김경종
29 KiB
The Cauchy ("true") stress components are defined from the strain energy potential as follows. From the virtual work principal the internal energy variation is
\delta W _ {I} = \int_ {V} \pmb {\sigma}: \delta \mathbf {D} d V = \int_ {V ^ {0}} J \pmb {\sigma}: \delta \mathbf {D} d V ^ {0},
where ¾ are the components of the Cauchy ("true") stress, V is the current volume, and V ^ { 0 } is the reference volume.
We decompose the stress into the equivalent pressure stress,
p \stackrel {\mathrm{def}} {=} - \frac {1}{3} \mathbf {I}: \pmb {\sigma},
and the deviatoric stress,
\mathbf {S} \stackrel {\mathrm{def}} {=} \pmb {\sigma} + p \mathbf {I},
so that the internal energy variation can be written
\delta W _ {I} = \int_ {V ^ {0}} J (\mathbf {S}: \delta \mathbf {e} - p \delta \varepsilon^ {\mathrm{vol}}) d V ^ {0}.
For isotropic, compressible materials the strain energy, U , is a function of \overline { { I } } _ { 1 } , \overline { { I } } _ { 2 } , and J :
U = U (\overline {{I}} _ {1}, \overline {{I}} _ {2}, J),
so that
\delta U = \frac {\partial U}{\partial \overline {{I}} _ {1}} \delta \overline {{I}} _ {1} + \frac {\partial U}{\partial \overline {{I}} _ {2}} \delta \overline {{I}} _ {2} + \frac {\partial U}{\partial J} \delta J.
Hence, using Equation 4.6.1-3, Equation 4.6.1-4, and Equation 4.6.1-5,
Equation 4.6.1-6
\delta U = 2 \left[ \left(\frac {\partial U}{\partial \overline {{{I}}} _ {1}} + \overline {{{I}}} _ {1} \frac {\partial U}{\partial \overline {{{I}}} _ {2}}\right) \overline {{{\mathbf {B}}}} - \frac {\partial U}{\partial \overline {{{I}}} _ {2}} \overline {{{\mathbf {B}}}} \cdot \overline {{{\mathbf {B}}}} \right]: \delta \mathbf {e} + J \frac {\partial U}{\partial J} \delta \varepsilon^ {\mathrm{vol}}.
Since the variation of the strain energy potential is, by definition, the internal virtual work per reference volume, ±WI , we have
Equation 4.6.1-7
\delta W _ {I} = \int_ {V ^ {0}} J (\mathbf {S}: \delta \mathbf {e} - p \delta \varepsilon^ {\mathrm{vol}}) d V ^ {0} = \int_ {V ^ {0}} \delta U d V ^ {0}.
For a compressible material the strain variations are arbitrary, so this equation defines the stress components for such a material as
\mathbf {S} = \frac {2}{J} \mathrm{DEV} \left[ \left(\frac {\partial U}{\partial \overline {{I}} _ {1}} + \overline {{I}} _ {1} \frac {\partial U}{\partial \overline {{I}} _ {2}}\right) \overline {{\mathbf {B}}} - \frac {\partial U}{\partial \overline {{I}} _ {2}} \overline {{\mathbf {B}}} \cdot \overline {{\mathbf {B}}} \right]
Equation 4.6.1-8
and
Equation 4.6.1-9
p = - \frac {\partial U}{\partial J}.
When the material response is almost incompressible, the pure displacement formulation, in which the strain invariants are computed from the kinematic variables of the finite element model, can behave poorly. One difficulty is that from a numerical point of view the stiffness matrix is almost singular because the effective bulk modulus of the material is so large compared to its effective shear modulus, thus causing difficulties with the solution of the discretized equilibrium equations. Similarly, in ABAQUS/Explicit the high bulk modulus increases the dilatational wave speed, thus reducing the stable time increment substantially. Another problem is that, unless reduced-integration techniques are used, the stresses calculated at the numerical integration points show large oscillations in the pressure stress values, because--in general--the elements cannot respond accurately and still have small volume changes at all numerical integration points. To avoid such problems, ABAQUS/Standard offers a "mixed" formulation for such cases. The concept is to introduce a variable, { \hat { J } } , that is used in place of the volume change, J , in the definition of the strain energy potential. The internal energy integral, W _ { I } , is augmented with the constraint that J - \hat { J } = 0 , imposed by the use of a Lagrange multiplier, \hat { p } , and integrated over the volume:
(W _ {I}) ^ {A} \stackrel {\mathrm{def}} {=} \int_ {V ^ {0}} \left[ U (\overline {{I}} _ {1}, \overline {{I}} _ {2}, \hat {J}) - \hat {p} (J - \hat {J}) \right] d V ^ {0}.
