김경종
26 KiB
T _ {S} = \frac {\partial U}{\partial \gamma} = \sum_ {j = 1} ^ {2} \biggl \{\frac {2 \gamma}{2 (\lambda_ {j} ^ {2} - 1) - \gamma^ {2}} \sum_ {i = 1} ^ {N} \frac {\mu_ {i}}{\alpha_ {i}} (\lambda_ {j} ^ {\alpha_ {i}} - 1) \biggr \},
where \lambda _ { j } are the principal stretches in the plane of shearing, related to the shear strain \gamma as follows:
\lambda_ {1, 2} = \sqrt {1 + \frac {\gamma^ {2}}{2} \pm \gamma \sqrt {1 + \frac {\gamma^ {2}}{4}}}.
The stretch in the direction perpendicular to the plane of shearing is \lambda _ { 3 } = 1 .
The transverse stress T _ { T } developed during simple shear deformation (as a result of the Poynting effect) is
T _ {T} = \frac {\partial U}{\partial \varepsilon} = \sum_ {j = 1} ^ {2} \left\{\frac {2 (\lambda_ {j} ^ {2} - 1)}{2 \lambda_ {j} ^ {4} - \lambda_ {j} ^ {2} (\gamma^ {2} + 2)} \sum_ {i = 1} ^ {N} \frac {\mu_ {i}}{\alpha_ {i}} (\lambda_ {j} ^ {\alpha_ {i}} - 1) \right\}.
Volumetric mode
The volumetric deformation is described as
\lambda_ {1} = \lambda_ {2} = \lambda_ {3} = \lambda_ {V}, J = \lambda_ {V} ^ {3}.
The pressure p is related to volume ratio J through
p = - \frac {\partial U}{\partial J} = - \frac {2}{J} \sum_ {i = 1} ^ {N} \frac {\mu_ {i}}{\alpha_ {i}} (J ^ {\frac {1}{3} \alpha_ {i}} - J ^ {- \alpha_ {i} \beta_ {i}}).
Least squares fit
Given experimental data, the material constants are determined through a least-squares-fit procedure, which minimizes the relative error in stress. For the n nominal-stress-nominal-strain data pairs, the relative error measure E is minimized,
E = \sum_ {i = 1} ^ {n} \left(1 - T _ {i} ^ {\mathrm{th}} / T _ {i} ^ {\mathrm{test}}\right) ^ {2},
where T _ { i } ^ { \mathrm { t e s t } } is a stress value from the test data and T _ { i } ^ { \mathrm { t h } } comes from one of the nominal stress expressions derived above.
The polynomial potential is linear in terms of the constants C _ { i j } \mathrm { ; } ; therefore, a linear least-squares procedure can be used. The Ogden, the Arruda-Boyce, and the Van der Waals potential are nonlinear in some of their coefficients, thus necessitating the use of a nonlinear least-squares procedure.
Linear least squares fit for the polynomial model
For the full polynomial model we can rewrite the expressions for the T _ { k } ^ { \mathrm { t h } } derived above as
T _ {k} ^ {\mathrm{th}} = \sum_ {i + j = 1} ^ {N} C _ {i j} X _ {i j} (\lambda_ {k}), \quad k = 1 \dots n,
where the X _ { i j } ( \lambda _ { k } ) are functions that depend on the stress state (uniaxial, biaxial, or planar), as explained above. N = 1 for the first-order polynomial (or Mooney-Rivlin form), and N = 2 for the second-order polynomial. To minimize the relative error, we need to set
\frac {\partial E}{\partial C _ {i j}} = 0,
which leads to the following set of \begin{array} { r } { M = \frac { 1 } { 2 } N ( N + 3 ) } \end{array} equations:
\sum_ {k = 1} ^ {n} \sum_ {i + j = 1} ^ {N} \frac {X _ {i j} (\lambda_ {k}) X _ {l m} (\lambda_ {k})}{(T _ {k} ^ {\mathrm{test}}) ^ {2}} C _ {i j} = \sum_ {k = 1} ^ {n} \frac {X _ {l m} (\lambda_ {k})}{T _ {k} ^ {\mathrm{test}}}, \quad l + m = 1 \dots N.
