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18 KiB
parameter, a , will lead to Drucker instability in the tensile range. Realistic values of the global interaction parameter, ^ { a , } will contribute to the characteristic S-shape of tensile stress-strain curves of rubber in the middle strain range before the final upturn as the locking stretch is approached, without causing instability.
The parameter \beta represents a linear mixture parameter combining both invariants \overline { { I } } _ { 1 } and \overline { { I } } _ { 2 } into { \tilde { I } } ; for \beta = 0 . 0 , the Van der Waals potential will be dependent on the first invariant only. Admissible values for this parameter are 0 . 0 \leq \beta \leq 1 . 0 .
The user can define the Van der Waals potential by specifying the VAN DER WAALS parameter on the *HYPERELASTIC option and defining the coefficients as functions of temperature. Alternatively, the parameters can be fitted from test data. The data fitting procedure will not necessarily yield a value of \cdot \beta between zero and one. If during the curve fitting procedure the parameter \beta leaves the admissible range, the curve fitting procedure is aborted and restarted with a fixed value of \beta = 0 . 0 . The user can enforce other values of \beta by using the BETA parameter on the *HYPERELASTIC option.
Strain energy potential for highly compressible elastomers
While the previous forms are intended for incompressible or almost incompressible materials, the elastic foam energy function is designed for describing highly compressible elastomers (Storåkers, 1986). This energy function has the form
Equation 4.6.1-15
U \stackrel {\mathrm{def}} {=} \sum_ {i = 1} ^ {N} \frac {2 \mu_ {i}}{\alpha_ {i} ^ {2}} \left[ \hat {\lambda} _ {1} ^ {\alpha_ {i}} + \hat {\lambda} _ {2} ^ {\alpha_ {i}} + \hat {\lambda} _ {3} ^ {\alpha_ {i}} - 3 + \frac {1}{\beta_ {i}} (J _ {e \ell} ^ {- \alpha_ {i} \beta_ {i}} - 1) \right],
where
\hat {\lambda} _ {i} = J _ {t h} ^ {- \frac {1}{3}} \lambda_ {i} \rightarrow \hat {\lambda} _ {1} \hat {\lambda} _ {2} \hat {\lambda} _ {3} = J _ {e \ell}.
The volumetric and the deviatoric contributions are coupled in this expression, which can be demonstrated clearly by writing the expression in the form
Equation 4.6.1-16
U = \sum_ {i = 1} ^ {N} \frac {2 \mu_ {i}}{\alpha_ {i} ^ {2}} \left[ J _ {e \ell} ^ {\frac {1}{3} \alpha_ {i}} (\overline {{\lambda}} _ {1} ^ {\alpha_ {i}} + \overline {{\lambda}} _ {2} ^ {\alpha_ {i}} + \overline {{\lambda}} _ {3} ^ {\alpha_ {i}} - 3) + 3 (J _ {e \ell} ^ {\frac {1}{3} \alpha_ {i}} - 1) + \frac {1}{\beta_ {i}} (J _ {e \ell} ^ {- \alpha_ {i} \beta_ {i}} - 1) \right].
Series expansion of the last two terms in terms of J _ { e \ell } - 1 shows that the first-order terms vanish and that the coefficients of the second-order terms are equal to \textstyle \frac { 1 } { 2 } \alpha _ { i } ^ { 2 } ( \frac { 1 } { 3 } + \beta _ { i } ) ( J _ { e \ell } - 1 ) ^ { 2 } . Hence, a stable material is obtained if \beta _ { i } > - \frac { 1 } { 3 } . The value of N and tables giving the \mu _ { i } , \alpha _ { i } . , and \beta _ { i } values as functions of temperature are defined in the *HYPERFOAM material option. If all \beta _ { i } are equal to a constant value \beta , one can define the effective Poisson's ratio
\nu = \frac {\beta}{1 + 2 \beta}.
