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Equation 4.7.2-14
\hat {\pmb {\tau}} _ {0} ^ {D} (t + \Delta t) = \mathrm{SYM} \left[ \Delta \mathbf {F} \cdot \mathbf {F} _ {t} ^ {- 1} (t + \Delta t) \cdot \pmb {\tau} _ {0} ^ {D} (t + \Delta t) \cdot \mathbf {F} _ {t} (t + \Delta t) \cdot \Delta \mathbf {F} ^ {- 1} \right] = \pmb {\tau} _ {0} ^ {D} (t + \Delta t).
Then we can also introduce
Equation 4.7.2-15
\hat {\pmb {\tau}} _ {i} ^ {D} (t) = \mathrm{SYM} \left[ \Delta \mathbf {F} \cdot \pmb {\tau} _ {i} ^ {D} (t) \cdot \Delta \mathbf {F} ^ {- 1} \right].
Substitution of Equation 4.7.2-5, Equation 4.7.2-12, and Equation 4.7.2-15 into Equation 4.7.2-11 yields
Equation 4.7.2-16
\pmb {\tau} _ {i} ^ {D} (t + \Delta t) = \frac {g _ {i}}{\tau_ {i}} e ^ {- \frac {\Delta \tau}{\tau_ {i}}} \int_ {- \Delta \tau} ^ {0} \hat {\pmb {\tau}} _ {0} ^ {D} (t - \overline {{t}}) e ^ {- \frac {\overline {{\tau}}}{\tau_ {i}}} d \overline {{\tau}} + e ^ {- \frac {\Delta \tau}{\tau_ {i}}} \hat {\pmb {\tau}} _ {i} ^ {D} (t).
To integrate the first integral in Equation 4.7.2-16, we assume that \hat { \pmb { \tau } } _ { 0 } ^ { D } ( t - \overline { { t } } ) varies linearly with the reduced time \overline { { \tau } } over the increment:
\hat {\pmb {\tau}} _ {0} ^ {D} (t - \overline {{{t}}}) = (1 + \frac {\overline {{{\tau}}}}{\Delta \tau}) \hat {\pmb {\tau}} _ {0} ^ {D} (t) - \frac {\overline {{{\tau}}}}{\Delta \tau} \hat {\pmb {\tau}} _ {0} ^ {D} (t + \Delta t) \qquad - \Delta \tau \leq \overline {{{\tau}}} \leq 0,
which with Equation 4.7.2-14 becomes
Equation 4.7.2-17
\hat {\pmb {\tau}} _ {0} ^ {D} (t - \overline {{t}}) = (1 + \frac {\overline {{\tau}}}{\Delta \tau}) \hat {\pmb {\tau}} _ {0} ^ {D} (t) - \frac {\overline {{\tau}}}{\Delta \tau} \pmb {\tau} _ {0} ^ {D} (t + \Delta t) \qquad - \Delta \tau \leq \overline {{\tau}} \leq 0.
Equation 4.7.2-16 and Equation 4.7.2-17 for the deviatoric stress have exactly the same form as Equation 4.7.2-7 and Equation 4.7.2-8 for the hydrostatic stress. Hence, after integration we obtain
Equation 4.7.2-18
\pmb {\tau} _ {i} ^ {D} (t + \Delta t) = \alpha_ {i} g _ {i} \pmb {\tau} _ {0} ^ {D} (t + \Delta t) + \beta_ {i} g _ {i} \hat {\pmb {\tau}} _ {0} ^ {D} (t) + \gamma_ {i} \hat {\pmb {\tau}} _ {i} ^ {D} (t)
with
\gamma_ {i} = e ^ {- \frac {\Delta \tau}{\tau_ {i}}}, \quad \alpha_ {i} = 1 - \frac {\tau_ {i}}{\Delta \tau} (1 - \gamma_ {i}), \quad \beta_ {i} = \frac {\tau_ {i}}{\Delta \tau} (1 - \gamma_ {i}) - \gamma_ {i}.
Equation 4.7.2-13, Equation 4.7.2-15, and Equation 4.7.2-18, thus, provide a straightforward integration scheme.