This linear set of M equations can be solved readily to define the coefficients C _ { i j }
To fit the volumetric coefficients, one needs to solve the system of N equations
\sum_ {k = 1} ^ {n} \sum_ {i = 1} ^ {N} \frac {X _ {i} (J _ {k}) X _ {j} (J _ {k})}{(p _ {k} ^ {\mathrm{test}}) ^ {2}} \frac {1}{D _ {i}} = \sum_ {k = 1} ^ {n} \frac {X _ {j} (J _ {k})}{p _ {k} ^ {\mathrm{test}}}, \quad j = 1 \dots N,
where
X _ {i} (J _ {k}) = - 2 i (J _ {k} - 1) ^ {2 i - 1},
and
J _ {k} = \frac {V _ {k}}{V _ {0}}
is given by the user. This system of equations can be solved readily for D _ { i } .
Linear least squares fit for the reduced polynomial model
For the reduced polynomial model we can rewrite the expressions for T _ { k } ^ { \mathrm { t h } } derived above as follows:
T _ {k} ^ {\mathrm{th}} = \sum_ {i = 1} ^ {N} C _ {i 0} X _ {i} (\lambda_ {k}),
where again the functions X _ { i } ( \lambda _ { k } ) depend on the stress state and the stretch, as outlined above, and N is the order of the reduced polynomial, which can take values up to N = 6 . The following also applies to the Yeoh and neo-Hookean forms since these models are special cases of the reduced polynomial, with N = 3 and N = 1, respectively.
Following the same arguments as for the full polynomial, we arrive at the system of N equations
\sum_ {k = 1} ^ {n} \sum_ {i = 1} ^ {N} \frac {X _ {i} (\lambda_ {k}) X _ {j} (\lambda_ {k})}{(T _ {k} ^ {\mathrm{test}}) ^ {2}} C _ {i 0} = \sum_ {k = 1} ^ {n} \frac {X _ {j} (\lambda_ {k})}{T _ {k} ^ {\mathrm{test}}}, \quad j = 1 \dots N.
This system of equations can be solved readily for the coefficients C _ { i 0 } . The volumetric coefficients are fitted with the same procedure as used for the general polynomial models.
Nonlinear least squares fit
The Ogden, Arruda-Boyce, and Van der Waals potentials are nonlinear in some of their coefficients; hence, a nonlinear least-squares-fit procedure is required. We use the Marquard-Levenberg algorithm in the formulation by Twizell and Ogden (1986). Let a _ { i } , i = 1 \ldots m be the coefficients of these hyperelastic models, where m is the number of coefficients contributing to the deviatoric behavior. Specifically, m = 2 N for the Ogden model, m = 2 for the Arruda-Boyce model, and m = 4 for the Van der Waals model. The coefficients are found by iterating the equation
a _ {i} ^ {(r + 1)} = a _ {i} ^ {(r)} - \sum_ {j = 1} ^ {m} \sum_ {k = 1} ^ {n} \left[ P _ {i k} ^ {(r)} P _ {j k} ^ {(r)} + \gamma \delta_ {i j} \right] ^ {- 1} P _ {j k} ^ {(r)} E _ {k} ^ {(r)},
where r is the iteration count, n is the number of data points,
E _ {k} = \frac {T _ {k} ^ {\mathrm{test}} - T _ {k} ^ {\mathrm{th}}}{T _ {k} ^ {\mathrm{test}}}
is the vector of relative errors, and
P _ {i k} = \frac {\partial E _ {k}}{\partial a _ {i}} = - \frac {1}{T _ {k} ^ {\mathrm{test}}} \frac {\partial T _ {k} ^ {\mathrm{th}}}{\partial a _ {i}}
is the derivative of the vector of relative errors with respect to the coefficients a _ { i }
For \gamma = 0 , the Newton algorithm is obtained; for very large values of \gamma , the steepest descent method is obtained. Thus, the Marquard-Levenberg algorithm represents a compromise between these two approaches: the value of \gamma is increased if the error grows and is reduced otherwise.
