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\mathbf {S} = 2 G _ {0} \left(\mathbf {e} - \sum_ {i = 1} ^ {n} \alpha_ {i} \mathbf {e} _ {i}\right),
where \begin{array} { r } { G _ { 0 } = G _ { \infty } + \sum _ { i = 1 } ^ { n } G _ { i } } \end{array} is the instantaneous shear modulus, \alpha _ { i } = G _ { i } / G _ { 0 } is the relative modulus of term i , and
Equation 4.7.1-2
\mathbf {e} _ {i} = \int_ {0} ^ {\tau} \left(1 - e ^ {(\tau^ {\prime} - \tau) / \tau_ {i}}\right) \frac {d \mathbf {e}}{d \tau^ {\prime}} d \tau^ {\prime}
is the viscous (creep) strain in each term of the series. For finite element analysis this equation must be integrated over a finite increment of time. To perform this integration, we will assume that during the increment e varies linearly with \tau { ; } hence, d { \bf e } / d \tau ^ { \prime } = \Delta { \bf e } / \Delta \tau . To use this relation, we break up Equation 4.7.1-2 into two parts:
\begin{array}{l} \mathbf {e} _ {i} ^ {n + 1} = \int_ {0} ^ {\tau^ {n}} \left(1 - e ^ {(\tau^ {\prime} - \tau^ {n + 1}) / \tau_ {i}}\right) \frac {d \mathbf {e}}{d \tau^ {\prime}} d \tau^ {\prime} \\ + \int_ {\tau^ {n}} ^ {\tau^ {n + 1}} \left(1 - e ^ {(\tau^ {\prime} - \tau^ {n + 1}) / \tau_ {i}}\right) \frac {d \mathbf {e}}{d \tau^ {\prime}} d \tau^ {\prime}. \\ \end{array}
Now observe that
1 - e ^ {(\tau^ {\prime} - \tau^ {n + 1}) / \tau_ {i}} = 1 - e ^ {- \Delta \tau / \tau_ {i}} + e ^ {- \Delta \tau / \tau_ {i}} \left(1 - e ^ {(\tau^ {\prime} - \tau^ {n}) / \tau_ {i}}\right).
Use of this expression and the approximation for d \mathbf { e } / d \tau ^ { \prime } during the increment yields
\begin{array}{l} \mathbf {e} _ {i} ^ {n + 1} = \left(1 - e ^ {- \Delta \tau / \tau_ {i}}\right) \int_ {0} ^ {\tau^ {n}} \frac {d \mathbf {e}}{d \tau^ {\prime}} d \tau^ {\prime} \\ + e ^ {- \Delta \tau / \tau_ {i}} \int_ {0} ^ {\tau^ {n}} \left(1 - e ^ {(\tau^ {\prime} - \tau^ {n}) / \tau_ {i}}\right) \frac {d \mathbf {e}}{d \tau^ {\prime}} d \tau^ {\prime} \\ + \frac {\Delta \mathbf {e}}{\Delta \tau} \int_ {\tau^ {n}} ^ {\tau^ {n + 1}} \left(1 - e ^ {(\tau^ {\prime} - \tau^ {n + 1}) / \tau_ {i}}\right) d \tau^ {\prime}. \\ \end{array}
The first and last integrals in this expression are readily evaluated, whereas from Equation 4.7.1-2 follows that the second integral represents the viscous strain in the i ^ { \mathrm { t h } } term at the beginning of the increment. Hence, the change in the i ^ { \mathrm { t h } } viscous strain is
Equation 4.7.1-3
\begin{array}{l} \Delta \mathbf {e} _ {i} = \left(1 - e ^ {- \Delta \tau / \tau_ {i}}\right) \mathbf {e} ^ {n} + (e ^ {- \Delta \tau / \tau_ {i}} - 1) \mathbf {e} _ {i} ^ {n} + \left(\Delta \tau - \tau_ {i} \left(1 - e ^ {- \Delta \tau / \tau_ {i}}\right)\right) \frac {\Delta \mathbf {e}}{\Delta \tau} \\ = \frac {\tau_ {i}}{\Delta \tau} \left(\frac {\Delta \tau}{\tau_ {i}} + e ^ {- \Delta \tau / \tau_ {i}} - 1\right) \Delta \mathbf {e} + \left(1 - e ^ {- \Delta \tau / \tau_ {i}}\right) \left(\mathbf {e} ^ {n} - \mathbf {e} _ {i} ^ {n}\right). \\ \end{array}
If \Delta \tau / \tau _ { i } approaches zero, this expression can be approximated by
Equation 4.7.1-4
\Delta \mathbf {e} _ {i} = \frac {\Delta \tau}{\tau_ {i}} \left(\frac {1}{2} \Delta \mathbf {e} + \mathbf {e} ^ {n} - \mathbf {e} _ {i} ^ {n}\right).
The last form is used in the computations if \Delta \tau / \tau _ { i } < 1 0 ^ { - 7 } .