The total stress at the end of the increment becomes
Equation 4.7.2-19
\pmb {\tau} (t + \Delta t) = \pmb {\tau} _ {0} (t + \Delta t) - \sum_ {i = 1} ^ {N} \pmb {\tau} _ {i} ^ {D} (t + \Delta t) - \sum_ {i = 1} ^ {N} \pmb {\tau} _ {i} ^ {H} (t + \Delta t),
which with Equation 4.7.2-9 and Equation 4.7.2-18 can also be written as
Equation 4.7.2-20
\begin{array}{l} \pmb {\tau} (t + \Delta t) = \left(1 - \sum_ {i = 1} ^ {N} \alpha_ {i} g _ {i}\right) \pmb {\tau} _ {0} ^ {D} (t + \Delta t) + \left(1 - \sum_ {i = 1} ^ {N} \alpha_ {i} k _ {i}\right) \pmb {\tau} _ {0} ^ {H} (t + \Delta t) \\ - \sum_ {i = 1} ^ {N} \beta_ {i} g _ {i} \hat {\pmb {\tau}} _ {0} ^ {D} (t) - \sum_ {i = 1} ^ {N} \beta_ {i} k _ {i} \pmb {\tau} _ {0} ^ {H} (t) - \sum_ {i = 1} ^ {N} \gamma_ {i} \hat {\pmb {\tau}} _ {i} ^ {D} (t) - \sum_ {i = 1} ^ {N} \gamma_ {i} \pmb {\tau} _ {i} ^ {H} (t). \\ \end{array}
Rate equation
To solve the system of nonlinear equations generated by the constitutive equations, we need to generate the corotational constitutive rate equations. From Equation 4.7.2-20 it follows
Equation 4.7.2-21
\begin{array}{l} \check {\pmb {\tau}} (t + \Delta t) = \left(1 - \sum_ {i = 1} ^ {N} \alpha_ {i} g _ {i}\right) \check {\pmb {\tau}} _ {0} ^ {D} (t + \Delta t) + \left(1 - \sum_ {i = 1} ^ {N} \alpha_ {i} k _ {i}\right) \check {\pmb {\tau}} _ {0} ^ {H} (t + \Delta t) \\ - \sum_ {i = 1} ^ {N} \beta_ {i} g _ {i} \check {\pmb {\tau}} _ {0} ^ {D} (t) - \sum_ {i = 1} ^ {N} \gamma_ {i} \check {\pmb {\tau}} _ {i} ^ {D} (t), \\ \end{array}
where \check { \tau } ( t ) is the corotational (Jaumann) stress rate. Since \tau _ { 0 } ^ { H } ( t ) and \pmb { \tau } _ { i } ^ { H } ( t ) in Equation 4.7.2-20 are independent of the increment size, their derivatives vanish. The derivatives \check { \pmb { \tau } } _ { 0 } ^ { D } ( t + \Delta t ) and \check { \tau } _ { 0 } ^ { H } ( t + \Delta t ) follow from the hyperelastic equations being used and, thus, do not need to be considered here.
With Equation 4.7.2-13 it follows that
Equation 4.7.2-22
\begin{array}{l} \dot {\pmb {\tau}} _ {0} ^ {D} (t) = \mathrm{SYM} \left[ \Delta \dot {\mathbf {F}} \cdot \pmb {\tau} _ {0} ^ {D} (t) \cdot \Delta \mathbf {F} ^ {- 1} + \Delta \mathbf {F} \cdot \pmb {\tau} _ {0} ^ {D} (t) \cdot \Delta \dot {\mathbf {F}} ^ {- 1} \right] \\ = \mathrm{SYM} \left[ \Delta \dot {\mathbf {F}} \cdot \Delta \mathbf {F} ^ {- 1} \cdot \Delta \mathbf {F} \cdot \pmb {\tau} _ {0} ^ {D} (t) \cdot \Delta \mathbf {F} ^ {- 1} - \Delta \mathbf {F} \cdot \pmb {\tau} _ {0} ^ {D} (t) \cdot \Delta \mathbf {F} ^ {- 1} \cdot \Delta \dot {\mathbf {F}} \cdot \Delta \mathbf {F} ^ {- 1} \right] \\ = \mathrm{SYM} \left[ \mathbf {L} \cdot \hat {\pmb {\tau}} _ {0} ^ {D} (t) - \hat {\pmb {\tau}} _ {0} ^ {D} (t) \cdot \mathbf {L} \right], \\ \end{array}
where
\mathbf {L} \equiv \Delta \dot {\mathbf {F}} \cdot \Delta \mathbf {F} ^ {- 1}
is the velocity gradient.