Nonlinear least squares fit for the Ogden model
After initializing the \alpha _ { i } ^ { ( 0 ) } , the parameters \mu _ { i } ^ { ( 0 ) } are found with a linear least squares fit. In the iterative procedure outlined above, the following derivatives are used:
\frac {\partial T _ {k} ^ {\mathrm{th}}}{\partial \mu_ {i}} = \frac {2}{\alpha_ {i}} \left(\lambda^ {\alpha_ {i} - 1} - \lambda^ {c \alpha_ {i} - 1}\right),
\frac {\partial T _ {k} ^ {\mathrm{th}}}{\partial \alpha_ {i}} = - \frac {2 \mu_ {i}}{\alpha_ {i} ^ {2}} (\lambda^ {\alpha_ {i} - 1} - \lambda^ {c \alpha_ {i} - 1}) - \frac {2 \mu_ {i}}{\alpha_ {i}} (\lambda^ {\alpha_ {i} - 1} - c \lambda^ {c \alpha_ {i} - 1}) \ln \lambda ,
where
c = \left\{ \begin{array}{l l} - \frac {1}{2}, & \mathrm{ifuniaxial;} \\ - 2, & \mathrm{ifbiaxial;} \\ - 1, & \mathrm{ifplanar.} \end{array} \right.
Nonlinear least squares fit for the Arruda-Boyce model
The Arruda-Boyce model is linear in the shear modulus \mu but nonlinear in the locking stretch \lambda _ { m } . The locking stretch is initialized as \lambda _ { m } ^ { ( 0 ) } = \operatorname* { m a x } ( 7 . 0 , 3 . 0 \times \lambda _ { \operatorname* { m a x } } ) , where \lambda _ { \mathrm { m a x } } is the maximum stretch in the user-specified test data. Given this locking stretch, the initial shear modulus, \mu ^ { ( 0 ) } , is obtained with a linear least squares fit.
In the iterative procedure outlined above, the following derivatives are used:
\frac {\partial T _ {k} ^ {\mathrm{th}}}{\partial \mu} = \left\{ \begin{array}{l l} 2 (\lambda_ {U} - \lambda_ {U} ^ {- 2}) \sum_ {i = 1} ^ {5} \frac {i C _ {i}}{\lambda_ {m} ^ {2 i - 2} \overline {{I}} _ {1} ^ {i - 1}}, & \mathrm{ifuniaxial;} \\ 2 (\lambda_ {B} - \lambda_ {B} ^ {- 5}) \sum_ {i = 1} ^ {5} \frac {i C _ {i}}{\lambda_ {m} ^ {2 i - 2} \overline {{I}} _ {1} ^ {i - 1}}, & \mathrm{ifbiaxial;} \\ 2 (\lambda_ {S} - \lambda_ {S} ^ {- 3}) \sum_ {i = 1} ^ {5} \frac {i C _ {i}}{\lambda_ {m} ^ {2 i - 2} \overline {{I}} _ {1} ^ {i - 1}}, & \mathrm{ifplanar.} \end{array} \right.
\frac {\partial T _ {k} ^ {\mathrm{th}}}{\partial \lambda_ {m}} = \left\{ \begin{array}{l l} 2 \mu (\lambda_ {U} - \lambda_ {U} ^ {- 2}) \sum_ {i = 2} ^ {5} (2 - 2 i) \frac {i C _ {i}}{\lambda_ {m} ^ {2 i - 1} \overline {{I}} _ {1} ^ {i - 1}}, & \mathrm{ifuniaxial;} \\ 2 \mu (\lambda_ {B} - \lambda_ {B} ^ {- 5}) \sum_ {i = 2} ^ {5} (2 - 2 i) \frac {i C _ {i}}{\lambda_ {m} ^ {2 i - 1} \overline {{I}} _ {1} ^ {i - 1}}, & \mathrm{ifbiaxial;} \\ 2 \mu (\lambda_ {S} - \lambda_ {S} ^ {- 3}) \sum_ {i = 2} ^ {5} (2 - 2 i) \frac {i C _ {i}}{\lambda_ {m} ^ {2 i - 1} \overline {{I}} _ {1} ^ {i - 1}}, & \mathrm{ifplanar.} \end{array} \right.
Nonlinear least squares fit for the Van der Waals model
The Van der Waals model is linear in the shear modulus \mu but nonlinear in the locking stretch \lambda _ { m } , the global interaction parameter ^ { a , } and the mixture parameter \beta . . The locking stretch is initialized as \lambda _ { m } ^ { ( 0 ) } = \operatorname* { m a x } ( 1 0 . 0 , 3 . 0 \times \lambda _ { \operatorname* { m a x } } ) , where \lambda _ { \mathrm { m a x } } is the maximum stretch in the user-specified test data.
Given this guess for the locking stretch, we make use of an expression proposed by Kilian et al. (1986) to initialize the global interaction parameter
a ^ {(0)} = \frac {(\lambda_ {m} ^ {(0)}) ^ {2}}{(\lambda_ {m} ^ {(0)}) ^ {3} - 1}.