This Poisson's ratio is valid for finite values of the logarithmic principal strains e _ { 1 } , \ e _ { 2 } , \ e _ { 3 } ;
e _ { 2 } = e _ { 3 } = - \nu e _ { 1 } in uniaxial tension. For \beta _ { i } = 0 there is no Poisson's effect. The initial shear modulus, \mu _ { 0 } , again follows from
\mu_ {0} = \sum_ {i = 1} ^ {N} \mu_ {i},
and the initial bulk modulus follows from
k _ {0} = \sum_ {i = 1} ^ {N} 2 \mu_ {i} (\frac {1}{3} + \beta_ {i}).
If Poisson's ratio is constant and known, ABAQUS can calculate the \mu _ { i } and \alpha _ { i } from measurements of nominal stress and stretch as before. If Poisson's ratio depends on the level of straining, ABAQUS can also calculate the \beta _ { i } from the nominal lateral strains.
Subroutine UHYPER
ABAQUS/Standard also allows other forms of strain energy potentials to be defined for isotropic materials via user subroutine UHYPER by programming the first and second derivatives of U with respect to \overline { { I } } _ { 1 } , \overline { { I } } _ { 2 } , and J in that subroutine.
4.6.2 Fitting of hyperelastic and hyperfoam constants
In this section we will derive the equations needed for fitting the hyperelastic (polynomial, Ogden, Arruda-Boyce, and Van der Waals form) and hyperfoam constants to experimental test data. In addition, the procedures for checking the material stability using the Drucker criterion will be described.
For the hyperelastic models full incompressibility is assumed in fitting the hyperelastic constants to the test data, except in the volumetric test.
Stress-strain relations for the polynomial strain energy potential
The hyperelastic polynomial form can be fitted by ABAQUS up to order N = 2. Since the Mooney-Rivlin potential corresponds to the case N = 1, these remarks also apply by setting the higher-order coefficients to zero. The energy potential is as follows:
\begin{array}{l} U = C _ {1 0} (\overline {{I}} _ {1} - 3) + C _ {0 1} (\overline {{I}} _ {2} - 3) + C _ {2 0} (\overline {{I}} _ {1} - 3) ^ {2} + C _ {1 1} (\overline {{I}} _ {1} - 3) (\overline {{I}} _ {2} - 3) + C _ {0 2} (\overline {{I}} _ {2} - 3) ^ {2} \\ + \sum_ {i = 1} ^ {2} \frac {1}{D _ {i}} (J _ {e \ell} - 1) ^ {2 i}. \\ \end{array}
The deformation modes are characterized in terms of the principal stretches. The nominal stress-strain relations are now derived for the polynomial form with N = 2.
Uniaxial mode
\lambda_ {1} = \lambda_ {U}, \quad \lambda_ {2} = \lambda_ {3} = \lambda_ {U} ^ {- \frac {1}{2}}, \quad \lambda_ {U} = 1 + \epsilon_ {U}
The deviatoric strain invariants are
\overline {{I}} _ {1} = \lambda_ {U} ^ {2} + 2 \lambda_ {U} ^ {- 1}, \qquad \overline {{I}} _ {2} = \lambda_ {U} ^ {- 2} + 2 \lambda_ {U}.