Hence, in an increment, Equation 4.7.1-3 or Equation 4.7.1-4 is used to calculate the new value of the viscous strains. Equation 4.7.1-1 is then used subsequently to obtain the new value of the stresses.
The tangent modulus is readily derived from these equations by differentiating the deviatoric stress increment, which is
\Delta \mathbf {S} = 2 G _ {0} \left(\Delta \mathbf {e} - \sum_ {i = 1} ^ {n _ {G}} \alpha_ {i} (\mathbf {e} _ {i} ^ {n + 1} - \mathbf {e} _ {i} ^ {n})\right)
with respect to the deviatoric strain increment \Delta \mathbf { e } . Since the equations are linear, the modulus depends only on the reduced time step:
G ^ {T} = \left\{ \begin{array}{l l} G _ {0} \left[ 1 - \sum_ {i = 1} ^ {n} \alpha_ {i} \frac {\tau_ {i}}{\Delta \tau} \left(\frac {\Delta \tau}{\tau_ {i}} + e ^ {- \Delta \tau / \tau_ {i}} - 1\right) \right] & \mathrm{if} \Delta \tau / \tau_ {i} > 1 0 ^ {- 7} \\ G _ {0} \left[ 1 - \sum_ {i = 1} ^ {n} \frac {1}{2} \alpha_ {i} \frac {\Delta \tau}{\tau_ {i}} \right] & \mathrm{if} \Delta \tau / \tau_ {i} < 1 0 ^ {- 7} \end{array} \right.
The energy dissipation follows from
\begin{array}{l} P _ {D} = \frac {1}{2} (\mathbf {S} ^ {n + 1} + \mathbf {S} ^ {n}): \sum_ {i = 1} ^ {n _ {G}} \alpha_ {i} \Delta \mathbf {e _ {i}} \\ = \frac {1}{2} (\mathbf {S} ^ {n + 1} + \mathbf {S} ^ {n}): \left(\Delta \mathbf {e} - \frac {1}{2 G _ {0}} (\mathbf {S} ^ {n + 1} - \mathbf {S} ^ {n})\right) \\ = P - P _ {E} \\ \end{array}
with the total work
P = \frac {1}{2} (\mathbf {S} ^ {n + 1} + \mathbf {S} ^ {n}): \Delta \mathbf {e}
and the elastic energy increase
P _ {E} = \frac {1}{4 G _ {0}} \left(\mathbf {S} ^ {n + 1}: \mathbf {S} ^ {n + 1} - \mathbf {S} ^ {n}: \mathbf {S} ^ {n}\right).
Finally, one needs a relation between the reduced time increment, \Delta \tau _ { ; } , and the actual time increment, \Delta t . . To do this, we observe that A _ { \theta } varies very nonlinearly with temperature; hence, any direct approximation of A _ { \theta } is likely to lead to large errors. On the other hand, h ( \theta ) will generally be a smoothly varying function of temperature that is well approximated by a linear function of temperature over an increment. If we further assume that incrementally the temperature µ is a linear function of
time t, one finds the relation
h (\theta) = - \ln A _ {\theta} (\theta (t)) = a + b t
or
A _ {\theta} ^ {- 1} (\theta (t)) = e ^ {a + b t}
with
a = \frac {1}{\Delta t} \left[ t ^ {n + 1} h (\theta^ {n}) - t ^ {n} h (\theta^ {n + 1}) \right]
b = \frac {1}{\Delta t} \left[ h (\theta^ {n + 1}) - h (\theta^ {n}) \right].
This yields the relation
\begin{array}{l} \Delta \tau = \int_ {t ^ {n}} ^ {t ^ {n + 1}} e ^ {a + b t} d t \\ = \frac {1}{b} \left(e ^ {a + b t ^ {n + 1}} - e ^ {a + b t ^ {n}}\right). \\ \end{array}
This expression can also be written as
\Delta \tau = \frac {A _ {\theta} ^ {- 1} (\theta^ {n + 1}) - A _ {\theta} ^ {- 1} (\theta^ {n})}{h (\theta^ {n + 1}) - h (\theta^ {n})} \Delta t.
Reduced states of stress
So far, we have discussed full triaxial stress states. If the stress state is reduced (i.e., plane stress or uniaxial stress), the equations derived here cannot be used directly because only the total stress state is reduced, not the individual terms in the series. Therefore, we use the following procedure.
For plane stress let the third component be the zero stress component. At the beginning of the increment we presumably know the volumetric elastic strain \phi _ { e } ^ { n } , the volumetric viscous strain \phi _ { c } ^ { n } , and the volumetric viscous strains \phi _ { i } ^ { n } associated with the Prony series. The total volumetric strain can be obtained by adding together the elastic volumetric strain and the volumetric viscous strain
Equation 4.7.1-5
\phi^ {n} = \phi_ {e} ^ {n} + \phi_ {c} ^ {n}.