Using the definition of the corotational (Jaumann) rate, it follows that
\check {\pmb {\tau}} _ {0} ^ {D} (t) = \dot {\hat {\pmb {\tau}}} _ {0} ^ {D} (t) - \pmb {\omega} \cdot \hat {\pmb {\tau}} _ {0} ^ {D} (t) - \hat {\pmb {\tau}} _ {0} ^ {D} (t) \cdot \pmb {\omega} ^ {T},
Equation 4.7.2-23
where ! is the spin tensor following from the increment. Note that
Equation 4.7.2-24
\mathbf {L} = \mathbf {D} + \boldsymbol {\omega};
hence, substitution of Equation 4.7.2-23 and Equation 4.7.2-24 into Equation 4.7.2-22 yields
\check {\hat {\pmb {\tau}}} _ {0} ^ {D} (t) = \mathrm{SYM} \left[ \mathbf {D} \cdot \hat {\pmb {\tau}} _ {0} ^ {D} (t) - \hat {\pmb {\tau}} _ {0} ^ {D} (t) \cdot \mathbf {D} \right] = \mathbf {0}
Equation 4.7.2-25
since both D and \hat { \pmb { \tau } } _ { 0 } ^ { D } ( t ) are symmetric. Similarly for \check { \hat { \tau } } _ { i } ^ { D } ( t ) ,
\check {\hat {\pmb {\tau}}} _ {i} ^ {D} (t) = \mathrm{SYM} \left[ \pmb {\mathbf {D}} \cdot \hat {\pmb {\tau}} _ {i} ^ {D} (t) - \hat {\pmb {\tau}} _ {i} ^ {D} (t) \cdot \pmb {\mathbf {D}} \right] = \pmb {\mathbf {0}}.
Equation 4.7.2-26
Equation 4.7.2-21 then simplifies to
\check {\pmb {\tau}} (t + \Delta t) = \left(1 - \sum_ {i = 1} ^ {N} \alpha_ {i} g _ {i}\right) \check {\pmb {\tau}} _ {0} ^ {D} (t + \Delta t) + \left(1 - \sum_ {i = 1} ^ {N} \alpha_ {i} k _ {i}\right) \check {\pmb {\tau}} _ {0} ^ {H} (t + \Delta t).
Equation 4.7.2-27
Cauchy versus Kirchhoff stress
All equations have been worked out in terms of the Kirchhoff stress. However, the implementation in ABAQUS uses the Cauchy stress. To transform to Cauchy stress, we use the relations
\mathbf {S} (t) = \pmb {\tau} ^ {D} (t) / J (t),
p (t) = - \frac {1}{3} \mathbf {I}: \pmb {\tau} ^ {H} (t) / J (t).
With \Delta J \equiv J ( t + \Delta t ) / J ( t ) , this allows us to write Equation 4.7.2-9, Equation 4.7.2-13, Equation 4.7.2-15, Equation 4.7.2-18, Equation 4.7.2-19, and Equation 4.7.2-27 in the following form:
p _ {i} (t + \Delta t) = \alpha_ {i} k _ {i} p _ {0} (t + \Delta t) + \frac {\beta_ {i} k _ {i} p _ {0} (t) + \gamma_ {i} p _ {i} (t)}{\Delta J},
\hat {\mathbf {S}} _ {0} (t) = \mathrm{SYM} \left[ \Delta \mathbf {F} \cdot \mathbf {S} _ {0} (t) \cdot \Delta \mathbf {F} ^ {- 1} \right],
\hat {\mathbf {S}} _ {i} (t) = \mathrm{SYM} \left[ \Delta \mathbf {F} \cdot \mathbf {S} _ {i} (t) \cdot \Delta \mathbf {F} ^ {- 1} \right],
\mathbf {S} _ {i} (t + \Delta t) = \alpha_ {i} g _ {i} \mathbf {S} _ {0} (t + \Delta t) + \frac {\beta_ {i} g _ {i} \hat {\mathbf {S}} _ {0} (t) + \gamma_ {i} \hat {\mathbf {S}} _ {i} (t)}{\Delta J},
\begin{array}{l} \pmb {\sigma} (t + \Delta t) = \pmb {\sigma} _ {0} (t + \Delta t) - \sum_ {i = 1} ^ {N} \mathbf {S} _ {i} (t + \Delta t) + \sum_ {i = 1} ^ {N} p _ {i} (t + \Delta t) \mathbf {I}, \\ \check {\pmb {\sigma}} (t + \Delta t) = \left(1 - \sum_ {i = 1} ^ {N} \alpha_ {i} g _ {i}\right) \check {\mathbf {S}} _ {0} (t + \Delta t) - \left(1 - \sum_ {i = 1} ^ {N} \alpha_ {i} k _ {i}\right) \dot {p} _ {0} (t + \Delta t) \mathbf {I}. \\ \end{array}
The virtual work and rate of virtual work equations are written with respect to the current volume. Therefore, the corotational stress rates are rates of Kirchhoff stress mapped into the current configuration and transformed in the same way as the stresses themselves.
