Kimura, Masato

Takaishi, Takeshi

Kyoto University Research Information Repository

TitlePhase field model of mode III-type crack evolution and itsenergy estimation (Recent Developments of NumericalAnalysis and Numerical Computation Algorithms)Author(s)Takaishi, Takeshi; Kimura, MasatoCitation数理解析研究所講究録 (2010), 1719: 21-25Issue Date2010-11URL http://hdl.handle.net/2433/170365RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityPhase field model of mode III-type crack evolution and itsenergy estimation1,*)Takeshi Takaishi , 2)Masato Kimural )Faculty of Information Design, Hiroshima Kokusai GakuinUniversity2)Faculty of Mathematics, Kyushu University*E-mail: t.takaishi@hkg.ac.jpKeywords: crack evolution, phase field model, numerical simulation1 Model for Crack EvolutionMany methods are proposed for numerical computations on crack evolutionproblems, because of the difficulty of the singularity arising from the crack tip,change of computational boundary arising from the new crack surface, lack ofthe explicit method to determine the direction of expand the crack. We showthat the mode III (anti-plane shear mode) crack growth on a plate is described asreaction-diffusion system that is consisted by the of the anti-plane displacementand the phase field that describes the crack[2]. This system is derived from theenergy description introduced by Francfort and Marigo[l], We show that thisreaction-diffusion system make the computation of the crack problem easy.(a) (b) (c)Figure 1: 3 modes of the crack evolution on a plate (a) mode I, (b) mode II, (c)mode III.We derive equation for the crack evolution of the plate, that is expandedby the small anti-plane displacement (Figure 1 $(c)$ ). Let $\Gamma$ be a bounded twodimensional domain with a piecewise smooth boundary $\Gamma_{N}$ , and let $\Gamma_{D}$ be anonempty open portion of $\Gamma_{N}$ which consists of a finite number of connected1719 2010 21-25 21components. We define $\Gamma_{N}$ $:=\Gamma\backslash \Gamma_{D}$ . For $t>0$ , we consider the equations:$\{\begin{array}{ll}\alpha_{1}\frac{\partial u}{\partial t} = div((1-z)^{2}\nabla u) x\in\Omega, t>0\alpha_{2}\frac{\partial z}{\partial t} = (\epsilon\triangle z-\frac{\gamma^{2}}{\epsilon}z+|\nabla u|^{2}(1-z))_{+} x\in\Omega, t>0u(x, t) = g(x, t) x\in\Gamma_{D}, t>0\frac{\partial u}{\partial n}=0 x\in\Gamma_{N}, t>0, \frac{\partial z}{\partial n}=0 x\in\Gamma, t>0u(x, 0)=u_{0}(x), z(x, 0)=z_{0}(x) x\in\Omega\end{array}$ (1)where $u(x, t)$ represents the small anti-plane displacement at the position $x\in\overline{\Omega}$and time $t\geq 0$ , and $g(x, t)$ is a given anti-plane displacement on the boundary$\Gamma_{D}$ . The variable $z(x, t)$ satisfies $0\leq z(x, t)\leq 1$ in $\Omega$ and represents thecrack shape, as $z\approx O$ in the region without crack and $z\approx 1$ near the crack. Theminimum length scale of $z$ is given as $O(\epsilon)$ with a small regularization parameter$\epsilon>0$ for the numerical stability. The function $z(x, t)$ is called the phase fieldfor the crack shape. For stable numerical simulations, we also introduce smalltime relaxation parameters $\alpha_{1}\geq 0$ and $\alpha_{2}>0$ .The first equation of (1) expresses the force balance in the uncracked region$(z\approx 0)$ , and the second equation expresses the crack evolution due to themodulus of the stress Vu $|$ . The material constant $\gamma>0$ is called the fracturetoughness, which prescribes the critical value of the energy release rate in theGriffith $s$ criterion. It is harder for the crack to grow, if the value of $\gamma$ is larger.A crack once generated can be no longer repaired. We put $($ $)_{+}$ to the righthand side of the second equation, where $(a)_{+}= \max(a, 0)$ . It guarantees thenon-repair condition for the crack: $\frac{\partial z}{\partial t}\geq 0$ .This model has advantages for numerical simulation of crack evolution asfollowings: i) automatic path selection of the crack that means the possibility ofcalculation of the outbreak of new crack and sub-crack, ii) possibility to use theordinal method (for example, FDM, FVM, and FEM) for numerical simulationbecause of the PDE model on fixed domain, iii) numerical