The deviatoric strain in the 3-direction follows from the relation \phi = \varepsilon _ { 1 } + \varepsilon _ { 2 } + \varepsilon _ { 3 } , which yields:
e _ {3} ^ {n} = \varepsilon_ {3} ^ {n} - \frac {1}{3} \phi^ {n} = \frac {2}{3} \phi^ {n} - \varepsilon_ {1} ^ {n} - \varepsilon_ {2} ^ {n}.
The out-of-plane deviatoric stress at the end of the increment is
s _ {3} ^ {n + 1} = 2 G _ {0} \left(e _ {3} ^ {n + 1} - \sum_ {i = 1} ^ {n _ {G}} \alpha_ {i} ^ {G} e _ {3 i} ^ {n + 1}\right).
Substituting Equation 4.7.1-3 for e _ { 3 i } ^ { n + 1 } , letting e _ { 3 } ^ { n + 1 } = e _ { 3 } ^ { n } + \Delta e _ { 3 } , and collecting terms gives
Equation 4.7.1-6
\begin{array}{l} s _ {3} ^ {n + 1} = 2 G ^ {T} \Delta e _ {3} + 2 G _ {0} e _ {3} ^ {n} \left[ 1 - \sum_ {i = 1} ^ {n _ {G}} \alpha_ {i} ^ {G} (1 - e ^ {- \Delta \tau / \tau_ {i}}) \right] \\ - 2 G _ {0} \sum_ {i = 1} ^ {n _ {G}} \alpha_ {i} ^ {G} e ^ {- \Delta \tau / \tau_ {i}} e _ {3 i} ^ {n}. \\ \end{array}
The hydrostatic stress is derived similarly as
Equation 4.7.1-7
- p ^ {n + 1} = K ^ {T} \Delta \phi + K _ {0} \phi^ {n} \left[ 1 - \sum_ {i = 1} ^ {n _ {K}} \alpha_ {i} ^ {K} (1 - e ^ {- \Delta \tau / \tau_ {i}}) \right]
- K _ {0} \sum_ {i = 1} ^ {n _ {K}} \alpha_ {i} ^ {K} e ^ {- \Delta \tau / \tau_ {i}} \phi_ {i} ^ {n}.
We can write these equations in the form
s _ {3} ^ {n + 1} = 2 G ^ {T} \Delta e _ {3} + \bar {s} _ {3}
- p ^ {n + 1} = K ^ {T} \Delta \phi - \bar {p}.
In the third direction the deviatoric stress minus the hydrostatic pressure is zero; hence,
Equation 4.7.1-8
2 G ^ {T} \Delta e _ {3} + K ^ {T} \Delta \phi + \bar {s} _ {3} - \bar {p} = 0.
Since \begin{array} { r } { \Delta e _ { 3 } = \frac { 2 } { 3 } \Delta \phi - \Delta \varepsilon _ { 1 } - \Delta \varepsilon _ { 2 } } \end{array} , it follows that
(K ^ {T} + \frac {4}{3} G ^ {T}) \Delta \phi = 2 G ^ {T} (\Delta \varepsilon_ {1} + \Delta \varepsilon_ {2}) - \bar {s} _ {3} + \bar {p},
from which \Delta \phi can be solved. One can then also calculate \Delta e _ { 1 } and \Delta e _ { 2 } , and with Equation 4.7.1-3 or Equation 4.7.1-4 one can update the deviatoric viscous strains \mathbf { e } _ { i } ^ { n + 1 } . The volumetric strains \phi _ { i } ^ { n + 1 } are obtained with a relation similar to Equation 4.7.1-3.
For uniaxial stress states a similar procedure is used. As before, \phi ^ { n } follows from Equation 4.7.1-5 and e _ { 3 } ^ { n } and e _ { 2 } ^ { n } follow from \varepsilon _ { 1 } + 2 \varepsilon _ { 3 } = \phi \colon
Equation 4.7.1-9
e _ {3} ^ {n} = e _ {2} ^ {n} = \varepsilon_ {3} ^ {n} - \frac {1}{3} \phi^ {n} = \frac {1}{6} \phi^ {n} - \frac {1}{2} \varepsilon_ {1} ^ {n}.
Equation 4.7.1-6 and Equation 4.7.1-7 can be used to calculate { \bar { s } } _ { 3 } and { \bar { p } } , which again leads to Equation 4.7.1-8. Applying Equation 4.7.1-9 for \Delta e _ { 3 } ;
(K ^ {T} + \frac {1}{3} G ^ {T}) \Delta \phi = G ^ {T} \Delta \varepsilon_ {1} - \bar {s} _ {3} + \bar {p}.
After this, one can follow the same procedure as for plane stress.