We invoke the principle of virtual work to derive the nominal stress-strain relationship,
\delta U = T _ {U} \delta \lambda_ {U} = \frac {\partial U}{\partial \lambda_ {U}} \delta \lambda_ {U},
and it follows that
\begin{array}{l} T _ {U} = \frac {\partial U}{\partial \lambda_ {U}} = \frac {\partial U}{\partial \overline {{I}} _ {1}} \frac {\partial \overline {{I}} _ {1}}{\partial \lambda_ {U}} + \frac {\partial U}{\partial \overline {{I}} _ {2}} \frac {\partial \overline {{I}} _ {2}}{\partial \lambda_ {U}} \\ = 2 (1 - \lambda_ {U} ^ {- 3}) \left(\lambda_ {U} \frac {\partial U}{\partial \overline {{I}} _ {1}} + \frac {\partial U}{\partial \overline {{I}} _ {2}}\right) \\ = 2 (1 - \lambda_ {U} ^ {- 3}) \bigg [ C _ {1 0} \lambda_ {U} + C _ {0 1} + 2 C _ {2 0} \lambda_ {U} (\overline {{{I}}} _ {1} - 3) + C _ {1 1} \big (\overline {{{I}}} _ {1} - 3 + \lambda_ {U} (\overline {{{I}}} _ {2} - 3) \big) + 2 C _ {0 2} (\overline {{{I}}} _ {2} - 3) \bigg ]. \\ \end{array}
Equibiaxial mode
\lambda_ {1} = \lambda_ {2} = \lambda_ {B}, \lambda_ {3} = \lambda_ {B} ^ {- 2}, \lambda_ {B} = 1 + \epsilon_ {B}
The deviatoric strain invariants are
\overline {{I}} _ {1} = 2 \lambda_ {B} ^ {2} + \lambda_ {B} ^ {- 4}, \qquad \overline {{I}} _ {2} = 2 \lambda_ {B} ^ {- 2} + \lambda_ {B} ^ {4}.
From virtual work
\delta U = 2 T _ {B} \delta \lambda_ {B} = \frac {\partial U}{\partial \lambda_ {B}} \delta \lambda_ {B},
and it follows that,
\begin{array}{l} T _ {B} = \frac {1}{2} \frac {\partial U}{\partial \lambda_ {B}} = 2 \left(\lambda_ {B} - \lambda_ {B} ^ {- 5}\right) \left(\frac {\partial U}{\partial \overline {{I}} _ {1}} + \lambda_ {B} ^ {2} \frac {\partial U}{\partial \overline {{I}} _ {2}}\right) \\ = 2 \left(\lambda_ {B} - \lambda_ {B} ^ {- 5}\right) \left[ C _ {1 0} + C _ {0 1} \lambda_ {B} ^ {2} + 2 C _ {2 0} (\bar {I} _ {1} - 3) + C _ {1 1} \left(\lambda_ {B} ^ {2} (\bar {I} _ {1} - 3) + (\bar {I} _ {2} - 3)\right) \right. \\ \left. + 2 C _ {0 2} \lambda_ {B} ^ {2} \left(\overline {{I}} _ {2} - 3\right) \right]. \\ \end{array}
Planar (pure shear) mode
\lambda_ {1} = \lambda_ {S}, \lambda_ {2} = 1, \lambda_ {3} = \lambda_ {S} ^ {- 1}, \lambda_ {S} = 1 + \epsilon_ {S}
The deviatoric strain invariants are
\overline {{I}} _ {1} = \overline {{I}} _ {2} = \lambda_ {S} ^ {2} + \lambda_ {S} ^ {- 2} + 1.
From virtual work
\delta U = T _ {S} \delta \lambda_ {S} = \frac {\partial U}{\partial \lambda_ {S}} \delta \lambda_ {S},
and it follows that,
\begin{array}{l} T _ {S} = \frac {\partial U}{\partial \lambda_ {S}} = 2 (\lambda_ {S} - \lambda_ {S} ^ {- 3}) \left(\frac {\partial U}{\partial \overline {{I}} _ {1}} + \frac {\partial U}{\partial \overline {{I}} _ {2}}\right) \\ = 2 \left(\lambda_ {S} - \lambda_ {S} ^ {- 3}\right) \left[ C _ {1 0} + C _ {0 1} + 2 \left(C _ {2 0} + C _ {1 1} + C _ {0 2}\right) \left(\bar {I} _ {1} - 3\right) \right]. \\ \end{array}
Volumetric mode
\lambda_ {1} = \lambda_ {2} = \lambda_ {3} = \lambda_ {V}, \quad J = \lambda_ {V} ^ {3}
From virtual work
\delta U = - p \delta J = \frac {\partial U}{\partial J} \delta J,
and it follows that,
p = - \frac {\partial U}{\partial J},
p = - \sum_ {i = 1} ^ {N} 2 i \frac {1}{D _ {i}} (J - 1) ^ {2 i - 1}.
