A Phase-Field Approach to Conchoidal Fracture

Noname manuscript No. (will be inserted by the editor) A phase-field approach to conchoidal fracture Carola Bilgen · Alena Kopanicˇakov´ a´ · Rolf Krause · Kerstin Weinberg ?? Received: date / Accepted: date Abstract Crack propagation involves the creation of new mussel shell like shape and faceted surfaces of fracture and internal surfaces of a priori unknown paths. A first chal- show that our approach can accurately capture the specific lenge for modeling and simulation of crack propagation is details of cracked surfaces, such as the rippled breakages of to identify the location of the crack initiation accurately, conchoidal fracture. Moreover, we show that using our ap- a second challenge is to follow the crack paths accurately. proach the arising systems can also be solved efficiently in Phase-field models address both challenges in an elegant parallel with excellent scaling behavior. way, as they are able to represent arbitrary crack paths by Keywords phase field, multigrid method, brittle fracture, means of a damage parameter. Moreover, they allow for the crack initiation, conchoidal fracture representation of complex crack patterns without changing the computational mesh via the damage parameter - which however comes at the cost of larger spatial systems to be 1 Introduction solved. Phase-field methods have already been proven to predict complex fracture patterns in two and three dimen- The prediction of crack nucleation and fragmentation pat- sional numerical simulations for brittle fracture. In this pa- tern is one of the main challenges in solid mechanics. Ev- per, we consider phase-field models and their numerical sim- ery crack in a solid forms a new surface of an a priori un- ulation for conchoidal fracture. The main characteristic of known position which needs to be identified. In order to conchoidal fracture is that the point of crack initiation is typ- compute such moving boundary problems, phase-field ap- ically located inside of the body. We present phase-field ap- proaches have recently gained attention , see, e.g.,[17,21,24, proaches for conchoidal fracture for both, the linear-elastic 36,6,10]. The main idea behind this approach is to mark the case as well as the case of finite deformations. We more- material’s state by an order parameter field z(x;t), and to let over present and discuss efficient methods for the numeri- it evolve in space and time. The order parameter -or phase- cal simulation of the arising large scale non-linear systems. field is by definition a continuous field and thus, the moving Here, we propose to use multigrid methods as solution tech- crack boundaries are ’smeared’ over a small but finite length. nique, which leads to a solution method of optimal complex- Therefore, phase field models constitute so-called diffuse- ity. We demonstrate the accuracy and the robustness of our interface formulations. This raises the question of how de- approach for two and three dimensional examples related to tailed the method can capture the specific details of cracked surfaces, like, e.g., the bend and rippled breakages of con- Prof. Dr. Kerstin Weinberg (??) · Carola Bilgen choidal fracture. Lehrstuhl fur¨ Festkorpermechanik,¨ Department Maschinenbau, Conchoidal fracture is a specific type of brittle fracture Universitat¨ Siegen, which is observed in fine-grained or amorphous materials Paul-Bonatz-Strae 9-11, 57068 Siegen, Germany E-mail: [email protected] such as rocks, minerals and glasses. The solid material breaks by cleavage but does not follow any natural planes of sep- Prof. Dr. Rolf Krause · Alena Kopanicˇakov´ a´ Chair of Advanced Scientific Computing, Institute of Computational aration. In contrast to the faceted fractures often seen in Science, USI - Universita´ della Svizzera italiana crystalline materials, conchoidal fracture typically results in Via Giuseppe Buffi 13, 6900 Lugano, Switzerland a curved breakage, see Fig. 1, that resembles the rippling, gradual curves of a sea shell, so called Wallner lines [37] 2 C. Bilgen, A. Kopanicˇakov´ a,´ R. Krause, K. Weinberg The fracture-energy density Gc quantifies the material’s re- sistance to cracking, for brittle fracture it corresponds to Griffith’s critical energy release rate. However, the energy functional (1) cannot be optimized in general and even an incremental approach is challenging because of the moving boundaries G (t). Sophisticated discretization techniques Fig. 1 The typical curved breakages and conch-like surfaces of con- have been developed to solve such problems, e.g. cohesive choidal fracture can nicely be observed in fragments of volcanic glass. zone models [40,28,30], the extended finite element method Source of the original photo is [1]. [23,35], eroded finite elements or recently developed eigen- fracture strategies [33,29]. - which gives the name “conchoid” (like a conch or shell). In a phase-field approach to fracture, the additional field Since in conchoidal fracture the shape of the broken surface z(x;t) with z 2 [0;1] characterizes the state of the material, is controlled only by the stresses state, and not by some pre- whereby z = 0 indicates the solid and z = 1 the broken state. ferred orientation of the material, it provides a record of the The set of evolving crack boundaries is now replaced by a stress state at the time of failure. The point of crack initia- surface-density function g = g(z(x;t)) and an approximation tion is typically located inside the body and not on the struc- of the form ture’s surface. Therefore, it will serve us here as a bench- Z Z dG ≈ g(z(x;t)) dW (2) mark to test the ability of the phase-field approach to predict G (t) W crack nucleation and fracture in a brittle material, without any notch, kerf, flaw or initial crack. which allows to re-write the total potential energy of a crack- In the remaining paper we will proceed as follows: In ing solid and to formulate the optimization problem locally. the next section we present the basic equations of the phase- Z mat field approach to fracture for both, linear-elastic fracture and E = Y + Gcg dW ! optimum (3) W finite deformations. In sect. 3 the temporal and spatial discretization is provided. A particular solution technique based In potential (3) the material’s energy is composed of two mat on a multigrid method is formulated in sect. 4. In sect. 5 we terms, a bulk energy density Y and a surface energy con- present a two-dimensional numerical example for the sim- tribution Gcg. By definition g is only different from zero ple but typical problem of an elastic bloc pulled on a part of along cracks. its boundary. This problem is related to the Hertzian contact of a rigid punch on a half-space and therefore, an analyti- cal solution of the stress-state at crack initiation is known. We will use this as a reference to evaluate the numerical solution for varying model parameters. Finally, in sect. 6 we analyze the three-dimensional problem of conchoidal fracture below the surface of a pulled stone block and in sect. 7 we add a study on the performance of our multigrid solver in this benchmark problem. Our results show, that a phase-field approach to fracture is able to capture the essential features of conchoidal fracture in both, the linearized and the finite deformation regime. Fig. 2 Uniaxial model with a crack at x = 0 and with a continuous phase field z 2 [0;1]; phase field approximation for a second order and 2 Phase-field approach a fourth order approximation of crack density function g. Let us consider a solid with domain W ⊂ R3 and boundary There is no unique way to choose the crack density func- ¶W ≡ G ⊂ R2 deforming within a time interval t 2 [0;t ] ⊂ tot tion; a typical form is the the second-order phase-field ap- R+. Crack growth corresponds to the creation of new bound- proach: ary surfaces G (t). Hence the total potential energy of a ho- mogenous but cracking solid is composed of its material’s 1 2 lc 2 mat g(z;∇z) = z + j∇zj (4) energy with free Helmholtz energy density Y and of sur- 2lc 2 face energy contributions from growing crack boundaries. At a crack, the first term in equation (4) describes a jump Z Z mat E = Y dW + Gc dG (1) and the gradient regularizes the discontinuity; the length- W G (t) scale parameter lc weights the influence of the gradient term. A phase-field approach to conchoidal fracture 3 Inserted into functional (3) it forms a potential which cor- The compression-tension anisotropy of fracture, i.e., the responds to the well known Ambrosio-Tortorelli functional fact, that only tensile stress states contribute to the crack of continuum damage mechanics [3]. Thus, the phase-field propagation, makes an appropriate decomposition of the elas- fracture approach can be seen as a gradient damage model tic strain-energy necessary. The energy (6) needs to be split with the major difference, that the order parameter z indi- into a tensile and a compressive part whereby only the first cates the material to be either intakt (z = 0) or broken (z = is influenced by the phase-field parameter: 1), intermediated states are not physically meaningful. The e( ;z) = g(z) e+ + e− length lc in (4) is a measure for the width of the diffuse tran- Y e Y Y (9) sition zone, see Fig. 2, and additional high order terms may with e± = 1 ±(u) : C : ±(u) and ±(u) being the positive improve the crack approximation properties of function (4), Y 2 e e e cf. [7,39]. and the negative part strain tensor.

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