This set of equations--combined with the expressions for \alpha _ { i } , \beta _ { i } , and \gamma _ { i } - -describe the full implementation of the hyper-viscoelasticity model in a displacement formulation.
The rate equations can be written in a form similar to ``Hyperelastic material behavior,'' Section 4.6.1. Introduce
\mathbf {C} _ {v} ^ {S} = \left(1 - \sum_ {i = 1} ^ {N} \alpha_ {i} g _ {i}\right) \mathbf {C} _ {0} ^ {S}
and
K _ {v} = \left(1 - \sum_ {i = 1} ^ {N} \alpha_ {i} k _ {i}\right) K _ {0},
where \mathbf { C } _ { 0 } ^ { S } and K _ { 0 } are the instantaneous moduli, corresponding to \mathbf { C } ^ { S } and K of ``Hyperelastic material behavior,'' Section 4.6.1. Thus, all rate equations can be obtained by substitution of \mathbf { C } _ { 0 } ^ { S } by \mathbf { C } _ { v } ^ { S } and K _ { 0 } by K _ { v } .
Reduced states of stress: plane stress
The in-plane deformation produces u _ { 1 } and u _ { 2 } , from which we can calculate only F _ { 1 1 } , F _ { 1 2 } , F _ { 2 1 } , and F _ { 2 2 . } F _ { 1 3 } , F _ { 2 3 } , F _ { 3 1 } and F _ { 3 2 } are zero. The deformation in the third direction, characterized by F _ { 3 3 } , is derived from the plane stress condition
\sigma_ {3 3} = 0 \qquad \mathrm{or} \qquad \tau_ {3 3} = 0.
Applying the condition to Equation 4.7.2-20 yields
Equation 4.7.2-28
\begin{array}{l} \boldsymbol {\tau} (t + \Delta t) | _ {3 3} = \left(1 - \sum_ {i = 1} ^ {N} \alpha_ {i} g _ {i}\right) \boldsymbol {\tau} _ {0} ^ {D} (t + \Delta t) | _ {3 3} + \left(1 - \sum_ {i = 1} ^ {N} \alpha_ {i} k _ {i}\right) \boldsymbol {\tau} _ {0} ^ {H} (t + \Delta t) | _ {3 3} \\ - \sum_ {i = 1} ^ {N} \beta_ {i} g _ {i} \hat {\pmb {\tau}} _ {0} ^ {D} (t) | _ {3 3} - \sum_ {i = 1} ^ {N} \beta_ {i} k _ {i} \pmb {\tau} _ {0} ^ {H} (t) | _ {3 3} - \sum_ {i = 1} ^ {N} \gamma_ {i} \left(\hat {\pmb {\tau}} _ {i} ^ {D} (t) + \pmb {\tau} _ {i} ^ {H} (t)\right) | _ {3 3} = 0, \\ \end{array}
where \left| _ { 3 3 } \right. stands for the projection along the 33 component. In the derivations it is convenient to express kinematic variables in terms of incremental values, such as \Delta \mathbf { F } and \Delta J .
Incompressible materials
In this case \Delta J = \operatorname* { d e t } \Delta \mathbf { F } = 1 or
\Delta F _ {3 3} = \left(\Delta F _ {1 1} \Delta F _ {2 2} - \Delta F _ {1 2} \Delta F _ {2 1}\right) ^ {- 1},
where \Delta \mathbf { F } = \mathbf { F } _ { t } ( t + \Delta t ) , from which \hat { \tau } _ { 0 } ^ { D } ( t ) and \hat { \pmb { \tau } } _ { i } ^ { D } ( t ) can be derived.
The rate-independent constitutive equations, based on F, produce
\pmb {\tau} _ {0} ^ {D} (t + \Delta t);
and then we can solve Equation 4.7.2-28 directly for \pmb { \tau } _ { 0 } ^ { H } ( t + \Delta t ) | _ { 3 3 } :
\pmb {\tau} _ {0} ^ {H} (t + \Delta t) | _ {3 3} = - \left(1 - \sum_ {i = 1} ^ {N} \alpha_ {i} g _ {i} \pmb {\tau} _ {0} ^ {D} (t + \Delta t) | _ {3 3}\right) + \sum_ {i = 1} ^ {N} \beta_ {i} g _ {i} \hat {\pmb {\tau}} _ {0} ^ {D} (t) | _ {3 3} + \sum_ {i = 1} ^ {N} \gamma_ {i} \hat {\pmb {\tau}} _ {i} ^ {D} (t) | _ {3 3}.