The invariant mixture parameter is initialized to \beta ^ { ( 0 ) } = 0 . Given these initial values, the shear
modulus, \mu ^ { ( 0 ) } , is initialized using a linear least-squares-fit procedure.
In the iterative procedure outlined above, the following derivatives are used:
\frac {\partial T _ {k} ^ {\mathrm{th}}}{\partial \mu} = \left\{ \begin{array}{l l} (1 - \lambda_ {U} ^ {- 3}) \bigg (\frac {1}{1 - \eta} - a \sqrt {\frac {\tilde {I} - 3}{2}} \bigg) \bigg [ \lambda_ {U} (1 - \beta) + \beta \bigg ], & \mathrm{ifuniaxial;} \\ (\lambda_ {B} - \lambda_ {B} ^ {- 5}) \bigg (\frac {1}{1 - \eta} - a \sqrt {\frac {\tilde {I} - 3}{2}} \bigg) \bigg (1 - \beta + \beta \lambda_ {B} ^ {2} \bigg), & \mathrm{ifbiaxial;} \\ (\lambda_ {S} - \lambda_ {S} ^ {- 3}) \bigg (\frac {1}{1 - \eta} - a \sqrt {\frac {\tilde {I} - 3}{2}} \bigg), & \mathrm{ifplanar.} \end{array} \right.
\frac {\partial T _ {k} ^ {\mathrm{th}}}{\partial \lambda_ {m}} = \left\{ \begin{array}{l l} - \mu (1 - \lambda_ {U} ^ {- 3}) \frac {\eta \lambda_ {m}}{(1 - \eta) ^ {2} (\lambda_ {m} ^ {2} - 3)} \bigg [ \lambda_ {U} (1 - \beta) + \beta \bigg ], & \mathrm{ifuniaxial;} \\ - \mu (\lambda_ {B} - \lambda_ {B} ^ {- 5}) \frac {\eta \lambda_ {m}}{(1 - \eta) ^ {2} (\lambda_ {m} ^ {2} - 3)} \bigg (1 - \beta + \beta \lambda_ {B} ^ {2} \bigg), & \mathrm{ifbiaxial;} \\ - \mu (\lambda_ {S} - \lambda_ {S} ^ {- 3}) \frac {\eta \lambda_ {m}}{(1 - \eta) ^ {2} (\lambda_ {m} ^ {2} - 3)}, & \mathrm{ifplanar.} \end{array} \right.
\frac {\partial T _ {k} ^ {\mathrm{th}}}{\partial a} = \left\{ \begin{array}{l l} - \mu (1 - \lambda_ {U} ^ {- 3}) \sqrt {\frac {\tilde {I} - 3}{2}} \bigg [ \lambda_ {U} (1 - \beta) + \beta \bigg ], & \mathrm{ifuniaxial;} \\ - \mu (\lambda_ {B} - \lambda_ {B} ^ {- 5}) \sqrt {\frac {\tilde {I} - 3}{2}} \bigg (1 - \beta + \beta \lambda_ {B} ^ {2} \bigg), & \mathrm{ifbiaxial;} \\ - \mu (\lambda_ {S} - \lambda_ {S} ^ {- 3}) \sqrt {\frac {\tilde {I} - 3}{2}}, & \mathrm{ifplanar.} \end{array} \right.
\frac {\partial T _ {k} ^ {\mathrm{th}}}{\partial \beta} = \left\{ \begin{array}{l l} 2 (1 - \lambda_ {U} ^ {- 3}) \bigg [ \frac {\partial U ^ {\prime}}{\partial \beta} \bigg (\lambda_ {U} (1 - \beta) + \beta \bigg) + U ^ {\prime} (1 - \lambda_ {U}) \bigg ], & \mathrm{ifuniaxial;} \\ 2 (\lambda_ {B} - \lambda_ {B} ^ {- 5}) \bigg [ \frac {\partial U ^ {\prime}}{\partial \beta} \bigg (1 - \beta + \beta \lambda_ {B} ^ {2} \bigg) + U ^ {\prime} (\lambda_ {B} ^ {2} - 1) \bigg ], & \mathrm{ifbiaxial;} \\ 0, & \mathrm{ifplanar.} \end{array} \right.