Taking the variation of this definition,
\delta (W _ {I}) ^ {A} = \int_ {V ^ {0}} \left[ J \mathbf {S}: \delta \mathbf {e} + \left(\frac {\partial U}{\partial \hat {J}} + \hat {p}\right) \delta \hat {J} - J \hat {p} \delta \varepsilon^ {\mathrm{vol}} - (J - \hat {J}) \delta \hat {p} \right] d V ^ {0}.
Since \delta \hat { J } is an independent variation in this expression, the Lagrange multiplier is
\hat {p} = - \frac {\partial U}{\partial \hat {J}},
and its variation is
\delta \hat {p} = - \hat {\mathbf {Q}}: \delta \mathbf {e} + \frac {\partial \hat {p}}{\partial \hat {J}} \delta \hat {J},
where
\hat {\mathbf {Q}} \stackrel {\mathrm{def}} {=} J \frac {\partial \mathbf {S}}{\partial \hat {J}}.
These results allow us to write the augmented internal energy variation as
Equation 4.6.1-10
\delta (W _ {I}) ^ {A} = \int_ {V ^ {0}} \left[ \left(J \mathbf {S} + (J - \hat {J}) \hat {\mathbf {Q}}\right): \delta \mathbf {e} - J \hat {p} \delta \varepsilon^ {\mathrm{vol}} - (J - \hat {J}) \frac {\partial \hat {p}}{\partial \hat {J}} \delta \hat {J} \right] d V ^ {0}.
This augmented formulation can be used for any value of compressibility except fully incompressible behavior. \hat { J } is interpolated independently in each element: ABAQUS uses constant \hat { J } in first-order elements and linear variation of \hat { J } with respect to position in second-order elements, which implies that \hat { J } is discontinuous between elements: continuity in such variables is not a requirement.
When the material is fully incompressible, U is a function of the first and second strain invariants-- \overline { { I } } _ { 1 } and \overline { { I } } _ { 2 } \mathrm { - } \mathsf { o n l y } , and we write the internal energy in the augmented form,
(W _ {I}) ^ {A} \stackrel {\mathrm{def}} {=} \int_ {V ^ {0}} \left[ U - \hat {p} (J - 1) \right] d V ^ {0},
where \hat { p } is again a Lagrange multiplier introduced to impose the constraint J - 1 = 0 in such a way that the variation of ( W _ { I } ) ^ { A } can be taken with respect to all kinematic variables, thus giving
Equation 4.6.1-11
\delta (W _ {I}) ^ {A} = \int_ {V ^ {0}} \left[ J \mathbf {S}: \delta \mathbf {e} - J \hat {p} \delta \varepsilon^ {\mathrm{vol}} - (J - 1) \delta \hat {p} \right] d V ^ {0}.
\hat { p } is interpolated in the same way as \hat { J } is interpolated in the augmented formulation for almost incompressible behavior; that is, \hat { p } is assumed to be constant in first-order elements and to vary linearly with respect to position in second-order elements.
Rate of change of the internal virtual work
The rate of change of the internal virtual work is required for use in the Newton method, which is generally used in ABAQUS/Standard to solve the nonlinear equilibrium equations (after discretization by finite elements). It will also be used when we extend the elasticity model to viscoelastic behavior for small (linearized) vibrations about a predeformed state.