stability introducedby the regularization parameter $\epsilon>0$ , iv) potential to adopt the spacial profileor hysteresis of the fracture toughness $\gamma$ .2 Energy estimationGriffith focused on the energy balance of the material that includes the crack,because he thinks that the rupture strength is mainly dominated by the stressconcentration at the tip of the crack. He introduced the relation between theenergy for making new crack surface and that for releasing by the crack evolutionunder the quasi-static condition Griffith’s fracture criterion).22Starting from the internal energy proposed by Francfort-Marigo [1], we treatthe following energy description:$\{\begin{array}{ll}\mathcal{E}(u, z) := \mathcal{E}_{1}(u, z)+\mathcal{E}_{2}(z)\mathcal{E}_{1}(u, z) ;= \frac{\mu}{2}\int_{\Omega}(1-z)^{2}|\nabla u|^{2}dx-\int_{\Omega}fudx\mathcal{E}_{2}(z) := \frac{1}{2}\int_{\Omega}\gamma(x)(\epsilon|\nabla z|^{2}+\frac{1}{\epsilon}z^{2})dx\end{array}$ (2)where $\mathcal{E}_{1}(u, z)$ is the regularized elastic energy of the plate, and $\mathcal{E}_{2}(z)$ is thesurface energy of the crack. $\gamma(x)>0$ is fracture toughness on $x\in\Omega$ . In [1],they proposed this energy and investigated precisely. Bourdin et al. and Buligamade some numerical simulations of crack evolution that minimize the energysimilar to (2).We set the total energy $\mathcal{E}$ as a free energy of Ginzburg-Landau Theory, andderive the temporal evolution equations of the displacement and the phase field.Though the detailed derivation of (1) is written in [3], the reaction-diffusionequation on two scalar variable (1) is given when we set $f=0,$ $\gamma(x)\equiv\gamma>$$0,$ $\mu=1,$ $\epsilon\equiv\epsilon\gamma$ .$t=0$ $t=0.5$ $t=1$Figure 2: Birdsview of $u$ (top), $u$ (middle) and z(bottom).23We studied that crack evolution phenomena can be developed by the numer-ical simulations of (1) with $g(x, t)$ ([2],[3]). In this model, quasi-static energyrelaxation that leads the crack evolution is assumed, then, we confirm that thisassumption is kept in the numerical simulation.Set the initial crack at $t=0$ , we make a numerical simulation fixed theboundary condition as $g(x, t)=g(x)$ . In the following simulations, we put$\epsilon=10^{-3},$ $\alpha_{1}=0,$ $\alpha_{2}=10^{-3},$ $\gamma=0.5$ in (1), and set the computational domainas $\Omega=(-1,1)\cross(-1,1)$ , with $\Gamma_{D}=\{(x_{1}, x_{2})|x_{1}\in(-1,1), x_{2}=\pm 1\}$ . Theboundary condition for $u$ is given as $g(x, t)=5x_{2}(x\in\Gamma_{D}, t\geq 0)$ .TimeFigure 3: Temporal evolution of $\mathcal{E}$ (solid line), $\mathcal{E}_{1}$ (dashed line), $\mathcal{E}_{2}$ (dotted line).The temporal evolution of $u$ (Figure 2) shows the crack evolution, however,velocity of the crack expansion becomes slower by time. From the results ofnumerical simulation, we calculate the energy of system (2). Figure 3 showsthe temporal evolution of energy that the elastic energy $\mathcal{E}_{1}$ is decaying, surfaceenergy $\mathcal{E}_{2}$ is growing, and total energy is decaying slowly as the crack growth tillthe material is fractured $(t\sim 1)$ . We confirm that these numerical results followour model and describe the crack evolution phenomena, The physical character-istics of the material can be estimated by calculating the stress intensity factorfrom these numerical results.References[1] G. A. Francfort and J.-J. Marigo, Revisiting brittle fracture as an energyminimization problem. J. Mech. Phys. Solids, 46 $(1998),1319-1342$ .[2] T. Takaishi and M. Kimura, Phase field model for mode III crack growth,Kybernetika 45 (2009), pp605-614.24[3] Takeshi Takaishi, Mode-III kiretsushinten no phase field model to sonosuuchikeisan, Nihon ouyousuurigakkai ronbunshi 19(2009), pp.351-369.25

Phase field model of mode III-type crack evolution and its energy estimation (Recent Developments of Numerical Analysis and Numerical Computation Algorithms)

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Phase field model of mode III-type crack evolution and its energy estimation (Recent Developments of Numerical Analysis and Numerical Computation Algorithms)

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