Stress-strain relations for the reduced polynomial strain energy potential
The hyperelastic reduced polynomial form can be fitted by ABAQUS up to order N = 6. For N = 3 the reduced polynomial is identical to the Yeoh model, and for N = 1 the neo-Hookean model is retained; hence, the following also applies to these forms. The reduced polynomial energy potential is as follows:
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 the arguments in the previous section, we derive the nominal stress-strain relations for the reduced polynomial.
Uniaxial mode
\lambda_ {1} = \lambda_ {U}, \quad \lambda_ {2} = \lambda_ {3} = \lambda_ {U} ^ {- \frac {1}{2}}, \quad \lambda_ {U} = 1 + \epsilon_ {U}
T _ {U} = 2 (\lambda_ {U} - \lambda_ {U} ^ {- 2}) \sum_ {i = 1} ^ {N} i C _ {i 0} (\overline {{I}} _ {1} - 3) ^ {i - 1}.
Equibiaxial mode
\lambda_ {1} = \lambda_ {2} = \lambda_ {B}, \lambda_ {3} = \lambda_ {B} ^ {- 2}, \lambda_ {B} = 1 + \epsilon_ {B}
T _ {B} = 2 (\lambda_ {B} - \lambda_ {B} ^ {- 5}) \sum_ {i = 1} ^ {N} i C _ {i 0} (\overline {{I}} _ {1} - 3) ^ {i - 1}.
Planar (pure shear) mode
\lambda_ {1} = \lambda_ {S}, \lambda_ {2} = 1, \lambda_ {3} = \lambda_ {S} ^ {- 1}, \quad \lambda_ {S} = 1 + \epsilon_ {S}
T _ {S} = 2 (\lambda_ {S} - \lambda_ {S} ^ {- 3}) \sum_ {i = 1} ^ {N} i C _ {i 0} (\overline {{I}} _ {1} - 3) ^ {i - 1}.
Volumetric mode
\lambda_ {1} = \lambda_ {2} = \lambda_ {3} = \lambda_ {V}, \quad J = \lambda_ {V} ^ {3}
p = - \sum_ {i = 1} ^ {N} 2 i \frac {1}{D _ {i}} (J - 1) ^ {2 i - 1}.
Stress-strain relations for the hyperelastic Ogden strain energy potential
The hyperelastic Ogden form can be fitted up to order N = 6:
U = \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}.
Following the same approach as for the polynomial form, we can derive the nominal stress-strain equations for the Ogden form.
Uniaxial mode
\lambda_ {1} = \lambda_ {U}, \quad \lambda_ {2} = \lambda_ {3} = \lambda_ {U} ^ {- \frac {1}{2}}, \quad \lambda_ {U} = 1 + \epsilon_ {U}
T _ {U} = \sum_ {i = 1} ^ {N} \frac {2 \mu_ {i}}{\alpha_ {i}} (\lambda_ {U} ^ {\alpha_ {i} - 1} - \lambda_ {U} ^ {- \frac {1}{2} \alpha_ {i} - 1}).
Equibiaxial mode
\lambda_ {1} = \lambda_ {2} = \lambda_ {B}, \quad \lambda_ {3} = \lambda_ {B} ^ {- 2}, \quad \lambda_ {B} = 1 + \epsilon_ {B}
T _ {B} = \sum_ {i = 1} ^ {N} \frac {2 \mu_ {i}}{\alpha_ {i}} (\lambda_ {B} ^ {\alpha_ {i} - 1} - \lambda_ {B} ^ {- 2 \alpha_ {i} - 1}).