When the pure displacement formulation is used for the compressible case, the deviatoric stress components, S, are defined by Equation 4.6.1-8, from which we can show that
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}),
where the "effective deviatoric elasticity" of the material, \mathbf { C } ^ { S } , is defined as
\begin{array}{l} \mathbf {C} ^ {S} \stackrel {\mathrm{def}} {=} \frac {2}{J} \left(\frac {\partial U}{\partial \overline {{I}} _ {1}} + \overline {{I}} _ {1} \frac {\partial U}{\partial \overline {{I}} _ {2}}\right) \mathbf {H} _ {1} - \frac {2}{J} \frac {\partial U}{\partial \overline {{I}} _ {2}} \mathbf {H} _ {2} + \frac {4}{J} \left(\frac {\partial^ {2} U}{\partial \overline {{I}} _ {1} ^ {2}} + \frac {\partial U}{\partial \overline {{I}} _ {2}} + 2 \overline {{I}} _ {1} \frac {\partial^ {2} U}{\partial \overline {{I}} _ {1} \partial \overline {{I}} _ {2}} + \overline {{I}} _ {1} ^ {2} \frac {\partial^ {2} U}{\partial \overline {{I}} _ {2} ^ {2}}\right) \overline {{\mathbf {B}}} \overline {{\mathbf {B}}} \\ - \frac {4}{J} \left(\frac {\partial^ {2} U}{\partial \overline {{I}} _ {1} \partial \overline {{I}} _ {2}} + \overline {{I}} _ {1} \frac {\partial^ {2} U}{\partial \overline {{I}} _ {2} ^ {2}}\right) \left(\overline {{\mathbf {B}}} \cdot \overline {{\mathbf {B}}} \overline {{\mathbf {B}}} + \overline {{\mathbf {B}}} \overline {{\mathbf {B}}} \cdot \overline {{\mathbf {B}}}\right) + \frac {4}{J} \frac {\partial^ {2} U}{\partial \overline {{I}} _ {2} ^ {2}} \overline {{\mathbf {B}}} \cdot \overline {{\mathbf {B}}} \overline {{\mathbf {B}}} \cdot \overline {{\mathbf {B}}} \\ - \frac {4}{3 J} \left[ \frac {\partial U}{\partial \overline {{I}} _ {1}} + 2 \overline {{I}} _ {1} \frac {\partial U}{\partial \overline {{I}} _ {2}} + \overline {{I}} _ {1} \frac {\partial^ {2} U}{\partial \overline {{I}} _ {1} ^ {2}} + \left(\overline {{I}} _ {1} ^ {2} + 2 \overline {{I}} _ {2}\right) \frac {\partial^ {2} U}{\partial \overline {{I}} _ {1} \partial \overline {{I}} _ {2}} + 2 \overline {{I}} _ {1} \overline {{I}} _ {2} \frac {\partial^ {2} U}{\partial \overline {{I}} _ {2} ^ {2}} \right] \left(\mathbf {I} \overline {{\mathbf {B}}} + \overline {{\mathbf {B}}} \mathbf {I}\right) \\ + \frac {4}{3 J} \left(2 \frac {\partial U}{\partial \overline {{I}} _ {2}} + \overline {{I}} _ {1} \frac {\partial^ {2} U}{\partial \overline {{I}} _ {1} \partial \overline {{I}} _ {2}} + 2 \overline {{I}} _ {2} \frac {\partial^ {2} U}{\partial \overline {{I}} _ {2} ^ {2}}\right) \left(\mathbf {I} \overline {{\mathbf {B}}} \cdot \overline {{\mathbf {B}}} + \overline {{\mathbf {B}}} \cdot \overline {{\mathbf {B}}} \mathbf {I}\right), \\ \end{array}
and the deviatoric stress rate-volumetric strain rate coupling term, Q, is
\mathbf {Q} \stackrel {\mathrm{def}} {=} \frac {\partial (J \mathbf {S})}{\partial J} = 2 \left(\frac {\partial^ {2} U}{\partial \overline {{I}} _ {1} \partial J} + \overline {{I}} _ {1} \frac {\partial^ {2} U}{\partial \overline {{I}} _ {2} \partial J}\right) \overline {{\mathbf {B}}} - 2 \frac {\partial^ {2} U}{\partial \overline {{I}} _ {2} \partial J} \overline {{\mathbf {B}}} \cdot \overline {{\mathbf {B}}} - \frac {2}{3} \left(\overline {{I}} _ {1} \frac {\partial^ {2} U}{\partial \overline {{I}} _ {1} \partial J} + 2 \overline {{I}} _ {2} \frac {\partial^ {2} U}{\partial \overline {{I}} _ {2} \partial J}\right) \mathbf {I}.
