Modeling dynamic fracture of solids with a phase-field regularized cohesive zone model Vinh Phu Nguyena , Jian-Ying Wub, b State
a Department of Civil Engineering, Monash University, Clayton, Victoria 3800, Australia. Key Laboratory of Subtropical Building Science, South China University of Technology, 510641 Guangzhou, China.
Abstract Being able to seamlessly deal with complex crack patterns like branching, merging and even fragmentation, the phasefield model, amongst several alternatives, is promising in the computational modeling of dynamic fracture in solids. This paper presents an extension of our recently introduced phase-field cohesive zone model for static fracture to dynamic fracture in brittle and quasi-brittle solids. The model performance is tested with several benchmarks for dynamic brittle and cohesive fracture. Good agreement is achieved with existing findings and experimental results; and particularly the results are independent to the discretization resolution and the incorporated length scale parameter. The latter is in contrast to existing phase-field models. Keywords: Phase-field model; cohesive zone model; dynamic fracture; damage; concrete. Highlights: Dynamic fracture is modeled by a phase-field regularized cohesive zone model. Both brittle fracture and quasi-brittle failure under dynamic loadings are considered; The results are independent of the length scale parameter regularizing the sharp cracks; Numerical results are in good agreement with existing computational/experimental findings.
1. Introduction Fracture is one of the most commonly encountered failure modes of engineering materials and structures. The prevention of fracture-induced failure is, therefore, a major concern in engineering designs. As with many other physical phenomena, computational modeling of the crack initiation and propagation in solids constitutes an indispensable tool not only to predict he failure of cracked structures but also to shed insights into the mechanism of the fracture processes of many materials such as concrete, rock, ceramic, metals, biological soft tissues etc. As mentioned in Tel.:
(+86) 20-87112787 Email address:
[email protected] (Jian-Ying Wu)
Preprint submitted to Computer Methods in Applied Mechanics and Engineering
June 11, 2018
Cox et al. (2005), at some scale all fracture is dynamic due to the dynamic process of the bond rupture and thus ‘the dynamic fracture problem is the most fundamental in the science of fracture’. Within the context of continuum mechanics (Malvern, 1969), this work addresses the dynamic fracture problem of brittle and quasi-brittle solids from a computational modeling point of view. One of our aims is to understand and anticipate the crack path and the crack speed under the influence of stress waves. Dynamic fracture has been extensively studied from different angles by researchers from different communities. For the theoretical and experimental aspects the readers are referred to the textbooks of Freund (1998); Ravi-Chandar (2004) and recent reviews in Cox et al. (2005); Fineberg and Bouchbinder (2015). Theoretically, the Rayleigh wave speed cR is the limiting velocity of a mode-I propagating crack (Freund, 1998). However, experiments have shown that the limiting velocity of a crack reaches just about half of the predicted value (Ravi-Chandar, 2004; Freund, 1998). An additional issue in dynamic fracture is crack branching. That is, a single crack will, under certain conditions, bifurcate to two growing cracks (Ramulu and Kobayashi, 1985a). In what follows we present a brief literature review on the computational modeling of dynamic fracture using the finite element method (FEM) (Zienkiewicz and Taylor, 2006; Hughes, 1987) which is arguably the most widely used numerical method in engineering. Note that FEM is also the most widely used numerical method for phase-field models – the ones being used in this paper – at the time of this writing. Meshless/meshfree methods are therefore intentionally not touched upon, and we refer to e.g. Belytschko et al. (1996); Nguyen et al. (2008) and references therein. A more comprehensive review on numerical modeling of brittle/cohesive fracture can be found in Rabczuk (2013). Fracture of solids can be modelled using either a discontinuous approach (also referred to as a discrete approach) or a continuous one (sometimes also referred to as a smeared one). In the former, the displacement field is allowed to be discontinuous across the fracture surfaces whereas in the latter the displacements are continuous everywhere but the stresses are gradually reduced to model the degradation process using some softening material models. The most well-known theories behind the discontinuous approaches are the linear elastic fracture mechanics (LEFM) (Griffith, 1920; Irwin, 1957), and the cohesive zone model (CZM) pioneered in Dugdale (1960); Barenblatt (1959); Hillerborg et al. (1976). Continuum damage mechanics (CDM) is probably the most widely used theory categorized in the continuous approach to fracture (Kachanov, 1958; Krajcinovic, 2003). Among various discrete approaches to dynamic fracture one can count the extended finite element method (XFEM), see for example, Belytschko et al. (2003); R´ethor´e et al. (2004); Song et al. (2008); Remmers et al. (2003); Combescure et al. (2008), the embedded strong discontinuity methods (Armero and Linder, 2009), the cohesive interface element technique (Xu and Needleman, 1994; Nguyen, 2014; Park et al., 2012a; Zhou et al., 2005; Spring and Paulino, 2018), the cracking node method (Song and Belytschko, 2009) and the space-time discontinuous Galerkin method (Abedi et al., 2017, 2006). In the embedded strong discontinuity method and XFEM, the kinematics of an element is enhanced or enriched in order to represent a discontinuous displacement field. To this end, the discontinuity surface has to be explicitly tracked, which is, sometimes, an intractable task particularly for those problems with arbitrary and complex crack paths in three dimensions. Except the formulation where interface elements are inserted a priori (Xu 2
and Needleman, 1994; Nguyen, 2014) and the cracking node method (Song and Belytschko, 2009), all other methods require a branching criterion, quite often just an ad hoc one, and usually limited to two crack branches. All aforementioned methods, except Zhou et al. (2005), employed velocity-independent fracture energy and yet are able to reproduce the experimentally observed phenomenon of dynamic crack branching, a limiting crack speed well below cR , and dynamic crack instability. Note that in all the aforementioned discontinuous approaches employing the CZM, the traction–separation law of linear softening is adopted. Modeling dynamic fracture using a CDM model is rather scarce, we can cite the ones reported in Wolff et al. (2015) using a non-local integral damage model (Pijaudier-Cabot and Baˇzant, 1987), in Pereira et al. (2017) using a non-local stress based damage model, in Oˇzbolt et al. (2013) with a microplane damage model regularized using the crack band method (Baˇzant and Oh, 1983), and in Karamnejad et al. (2013) using the gradient-enhanced damage model (Peerlings et al., 1996). Among them, Pereira et al. (2017); Oˇzbolt et al. (2013) reported successful modeling of dynamic crack branching of concrete samples. Peridynamics, originated by Silling (2000); Silling et al. (2007), is a novel non-local continuum mechanics without spatial derivatives. Accordingly, it can handle topologically complex fractures such as intersecting and branching cracks in both two and three dimensions. Moreover, the dual-horizon peridynamics (Ren et al., 2016, 2018) can effectively suppress spurious wave reflections. The governing equations of peridynamics are integro-differential equations which result in a natural meshless discretisation (Silling and Askari, 2005; Ganzenmuller et al., 2015). Therefore, modelling dynamic fracture under extreme loading (impact, explosive, etc.) of large structures is also feasible (Ha and Bobaru, 2010; Bobaru and Zhang, 2015). Phase-field fracture/damage models (Francfort and Marigo, 1998; Aranson et al., 2000) are relatively new compared to LEFM, CZM and CDM. The model presented in Francfort and Marigo (1998) aims to seek for, at the same time, the displacement field and the cracks by globally minimising a potential energy that consists of the stored bulk energy, the work of external forces, and the surface energy dissipated during fracture process. It was coined the variational approach to fracture that generalizes Griffith’s energetic theory for brittle fracture. The numerical implementation was set forth in Bourdin et al. (2000) where sharp cracks are regularized by a diffuse phase-field (or damage) approximation. The formulation introduced a small positive length scale parameter and when it approaches to zero, the solution converges to the one of the original problem according to
-convergence theorems1 . The gov-
erning equations, i.e., the conventional equilibrium equation and an extra evolution law of gradient type similar to the screened Poisson equation (Areias et al., 2016a, 2018) for the crack phase-field, are usually derived from the variational approach (Bourdin et al., 2000, 2008). Phase-field models can seamlessly handle topologically complex fractures such as intersecting cracks, branching cracks in both two dimensional and three dimensional solids with a relatively simple computer implementation. Recently, the application has been extended from fracture in solids with infinitesimal deformations to plates, shells and composites under finite strains (Areias et al., 2016b; Reinoso and 1 We
recall that
convergence is a notion of variational convergence that implies convergence of minimizers.