Planar (pure shear) mode
\lambda_ {1} = \lambda_ {S}, \lambda_ {2} = 1, \lambda_ {3} = \lambda_ {S} ^ {- 1}, \lambda_ {S} = 1 + \epsilon_ {S}
T _ {S} = \sum_ {i = 1} ^ {N} \frac {2 \mu_ {i}}{\alpha_ {i}} (\lambda_ {S} ^ {\alpha_ {i} - 1} - \lambda_ {S} ^ {- \alpha_ {i} - 1}).
Volumetric mode
\lambda_ {1} = \lambda_ {2} = \lambda_ {3} = \lambda_ {V}, \quad J = \lambda_ {V} ^ {3}
p = - \sum_ {i = 1} ^ {N} 2 i \frac {1}{D _ {i}} (J - 1) ^ {2 i - 1}.
Stress-strain relations for the hyperelastic Arruda-Boyce strain energy potential
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}, \quad C _ {2} = \frac {1}{2 0}, \quad C _ {3} = \frac {1 1}{1 0 5 0}, \quad 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}.
Following the same approach as for the polynomial form, we can derive the nominal stress-strain equations for the Arruda-Boyce potential.
Uniaxial mode
\lambda_ {1} = \lambda_ {U}, \quad \lambda_ {2} = \lambda_ {3} = \lambda_ {U} ^ {- \frac {1}{2}}, \quad \lambda_ {U} = 1 + \epsilon_ {U}
T _ {U} = 2 \mu (\lambda_ {U} - \lambda_ {U} ^ {- 2}) \sum_ {i = 1} ^ {5} \frac {i C _ {i}}{\lambda_ {m} ^ {2 i - 2}} \overline {{I}} _ {1} ^ {i - 1}.
Equibiaxial mode
\lambda_ {1} = \lambda_ {2} = \lambda_ {B}, \lambda_ {3} = \lambda_ {B} ^ {- 2}, \lambda_ {B} = 1 + \epsilon_ {B}
T _ {B} = 2 \mu (\lambda_ {B} - \lambda_ {B} ^ {- 5}) \sum_ {i = 1} ^ {5} \frac {i C _ {i}}{\lambda_ {m} ^ {2 i - 2}} \overline {{I}} _ {1} ^ {i - 1}.
Planar (pure shear) mode
\lambda_ {1} = \lambda_ {S}, \lambda_ {2} = 1, \lambda_ {3} = \lambda_ {S} ^ {- 1}, \lambda_ {S} = 1 + \epsilon_ {S}
T _ {S} = 2 \mu (\lambda_ {S} - \lambda_ {S} ^ {- 3}) \sum_ {i = 1} ^ {5} \frac {i C _ {i}}{\lambda_ {m} ^ {2 i - 2}} \overline {{I}} _ {1} ^ {i - 1}.
Volumetric mode
\lambda_ {1} = \lambda_ {2} = \lambda_ {3} = \lambda_ {V}, \quad J = \lambda_ {V} ^ {3}
p = - \frac {1}{D} \left(J - \frac {1}{J}\right).
Stress-strain relations for the hyperelastic Van der Waals energy potential
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}}.
Following the same approach as for the polynomial form, we can derive the nominal stress-strain
relations for the Van der Waals form.
Uniaxial mode
\lambda_ {1} = \lambda_ {U}, \quad \lambda_ {2} = \lambda_ {3} = \lambda_ {U} ^ {- \frac {1}{2}}, \quad \lambda_ {U} = 1 + \epsilon_ {U}
T _ {U} = \mu (1 - \lambda_ {U} ^ {- 3}) \left(\frac {1}{1 - \eta} - a \sqrt {\frac {\tilde {I} - 3}{2}}\right) \left[ \lambda_ {U} (1 - \beta) + \beta \right].