Automatic time stepping procedure
To create an automatic time stepping procedure in ABAQUS/Standard, we want to compare viscous strain rates at the beginning and the end of the increment. The strain rates in the individual terms at the beginning of the increment can be obtained directly by taking the limit of the incremental strain:
\begin{array}{l} \dot {\mathbf {e}} _ {i} ^ {n} = \lim _ {\Delta t \to 0} \frac {\mathbf {e} _ {i} ^ {n + 1} - \mathbf {e} _ {i} ^ {n}}{\Delta t} \\ = \lim _ {\Delta t \to 0} \frac {\Delta \tau}{\Delta t \tau_ {i}} \left(\frac {1}{2} \Delta \mathbf {e} + \mathbf {e} ^ {n} - \mathbf {e} _ {i} ^ {n}\right) \\ = \frac {\mathbf {e} ^ {n} - \mathbf {e} _ {i} ^ {n}}{A _ {\theta} (\theta^ {n}) \tau_ {i}}. \\ \end{array}
A similar expression can be derived for the strain rate at the end of the increment:
\dot {\mathbf {e}} _ {i} ^ {n + 1} = \frac {\mathbf {e} ^ {n + 1} - \mathbf {e} _ {i} ^ {n + 1}}{A _ {\theta} (\theta^ {n + 1}) \tau_ {i}}.
If we use these expressions to calculate a difference in estimated total viscous strain increment, one finds
\begin{array}{l} \Delta \bar {\mathbf {e}} _ {V} = \Delta t \sum_ {i = 1} ^ {n _ {G}} \alpha_ {i} ^ {G} (\dot {\mathbf {e}} _ {i} ^ {n + 1} - \dot {\mathbf {e}} _ {i} ^ {n}) \\ = \Delta t \sum_ {i = 1} ^ {n _ {G}} \frac {\alpha_ {i} ^ {G}}{\tau_ {i}} \left(\frac {\mathbf {e} ^ {n + 1} - \mathbf {e} _ {i} ^ {n + 1}}{A _ {\theta} (\theta^ {n})} - \frac {\mathbf {e} ^ {n} - \mathbf {e} _ {i} ^ {n}}{A _ {\theta} (\theta^ {n})}\right). \\ \end{array}
This expression is readily evaluated. A similar expression can be calculated for volumetric strain \Delta \bar { \phi } _ { V } , and from these two quantities a suitable scalar measure can be constructed; for example,
\Delta \varepsilon^ {\mathrm{est}} = \sqrt {\frac {2}{3} \Delta \bar {\bf e} _ {V} : \Delta \bar {\bf e} _ {V} + \frac {1}{3} \Delta \bar {\phi} _ {V} ^ {2}}.
Comparison with the user-specified tolerance CETOL allows construction of an automatic time stepping scheme.
4.7.2 Finite-strain viscoelasticity
Integral formulation
The finite-strain viscoelasticity theory implemented in ABAQUS is a time domain generalization of either the hyperelastic or hyperfoam constitutive models. It is assumed that the instantaneous response of the material follows from the hyperelastic constitutive equations:
\pmb {\tau} _ {0} (t) = \pmb {\tau} _ {0} ^ {D} (\overline {{\mathbf {F}}} (t)) + \pmb {\tau} _ {0} ^ {H} (J (t))
for a compressible material and
\pmb {\tau} _ {0} (t) = \pmb {\tau} _ {0} ^ {D} (\overline {{\mathbf {F}}} (t)) + \pmb {\tau} _ {0} ^ {H} (t)
for an incompressible material. In the above, \pmb { \tau } _ { 0 } ^ { D } and \pmb { \tau } _ { 0 } ^ { H } are, respectively, the deviatoric and the hydrostatic parts of the instantaneous Kirchhoff stress { \pmb \tau _ { 0 } } . \overline { { \mathbf { F } } } is the "distortion gradient" related to the deformation gradient F by
\overline {{\mathbf {F}}} = \frac {\mathbf {F}}{J ^ {\frac {1}{3}}},
where
J = \det (\mathbf {F})
is the volume change.
Using integration by parts and a variable transformation, the basic hereditary integral formulation for linear isotropic viscoelasticity can be written in the form
\pmb {\sigma} (t) = 2 G _ {0} \mathbf {e} (t) + \int_ {0} ^ {\tau} 2 \dot {G} (\tau^ {\prime}) \mathbf {e} (t - t ^ {\prime}) d \tau^ {\prime} + \mathbf {I} \left(K _ {0} \phi (t) + \int_ {0} ^ {\tau} \dot {K} (\tau^ {\prime}) \phi (t - t ^ {\prime}) d \tau^ {\prime}\right)
or entirely in terms of stress quantities,
\pmb {\sigma} (t) = \mathbf {S} _ {0} (t) + \int_ {0} ^ {\tau} \frac {\dot {G} (\tau^ {\prime})}{G _ {0}} \mathbf {S} _ {0} (t - t ^ {\prime}) d \tau^ {\prime} + \mathbf {I} \left(p _ {0} (t) + \int_ {0} ^ {\tau} \frac {\dot {K} (\tau^ {\prime})}{K _ {0}} p (t - t ^ {\prime}) d \tau^ {\prime}\right),
where ¿ is the reduced time, \dot { G } ( \tau ^ { \prime } ) = d G ( \tau ^ { \prime } ) / d \tau ^ { \prime } , and \dot { K } ( \tau ^ { \prime } ) = d K ( \tau ^ { \prime } ) / d \tau ^ { \prime } . G _ { 0 } and K _ { 0 } are the instantaneous small-strain shear and bulk moduli, and G ( t ) and K ( t ) are the time-dependent small-strain shear and bulk relaxation moduli. Recall that the reduced time represents a shift in time with temperature and is related to the actual time through the differential equation
d \tau^ {\prime} = \frac {d t ^ {\prime}}{A _ {\theta} (\theta (t ^ {\prime}))},
where µ is the temperature and A _ { \theta } is the shift function.