To obtain the rate equation, we use the linearized expression
\Delta \dot {F} _ {3 3} = - \left(\Delta \dot {F} _ {1 1} \Delta F _ {2 2} + \Delta F _ {1 1} \Delta \dot {F} _ {2 2} - \Delta \dot {F} _ {1 2} \Delta F _ {2 1} - \Delta F _ {1 2} \Delta \dot {F} _ {2 1}\right) \Delta F _ {3 3} ^ {2}
to obtain the deformation rate D _ { 3 3 } . We then use D _ { 3 3 } (along with D _ { 1 1 } , D _ { 2 2 } ; and D _ { 1 2 } ) in the three-dimensional hyperelastic rate equation to calculate \check { \pmb { \tau } } _ { 0 } ^ { D } ( t + \Delta t ) in Equation 4.7.2-27.
Compressible materials
In this case Equation 4.7.2-28 becomes an implicit equation in \Delta F _ { 3 3 } that needs to be solved iteratively. We use the Newton method, for which the first variation of \pmb { \tau } ( t + \Delta t ) | _ { 3 3 } with respect to \Delta F _ { 3 3 } needs to be calculated
\begin{array}{l} \delta \pmb {\tau} (t + \Delta t) | _ {3 3} = \left(1 - \sum_ {i = 1} ^ {N} \alpha_ {i} g _ {i}\right) \delta \pmb {\tau} _ {0} ^ {D} (t + \Delta t) | _ {3 3} + \left(1 - \sum_ {i = 1} ^ {N} \alpha_ {i} k _ {i}\right) \delta \pmb {\tau} _ {0} ^ {H} (t + \Delta t) | _ {3 3} \\ - \sum_ {i = 1} ^ {N} \beta_ {i} g _ {i} \delta \hat {\pmb {\tau}} _ {0} ^ {D} (t) | _ {3 3} - \sum_ {i = 1} ^ {N} \gamma_ {i} \delta \hat {\pmb {\tau}} _ {i} ^ {D} (t) | _ {3 3}, \\ \end{array}
where use was made of \delta \pmb { \tau } _ { 0 } ^ { H } ( t ) = 0 and \delta \pmb { \tau } _ { i } ^ { H } ( t ) = 0 .
Similar to Equation 4.7.2-27, the last two terms vanish, which yields
Equation 4.7.2-29
\delta \pmb {\tau} (t + \Delta t) | _ {3 3} = \left(1 - \sum_ {i = 1} ^ {N} \alpha_ {i} g _ {i}\right) \delta \pmb {\tau} _ {0} ^ {D} (t + \Delta t) | _ {3 3} + \left(1 - \sum_ {i = 1} ^ {N} \alpha_ {i} k _ {i}\right) \delta \pmb {\tau} _ {0} ^ {H} (t + \Delta t) | _ {3 3},
where \delta \pmb { \tau } _ { 0 } ^ { D } ( t + \Delta t ) and \delta \pmb { \tau } _ { 0 } ^ { H } ( t + \Delta t ) are obtained directly from the rate-independent constitutive equations.