Equibiaxial mode
\lambda_ {1} = \lambda_ {2} = \lambda_ {B}, \lambda_ {3} = \lambda_ {B} ^ {- 2}, \lambda_ {B} = 1 + \epsilon_ {B}
T _ {B} = \mu (\lambda_ {B} - \lambda_ {B} ^ {- 5}) \left(\frac {1}{1 - \eta} - a \sqrt {\frac {\tilde {I} - 3}{2}}\right) \left(1 - \beta + \beta \lambda_ {B} ^ {2}\right).
Planar (pure shear) mode
\lambda_ {1} = \lambda_ {S}, \lambda_ {2} = 1, \lambda_ {3} = \lambda_ {S} ^ {- 1}, \quad \lambda_ {S} = 1 + \epsilon_ {S}
T _ {S} = \mu (\lambda_ {S} - \lambda_ {S} ^ {- 3}) \left(\frac {1}{1 - \eta} - a \sqrt {\frac {\tilde {I} - 3}{2}}\right).
Volumetric mode
\lambda_ {1} = \lambda_ {2} = \lambda_ {3} = \lambda_ {V}, \quad J = \lambda_ {V} ^ {3}
p = - \frac {1}{D} \left(J - \frac {1}{J}\right).
Stress-strain relations for the hyperfoam strain energy potential
The hyperfoam potential is a modified form of the Hill strain energy potential and can be fitted up to
order N = 6 :
U = \sum_ {i = 1} ^ {N} \frac {2 \mu_ {i}}{\alpha_ {i} ^ {2}} \bigg [ \hat {\lambda} _ {1} ^ {\alpha_ {i}} + \hat {\lambda} _ {2} ^ {\alpha_ {i}} + \hat {\lambda} _ {3} ^ {\alpha_ {i}} - 3 + \frac {1}{\beta_ {i}} (J _ {e \ell} ^ {- \alpha_ {i} \beta_ {i}} - 1) \bigg ].
The deformation modes are characterized in terms of the principal stretches and the volume ratio J . The elastomeric foams are not incompressible: J = \lambda _ { 1 } \lambda _ { 2 } \lambda _ { 3 } \neq 1 . The transverse stretches \lambda _ { 2 } and/or \lambda _ { 3 } are independently specified in the test data either as individual values depending on the lateral deformations or through the definition of an effective Poisson's ratio.
Uniaxial mode
\lambda_ {1} = \lambda_ {U}, \lambda_ {2} = \lambda_ {3}, J = \lambda_ {U} \lambda_ {2} ^ {2}, \lambda_ {U} = 1 + \epsilon_ {U}
Equibiaxial mode
\lambda_ {1} = \lambda_ {2} = \lambda_ {B}, J = \lambda_ {B} ^ {2} \lambda_ {3} ^ {2}, \lambda_ {B} = 1 + \epsilon_ {B}
Planar mode
\lambda_ {1} = \lambda_ {P}, \lambda_ {2} = 1, J = \lambda_ {P} \lambda_ {3}, \lambda_ {P} = 1 + \epsilon_ {P}
The common nominal stress-strain relation for the three deformation modes above is
T _ {j} = \frac {\partial U}{\partial \lambda_ {j}} = \frac {2}{\lambda_ {j}} \sum_ {i = 1} ^ {N} \frac {\mu_ {i}}{\alpha_ {i}} (\lambda_ {j} ^ {\alpha_ {i}} - J ^ {- \alpha_ {i} \beta_ {i}}),
where T _ { j } is the nominal stress and \lambda _ { j } is the stretch in the direction of loading.
Simple shear mode
The simple shear deformation is described in terms of the deformation gradient,
\mathbf {F} = \left[ \begin{array}{c c c} 1 & \gamma & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right],
where \gamma is the shear strain. Note also that for this deformation, J = \operatorname* { d e t } ( \mathbf { F } ) = 1 . The nominal shear stress T _ { S } is