part of the strain energy increases from zero. When the maximum strain that was attained earlier is exceeded upon reloading, further accumulation of damage energy occurs. # 22.7 Linear viscoelasticity • “Time domain viscoelasticity,” Section 22.7.1 • “Frequency domain viscoelasticity,” Section 22.7.2 # 22.7.1 TIME DOMAIN VISCOELASTICITY Products: Abaqus/Standard Abaqus/Explicit Abaqus/CAE # References • “Material library: overview,” Section 21.1.1 • “Elastic behavior: overview,” Section 22.1.1 • “Frequency domain viscoelasticity,” Section 22.7.2 • \*VISCOELASTIC • \*SHEAR TEST DATA • \*VOLUMETRIC TEST DATA • \*COMBINED TEST DATA • \*TRS • “Defining time domain viscoelasticity” in “Defining elasticity,” Section 12.9.1 of the Abaqus/CAE User’s Guide, in the HTML version of this guide # Overview The time domain viscoelastic material model: • describes isotropic rate-dependent material behavior for materials in which dissipative losses primarily caused by “viscous” (internal damping) effects must be modeled in the time domain; • assumes that the shear (deviatoric) and volumetric behaviors are independent in multiaxial stress states (except when used for an elastomeric foam); • can be used only in conjunction with “Linear elastic behavior,” Section 22.2.1; “Hyperelastic behavior of rubberlike materials,” Section 22.5.1; or “Hyperelastic behavior in elastomeric foams,” Section 22.5.2, to define the continuum elastic material properties; • can be used in Abaqus/Explicit with “Linear elastic traction-separation behavior” in “Defining the constitutive response of cohesive elements using a traction-separation description,” Section 32.5.6; • is active only during a transient static analysis (“Quasi-static analysis,” Section 6.2.5), a transient implicit dynamic analysis (“Implicit dynamic analysis using direct integration,” Section 6.3.2), an explicit dynamic analysis (“Explicit dynamic analysis,” Section 6.3.3), a steady-state transport analysis (“Steady-state transport analysis,” Section 6.4.1), a fully coupled temperature-displacement analysis (“Fully coupled thermal-stress analysis,” Section 6.5.3), a fully coupled thermal-electrical-structural analysis (“Fully coupled thermal-electrical-structural analysis,” Section 6.7.4), or a transient (consolidation) coupled pore fluid diffusion and stress analysis (“Coupled pore fluid diffusion and stress analysis,” Section 6.8.1); • can be used in large-strain problems; • can be calibrated using time-dependent creep test data, time-dependent relaxation test data, or frequency-dependent cyclic test data; and • can be used to couple viscous dissipation with the temperature field in a fully coupled temperature-displacement analysis (“Fully coupled thermal-stress analysis,” Section 6.5.3) or a fully coupled thermal-electrical-structural analysis (“Fully coupled thermal-electrical-structural analysis,” Section 6.7.4). # Defining the shear behavior Time domain viscoelasticity is available in Abaqus for small-strain applications where the rate-independent elastic response can be defined with a linear elastic material model and for large-strain applications where the rate-independent elastic response must be defined with a hyperelastic or hyperfoam material model. # Small strain Consider a shear test at small strain in which a time varying shear strain, $\gamma ( t )$ , is applied to the material. The response is the shear stress . The viscoelastic material model defines as $$ \tau (t) = \int_ {0} ^ {t} G _ {R} (t - s) \dot {\gamma} (s) d s, $$ where $G _ { R } ( t )$ is the time-dependent “shear relaxation modulus” that characterizes the material’s response. This constitutive behavior can be illustrated by considering a relaxation test in which a strain is suddenly applied to a specimen and then held constant for a long time. The beginning of the experiment, when the strain is suddenly applied, is taken as zero time, so that $$ \tau (t) = \int_ {0} ^ {t} G _ {R} (t - s) \dot {\gamma} (s) d s = G _ {R} (t) \gamma \quad (\text { since } \dot {\gamma} = 0 \text { for } t > 0), $$ where is the fixed strain. The viscoelastic material model is “long-term elastic” in the sense that, after having been subjected to a constant strain for a very long time, the response settles down to a constant stress; i.e., $G _ { R } ( t ) G _ { \infty }$ as . The shear relaxation modulus can be written in dimensionless form: $$ g _ {R} (t) = G _ {R} (t) / G _ {0}, $$ where $G _ { 0 } = G _ { R } ( 0 )$ is the instantaneous shear modulus, so that the expression for the stress takes the form $$ \tau (t) = G _ {0} \int_ {0} ^ {t} g _ {R} (t - s) \dot {\gamma} (s) d s. $$ The dimensionless relaxation function has the limiting values $g _ { R } ( 0 ) = 1$ and $g _ { R } ( \infty ) \ = \ G _ { \infty } / G _ { 0 }$ . Anisotropic elasticity in Abaqus/Explicit The equation for the shear stress can be transformed by using integration by parts: $$ \tau (t) = G _ {0} \left(\gamma + \int_ {0} ^ {t} \dot {g} _ {R} (s) \gamma (t - s) d s\right). $$ It is convenient to write this equation in the form $$ \tau (t) = \tau_ {0} (t) + \int_ {0} ^ {t} \dot {g} _ {R} (s) \tau_ {0} (t - s) d s, $$ where $\tau _ { 0 } ( t )$ is the instantaneous shear stress at time t. This can be generalized to multi-dimensions as $$ \boldsymbol {\tau} (t) = \boldsymbol {\tau} _ {0} (t) + \int_ {0} ^ {t} \dot {g} _ {R} (s) \boldsymbol {\tau} _ {0} (t - s) d s, $$ where $\tau ( t )$ is the deviatoric part of the stress tensor and $\tau _ { 0 } ( t )$ is the deviatoric part of the instantaneous stress tensor. Here the viscoelasticity is assumed to be isotropic; i.e., the relaxation function is independent of the loading direction. This form allows a straightforward generalization to anisotropic elastic deformations, where the deviatoric part of instantaneous stress tensor is computed as $\pmb { \tau } _ { 0 } ( t ) = \overline { { \mathbf { D } } } _ { 0 } : \mathbf { e }$ . Here $\overline { { \mathbf { D } } } _ { 0 }$ is the instantaneous deviatoric elasticity tensor, and is the deviatoric part of the strain tensor. # Large strain The above form also allows a straightforward generalization to nonlinear elastic deformations, where the deviatoric part of the instantaneous stress $\tau _ { 0 } ( t )$ is computed using a hyperelastic strain enery potential. This generalization yields a linear viscoelasticity model, in the sense that the dimensionless stress relaxation function is independent of the magnitude of the deformation. In the above equation the instantaneous stress, , applied at time influences the stress, , at time t. Therefore, to create a proper finite-strain formulation, it is necessary to map the stress that existed in the configuration at time into the configuration at time t. In Abaqus this is done by means of the “standard-push-forward” transformation with the relative deformation gradient $\mathbf { F } _ { t - s } ( t )$ : $$ \mathbf {F} _ {t - s} (t) = \frac {\partial \mathbf {x} (t)}{\partial \mathbf {x} (t - s)}, $$ which results in the following hereditary integral: $$ \pmb {\tau} = \pmb {\tau} _ {0} + \mathrm{dev} \left[ \int_ {0} ^ {t} \dot {g} _ {R} (s) \overline {{\mathbf {F}}} _ {t} ^ {- 1} (t - s) \cdot \pmb {\tau} _ {0} (t - s) \cdot \overline {{\mathbf {F}}} _ {t} ^ {- T} (t - s) d s \right], $$ where $\tau$ is the deviatoric part of the Kirchhoff stress. The finite-strain theory is described in more detail in “Finite-strain viscoelasticity,” Section 4.8.2 of the Abaqus Theory Guide. The volumetric behavior can be written in a form that is similar to the shear behavior: $$ p (t) = - \mathrm{K} _ {0} \int_ {0} ^ {t} k _ {R} (t - s) \dot {\varepsilon} ^ {v o l} (s) d s, $$ where $\pmb { p }$ is the hydrostatic pressure, $\mathrm { K } _ { 0 }$ is the instantaneous elastic bulk modulus, $k _ { R } ( t )$ is the dimensionless bulk relaxation modulus, and $\varepsilon ^ { v o l }$ is the volume strain. The above expansion applies to small as well as finite strain since the volume strains are generally small and there is no need to map the pressure from time $t - s$ to time t. # Defining viscoelastic behavior for traction-separation elasticity in Abaqus/Explicit Time domain viscoelasticity can be used in Abaqus/Explicit to model rate-dependent behavior of cohesive elements with traction-separation elasticity (“Defining elasticity in terms of tractions and separations for cohesive elements” in “Linear elastic behavior,” Section 22.2.1). In this case the evolution equation for the normal and two shear nominal tractions take the form: $$ t _ {n} (t) = t _ {n} ^ {0} (t) + \int_ {0} ^ {t} \dot {k} _ {R} (s) t _ {n} ^ {0} (t - s) d s, $$ $$ t _ {s} (t) = t _ {s} ^ {0} (t) + \int_ {0} ^ {t} \dot {g} _ {R} (s) t _ {s} ^ {0} (t - s) d s, $$ $$ t _ {t} (t) = t _ {t} ^ {0} (t) + \int_ {0} ^ {t} \dot {g} _ {R} (s) t _ {t} ^ {0} (t - s) d s, $$ where $t _ { n } ^ { 0 } ( t ) , t _ { s } ^ { 0 } ( t )$ , and $t _ { t } ^ { 0 } ( t )$ are the instantaneous nominal tractions at time t in the normal and the two local shear directions, respectively. The functions $g _ { R } ( t )$ and $k _ { R } ( t )$ now represent the dimensionless shear and normal relaxation moduli, respectively. Note the close similarity between the