In the derivatives of \partial T _ { k } ^ { \mathrm { t h } } / \partial \beta
U ^ {\prime} = \frac {\partial U}{\partial \tilde {I}} = \frac {\mu}{2} \left(\frac {1}{1 - \eta} - a \sqrt {\frac {\tilde {I} - 3}{2}}\right),
and
\frac {\partial U ^ {\prime}}{\partial \beta} = \mu \left[ \frac {1}{4} \frac {1}{y - 3} \frac {1}{\eta (1 - \eta^ {2})} - \frac {a}{8} \sqrt {\frac {2}{\tilde {I} - 3}} \right] (\overline {{I}} _ {1} - \overline {{I}} _ {2}).
In the planar case \overline { { I } } _ { 1 } = \overline { { I } } _ { 2 } ; ; hence, \partial T _ { k } ^ { \mathrm { t h } } / \partial \beta vanishes.
Drucker stability check
ABAQUS checks the Drucker stability of the material for the first three modes of deformation described above. The Drucker stability condition requires that the change in the Kirchhoff stress d¿ following from an infinitesimal change in the logarithmic strain d" satisfies the inequality
d \pmb {\tau}: d \pmb {\varepsilon} > 0.
Using d { \boldsymbol { \tau } } = \mathbf { D } \cdot d { \boldsymbol { \varepsilon } } , the inequality becomes
d \boldsymbol {\varepsilon} \cdot \mathbf {D}: d \boldsymbol {\varepsilon} > 0,
thus requiring the tangential material stiffness D to be positive definite for material stability to be satisfied.
For the isotropic elastic formulation considered here, the inequality can be represented in terms of the principal stresses and strains:
d \tau_ {1} d \varepsilon_ {1} + d \tau_ {2} d \varepsilon_ {2} + d \tau_ {3} d \varepsilon_ {3} > 0.
Polynomial form
With the incompressibility assumption for the two hyperelastic models, the Kirchhoff stress is equal to the Cauchy stress: ¿ = ¾ and, thus,
d \boldsymbol {\sigma}: d \boldsymbol {\varepsilon} > 0.
In addition, we can choose any value for the hydrostatic pressure without affecting the strains. For the stability calculation a convenient choice is \sigma _ { 3 } = d \sigma _ { 3 } = 0 , which gives us
d \sigma_ {1} d \varepsilon_ {1} + d \sigma_ {2} d \varepsilon_ {2} > 0.
The infinitesimal strain changes are related to the changes in stretch ratios by the equations
d \varepsilon_ {1} = \frac {d \lambda_ {1}}{\lambda_ {1}}, \quad d \varepsilon_ {2} = \frac {d \lambda_ {2}}{\lambda_ {2}}.
The stresses follow from the strain energy, which in turn follow from the changes in the strain invariants or in the stretches.
The relation between changes in the stress and changes in strain are described by the matrix equation
\left\{ \begin{array}{l} d \sigma_ {1} \\ d \sigma_ {2} \end{array} \right\} = \left[ \begin{array}{c c} D _ {1 1} & D _ {1 2} \\ D _ {2 1} & D _ {2 2} \end{array} \right] \left\{ \begin{array}{l} d \varepsilon_ {1} \\ d \varepsilon_ {2} \end{array} \right\},
where
D _ {1 1} = 4 (\lambda_ {1} ^ {2} + \lambda_ {3} ^ {2}) \left(\frac {\partial U}{\partial \overline {{I}} _ {1}} + \lambda_ {2} ^ {2} \frac {\partial U}{\partial \overline {{I}} _ {2}}\right) + 4 (\lambda_ {1} ^ {2} - \lambda_ {3} ^ {2}) ^ {2} \left(\frac {\partial^ {2} U}{\partial \overline {{I}} _ {1} ^ {2}} + 2 \lambda_ {2} ^ {2} \frac {\partial^ {2} U}{\partial \overline {{I}} _ {1} \partial \overline {{I}} _ {2}} + \lambda_ {2} ^ {4} \frac {\partial^ {2} U}{\partial \overline {{I}} _ {2} ^ {2}}\right),
D _ {2 2} = 4 (\lambda_ {2} ^ {2} + \lambda_ {3} ^ {2}) \left(\frac {\partial U}{\partial \overline {{I}} _ {1}} + \lambda_ {1} ^ {2} \frac {\partial U}{\partial \overline {{I}} _ {2}}\right) + 4 (\lambda_ {2} ^ {2} - \lambda_ {3} ^ {2}) ^ {2} \left(\frac {\partial^ {2} U}{\partial \overline {{I}} _ {1} ^ {2}} + 2 \lambda_ {1} ^ {2} \frac {\partial^ {2} U}{\partial \overline {{I}} _ {1} \partial \overline {{I}} _ {2}} + \lambda_ {1} ^ {4} \frac {\partial^ {2} U}{\partial \overline {{I}} _ {2} ^ {2}}\right),
D _ {1 2} = D _ {2 1} = 4 \lambda_ {3} ^ {2} \frac {\partial U}{\partial \overline {{I}} _ {1}} + 4 \lambda_ {3} ^ {- 2} \frac {\partial U}{\partial \overline {{I}} _ {2}} + 4 (\lambda_ {1} ^ {2} - \lambda_ {3} ^ {2}) (\lambda_ {2} ^ {2} - \lambda_ {3} ^ {2}) \left(\frac {\partial^ {2} U}{\partial \overline {{I}} _ {1} ^ {2}} + (\lambda_ {1} ^ {2} + \lambda_ {2} ^ {2}) \frac {\partial^ {2} U}{\partial \overline {{I}} _ {1} \partial \overline {{I}} _ {2}} + \lambda_ {1} ^ {2} \lambda_ {2} ^ {2} \frac {\partial^ {2} U}{\partial \overline {{I}} _ {2} ^ {2}}\right).