A suitable generalization to finite strain of the hereditary integral formulation is obtained as follows:
\pmb {\tau} (t) = \pmb {\tau} _ {0} (t) + \mathrm{SYM} \left[ \int_ {0} ^ {\tau} \mathbf {F} _ {t} ^ {- 1} (t - t ^ {\prime}) \cdot \left(\frac {\dot {G} (\tau^ {\prime})}{G _ {0}} \pmb {\tau} _ {0} ^ {D} (t - t ^ {\prime}) + \frac {\dot {K} (\tau^ {\prime})}{K _ {0}} \pmb {\tau} _ {0} ^ {H} (t - t ^ {\prime})\right) \cdot \mathbf {F} _ {t} (t - t ^ {\prime}) d \tau^ {\prime} \right]
where \mathbf { F } _ { t } ( t - t ^ { \prime } ) is the deformation gradient of the state at t - t ^ { \prime } relative to the state at t. A transformation is performed on the stress relating the state at time t - t ^ { \prime } to the state at time t. We also ensure the symmetry of the transformed stress. Observe that because \pmb { \tau } _ { 0 } ^ { H } is proportional to the identity tensor, we have
\mathbf {F} _ {t} ^ {- 1} (t - t ^ {\prime}) \cdot \pmb {\tau} _ {0} ^ {H} (t - t ^ {\prime}) \cdot \mathbf {F} _ {t} (t - t ^ {\prime}) = \pmb {\tau} _ {0} ^ {H} (t - t ^ {\prime}).
It can also be observed that
\operatorname{tr} \left(\mathbf {F} _ {t} ^ {- 1} (t - t ^ {\prime}) \cdot \pmb {\tau} _ {0} ^ {D} (t - t ^ {\prime}) \cdot \mathbf {F} _ {t} (t - t ^ {\prime})\right) = \mathbf {F} _ {t} ^ {- 1} (t - t ^ {\prime}): \left(\pmb {\tau} _ {0} ^ {D} (t - t ^ {\prime}) \cdot \mathbf {F} _ {t} (t - t ^ {\prime})\right) =
\pmb {\tau} _ {0} ^ {D} (t - t ^ {\prime}): \left(\mathbf {F} _ {t} (t - t ^ {\prime}) \cdot \mathbf {F} _ {t} ^ {- 1} (t - t ^ {\prime})\right) = \pmb {\tau} _ {0} ^ {D} (t - t ^ {\prime}): \mathbf {I} = 0,
since \tau _ { 0 } ^ { D } is deviatoric. Hence the deviatoric and volumetric parts can be separated into two hereditary integrals:
Equation 4.7.2-1
\pmb {\tau} ^ {D} (t) = \pmb {\tau} _ {0} ^ {D} (t) + \mathrm{SYM} \left[ \int_ {0} ^ {\tau} \frac {\dot {G} (\tau^ {\prime})}{G _ {0}} \mathbf {F} _ {t} ^ {- 1} (t - t ^ {\prime}) \cdot \pmb {\tau} _ {0} ^ {D} (t - t ^ {\prime}) \cdot \mathbf {F} _ {t} (t - t ^ {\prime}) d \tau^ {\prime} \right],
\pmb {\tau} ^ {H} (t) = \pmb {\tau} _ {0} ^ {H} (t) + \int_ {0} ^ {\tau} \frac {\dot {K} (\tau^ {\prime})}{K _ {0}} \pmb {\tau} _ {0} ^ {H} (t - t ^ {\prime}) d \tau^ {\prime}.
Implementation
As in small-strain viscoelasticity, we represent the relaxation moduli in terms of the Prony series
Equation 4.7.2-2
G (\tau) = G _ {0} \left(g _ {\infty} + \sum_ {i = 1} ^ {N _ {G}} g _ {i} e ^ {- \tau / \tau_ {i} ^ {G}}\right),
Equation 4.7.2-3
K (\tau) = K _ {0} \left(k _ {\infty} + \sum_ {i = 1} ^ {N _ {K}} k _ {i} e ^ {- \tau / \tau_ {i} ^ {K}}\right),
where g _ { i } and k _ { i } are the relative moduli of terms i. Note that \begin{array} { r } { g _ { \infty } + \sum _ { i = 1 } ^ { N _ { G } } g _ { i } = k _ { \infty } + \sum _ { i = 1 } ^ { N _ { K } } k _ { i } = 1 } \end{array} ABAQUS assumes that the relaxation times \tau _ { i } = \tau _ { i } ^ { K } = \tau _ { i } ^ { G } are the same so that from here on, we will sum on N terms for both bulk and shear behavior. In reality, the number of nonzero terms in bulk and shear, N _ { K } and N _ { G } , need not be equal, unless the instantaneous behavior is based on the *HYPERFOAM model. In the latter case, the two deformation modes are closely related and are then assumed to relax equally and simultaneously.