From Equation 4.6.1-9 it can be shown that
d (J p) = - J (\mathbf {Q}: d \mathbf {e} + K d \varepsilon^ {\mathrm{vol}}),
where K is the effective bulk modulus of the material,
K \stackrel {\mathrm{def}} {=} - \left(J \frac {\partial p}{\partial J} + p\right) = J \frac {\partial^ {2} U}{\partial J ^ {2}} + \frac {\partial U}{\partial J}.
Thus,
Equation 4.6.1-12
d \delta W _ {I} = \int_ {V} \left[ \left\lfloor \delta \mathbf {e} \quad \delta \varepsilon^ {\mathrm{vol}} \right\rfloor : \left[ \begin{array}{c c} \mathbf {C} ^ {S} & \mathbf {Q} \\ \mathbf {Q} & K \end{array} \right]: \left\{ \begin{array}{c} d \mathbf {e} \\ d \varepsilon^ {\mathrm{vol}} \end{array} \right\} - \pmb {\sigma}: (2 \delta \pmb {\varepsilon} \cdot d \pmb {\varepsilon} - \delta \mathbf {L} ^ {T} \cdot d \mathbf {L}) \right] d V,
since
\delta \mathbf {e}: (d \pmb {\Omega} \cdot \mathbf {S} - \mathbf {S} \cdot d \pmb {\Omega}) + \mathbf {S}: d \delta \mathbf {e} - p d \delta \varepsilon^ {\mathrm{vol}} = - \pmb {\sigma}: (2 \delta \pmb {\varepsilon} \cdot d \pmb {\varepsilon} - \delta \mathbf {L} ^ {T} \cdot d \mathbf {L}).
For the mixed formulation introduced for almost incompressible materials, the rate of change of the augmented variation of internal energy, Equation 4.6.1-10, is
\begin{array}{l} d \delta (W _ {I}) ^ {A} = \int_ {V} \left[ \begin{array}{c c c} \lfloor \delta \mathbf {e} & \delta \varepsilon^ {\mathrm{vol}} & \delta \hat {J} \rfloor \left[ \begin{array}{c c c} \mathbf {\ddot {C}} ^ {S} & \mathbf {\ddot {Q}} & \mathbf {\ddot {A}} \\ \mathbf {\dot {Q}} & - \hat {p} & - \partial \hat {p} / \partial \hat {J} \\ \mathbf {\hat {A}} & - \partial \hat {p} / \partial \hat {J} & - \tilde {K} \end{array} \right] & \left\{ \begin{array}{c} d \mathbf {e} \\ d \varepsilon^ {\mathrm{vol}} \\ d \hat {J} \end{array} \right\} \end{array} \right] \\ \left. - \left(\pmb {\sigma} + \frac {1}{J} (J - \hat {J}) \hat {\mathbf {Q}}\right): (2 \delta \pmb {\varepsilon} \cdot d \pmb {\varepsilon} - \delta \mathbf {L} ^ {T} \cdot d \mathbf {L}) \right] d V, \\ \end{array}
where
\tilde {\mathbf {C}} ^ {S} \stackrel {\mathrm{def}} {=} \mathbf {C} ^ {S} + (J - \hat {J}) \frac {\partial \mathbf {C} ^ {S}}{\partial \hat {J}},
\hat {\mathbf {Q}} \stackrel {\mathrm{def}} {=} J \frac {\partial \mathbf {S}}{\partial \hat {J}},
\hat {\mathbf {A}} \stackrel {\mathrm{def}} {=} \frac {1}{J} (J - \hat {J}) \frac {\partial \hat {\mathbf {Q}}}{\partial \hat {J}},
and
\tilde {K} \stackrel {\mathrm{def}} {=} \frac {1}{J ^ {2}} \left(\hat {K} - (J - \hat {J}) \frac {\partial \hat {K}}{\partial \hat {J}}\right),
in which
\hat {K} \stackrel {\mathrm{def}} {=} - J \frac {\partial \hat {p}}{\partial \hat {J}}.