For material stability D must be positive definite; thus, it is necessary that
D _ {1 1} + D _ {2 2} > 0,
D _ {1 1} D _ {2 2} - D _ {1 2} D _ {2 1} > 0,
for all relevant values of \lambda _ { 1 } , \lambda _ { 2 } , and \lambda _ { 3 } .
Ogden form
For the Ogden form we follow the same approach as the polynomial form. Using \lambda _ { 3 } = \lambda _ { 1 } ^ { - 1 } \lambda _ { 2 } ^ { - 1 } , we have
U = \sum_ {i = 1} ^ {N} \frac {2 \mu_ {i}}{\alpha_ {i} ^ {2}} (\lambda_ {1} ^ {\alpha_ {i}} + \lambda_ {2} ^ {\alpha_ {i}} + \lambda_ {1} ^ {- \alpha_ {i}} \lambda_ {2} ^ {- \alpha_ {i}} - 3),
\sigma_ {1} = \lambda_ {1} \frac {\partial U}{\partial \lambda_ {1}} = \sum_ {i = 1} ^ {N} \frac {2 \mu_ {i}}{\alpha_ {i}} (\lambda_ {1} ^ {\alpha_ {i}} - \lambda_ {1} ^ {- \alpha_ {i}} \lambda_ {2} ^ {- \alpha_ {i}}),
\sigma_ {2} = \lambda_ {2} \frac {\partial U}{\partial \lambda_ {2}} = \sum_ {i = 1} ^ {N} \frac {2 \mu_ {i}}{\alpha_ {i}} (\lambda_ {2} ^ {\alpha_ {i}} - \lambda_ {1} ^ {- \alpha_ {i}} \lambda_ {2} ^ {- \alpha_ {i}}),
and the material stiffness D that we check for positive definiteness is
\left[ \begin{array}{c c} D _ {1 1} & D _ {1 2} \\ D _ {2 1} & D _ {2 2} \end{array} \right] = \sum_ {i = 1} ^ {N} 2 \mu_ {i} \lambda_ {1} ^ {- \alpha_ {i}} \lambda_ {2} ^ {- \alpha_ {i}} \left[ \begin{array}{c c} \lambda_ {1} ^ {2 \alpha_ {i}} \lambda_ {2} ^ {\alpha_ {i}} + 1 & 1 \\ 1 & \lambda_ {1} ^ {\alpha_ {i}} \lambda_ {2} ^ {2 \alpha_ {i}} + 1 \end{array} \right].
Arruda-Boyce form
For positive values of the shear modulus, \mu , and the locking stretch, \lambda _ { m } , the Arruda-Boyce form is always stable. Hence, it suffices to check the coefficients to determine whether the material satisfies Drucker stability.
Van der Waals form
When the Van der Waals model is employed in its admissible stretch range given by \tilde { I } < \lambda _ { m } ^ { 2 } , its stability depends on the global interaction parameter, a, for positive values of the initial shear modulus, \mu , and the locking stretch, \lambda _ { m } . To verify the Drucker stability of the Van der Waals model, we can employ the equations derived for the polynomial models by making use of the fact that
\frac {\partial U}{\partial \overline {{I}} _ {1}} = (1 - \beta) \frac {\partial U}{\partial \tilde {I}} \quad \mathrm{and} \quad \frac {\partial U}{\partial \overline {{I}} _ {2}} = \beta \frac {\partial U}{\partial \tilde {I}}.
