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ABAQUS Analysis User's Manual 17.7.1 Time domain viscoelasticity
Products: ABAQUS/Standard ABAQUS/Explicit ABAQUS/CAE
References
? ? ? ? ? ? ? ?
“Material library: overview,” Section 16.1.1 “Elastic behavior: overview,” Section 17.1.1
“Frequency domain viscoelasticity,” Section 17.7.2 *VISCOELASTIC *SHEAR TEST DATA
*VOLUMETRIC TEST DATA *COMBINED TEST DATA *TRS
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 17.2.1; “Hyperelastic behavior of rubberlike materials,” Section 17.5.1; or “Hyperelastic behavior in elastomeric foams,” Section 17.5.2, to define the elastic material properties;
? 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 6.5.4), or a transient (consolidation) coupled pore fluid diffusion and stress analysis (“Coupled pore fluid diffusion and stress analysis,” Section 6.7.1);
?
can be used in large-strain problems; and
? can be calibrated using time-dependent creep test data,
time-dependent relaxation test data, or frequency-dependent cyclic test data.
?
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. 注释：小应变简化为弹性，大应变超弹性或超弹泡沫 Small strain（小应变）
Consider a shear test at small strain, in which a time varying shear strain,
, is applied to the material. The response is the shear stress The viscoelastic material model defines
as
.
where is the time-dependent “shear relaxation modulus”（剪切松弛模量）
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 （t=0,初始应变突然施加）
where is the fixed strain. The viscoelastic material model is “long-term elastic” （长期弹性）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.,
.
as
The shear relaxation modulus can be written in dimensionless form:
where
is the instantaneous shear modulus（瞬时剪切模量）, so that
the expression for the stress takes the form
The dimensionless relaxation function has the limiting values
.
Large strain（大应变）
The equation for the stress can be transformed by using integration by
parts:
and
It is convenient to write this equation in the form
where
is the instantaneous shear stress at time t. This form allows a
straightforward generalization（简单概括） to nonlinear elastic deformations by replacing the linear elastic relation
with the nonlinear elasticity relation
. 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 a mixed “push-forward” transformation with the relative deformation gradient
:
To ensure that the stress remains symmetric, ABAQUS uses the integral form:
where is the deviatoric part of the Kirchhoff stress.基尔莫夫应力偏量 The finite-strain theory is described in more detail in “Finite-strain viscoelasticity,” Section 4.8.2 of the ABAQUS Theory Manual.
Defining the volumetric behavior定义体
The volumetric behavior can be written in a form that is similar to the shear behavior:
where p is the hydrostatic pressure,（静水压力） bulk modulus, 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 to time t.
is the instantaneous elastic
is the
is the dimensionless bulk relaxation modulus, and