For the case of incompressible materials the rate of change of the augmented variation of internal energy is similarly obtained from Equation 4.6.1-11 as
\begin{array}{l} d \delta (W _ {I}) ^ {A} = \int_ {V} \left[ \begin{array}{l l l} \lfloor \delta \mathbf {e} & \delta \varepsilon^ {\mathrm{vol}} & \delta \hat {p} \rfloor \left[ \begin{array}{l l l} \mathbf {C} ^ {S} & 0 & 0 \\ 0 & - \hat {p} & - 1 \\ 0 & - 1 & 0 \end{array} \right] & \left\{ \begin{array}{l} d \mathbf {e} \\ d \varepsilon^ {\mathrm{vol}} \\ d \hat {p} \end{array} \right\} \end{array} \right] \\ \left. - \pmb {\sigma}: (2 \delta \pmb {\varepsilon} \cdot d \pmb {\varepsilon} - \delta \mathbf {L} ^ {T} \cdot d \mathbf {L}) \right] d V. \\ \end{array}
Particular forms of the strain energy potential
Several particular forms of the strain energy potential are available in ABAQUS. The incompressible or almost incompressible models make up:
- the polynomial form and its particular cases--the reduced polynomial form, the neo-Hookean form,
the Mooney-Rivlin form, and the Yeoh form;
- the Ogden form;
- the Arruda-Boyce form; and
- the Van der Waals form.
In addition, a hyperelastic model for highly compressible, elastic materials is offered.
Polynomial form and particular cases
Given isotropy and additive decomposition of the deviatoric and volumetric strain energy contributions in the presence of incompressible or almost incompressible behavior, we can write the potential, in general, as
U = f (\overline {{I}} _ {1} - 3, \overline {{I}} _ {2} - 3) + g (J _ {e \ell} - 1).
Setting \begin{array} { r } { g = \sum _ { i = 1 } ^ { N } \frac { 1 } { D _ { i } } ( J _ { e \ell } - 1 ) ^ { 2 i } } \end{array} and expanding f ( \overline { { I } } _ { 1 } - 3 , \overline { { I } } _ { 2 } - 3 ) in a Taylor series, we arrive at
Equation 4.6.1-13
U \stackrel {\mathrm{def}} {=} \sum_ {i + j = 1} ^ {N} C _ {i j} (\overline {{I}} _ {1} - 3) ^ {i} (\overline {{I}} _ {2} - 3) ^ {j} + \sum_ {i = 1} ^ {N} \frac {1}{D _ {i}} (J _ {e \ell} - 1) ^ {2 i}.
This form is the polynomial representation of the strain energy in ABAQUS. The parameter N can take values up to six; however, values of N greater than 2 are rarely used when both the first and second invariants are taken into account. C _ { i j } and D _ { i } are temperature-dependent material parameters. The value of N and tables giving the C _ { i j } and D _ { i } values as functions of temperature are defined on the *HYPERELASTIC material option if the POLYNOMIAL parameter is chosen. The elastic volume strain, J _ { e \ell ; \ell } , follows from the total volume strain, J , and the thermal volume strain, J _ { t h } , with the relation
J _ {e \ell} = \frac {J}{J _ {t h}},
and J _ { t h } follows from the linear thermal expansion, \varepsilon _ { t h } , with
J _ {t h} = (1 + \varepsilon_ {t h}) ^ {3},
where \varepsilon _ { t h } follows from the temperature and the isotropic thermal expansion coefficient defined in the *EXPANSION material option.
The D _ { i } values determine the compressibility of the material: if all the D _ { i } are zero, the material is taken as fully incompressible. If D _ { 1 } = 0 , all D _ { i } must be zero.
Regardless of the value of N , the initial shear modulus, \mu _ { 0 } ; and the bulk modulus, k _ { 0 } ; depend only on the polynomial coefficients of order N = 1:
\mu_ {0} = 2 (C _ {1 0} + C _ {0 1}), \qquad k _ {0} = \frac {2}{D _ {1}}.
If N = 1, so that only the linear terms in the deviatoric strain energy are retained, the Mooney-Rivlin form is recovered:
U = C _ {1 0} (\overline {{I}} _ {1} - 3) + C _ {0 1} (\overline {{I}} _ {2} - 3) + \frac {1}{D _ {1}} (J _ {e \ell} - 1) ^ {2}.
The Mooney-Rivlin form can be viewed as an extension of the neo-Hookean form (discussed below) in that it adds a term that depends on the second invariant of the left Cauchy-Green tensor. In some cases this form will give a more accurate fit to the experimental data than the neo-Hookean form; in general, however, both models give similar accuracy since they use only linear functions of the invariants. These functions do not allow representation of the "upturn" at higher strain levels in the stress-strain curve.