Substituting Equation 4.7.2-2 and Equation 4.7.2-3 in Equation 4.7.2-1, we obtain
Equation 4.7.2-4
\pmb {\tau} ^ {D} (t) = \pmb {\tau} _ {0} ^ {D} (t) - \mathrm{SYM} \left[ \sum_ {i = 1} ^ {N} \frac {g _ {i}}{\tau_ {i}} \int_ {0} ^ {\tau} \mathbf {F} _ {t} ^ {- 1} (t - t ^ {\prime}) \cdot \pmb {\tau} _ {0} ^ {D} (t - t ^ {\prime}) \cdot \mathbf {F} _ {t} (t - t ^ {\prime}) e ^ {- \frac {\tau^ {\prime}}{\tau_ {i}}} d \tau^ {\prime} \right],
\pmb {\tau} ^ {H} (t) = \pmb {\tau} _ {0} ^ {H} (t) - \sum_ {i = 1} ^ {N} \frac {k _ {i}}{\tau_ {i}} \int_ {0} ^ {\tau} \pmb {\tau} _ {0} ^ {H} (t - t ^ {\prime}) e ^ {- \frac {\tau^ {\prime}}{\tau_ {i}}} d \tau^ {\prime}.
Next, we introduce the internal stresses, associated with each term of the series
Equation 4.7.2-5
\pmb {\tau} _ {i} ^ {D} (t) \equiv \mathrm{SYM} \left[ \frac {g _ {i}}{\tau_ {i}} \int_ {0} ^ {\tau} \mathbf {F} _ {t} ^ {- 1} (t - t ^ {\prime}) \cdot \pmb {\tau} _ {0} ^ {D} (t - t ^ {\prime}) \cdot \mathbf {F} _ {t} (t - t ^ {\prime}) e ^ {- \frac {\tau^ {\prime}}{\tau_ {i}}} d \tau^ {\prime} \right],
Equation 4.7.2-6
\pmb {\tau} _ {i} ^ {H} (t) \equiv \frac {k _ {i}}{\tau_ {i}} \int_ {0} ^ {\tau} \pmb {\tau} _ {0} ^ {H} (t - t ^ {\prime}) e ^ {- \frac {\tau^ {\prime}}{\tau_ {i}}} d \tau^ {\prime}.
These stresses are stored at each material point and are integrated forward in time. We will assume that the solution is known at time t, and we need to construct the solution at time t + \Delta t .
Integration of the hydrostatic stress
The internal hydrostatic stresses \pmb { \tau } _ { i } ^ { H } at time t + \Delta t follow from
\pmb {\tau} _ {i} ^ {H} (t + \Delta t) = \frac {k _ {i}}{\tau_ {i}} \int_ {0} ^ {\tau + \Delta \tau} \pmb {\tau} _ {0} ^ {H} (t + \Delta t - t ^ {\prime}) e ^ {- \frac {\tau^ {\prime}}{\tau_ {i}}} d \tau^ {\prime}.
With \overline { { \tau } } = \tau ^ { \prime } - \Delta \tau and \overline { { t } } = t ^ { \prime } - \Delta t , it follows that
\begin{array}{l} \pmb {\tau} _ {i} ^ {H} (t + \Delta t) = \frac {k _ {i}}{\tau_ {i}} e ^ {- \frac {\Delta \tau}{\tau_ {i}}} \int_ {- \Delta \tau} ^ {\tau} \pmb {\tau} _ {0} ^ {H} (t - \overline {{t}}) e ^ {- \frac {\overline {{\tau}}}{\tau_ {i}}} d \overline {{\tau}} \\ = \frac {k _ {i}}{\tau_ {i}} e ^ {- \frac {\Delta \tau}{\tau_ {i}}} \int_ {- \Delta \tau} ^ {0} \pmb {\tau} _ {0} ^ {H} (t - \overline {{t}}) e ^ {- \frac {\overline {{\tau}}}{\tau_ {i}}} d \overline {{\tau}} + \frac {k _ {i}}{\tau_ {i}} e ^ {- \frac {\Delta \tau}{\tau_ {i}}} \int_ {0} ^ {\tau} \pmb {\tau} _ {0} ^ {H} (t - \overline {{t}}) e ^ {- \frac {\overline {{\tau}}}{\tau_ {i}}} d \overline {{\tau}}, \\ \end{array}
which yields with Equation 4.7.2-6
Equation 4.7.2-7
\pmb {\tau} _ {i} ^ {H} (t + \Delta t) = \frac {k _ {i}}{\tau_ {i}} e ^ {- \frac {\Delta \tau}{\tau_ {i}}} \int_ {- \Delta \tau} ^ {0} \pmb {\tau} _ {0} ^ {H} (t - \overline {{t}}) e ^ {- \frac {\overline {{\tau}}}{\tau_ {i}}} d \overline {{\tau}} + e ^ {- \frac {\Delta \tau}{\tau_ {i}}} \pmb {\tau} _ {i} ^ {H} (t).