In the ABAQUS implementation we use Cauchy stresses instead of Kirchhoff stresses. The stresses can easily be mapped by dividing by J. Equation 4.7.2-28 and Equation 4.7.2-29 transform into
\begin{array}{l} \pmb {\sigma} (t + \Delta t) | _ {3 3} = \left(1 - \sum_ {i = 1} ^ {N} \alpha_ {i} g _ {i}\right) \mathbf {S} _ {0} (t + \Delta t) | _ {3 3} - \left(1 - \sum_ {i = 1} ^ {N} \alpha_ {i} k _ {i}\right) p _ {0} (t + \Delta t) - \\ \frac {1}{\Delta J} \left[ \sum_ {i = 1} ^ {N} \beta_ {i} g _ {i} \hat {\mathbf {S}} _ {0} (t) | _ {3 3} - \sum_ {i = 1} ^ {N} \beta_ {i} k _ {i} p _ {0} (t) + \sum_ {i = 1} ^ {N} \gamma_ {i} \left(\hat {\mathbf {S}} _ {i} (t) | _ {3 3} - p _ {i} (t)\right) \right] = 0, \\ \delta \pmb {\sigma} (t + \Delta t) | _ {3 3} = \left(1 - \sum_ {i = 1} ^ {N} \alpha_ {i} g _ {i}\right) \delta \mathbf {S} _ {0} (t + \Delta t) | _ {3 3} - \left(1 - \sum_ {i = 1} ^ {N} \alpha_ {i} k _ {i}\right) \delta p _ {0} (t + \Delta t). \\ \end{array}
To obtain the rate equation, we use the constraint
\dot {\pmb {\sigma}} (t + \Delta t) | _ {3 3} = \left(1 - \sum_ {i = 1} ^ {N} \alpha_ {i} g _ {i}\right) \dot {\mathbf {S}} _ {0} (t + \Delta t) | _ {3 3} - \left(1 - \sum_ {i = 1} ^ {N} \alpha_ {i} k _ {i}\right) \dot {p} _ {0} (t + \Delta t) = 0
to express \Delta \dot { F } _ { 3 3 } in terms of \Delta \dot { F } _ { 3 3 } , \Delta \dot { F } _ { 3 3 } , \Delta \dot { F } _ { 3 3 } , and \Delta \dot { F } _ { 3 3 } , which is again used in Equation 4.7.2-27 to calculate \check { \tau } _ { 0 } ^ { D } ( t + \Delta t ) .
4.7.3 Frequency domain viscoelasticity
Many applications of elastomers involve dynamic loading in the form of steady-state vibration, and often in such cases the dissipative losses in the material (the "viscous" part of the material's viscoelastic behavior) must be modeled to obtain useful results. In most problems of this class the structure is first preloaded statically, and this preloading generally involves large deformation of the elastomers. The response to that preloading is computed on the basis of purely elastic behavior in the elastomeric parts of the model--that is, we assume that the preloading is applied for a sufficiently long time so that any viscous response in the material has time to decay away.
The dynamic analysis problem in this case is, therefore, to investigate the dynamic, viscoelastic response about a predeformed elastic state. In some such cases we can reasonably assume that the vibration amplitude is sufficiently small that both the kinematic and material response in the dynamic phase of the problem can be treated as linear perturbations about the predeformed state. The small amplitude viscoelastic vibration capability provided in ABAQUS/Standard, which is described in
Morman and Nagtegaal (1983) and uses the procedure described in ``Direct steady-state dynamic analysis,'' Section 2.6.1, is based on such a linearization. Its appropriateness to a particular application will depend on the magnitude of the vibration with respect to possible kinematic nonlinearities (the additional strains and rotations that occur during the dynamic loading must be small enough so that the linearization of the kinematics is reasonable) and with respect to possible nonlinearities in the material response, and on the particular constitutive assumptions incorporated in the viscoelastic model described in this section--in particular, the assumption of separation of prestrain and time effects described below.
In ``Hyperelastic material behavior,'' Section 4.6.1, it is shown that the rate of change of the true (Cauchy) stress in an elastomeric material with a strain energy potential is given by
Equation 4.7.3-1
d (J \mathbf {S}) = J (\mathbf {C} ^ {S}: d \mathbf {e} + \mathbf {Q} d \varepsilon^ {\mathrm{vol}} + d \mathbf {\Omega} \cdot \mathbf {S} - \mathbf {S} \cdot d \mathbf {\Omega})
for the deviatoric part of the stress and
Equation 4.7.3-2
d p = - \mathbf {Q}: d \mathbf {e} - K d \varepsilon^ {\mathrm{vol}}
for the equivalent pressure stress in a compressible material. The various quantities in these equations are defined in ``Hyperelastic material behavior,'' Section 4.6.1. For a fully incompressible material all components of \mathbf { Q } are zero and the equivalent pressure stress is defined only by the loading of the structure, so that the second equation is not applicable.
For small viscoelastic vibrations about a predeformed state we linearize the additional motions that occur during the vibration so that the differential of a quantity in Equation 4.7.3-1 and Equation 4.7.3-2 can be interpreted as the additional incremental value,
d (f) \rightarrow \Delta (f) \stackrel {\mathrm{def}} {=} f | _ {t} - f | _ {0},
for any quantity f , , where f | _ { t } is the current value of f at some time during the vibration and f | _ { 0 } is the reference value of f ; that is, f | _ { 0 } is the value of f at the end of the static (long term) preloading, about which f is fluctuating during the vibration.