Particular forms of the polynomial model can also be obtained by setting specific coefficients to zero. If all C _ { i j } with j \neq 0 are set to zero, the reduced polynomial form is obtained:
U = \sum_ {i = 1} ^ {N} C _ {i 0} (\overline {{I}} _ {1} - 3) ^ {i} + \sum_ {i = 1} ^ {N} \frac {1}{D _ {i}} (J _ {e \ell} - 1) ^ {2 i}.
Following Yeoh (1993) the justification for reducing the general polynomial series expansion by omitting the dependence on the second invariant arises from the following observations. The sensitivity of the strain energy function to changes in the second invariant is generally much smaller than the sensitivity to changes in the first invariant. In addition, the \overline { { I } } _ { 2 } -dependence is difficult to measure, so it might be preferable to neglect it rather than to calculate it based on potentially inaccurate measurements. Finally, it appears that omitting the dependence on the second invariant if only data for a particular mode of deformation are known might enhance the prediction for other deformation states. This conjecture is supported by investigating the so-called reduced stresses in the presence of almost incompressible behavior:
\begin{array}{l} \frac {\sigma_ {1} - \sigma_ {2}}{\lambda_ {1} ^ {2} - \lambda_ {2} ^ {2}} = 2 \left(\frac {\partial U}{\partial \overline {{I}} _ {1}} + \lambda_ {3} ^ {2} \frac {\partial U}{\partial \overline {{I}} _ {2}}\right), \\ \frac {\sigma_ {1} - \sigma_ {3}}{\lambda_ {1} ^ {2} - \lambda_ {3} ^ {2}} = 2 \left(\frac {\partial U}{\partial \overline {{I}} _ {1}} + \lambda_ {2} ^ {2} \frac {\partial U}{\partial \overline {{I}} _ {2}}\right), \\ \frac {\sigma_ {2} - \sigma_ {3}}{\lambda_ {2} ^ {2} - \lambda_ {3} ^ {2}} = 2 \left(\frac {\partial U}{\partial \overline {{I}} _ {1}} + \lambda_ {1} ^ {2} \frac {\partial U}{\partial \overline {{I}} _ {2}}\right), \\ \end{array}
where the \sigma _ { i } , i = 1 \ldots 3 represent the principal Cauchy ( " \mathrm { t r u e " } ) stresses. If the derivatives with respect to \overline { { I } } _ { 2 } are omitted and different stress states--uniaxial, biaxial, and planar--are considered, the reduced stresses have the same form regardless of the stress state.
Measurements of the \overline { { I } } _ { 2 } { \mathrm { - d e p e n d e n c e } } of carbon-black reinforced rubber vulcanizates confirming these
findings can be found in Kawabata, Yamashita, et al. (1995). The paper of Kaliske and Rothert (1997) also supports the notion that often the prediction of general deformation states based on a uniaxial measurement can be enhanced only by ignoring the \overline { { I } } _ { 2 } { \mathrm { - d e p e n d e n c e } } . .
In this context it is worth noting that the mathematical structure of the Arruda-Boyce model can be viewed as a fifth-order reduced polynomial, where the five coefficients C _ { 1 0 } \ldots C _ { 5 0 } are implicit nonlinear functions of the two parameters \mu and \lambda _ { m } in the Arruda-Boyce form. However, the Arruda-Boyce model offers a physical interpretation of the parameters, which the general fifth-order reduced polynomial fails to provide.
The Yeoh form (Yeoh, 1993) can be viewed as a special case of the reduced polynomial with N = 3 \cdot :
U = \sum_ {i = 1} ^ {3} C _ {i 0} (\overline {{I}} _ {1} - 3) ^ {i} + \sum_ {i = 1} ^ {3} \frac {1}{D _ {i}} (J _ {e \ell} - 1) ^ {2 i}.