To integrate the first integral in Equation 4.7.2-7, we assume that \tau _ { 0 } ^ { H } ( t - \overline { { t } } ) varies linearly with the reduced time \overline { { \tau } } over the increment
Equation 4.7.2-8
\pmb {\tau} _ {0} ^ {H} (t - \overline {{t}}) = (1 + \frac {\overline {{\tau}}}{\Delta \tau}) \pmb {\tau} _ {0} ^ {H} (t) - \frac {\overline {{\tau}}}{\Delta \tau} \pmb {\tau} _ {0} ^ {H} (t + \Delta t) \qquad - \Delta \tau \leq \overline {{\tau}} \leq 0.
Substituting into Equation 4.7.2-7 yields
\begin{array}{l} \pmb {\tau} _ {i} ^ {H} (t + \Delta t) = \left[ \frac {k _ {i}}{\tau_ {i}} e ^ {- \frac {\Delta \tau}{\tau_ {i}}} \int_ {- \Delta \tau} ^ {0} - \frac {\overline {{\tau}}}{\Delta \tau} e ^ {- \frac {\overline {{\tau}}}{\tau_ {i}}} d \overline {{\tau}} \right] \pmb {\tau} _ {0} ^ {H} (t + \Delta t) \\ + \left[ \frac {k _ {i}}{\tau_ {i}} e ^ {- \frac {\Delta \tau}{\tau_ {i}}} \int_ {- \Delta \tau} ^ {0} (1 + \frac {\overline {{\tau}}}{\Delta \tau}) e ^ {- \frac {\overline {{\tau}}}{\tau_ {i}}} d \overline {{\tau}} \right] \pmb {\tau} _ {0} ^ {H} (t) + e ^ {- \frac {\Delta \tau}{\tau_ {i}}} \pmb {\tau} _ {i} ^ {H} (t). \\ \end{array}
The integrals are readily evaluated, providing the solution at the end of the increment
\begin{array}{l} \pmb {\tau} _ {i} ^ {H} (t + \Delta t) = \left[ 1 - \frac {\tau_ {i}}{\Delta \tau} (1 - e ^ {- \frac {\Delta \tau}{\tau_ {i}}}) \right] k _ {i} \pmb {\tau} _ {0} ^ {H} (t + \Delta t) \\ + \left[ \frac {\tau_ {i}}{\Delta \tau} (1 - e ^ {- \frac {\Delta \tau}{\tau_ {i}}}) - e ^ {- \frac {\Delta \tau}{\tau_ {i}}} \right] k _ {i} \pmb {\tau} _ {0} ^ {H} (t) + e ^ {- \frac {\Delta \tau}{\tau_ {i}}} \pmb {\tau} _ {i} ^ {H} (t) \\ \end{array}
or, in a slightly different form
Equation 4.7.2-9
\pmb {\tau} _ {i} ^ {H} (t + \Delta t) = \alpha_ {i} k _ {i} \pmb {\tau} _ {0} ^ {H} (t + \Delta t) + \beta_ {i} k _ {i} \pmb {\tau} _ {0} ^ {H} (t) + \gamma_ {i} \pmb {\tau} _ {i} ^ {H} (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}.
Observe that for \Delta t = \Delta \tau = 0 , \gamma _ { i } = 1 and \alpha _ { i } = \beta _ { i } = 0 . For \Delta t = \Delta \tau = \infty , \alpha _ { i } = 1 and \gamma _ { i } = \beta _ { i } = 0 .
Integration of the deviatoric stress
The internal deviatoric stresses \pmb { \tau } _ { i } ^ { D } at time t + \Delta t follow from
Equation 4.7.2-10
\pmb {\tau} _ {i} ^ {D} (t + \Delta t) =
\mathrm{SYM} \left[ \frac {g _ {i}}{\tau_ {i}} \int_ {0} ^ {\tau + \Delta \tau} \mathbf {F} _ {t + \Delta t} ^ {- 1} (t + \Delta t - t ^ {\prime}) \cdot \pmb {\tau} _ {0} ^ {D} (t + \Delta t - t ^ {\prime}) \cdot \mathbf {F} _ {t + \Delta t} (t + \Delta t - t ^ {\prime}) e ^ {- \frac {\tau^ {\prime}}{\tau_ {i}}} d \tau^ {\prime} \right].
