Isochoric Hyperelasticity#
Ratel solves the momentum balance equations using unstructured high-order finite/spectral element spatial discretizations.
Initial configuration#
In the total Lagrangian approach for the incompressible Neo-Hookean hyperelasticity problem, the discrete equations are formulated with respect to the initial configuration. In this formulation, we solve for displacement and pressure \(\bm{u} \left( \bm{X} \right), p \) in the reference frame \(\bm{X}\). The notation for elasticity at finite strain is inspired by [Hol00] to distinguish between the current and initial configurations. As explained in the Continuum Mechanics section, we denote by capital letters the reference frame and by small letters the current one.
Weak form#
We multiply the following strong form
by test functions \((\bm{v}, q)\) and integrate by parts to obtain the weak form for finite-strain incompressible hyperelasticity: find \((\bm{u}, p) \in \mathcal{V} \times \mathcal{Q} \subset H^1 \left( \Omega_0 \right) \times L^2 \left( \Omega_0 \right)\) such that
where \(\bm{P} \cdot \hat{\bm{N}}|_{\partial\Omega}\) is replaced by any prescribed force/traction boundary condition written in terms of the initial configuration. This equation contains material/constitutive nonlinearities in defining \(\bm{S}(\bm{E})\), as well as geometric nonlinearities through \(\bm{P} = \bm{F}\, \bm{S}\), \(\bm{E}(\bm{F})\), and the body force \(\bm{g}\), which must be pulled back from the current configuration to the initial configuration.
Newton linearization#
To derive a Newton linearization, we need the Jacobian form of the (108): find \((\diff \bm{u}, \diff p) \in \mathcal{V} \times \mathcal{Q}\) such that
where \(K = -\frac{1}{2 J} (J^2 -1) - \frac{p}{\kappa} \) and
with \(\diff \bm{F} = \nabla_X\diff \bm{u}\) and
The linearization of the second equation of (110) is
The linearization of the second Piola-Kirchhoff stress tensor, \(\diff \bm{S}\), depends upon the material model.
Deriving \(\diff\bm{S}\) for isochoric Neo-Hookean material
For the Neo-Hookean model (44), we derive split
then,
and
where
Note
If we use single field isochoric model (50) the linearization of the volumetric stress becomes
\(\diff \bm{S}_{p} = 0\) and we only need to solve the first equation of (108).
Deriving \(\diff\bm{S}\) for isochoric Mooney-Rivlin material
For the Mooney-Rivlin model (58), we derive split
where \(\diff \bm{S}_{p}\) and \(\diff \bm{S}_{vol}\) are the same as (114), (113) (or (116) if we use single field stress (75)) and the isochoric part is
where
Deriving \(\diff\bm{S}\) for isochoric Ogden material
Similar to the Neo-Hookean model we have
where \(\diff \bm{S}_{p}\) and \(\diff \bm{S}_{vol}\) are similar to Neo-Hookean and Mooney-Rivlin models and the isochoric part is
For example computing \(\diff S_1^{iso}\) from (71) gives
To compute \(\diff \beta_i\) we differentiate \(\bm{C} = \sum_{i=1}^3 \beta_i^2 \hat{\bm{N}_i} \hat{\bm{N}_i}^T\) as
and used the fact that the eigenvectors are orthonormal i.e., \(\langle \hat{\bm{N}_i}, \hat{\bm{N}_j} \rangle = \delta_{ij}\) so that \(\langle \hat{\bm{N}_i}, \diff \hat{\bm{N}_i} \rangle = 0\), we can multiply (123) from the left and right by \(\hat{\bm{N}_i}\)
By differentiating \(\bm{E} \hat{\bm{N}_i} = \beta_i^E \hat{\bm{N}_i}\) we will arrive at
taking the inner product of above equation with \(\hat{\bm{N}_j}, \, j\neq i\) simplifies to
and finally the linearization of \(\ell_i = \log \beta_i\) is
which complete \(\diff\bm{S}\) for Ogden model.
Current configuration#
Similar to what we have shown in (85), we can write the first equation of (108) in the current configuration (see (86) ) which yields to the Jacobian form
where the second equation is similar to (109) and we can use (92)
to compute the linearization of the first equation in current configuration for different material models.
Representation of \(\bm F \diff\bm S \bm F^T\) for isochoric Neo-Hookean material
Based on the split (112), we can derive
where \(\diff \bm \epsilon\) is defined in (90) and we have used (97) and (98). The isochoric part can be simplified as
where \(\mathbb{{I}_1} = \trace(\bm{b})\).
Note
If we use single field isochoric model the linearization of the volumetric stress becomes
\(\diff \bm{S}_{p} = 0\) and we only need to solve the first equation of (125).
Representation of \(\bm F \diff\bm S \bm F^T\) for isochoric Mooney-Rivlin material
The \(\bm{F} \diff \bm{S}_{p} \bm{F}^T\) and \(\bm{F} \diff \bm{S}_{vol} \bm{F}^T\) in (117) are the same as (127), (126) (or (129) if we use single field formulation ). The isochoric part can be derived as
where
and we have used (100), (101), and
Representation of \(\bm F \diff\bm S \bm F^T\) for isochoric Ogden material
Based on the split (120), the \(\bm{F} \diff \bm{S}_{p} \bm{F}^T\) and \(\bm{F} \diff \bm{S}_{vol} \bm{F}^T\) are the same as (127), (126) (or (129) if we use single field formulation ) and the isochoric part is
where for example \(\diff \tau_1^{iso}\) can be written
and
where \(\tau_i^{iso}\) is defined in (74) and we have used \(\bm{F} \hat{\bm{N}_i} = \beta_i \hat{\bm{n}_i}\) and \(\tau_i^{iso} = \beta_i^2 S_i^{iso}\). Note that the linearization of principal stretches, \(\beta_i\), and eigenvectors, \(\hat{\bm{n}_i}\), are similar to initial configuration but are written in terms of Green-Euler tensor i.e.,