Phase Field

Phase Field#

The phase-field method represents a sharp crack by a smooth damage field over a narrow band. This avoids explicitly tracking a moving crack surface and allows complex crack paths, branching, and coalescence to emerge from the solution. In this section, \(d=0\) denotes intact material and \(d=1\) fully developed fracture. The formulation is a standard variational model for quasi-brittle or brittle fracture; the precise degradation law and material split used in a computation are model choices.

Regularized fracture energy#

For linear elastic material, a representative total potential energy is

\[\Pi(\mathbf{u},d) =\int_{\Omega} \left[g(d)\psi_e^+(\boldsymbol{\varepsilon}) +\psi_e^-(\boldsymbol{\varepsilon})\right]\,\mathrm{d}\Omega +\int_{\Omega}G_c\,\gamma_\ell(d,\nabla d)\,\mathrm{d}\Omega -W_{\mathrm{ext}},\]

where \(\mathbf{u}\) is displacement, \(\boldsymbol{\varepsilon}\) is strain, \(\psi_e^+\) and \(\psi_e^-\) are the tensile and compressive parts of the elastic energy, \(G_c\) is fracture energy per unit crack area, and \(W_{\mathrm{ext}}\) is external work. Separating tensile and compressive energy prevents compressive states from spuriously driving crack growth.

For the common AT2 crack-density function and quadratic degradation law,

\[\gamma_\ell(d,\nabla d) =\frac{d^2}{2\ell}+\frac{\ell}{2}|\nabla d|^2, \qquad g(d)=(1-d)^2+\kappa,\]

where \(\ell\) is the regularization length and \(\kappa\) is a small residual stiffness used to avoid a singular elastic operator. The length \(\ell\) controls the width of the diffused crack and must be resolved by the mesh. Other crack-density functions, such as AT1, lead to different evolution equations and nucleation behavior.

Mechanical equilibrium and damage evolution#

Quasi-static mechanical equilibrium is expressed as

\[\nabla\cdot\boldsymbol{\sigma}+\mathbf{b}=\mathbf{0}, \qquad \boldsymbol{\sigma} =g(d)\frac{\partial\psi_e^+}{\partial\boldsymbol{\varepsilon}} +\frac{\partial\psi_e^-}{\partial\boldsymbol{\varepsilon}},\]

where \(\boldsymbol{\sigma}\) is Cauchy stress and \(\mathbf{b}\) is body force per unit volume. For the AT2 choice, a common phase-field equation is

\[\frac{G_c}{\ell}d-G_c\ell\nabla^2d -2(1-d)H=0,\]

where \(H\) is a history field that stores the maximum tensile elastic energy attained at a material point. Using a non-decreasing history field enforces the irreversibility of fracture in a staggered solution; equivalently, the damage is constrained not to decrease with loading. Boundary conditions for \(d\) and mechanical displacement must be selected consistently with the physical crack and loading problem.

Numerical interpretation#

The regularization length is a material/model scale, not simply a mesh size. The mesh should contain multiple elements across the diffused crack band, and results should be checked for sensitivity to \(\ell\), mesh resolution, and residual stiffness. Staggered solution alternates between mechanical equilibrium and damage evolution; monolithic solution solves both fields together. These are numerical strategies for the coupled equations, not different fracture-energy definitions.

The equations shown are a reference AT2-type formulation. They provide theoretical context for the Phase Field module and should not be interpreted as a complete listing of every constitutive option or solver detail in OpenFDEM.

Phase Field

Contents

Phase Field#

Regularized fracture energy#

Mechanical equilibrium and damage evolution#

Numerical interpretation#

Further reading#