Observe that
\mathbf {F} _ {t + \Delta t} (t - t ^ {\prime}) = \mathbf {F} _ {t} (t - t ^ {\prime}) \cdot \mathbf {F} _ {t + \Delta t} (\tau),
and the inverse of this is
\mathbf {F} _ {t + \Delta t} ^ {- 1} (t - t ^ {\prime}) = \mathbf {F} _ {t + \Delta t} ^ {- 1} (t) \cdot \mathbf {F} _ {t} ^ {- 1} (t - t ^ {\prime}) = \mathbf {F} _ {t} (t + \Delta t) \cdot \mathbf {F} _ {t} ^ {- 1} (t - t ^ {\prime}),
which--when substituted into Equation 4.7.2-10, with \Delta \mathbf { F } \equiv \mathbf { F } _ { t } ( t + \Delta t ) and
\Delta \mathbf {F} ^ {- 1} \equiv \mathbf {F} _ {t + \Delta t} (t) \text {- - gives}
\pmb {\tau} _ {i} ^ {D} (t + \Delta t) =
\mathrm{SYM} \left[ \frac {g _ {i}}{\tau_ {i}} \Delta \mathbf {F} \cdot \int_ {0} ^ {\tau + \Delta \tau} \mathbf {F} _ {t} ^ {- 1} (t + \Delta t - t ^ {\prime}) \cdot \pmb {\tau} _ {0} ^ {D} (t + \Delta t - t ^ {\prime}) \cdot \mathbf {F} _ {t} (t + \Delta t - t ^ {\prime}) e ^ {- \frac {\tau^ {\prime}}{\tau_ {i}}} d \tau^ {\prime} \cdot \Delta \mathbf {F} ^ {- 1} \right].
With \overline { { \tau } } = \tau ^ { \prime } - \Delta \tau and \overline { { t } } = t ^ { \prime } - \Delta t . , it follows:
Equation 4.7.2-11
\pmb {\tau} _ {i} ^ {D} (t + \Delta t) = \mathrm{SYM} \left[ \frac {g _ {i}}{\tau_ {i}} e ^ {- \frac {\Delta \tau}{\tau_ {i}}} \Delta \mathbf {F} \cdot \int_ {- \Delta \tau} ^ {\tau} \mathbf {F} _ {t} ^ {- 1} (t - \overline {{t}}) \cdot \pmb {\tau} _ {0} ^ {D} (t - \overline {{t}}) \cdot \mathbf {F} _ {t} (t - \overline {{t}}) e ^ {- \frac {\overline {{\tau}}}{\tau_ {i}}} d \overline {{\tau}} \cdot \Delta \mathbf {F} ^ {- 1} \right]
= \mathrm{SYM} \left[ \frac {g _ {i}}{\tau_ {i}} e ^ {- \frac {\Delta \tau}{\tau_ {i}}} \int_ {- \Delta \tau} ^ {0} \Delta \mathbf {F} \cdot \mathbf {F} _ {t} ^ {- 1} (t - \overline {{t}}) \cdot \pmb {\tau} _ {0} ^ {D} (t - \overline {{t}}) \cdot \mathbf {F} _ {t} (t - \overline {{t}}) \cdot \Delta \mathbf {F} ^ {- 1} e ^ {- \frac {\overline {{\tau}}}{\tau_ {i}}} d \overline {{\tau}} \right]
+ \mathrm{SYM} \left[ \frac {g _ {i}}{\tau_ {i}} e ^ {- \frac {\Delta \tau}{\tau_ {i}}} \Delta \mathbf {F} \cdot \int_ {0} ^ {\tau} \mathbf {F} _ {t} ^ {- 1} (t - \overline {{{t}}}) \cdot \pmb {\tau} _ {0} ^ {D} (t - \overline {{{t}}}) \cdot \mathbf {F} _ {t} (t - \overline {{{t}}}) e ^ {- \frac {\overline {{{\tau}}}}{\tau_ {i}}} d \overline {{{\tau}}} \cdot \Delta \mathbf {F} ^ {- 1} \right].
Now introduce the variable
Equation 4.7.2-12
\hat {\pmb {\tau}} _ {0} ^ {D} (t - \overline {{{t}}}) \equiv \mathrm{SYM} \left[ \Delta \mathbf {F} \cdot \mathbf {F} _ {t} ^ {- 1} (t - \overline {{{t}}}) \cdot \pmb {\tau} _ {0} ^ {D} (t - \overline {{{t}}}) \cdot \mathbf {F} _ {t} (t - \overline {{{t}}}) \cdot \Delta \mathbf {F} ^ {- 1} \right].
Note that
Equation 4.7.2-13
\hat {\pmb {\tau}} _ {0} ^ {D} (t) = \mathrm{SYM} \left[ \Delta \mathbf {F} \cdot \mathbf {F} _ {t} ^ {- 1} (t) \cdot \pmb {\tau} _ {0} ^ {D} (t) \cdot \mathbf {F} _ {t} (t) \cdot \Delta \mathbf {F} ^ {- 1} \right] = \mathrm{SYM} \left[ \Delta \mathbf {F} \cdot \pmb {\tau} _ {0} ^ {D} (t) \cdot \Delta \mathbf {F} ^ {- 1} \right]
and