Typically, if C _ { 1 0 } = O ( 1 ) , the second coefficient will be negative and one to two orders of magnitude smaller \mathrm { [ i . e . , } C _ { 2 0 } \mathrm { i s } - O ( 0 . 1 ) \mathrm { t o } - O ( 0 . 0 1 ) \mathrm { l } , while the third coefficient C _ { 3 0 } is again one to two orders of magnitude smaller but positive [i.e., C _ { 3 0 } ~ \mathrm { i s } + O ( 1 . E - 2 ) ~ \mathrm { t o } + O ( 1 . E - 4 ) ] . These magnitudes will create the typical S-shape of the stress-strain behavior of rubber; at low strains C _ { 1 0 } represents the initial shear modulus, which softens at moderate strains due to the effect of the negative second coefficient C _ { 2 0 } and is followed by an upturn at large strains due to the positive third coefficient C _ { 3 0 } . Thus, the Yeoh model often provides an accurate fit over a large strain range.
If the reduced-polynomial strain-energy function is simplified further by setting N = 1 , the neo-Hookean form is obtained:
U = C _ {1 0} (\overline {{I}} _ {1} - 3) + \frac {1}{D _ {1}} (J _ {e \ell} - 1) ^ {2}.
This form is the simplest hyperelastic model and often serves as a prototype for elastomeric materials in the absence of accurate material data. It also has some theoretical relevance since the mathematical representation is analogous to that of an ideal gas: the neo-Hookean potential represents the Helmholtz free energy of a molecular network with Gaussian chain-length distribution (see Treloar, 1975).
The user can request that ABAQUS calculate the C _ { i j } and D _ { i } values from measurements of nominal stress and strain in simple experiments. The basis of this calculation is described in ``Fitting of hyperelastic and hyperfoam constants,'' Section 4.6.2, and ``Hyperelastic behavior,'' Section 10.5.1 of the ABAQUS/Standard User's Manual and Section 9.3.1 of the ABAQUS/Explicit User's Manual.
Ogden form
The Ogden strain energy potential is expressed in terms of the principal stretches. In ABAQUS the following formulation is used:
U \stackrel {\mathrm{def}} {=} \sum_ {i = 1} ^ {N} \frac {2 \mu_ {i}}{\alpha_ {i} ^ {2}} (\overline {{\lambda}} _ {1} ^ {\alpha_ {i}} + \overline {{\lambda}} _ {2} ^ {\alpha_ {i}} + \overline {{\lambda}} _ {3} ^ {\alpha_ {i}} - 3) + \sum_ {i = 1} ^ {N} \frac {1}{D _ {i}} (J _ {e \ell} - 1) ^ {2 i},
Equation 4.6.1-14
where
\overline {{\lambda}} _ {i} = J ^ {- \frac {1}{3}} \lambda_ {i} \rightarrow \overline {{\lambda}} _ {1} \overline {{\lambda}} _ {2} \overline {{\lambda}} _ {3} = 1.
Hence, the first part of Ogden's strain energy function depends only on \overline { { I } } _ { 1 } and \overline { { I } } _ { 2 } . Ogden's energy function cannot be written explicitly in terms of \overline { { I } } _ { 1 } and \overline { { I } } _ { 2 } . It is, however, possible to obtain closed-form expressions for the derivatives of U with respect to \overline { { I } } _ { 1 } and \overline { { I } } _ { 2 } .
The value of N and tables giving the \mu _ { i } and \alpha _ { i } values as functions of temperature are defined in the *HYPERELASTIC material option if the OGDEN parameter is chosen. If N = 2 , \alpha _ { 1 } = 2 , and \alpha _ { 2 } = - 2 , the Mooney-Rivlin model is obtained. If N = 1 and \alpha _ { 1 } = 2 , Ogden's model degenerates to the neo-Hookean material model. In the Ogden form the initial shear modulus, \mu _ { 0 } , depends on all coefficients:
\mu_ {0} = \sum_ {i = 1} ^ {N} \mu_ {i},
and the initial bulk modulus, k _ { 0 } , depends on D _ { 1 } as before. The user can request that ABAQUS calculate the \mu _ { i } and \alpha _ { i } values from measurements of nominal stress and strain.
Arruda-Boyce form
The hyperelastic Arruda-Boyce potential has the following form:
U = \mu \sum_ {i = 1} ^ {5} \frac {C _ {i}}{\lambda_ {m} ^ {2 i - 2}} \left(\overline {{I}} _ {1} ^ {i} - 3 ^ {i}\right) + \frac {1}{D} \left(\frac {J _ {e \ell} ^ {2} - 1}{2} - \ln J _ {e \ell}\right),
where
C _ {1} = \frac {1}{2}, \qquad C _ {2} = \frac {1}{2 0}, \qquad C _ {3} = \frac {1 1}{1 0 5 0}, \qquad C _ {4} = \frac {1 9}{7 0 0 0}, \quad \text {and} \quad C _ {5} = \frac {5 1 9}{6 7 3 7 5 0}.