The incremental elastic constitutive behavior for small added motions defined by this interpretation of Equation 4.7.3-1 and Equation 4.7.3-2 is now generalized to include viscous dissipation as well as elastic response in the material, following Lianis (1965), to give
\Delta (J \mathbf {S}) = J \bigg (\mathbf {C} ^ {S} | _ {0}: \Delta \mathbf {e} + \mathbf {Q} | _ {0} \Delta \varepsilon^ {\mathrm{vol}} + \Delta \pmb {\Omega} \cdot \mathbf {S} | _ {0} - \mathbf {S} | _ {0} \cdot \Delta \pmb {\Omega} + \int_ {0} ^ {t} \pmb {\Phi} (| _ {0}, t - \tau): \dot {\mathbf {e}} (\tau) d \tau \bigg),
and, for a compressible material,
Mechanical Constitutive Theories
\Delta p = - \mathbf {Q} | _ {0}: \Delta \mathbf {e} - K | _ {0} \Delta \varepsilon^ {\mathrm{vol}} - \int_ {0} ^ {t} \kappa (| _ {0}, t - \tau) \dot {\varepsilon} ^ {\mathrm{vol}} (\tau) d \tau .
In these expressions f ( | _ { 0 } , t - \tau ) is meant to indicate that f depends on the elastic predeformation that has occurred prior to the small dynamic vibrations (the state at t = 0 ) and is evaluated at time t - \tau _ { \mathrm { { i } } } , between the start of the vibrations and the current time, t. © and ∙ are the functions that define the viscous part of the material's response: the notation is intended to imply that these are functions of the elastic predeformation and time. { \dot { f } } { \stackrel { \mathrm { d e f } } { = } } d f / d t is the time rate of change of a quantity.
The definitions of the viscous parts of the behavior, © and \kappa , provided in ABAQUS are simplified by assuming that this viscous behavior exhibits separation of time and prestrain effects; that is, that
\Phi (| _ {0}, t - \tau) = g (t - \tau) \mathbf {C} ^ {S} | _ {0}
and
\kappa (| _ {0}, t - \tau) = k (t - \tau) K | _ {0},
where \mathbf { C } ^ { S } \vert _ { 0 } and K | _ { 0 } are the "effective elasticity" of the material in its predeformed state, prior to the vibration. This assumption simply means that measurements of the viscous behavior during small motions of the material about a predeformed state depend only on the predeformation to the extent that the effective elasticity of the material also depends on that predeformation. There is experimental evidence that this simplification is appropriate for some practical materials (see Morman's (1979) discussion). With this assumption the definition of the viscous part of the material's behavior is reduced to finding the scalar functions of time, g and k (only g for fully incompressible materials), and the constitutive response to small perturbations is simplified to
\Delta (J \mathbf {S}) = J \left(\mathbf {C} ^ {S} | _ {0}: \left\{\Delta \mathbf {e} + \int_ {0} ^ {t} g (t - \tau) \dot {\mathbf {e}} (\tau) d \tau \right\} + \mathbf {Q} | _ {0} \Delta \varepsilon^ {\mathrm{vol}} + \Delta \mathbf {\Omega} \cdot \mathbf {S} | _ {0} - \mathbf {S} | _ {0} \cdot \Delta \mathbf {\Omega}\right),
and, for compressible materials,
\Delta p = - \mathbf {Q} | _ {0}: \Delta \mathbf {e} - K | _ {0} \left(\Delta \varepsilon^ {\mathrm{vol}} + \int_ {0} ^ {t} k (t - \tau) \dot {\varepsilon} ^ {\mathrm{vol}} (\tau) d \tau\right).
In ABAQUS this model is provided only for the *STEADY STATE DYNAMICS, DIRECT dynamic analysis option, in which we assume that the dynamic response is steady-state harmonic vibration, so that we can write
f | _ {t} - f | _ {0} = \left(\Re (\Delta f) + i \Im (\Delta f)\right) \exp (i \omega t),
where \Re ( \Delta f ) + i \Im ( \Delta f ) is the complex amplitude of a variable f .