viscoelastic formulation for the continuum elastic response discussed in the previous sections and the formulation for cohesive behavior with traction-separation elasticity after reinterpreting shear and bulk relaxation as shear and normal relaxation. For the case of uncoupled traction elasticity, the viscoelastic normal and shear behaviors are assumed to be independent. The normal relaxation modulus is defined as $$ k _ {R} (t) = K _ {n n} (t) / K _ {n n} ^ {0}, $$ where $K _ { n n } ^ { 0 }$ is the instantaneous normal moduli. The shear relaxation modulus is assumed to be isotropic and, therefore, independent of the local shear directions: $$ g _ {R} (t) = K _ {s s} (t) / K _ {s s} ^ {0} = K _ {t t} (t) / K _ {t t} ^ {0}, $$ where $K _ { s s } ^ { 0 }$ and $K _ { t t } ^ { 0 }$ are the instantaneous shear moduli. For the case of coupled traction-separation elasticity the normal and shear relaxation moduli must be the same, $g _ { R } ( t ) = k _ { R } ( t )$ , and you must use the same relaxation data for both behaviors. # Temperature effects The effect of temperature, , on the material behavior is introduced through the dependence of the instantaneous stress, , on temperature and through a reduced time concept. The expression for the linear-elastic shear stress is rewritten as $$ \tau (t) = G _ {0} (\theta) \int_ {0} ^ {t} g _ {R} \big (\xi (t) - \xi (s) \big) \dot {\gamma} (s) d s, $$ where the instantaneous shear modulus $G _ { 0 }$ is temperature dependent and is the reduced time, defined by $$ \xi (t) = \int_ {0} ^ {t} \frac {d s}{A (\theta (s))}, $$ where $A \left( \theta ( t ) \right)$ is a shift function at time t. This reduced time concept for temperature dependence is usually referred to as thermo-rheologically simple (TRS) temperature dependence. Often the shift function is approximated by the Williams-Landel-Ferry (WLF) form. See “Thermo-rheologically simple temperature effects” below, for a description of the WLF and other forms of the shift function available in Abaqus. The reduced time concept is also used for the volumetric behavior, the large-strain formulation, and the traction-separation formulation. # Numerical implementation Abaqus assumes that the viscoelastic material is defined by a Prony series expansion of the dimensionless relaxation modulus: $$ g _ {R} (t) = 1 - \sum_ {i = 1} ^ {N} \bar {g} _ {i} ^ {P} (1 - e ^ {- t / \tau_ {i} ^ {G}}), $$ where $N , \bar { g } _ { i } ^ { P }$ , and $\tau _ { i } ^ { G } , i = 1 , 2 , \dots , N$ , are material constants. For linear isotropic elasticity, substitution in the small-strain expression for the shear stress yields $$ \tau (t) = G _ {0} \left(\gamma - \sum_ {i = 1} ^ {N} \gamma_ {i}\right), $$ where $$ \gamma_ {i} = \frac {\bar {g} _ {i} ^ {P}}{\tau_ {i} ^ {G}} \int_ {0} ^ {t} e ^ {- s / \tau_ {i} ^ {G}} \gamma (t - s) d s. $$ The $\gamma _ { i }$ are interpreted as state variables that control the stress relaxation, and $$ \gamma^ {c r} = \sum_ {i = 1} ^ {N} \gamma_ {i} $$ is the “creep” strain: the difference between the total mechanical strain and the instantaneous elastic strain (the stress divided by the instantaneous elastic modulus). In Abaqus/Standard $\gamma ^ { c r }$ is available as the creep strain output variable CE (“Abaqus/Standard output variable identifiers,” Section 4.2.1). A similar Prony series expansion is used for the volumetric response, which is valid for both smalland finite-strain applications: $$ p = - K _ {0} \left(\varepsilon^ {v o l} - \sum_ {i = 1} ^ {N} \varepsilon_ {i} ^ {v o l}\right), $$ where $$ \varepsilon_ {i} ^ {v o l} = \frac {\bar {k} _ {i} ^ {P}}{\tau_ {i} ^ {K}} \int_ {0} ^ {t} e ^ {- s / \tau_ {i} ^ {K}} \varepsilon^ {v o l} (t - s) d s. $$ Abaqus assumes that $\tau _ { i } ^ { G } = \tau _ { i } ^ { K } = \tau _ { i }$ Ti 二Ti. For linear anisotropic elasticity, the Prony series expansion, in combination with the generalized small-strain expression for the deviatoric stress, yields $$ \boldsymbol {\tau} = \boldsymbol {\tau} _ {0} - \sum_ {i = 1} ^ {N} \boldsymbol {\tau} _ {i}, $$ where $$ \pmb {\tau} _ {i} = \frac {\bar {g} _ {i} ^ {P}}{\tau_ {i} ^ {G}} \int_ {0} ^ {t} e ^ {- s / \tau_ {i} ^ {G}} \pmb {\tau} _ {0} (t - s) d s. $$ The $\tau _ { i }$ are interpreted as state variables that control the stress relaxation. For finite strains, the Prony series expansion, in combination with the finite-strain expression for the shear stress, produces the following expression for the deviatoric stress: $$ \boldsymbol {\tau} = \boldsymbol {\tau} _ {0} - \sum_ {i = 1} ^ {N} \operatorname{dev} \left(\boldsymbol {\tau} _ {i}\right), $$ where