To determine the admissible stretch range, we need to find the two positive real-valued roots neighboring ¸ = 1 of the equation
\tilde {I} (\lambda) - \lambda_ {m} ^ {2} = 0
for each of the three stress states--uniaxial, biaxial, and planar--by using a simple bisection method.
Hyperfoam
The Kirchhoff stress-strain relation for the uniaxial, biaxial, planar, and volumetric deformation modes is
\tau_ {j} = \lambda_ {j} \frac {\partial U}{\partial \lambda_ {j}} = 2 \sum_ {i = 1} ^ {N} \frac {\mu_ {i}}{\alpha_ {i}} \left[ \lambda_ {j} ^ {\alpha_ {i}} - J ^ {- \alpha_ {i} \beta_ {i}} \right].
Taking the total differential of \tau _ { j } and using d \varepsilon _ { j } = d \lambda _ { j } / \lambda _ { j } ,
d \tau_ {j} = 2 \sum_ {i = 1} ^ {N} \mu_ {i} \bigg [ \lambda_ {j} ^ {\alpha_ {i}} d \varepsilon_ {i} + \beta_ {i} J ^ {- \alpha_ {i} \beta_ {i}} \left(d \varepsilon_ {1} + d \varepsilon_ {2} + d \varepsilon_ {3}\right) \bigg ], \quad j = 1, 2, 3.
Since we cannot use the incompressibility assumption, we have to use all three principal stress and strain components and a 3 £ 3 D matrix,
\left\{ \begin{array}{l} d \tau_ {1} \\ d \tau_ {2} \\ d \tau_ {3} \end{array} \right\} = \left[ \begin{array}{l l l} D _ {1 1} & D _ {1 2} & D _ {1 3} \\ D _ {2 1} & D _ {2 2} & D _ {2 3} \\ D _ {3 1} & D _ {3 2} & D _ {3 3} \end{array} \right] \left\{ \begin{array}{l} d \varepsilon_ {1} \\ d \varepsilon_ {2} \\ d \varepsilon_ {3} \end{array} \right\}.
Specifically,
Mechanical Constitutive Theories
\left\{ \begin{array}{l} d \tau_ {1} \\ d \tau_ {2} \\ d \tau_ {3} \end{array} \right\} = 2 \sum_ {i = 1} ^ {N} \mu_ {i} \left[ \begin{array}{c c c c c} \lambda_ {1} ^ {\alpha_ {i}} + A _ {i} & A _ {i} & A _ {i i} & \lambda_ {2} ^ {\alpha_ {i}} + A _ {i} & A _ {i} \\ A _ {i} & A _ {i} & \lambda_ {3} ^ {\alpha_ {i}} + A _ {i} \end{array} \right] \left\{ \begin{array}{l} d \varepsilon_ {1} \\ d \varepsilon_ {2} \\ d \varepsilon_ {3} \end{array} \right\},
where A _ { i } = \beta _ { i } J ^ { - \alpha _ { i } \beta _ { i } } ,
For the simple shear case the principal stretches \lambda _ { 1 } and \lambda _ { 2 } are computed from the shear strain ° (as given in an earlier expression). Thus, the same form of equations is used in checking material stability during simple shear deformation.
For material stability (i.e., for D to be positive definite) the following conditions must be satisfied:
D _ {1 1} + D _ {2 2} + D _ {3 3} > 0,
D _ {1 1} D _ {2 2} + D _ {2 2} D _ {3 3} + D _ {3 3} D _ {1 1} - D _ {2 3} ^ {2} - D _ {1 3} ^ {2} - D _ {1 2} ^ {2} > 0,
det \mathbf { \tau } ( \mathbf { D } ) > 0 .
4.7 Viscoelasticity
4.7.1 Viscoelasticity
The basic hereditary integral formulation for linear isotropic viscoelasticity is
\pmb {\sigma} (t) = \int_ {0} ^ {t} 2 G (\tau - \tau^ {\prime}) \dot {\bf e} d t ^ {\prime} + {\bf I} \int_ {0} ^ {t} K (\tau - \tau^ {\prime}) \dot {\phi} d t ^ {\prime}.