Defining the Fourier transforms of the viscous relaxation functions g ( t ) and k ( t ) as
\Re (g) + i \Im (g) \stackrel {\mathrm{def}} {=} \int_ {- \infty} ^ {\infty} g (t) \exp (- i \omega t) d t,
and
\Re (k) + i \Im (k) \stackrel {\mathrm{def}} {=} \int_ {- \infty} ^ {\infty} k (t) \exp (- i \omega t) d t,
allows the constitutive model to be written for such harmonic motions in the linear form
\begin{array}{l} \Re \bigl (\Delta (J \mathbf {S}) \bigr) = J \bigg (\mathbf {C} ^ {S} | _ {0}: \big \{\big (1 - \omega \Im (g) \big) \Re (\Delta \mathbf {e}) - \omega \Re (g) \Im (\Delta \mathbf {e}) \big \} \\ \left. + \mathbf {Q} | _ {0} \Re (\Delta \varepsilon^ {\mathrm{vol}}) + \Re (\Delta \boldsymbol {\Omega}) \cdot \mathbf {S} | _ {0} - \mathbf {S} | _ {0} \cdot \Re (\Delta \boldsymbol {\Omega})\right), \\ \end{array}
and
\begin{array}{l} \Im \bigl (\Delta (J \mathbf {S}) \bigr) = J \biggl (\mathbf {C} ^ {S} | _ {0}: \bigl \{\omega \Re (g) \Re (\Delta \mathbf {e}) + \bigl (1 - \omega \Im (g) \bigr) \Im (\Delta \mathbf {e}) \bigr \} \\ \left. + \mathbf {Q} | _ {0} \Im (\Delta \varepsilon^ {\mathrm{vol}}) + \Im (\Delta \boldsymbol {\Omega}) \cdot \mathbf {S} | _ {0} - \mathbf {S} | _ {0} \cdot \Im (\Delta \boldsymbol {\Omega})\right); \\ \end{array}
and, for compressible materials,
\Re (\Delta p) = - \mathbf {Q} | _ {0}: \Re (\Delta \mathbf {e}) - K | _ {0}: \left\{\left(1 - \omega \Im (k)\right) \Re (\Delta \varepsilon^ {\mathrm{vol}}) - \omega \Re (k) \Im (\Delta \varepsilon^ {\mathrm{vol}}) \right\},
and
\Im (\Delta p) = - \mathbf {Q} | _ {0}: \Im (\Delta \mathbf {e}) - K | _ {0}: \left\{\omega \Re (k) \Re (\Delta \varepsilon^ {\mathrm{vol}}) + \left(1 - \omega \Im (k)\right) \Im (\Delta \varepsilon^ {\mathrm{vol}}) \right\}.
The viscous behavior of the material is, thus, reduced to defining <(g), =(g), <(k), and =(k) as functions of frequency: the *VISCOELASTIC material option is provided for this purpose.
When the pure displacement formulation is used for a compressible material, the virtual work equation for dynamic response is
\delta W _ {I} - \delta W _ {D} - \delta W _ {E} = 0,
Equation 4.7.3-3
where
\delta W _ {I} = \int_ {V} \pmb {\sigma}: \delta \mathbf {D} d V
is the internal virtual work,
\delta W _ {D} = - \int_ {V ^ {0}} \rho^ {0} \delta \mathbf {u} \cdot \ddot {\mathbf {u}} d V ^ {0}
is the virtual work of the d'Alembert forces ( \rho ^ { 0 } is the mass density of the material in the original configuration), and
\delta W _ {E} = \int_ {S} \delta \mathbf {u} \cdot \mathbf {p} d S + \int_ {V} \delta \mathbf {u} \cdot \mathbf {f} d V
is the virtual work of externally prescribed surface tractions p per current surface area and body forces f per current volume.
For the linearized perturbations considered here we recast Equation 4.7.3-3 in incremental form, giving
Equation 4.7.3-4
\Delta \delta W _ {I} - \Delta \delta W _ {D} - \Delta \delta W _ {E} = 0,
where \Delta \delta W _ { I } is obtained from Equation 4.6.1-12 with the interpretation d ( f ) \to \Delta ( f ) ;
\Delta \delta W _ {D} \stackrel {\mathrm{def}} {=} \delta W _ {D}
and
\begin{array}{l} \Delta \delta W _ {E} = \int_ {S} \delta \mathbf {u} \cdot \left(\mathbf {P} _ {u} \cdot \Delta \mathbf {u} + \mathbf {p} _ {p} \Delta p\right) d S \\ + \int_ {V} \delta \mathbf {u} \cdot (\mathbf {F} _ {u} \cdot \Delta \mathbf {u} + \mathbf {f} _ {f} \Delta f) d V, \\ \end{array}
where
\mathbf {P} _ {u} = \frac {1}{J _ {A}} \frac {\partial (J _ {A} \mathbf {p})}{\partial \mathbf {u}},
in which
J _ {A} = \left| \frac {d A}{d A _ {0}} \right|
is the ratio of current to reference surface area;
\mathbf {p} _ {p} = \frac {\partial \mathbf {p}}{\partial p};