The deviatoric part of the strain energy density comes from Arruda and Boyce (1993). This model is also known as the eight-chain model, since it was developed starting out from a representative volume element where eight springs emanate from the center of a cube to its corners. The values of the coefficients C _ { 1 } \ldots C _ { 5 } arise from a series expansion of the inverse Langevin function, which arises in the statistical treatment of non-Gaussian chains. The series expansion is truncated after the fifth term. The coefficient \mu represents the initial shear modulus, and the coefficient \lambda _ { m } is referred to as the locking stretch. Approximately at this stretch the slope of the stress-strain curve will rise significantly.
The initial bulk modulus is obtained as K _ { 0 } = 2 / D . To the deviatoric part of the strain energy density we add a simplified representation of the volumetric strain energy density, which requires only one material parameter, so that all material parameters can be estimated easily even with limited knowledge of the material behavior. This volumetric representation has been used successfully by
Kaliske and Rothert (1997) and provides sufficient accuracy for most engineering elastomeric materials.
The Arruda-Boyce potential depends on the first invariant only. The physical interpretation is that the eight chains are stretched equally under the action of a general deformation state. The first invariant, \overline { { I } } _ { 1 } = \lambda _ { 1 } ^ { 2 } + \lambda _ { 2 } ^ { 2 } + \lambda _ { 3 } ^ { 2 } , directly represents this elongation.
The user can specify the Arruda-Boyce form by choosing the ARRUDA-BOYCE parameter and defining the coefficients as functions of temperature. Alternatively, ABAQUS can perform a fit of the test data specified by the user to determine the coefficients.
Van der Waals form
The hyperelastic Van der Waals potential, also known as the Kilian model, has the following form:
U = \mu \biggl \{- (\lambda_ {m} ^ {2} - 3) \biggl [ \ln (1 - \eta) + \eta \biggr ] - \frac {2}{3} a \left(\frac {\tilde {I} - 3}{2}\right) ^ {\frac {3}{2}} \biggr \} + \frac {1}{D} \left(\frac {J _ {e \ell} ^ {2} - 1}{2} - \ln J _ {e \ell}\right),
where
\tilde {I} = (1 - \beta) \overline {{I}} _ {1} + \beta \overline {{I}} _ {2} \quad \mathrm{and} \quad \eta = \sqrt {\frac {\tilde {I} - 3}{\lambda_ {m} ^ {2} - 3}}.
The name "Van der Waals" draws on the analogy in the thermodynamic interpretation of the equations of state for rubber and gas. While the neo-Hookean model can be compared with an ideal gas in that it starts out from a Gaussian network with no mutual interaction between the "quasi-particles" ( Kilian, 1981), the Van der Waals strain energy potential is analogous to the equations of state of a real gas. This introduces two additional material parameters: the locking stretch, \lambda _ { m } , and the global interaction parameter, a. (Similarly, the Van der Waals equation for a real gas introduces two parameters to account for excluded volume and modified exchange of momentum between the particles.)
The locking stretch, \lambda _ { m } , accounts for finite extendability of the non-Gaussian chain network. In contrast to the Arruda-Boyce model the mathematical structure of the Van der Waals potential is such that the strain energy tends to infinity as the locking stretch, \lambda _ { m } , is reached; more precisely, as \tilde { I } \lambda _ { m } ^ { 2 } . Thus, the Van der Waals potential cannot be used at stretches larger than the locking stretch.
The global interaction parameter, a, models the interaction between the chains; it is difficult to estimate. Kilian et al. (1986) point out that, given Mooney-Rivlin coefficients and a locking stretch \lambda _ { m } , a suitable value for the global interaction parameter is
a = \frac {2 C _ {0 1}}{3 \mu} + \frac {\lambda_ {m} ^ {2}}{\lambda_ {m} ^ {3} - 1},
where \mu is the initial shear modulus at low strains and C _ { 0 1 } is the second Mooney-Rivlin parameter. Given a positive initial shear modulus, \mu , and locking stretch, \lambda _ { m } , too large a positive interaction