Here e and \phi are the mechanical deviatoric and volumetric strains; K is the bulk modulus and G is the shear modulus, which are functions of the reduced time \tau ; and _ denotes differentiation with respect to t ^ { \prime } . .
The reduced time is related to the actual time through the integral differential equation
\tau = \int_ {0} ^ {t} \frac {d t ^ {\prime}}{A _ {\theta} (\theta (t ^ {\prime}))}, \quad \frac {d \tau}{d t} = \frac {1}{A _ {\theta} (\theta (t))},
where \theta is the temperature and A _ { \theta } is the shift function. (Hence, if A _ { \theta } = 1 , \tau = t . ) A commonly used shift function is the Williams-Landell-Ferry (WLF) equation, which has the following form:
- \log A _ {\theta} = h (\theta) = \frac {C _ {1} ^ {g} (\theta - \theta_ {g})}{C _ {2} ^ {g} + (\theta - \theta_ {g})},
where C _ { 1 } ^ { g } and C _ { 2 } ^ { g } are constants and \theta _ { g } is the "glass" transition temperature. This is the temperature at which, in principle, the behavior of the material changes from glassy to rubbery. If \theta \leq \theta _ { g } - C _ { 2 } ^ { g } ,
deformation changes will be elastic. C _ { 1 } ^ { g } and C _ { 2 } ^ { g } were once thought to be "universal" constants whose values were obtained at \theta _ { g } , but these constants have been shown to vary slightly from polymer to polymer.
ABAQUS allows the WLF equation to be used with any convenient temperature, other than the glass transition temperature, as the reference temperature. The form of the equation remains the same, but the constants are different. Namely,
- \log A _ {\theta} = h (\theta) = \frac {C _ {1} (\theta - \theta_ {0})}{C _ {2} + (\theta - \theta_ {0})},
where \theta _ { 0 } is the reference temperature at which the relaxation data are given, and C _ { 1 } and C _ { 2 } are the calibration constants at the reference temperature. The "universal" constants C _ { 1 } ^ { g } and C _ { 2 } ^ { g } are related to C _ { 1 } and C _ { 2 } as follows:
C _ {1} = \frac {C _ {1} ^ {g}}{1 + (\theta_ {0} - \theta_ {g}) / C _ {2} ^ {g}},
C _ {2} = C _ {2} ^ {g} + \theta_ {0} - \theta_ {g}.
Other forms of h ( \theta ) are also used, such as a power series in \theta - \theta _ { 0 } . ABAQUS allows a general definition of the shift function with user subroutine UTRS.
The relaxation functions K ( t ) and G ( t ) can be defined individually in terms of a series of exponentials known as the Prony series:
K (\tau) = K _ {\infty} + \sum_ {i = 1} ^ {n _ {K}} K _ {i} e ^ {- \tau / \tau_ {i} ^ {K}} \qquad G (\tau) = G _ {\infty} + \sum_ {i = 1} ^ {n _ {G}} G _ {i} e ^ {- \tau / \tau_ {i} ^ {G}},
where K _ { \infty } and G _ { \infty } represent the long-term bulk and shear moduli. In general, the relaxation times \tau _ { i } ^ { K } and \tau _ { i } ^ { G } need not equal each other; however, ABAQUS assumes that \tau _ { i } = \tau _ { i } ^ { K } = \tau _ { i } ^ { G } . On the other hand, the number of terms in bulk and shear, n _ { K } and n _ { G } , need not equal each other. In fact, in many practical cases it can be assumed that n _ { K } = 0 . Hence, we now concentrate on the deviatoric behavior. The equations for the volumetric terms can be derived in an analogous way.
The deviatoric integral equation is
\begin{array}{l} \mathbf {S} = \int_ {0} ^ {t} 2 \left(G _ {\infty} + \sum_ {i = 1} ^ {n _ {G}} G _ {i} e ^ {(\tau^ {\prime} - \tau) / \tau_ {i}}\right) \dot {\mathbf {e}} d t ^ {\prime} \\ = \int_ {0} ^ {\tau} 2 \left(G _ {\infty} + \sum_ {i = 1} ^ {n _ {G}} G _ {i} e ^ {(\tau^ {\prime} - \tau) / \tau_ {i}}\right) \frac {d \mathbf {e}}{d \tau^ {\prime}} d \tau^ {\prime}. \\ \end{array}
We rewrite this equation in the form
Equation 4.7.1-1