Energy-Controlled PFF Solvers#

This section describes energy-controlled phase-field fracture (PFF) solvers presented in Castillón et al.[1].

Note

Please view the examples related to phase-field fracture Energy-Controlled PFF Solvers.

Traditional PFF strategies, such as displacement or force control, often fail to capture the complete equilibrium path during instabilities like snap-back or snap-through behavior. Displacement-controlled solvers may skip stable equilibrium points, while force-controlled schemes may not trace the path entirely.

To address these issues, Castillón et al.[1] proposes two energy-controlled schemes that robustly trace the equilibrium path during crack propagation by using a monotonically increasing energy-like parameter. These schemes ensure stable progress through instabilities and accurately characterize the system’s response throughout the fracture process.

Variational scheme#

The first approach is a variational scheme driven by an energetic control function, \(\tau(t)\). This is achieved by introducing a constraint that equates \(\tau(t)\) to a weighted combination of crack surface energy and external work. A Lagrange multiplier, \(\lambda\), enforces this constraint:

\[c_1 \int_\Omega \left( \frac{1}{2l} \phi^2 + \frac{l}{2} |\nabla \phi|^2 \right) \,\mathrm{d}\Omega + c_2 \int_{\partial_N\Omega} \boldsymbol{t} \cdot \boldsymbol{u} \,\mathrm{d}S = \tau(t)\]

Here, \(c_1\) and \(c_2\) are numerical parameters for solver convergence and dimensional consistency.

The augmented functional for this constrained system, \(V\), is defined as:

\[\begin{split}\begin{aligned} V(\boldsymbol{u}, \phi, \lambda) &= \int_\Omega g(\phi)\psi(\boldsymbol{\epsilon}(\boldsymbol{u})) \,\mathrm{d}\Omega + G_c \int_\Omega \left( \frac{1}{2l} \phi^2 + \frac{l}{2} |\nabla \phi|^2 \right) \,\mathrm{d}\Omega - \int_{\partial_N\Omega} \boldsymbol{t} \cdot \boldsymbol{u} \,\mathrm{d}S \\ &\quad + \lambda \left[ c_1 \int_\Omega \left( \frac{1}{2l} \phi^2 + \frac{l}{2} |\nabla \phi|^2 \right) \,\mathrm{d}\Omega + c_2 \int_{\partial_N\Omega} \boldsymbol{t} \cdot \boldsymbol{u} \,\mathrm{d}S - \tau(t) \right] \end{aligned}\end{split}\]

The equilibrium equations are obtained by enforcing stationarity, \(\delta V = 0\). The resulting weak form is:

\[\int_\Omega g(\phi)\boldsymbol{\sigma}(\boldsymbol{\epsilon}(\boldsymbol{u})):\boldsymbol{\epsilon}(\delta \boldsymbol{u}) \,\mathrm{d}\Omega - (1 - \lambda c_2) \int_{\partial_N\Omega} \boldsymbol{t} \cdot \delta\boldsymbol{u} \,\mathrm{d}S = 0\]
\[\int_\Omega g'(\phi) \delta\phi \, \psi(\boldsymbol{\epsilon}(\boldsymbol{u})) \,\mathrm{d}\Omega + (G_c + \lambda c_1) \int_\Omega \left( \frac{1}{l} \phi \delta\phi + l \nabla\phi \cdot \nabla \delta \phi \right) \,\mathrm{d}\Omega = 0\]
\[\delta \lambda \left[ c_1 \int_\Omega \left( \frac{1}{2l} \phi^2 + \frac{l}{2} |\nabla \phi|^2 \right) \,\mathrm{d}\Omega + c_2 \int_{\partial_N\Omega} \boldsymbol{t} \cdot \boldsymbol{u} \,\mathrm{d}S - \tau(t) \right] = 0\]

From the solution \((\boldsymbol{u}, \phi, \lambda)\), the Lagrange multiplier provides key insights. A correction factor \(\alpha\) allows recovering the physical response for a constant \(G_c\):

\[\alpha = \sqrt{\frac{G_c}{G_c^{\text{eff}}}} = \frac{1}{\sqrt{1 + \frac{\lambda c_1}{G_c}}}\]
\[P_{G_c} = \alpha \cdot P_{\text{variational}}\]
\[u_{G_c} = \alpha \cdot u_{\text{variational}}\]
\[\psi_{G_c} = \alpha^2 \cdot \psi_{\text{variational}}\]

Non-variational scheme#

Alternatively, a non-variational energy-controlled scheme computes the physical equilibrium path for an effective constant \(G_c\) without post-processing. The governing equations are:

\[\int_\Omega g(\phi)\boldsymbol{\sigma}(\boldsymbol{\epsilon}(\boldsymbol{u})):\boldsymbol{\epsilon}(\delta \boldsymbol{u}) \,\mathrm{d}\Omega - (1 - \lambda c_2) \int_{\partial_N\Omega} \boldsymbol{t} \cdot \delta\boldsymbol{u} \,\mathrm{d}S = 0\]
\[\int_\Omega g'(\phi) \delta\phi \, \psi(\boldsymbol{\epsilon}(\boldsymbol{u})) \,\mathrm{d}\Omega + G_c \int_\Omega \left( \frac{1}{l} \phi \delta\phi + l \nabla\phi \cdot \nabla \delta \phi \right) \,\mathrm{d}\Omega = 0\]
\[\delta \lambda \left[ c_1 \int_\Omega \left( \frac{1}{2l} \phi^2 + \frac{l}{2} |\nabla \phi|^2 \right) \,\mathrm{d}\Omega + c_2 \int_{\partial_N\Omega} \boldsymbol{t} \cdot \boldsymbol{u} \,\mathrm{d}S - \tau(t) \right] = 0\]

The scalar \(\lambda\) in this scheme is not strictly a Lagrange multiplier but serves a similar role, ensuring that the material toughness \(G_c\) remains constant and allowing the solver to trace the physical force-displacement curve.

Both energy-controlled solvers trace the same physical equilibrium path. The choice between variational and non-variational schemes depends on computational preference, while numerical parameters \(c_1\) and \(c_2\) are used solely for convergence and do not affect physical results.