Fourth order phase field modeling of brittle fracture by Natural element method

Abstract

Contrary to the second-order Phase field model (PFM) of fracture, fourth-order PFM provides a more precise representation of the crack surface by incorporating higher-order derivatives (curvature) of the phase-field order parameter in the so-called crack density functional. As a result, in a finite element setting, the weak form of the phase-field governing differential equation requires $C^1$ continuity in the basis function. $C^0$ Sibson interpolants or Natural element interpolants are obtained by the ratio of area traced by the second-order Voronoi cell over the first-order Voronoi cells, which is based on the natural neighbor of a nodal point set. $C^1$ Sibson interpolants are obtained by degree elevating the evaluated $C^0$ interpolants in the Bernstein-Bezier patch of a cubic simplex. For better computational efficiency while accounting only for the tensile part for driving fracture, a hybrid PFM is adopted. In this work, the numerical implementation of higher-order PFM with $C^1$ Sibson interpolants along with some benchmark examples are presented to showcase the performance of this method for simulating fracture in brittle materials.

A Appendix

1.1 Weak form of fourth-order term

The fourth-order term in the Eq. (28) is separately written as,

$$\begin{aligned}&\frac{{\mathcal {G}}_c\ell ^3}{16} \int _{\Omega }w_\phi \nabla \cdot \nabla (\Delta \phi )~d\Omega = \frac{{\mathcal {G}}_c\ell ^3}{16} \int _{\Omega }w_\phi \varvec{J}_{PF}~d\Omega \nonumber \\&\quad \int _{\Omega }w_\phi \varvec{J}_{PF} ~d\Omega = \int _{\Omega }w_\phi \nonumber \\&\quad \left[ \frac{\partial ^4\phi }{\partial x^4}+\frac{2\partial ^4\phi }{\partial x^2\partial y^2}+\frac{\partial ^4\phi }{\partial y^4}\right] d\Omega =I_1+I_2+I_3. \end{aligned}$$

(40)

The first term is expanded as

$$\begin{aligned} I_1&= \int _{\Omega } w_\phi \frac{\partial ^4\phi }{\partial x^4} d\Omega =\oint _{\partial \Omega } w_\phi \frac{\partial ^3\phi }{\partial x^3} n_x \,dA\nonumber \\&\quad -\int _{\Omega }\frac{\partial w_\phi }{\partial x} \frac{\partial ^3\phi }{\partial x^3}\,d\Omega \nonumber \\&=\oint _{\partial \Omega } w_\phi \frac{\partial ^3\phi }{\partial x^3} n_x \,dA-\oint _{\partial \Omega }\frac{\partial w_\phi }{\partial x} \frac{\partial ^2\phi }{\partial x^2} n_x\,dA\nonumber \\&\quad +\int _{\Omega }\frac{\partial ^2w_\phi }{\partial x^2} \frac{\partial ^2\phi }{\partial x^2}\,d\Omega . \end{aligned}$$

(41)

The third term is expanded as

$$\begin{aligned}&I_3= \int _{\Omega }w_\phi \frac{\partial ^4\phi }{\partial y^4} d\Omega =\oint _{\partial \Omega } w_\phi \frac{\partial ^3\phi }{\partial y^3} n_y \,dA\nonumber \\&\quad -\int _{\Omega }\frac{\partial w_\phi }{\partial y} \frac{\partial ^3\phi }{\partial y^3}\,d\Omega \nonumber \\&\quad =\oint _{\partial \Omega } w_\phi \frac{\partial ^3\phi }{\partial y^3} n_y \,dA-\oint _{\partial \Omega }\frac{\partial w_\phi }{\partial y} \frac{\partial ^2\phi }{\partial y^2} n_y\,dA\nonumber \\&\quad +\int _{\Omega }\frac{\partial ^2 w_\phi }{\partial y^2} \frac{\partial ^2\phi }{\partial y^2}\,d\Omega . \end{aligned}$$

(42)

The second term is expanded as

$$\begin{aligned} I_2&= \int _{\Omega }w_\phi \frac{2\partial ^4\phi }{\partial x^2\partial y^2} d\Omega =\oint _{\partial \Omega } w_\phi \frac{\partial ^3\phi }{\partial x\partial y^2} n_x\,dA\nonumber \\&\quad +\oint _{\partial \Omega } w_\phi \frac{\partial ^3\phi }{\partial y\partial x^2} n_y\,dA\nonumber \\&\quad -\int _{\Omega }\frac{\partial w_\phi }{\partial x} \frac{\partial ^3\phi }{\partial x\partial y^2}\,d\Omega -\int _{\Omega }\frac{\partial w_\phi }{\partial y} \frac{\partial ^3\phi }{\partial y\partial x^2}\,d\Omega . \end{aligned}$$

(43)

$$\begin{aligned}&=\oint _{\partial \Omega } w_\phi \frac{\partial ^3\phi }{\partial x\partial y^2} n_x\,dA+\oint _{\partial \Omega } w_\phi \frac{\partial ^3\phi }{\partial y\partial x^2} n_y\,dA\nonumber \\&\quad - \oint _{\partial \Omega } \frac{\partial w_\phi }{\partial y} \frac{\partial ^2\phi }{\partial x\partial y} n_y\,dA -\oint _{\partial \Omega } \frac{\partial w_\phi }{\partial x} \frac{\partial ^2\phi }{\partial x\partial y} n_x\,dA\nonumber \\&\quad +\int _{\Omega }2\frac{\partial ^2w_\phi }{\partial x\partial y} \frac{\partial ^2\phi }{\partial x\partial y}\,d\Omega . \end{aligned}$$

(44)

1.2 Crack modeled numerically for the asymmetric bending test

The modeling of the initial crack/notch is carried out numerically by enforcing it either by making the phase field ($\phi$) to be one on those nodal values or by specifying significantly higher strain-history (${\mathcal {H}}^+=10^3$) value (present approach). This approach of modeling the crack certainly does not represent the correct way of geometric discontinuity, as well as it is represented by the physical crack.

It is observed that when the initial notch is modeled numerically [Fig. 26], the crack path deviates from the experimental observation (after reaching the first hole). But in the case of a physical notch, the crack pattern matches with the experiment (Bittencourt et al. 1996). This may be attributed to the fact that, when the crack is modeled numerically, it has more residual stiffness than when it is modeled physically. Another factor that contributes to this disparity could be the effective width of the initial notch (when modeled numerically) than when it is modeled physically.

Fig. 26

Crack pattern for asymmetrically notched three-point bending test ($a=5.0$ mm, $b=1.5$ mm). The initial notch is modeled numerically. Figure a, b, and c represent crack patterns at different load steps

In the case of an initial physical crack geometry, the crack evolution is obtained by only solving the two governing equations, and no numerical enforcement is required for the initial crack. However, while dealing with it numerically, the phase field ($\phi =1$) is enforced as nodal values or linearly varying history functions (${\mathcal {H}}^+=10^3$, which is a pure assumption) at the initial crack location at every time step inside the solver. This somewhat limits the pure variational nature of the phase field formulation. Because of this, while numerical modeling predicts the crack pattern in simple geometry effectively, it deviates from the original crack trajectory in somewhat complex geometry and initial crack locations. The phase field method is based on variational theory and the pure energy minimization principle, making it robust and very efficient in various complex crack trajectories. However, defining the initial discontinuity/ notch numerically (Defining high history value) may not be the best practice while using PFM. It may be an efficient way for the crack representation in 3D geometry where defining geometric discontinuity is not feasible and when meshless methods are used for numerical approximations (Borden et al. 2012).