3
Paggi, 2017; Carollo et al., 2017; Msekh et al., 2018). State-of-the-art reviews of phase-field models can be found in Ambati et al. (2015); Wu et al. (2018). As phase-field models for fracture and damage are related to both fracture mechanics and damage mechanics, they are herein referred to as phase-field damage, or phase-field fracture or even shortly as phase-field models. Note that we have skipped those phase-field models which are developed by the applied physics community since they are distant to the well accepted Griffith’s fracture theory and refer to Spatschek et al. (2011) for these works. Based on the assumption that the kinetic energy is not affected by the phase-field and the fracture energy is constant (i.e., independent of the crack velocity), phase-field models have been applied to dynamic fracture as reported in Larsen et al. (2010); Bourdin et al. (2011); Schl¨uter et al. (2014); Hofacker and Miehe (2012a, 2013); Bleyer et al. (2016); Borden et al. (2012); Steinke et al. (2016); Li et al. (2016). The phenomenon of dynamic crack branching experimentally observed can be captured and mesh independent fracture energies were obtained. Except Li et al. (2016), a majority of those work adopted some phase-field models without an elastic domain. Consequently, overestimation of the dissipation energy was reported (Borden et al., 2012). Moreover, most of these works focused on the numerical aspects of phase-field modeling of dynamic crack growth whereas Bleyer et al. (2016); Bleyer and Molinari (2017) presented some explanations of dynamic fracture behavior using the model of Li et al. (2016). The length scale parameter plays a significant role in phase-field models that regularize the sharp crack topology into a diffuse damage band with a small but finite length scale. In earlier phase-field models (Bourdin et al., 2000, 2008), the length scale is regarded as a numerical parameter and its values should be taken as small as possible such that the
-convergence to the original problem can be guaranteed. However, these phase-field models predict crack
nucleation at a critical stress inversely proportional to the square root of the length scale parameter. Accordingly, in the limit of a vanishing length scale, the deficiency of Griffith’s theory is inherited when dealing with scaling properties and crack nucleation (Tann´e et al., 2018). Consequently, it is generally assumed that, rather than a numerical parameter, the incorporated length scale should be a material property and fixed for a specific problem as in gradient-damage models. Usually, the length scale is determined from the tensile strength using the exact homogeneous solution under 1-D uniaxial traction (Pham et al., 2011; Borden et al., 2012). However, this value is often very large compared to the specimen size, see Zhang et al. (2017b) for concrete fracture and very small e.g., for polymethylmethacrylate (PMMA) materials (Pham et al., 2017). This has led these authors to scale down and up the length scale parameter quite arbitrarily. Still, good agreement with the experimental load-displacement data has been reported (Nguyen et al., 2016; Pham et al., 2017). Recently, Wu (2017, 2018b,a) proposed a unified phase-field theory with a generic crack geometric function and a new energetic/stress degradation function such that both brittle fracture and cohesive failure can be accounted for. The formulation is based purely on standard thermodynamics with internal variables. Moreover, the traction and the apparent displacement jump across the localization band mimics those of a traction – separation law (TSL) in Barenblatt’s CZM. With a set of optimal characteristic functions, those general softening laws frequently adopted for brittle/quasi-brittle materials, such as the linear, exponential, hyperbolic, Cornelissen et al. (1986) ones, etc., can 4
be reproduced or approximated with sufficient precision. Remarkably, all the involved model parameters can be determined from the standard material properties, i.e., Young’s modulus, the failure strength, the fracture energy, the initial slope and ultimate crack opening of the objective TSL. As the resulting model is equivalent to the well-known cohesive zone model (CZM), at least in one dimension, we have named it as a phase-field regularized CZM (Feng and Wu, 2018). It is emphasized that this model is different from the one elaborated in e.g., Bourdin et al. (2008); Verhoosel and de Borst (2013); May et al. (2015); Vignollet et al. (2014) which requires the introduction of an auxiliary field that defines the crack opening vector and only applies to a horizontal crack. Note that a similar phase-field for cohesive fracture was reported in Conti et al. (2016) which, however, gives numerical results heavily sensitive to the value of the length scale parameter (Freddi and Iurlano, 2017) similarly to those phase-field models for brittle fracture. Compared to other related models, the above phase-field regularized CZM is of several merits. On the one hand, this model can deal with arbitrary crack propagation with no need of explicit representation of the sharp crack surface and of cumbersome crack tracking algorithm as in the aforesaid CZM based discontinuous approaches. On the other hand, being equivalent to the CZM (at least in the 1-D case), it gives numerical results independent of the length scale parameter, so long as the damage field of high gradient within the localization band can be sufficiently resolved. The above two properties are both very useful for the modeling of complex crack nucleation and propagation, in particular, under dynamic loading. It is the aims of this paper to extend the above phase-field regularized CZM to dynamic fracture. In line with existing models, it is also assumed here that the kinetic energy is not affected by the crack phase-field and the fracture energy is constant. Well-known benchmark tests for dynamic fracture including a crack branching test, an edgecracked plate under impulsive loading test and a mixed-mode concrete fracture test are solved. Good agreement is achieved with existing findings and experimental results. Moreover, the numerical results are independent of the discretization resolution and the incorporated length scale for both brittle and quasi-brittle fracture. This property is in contrast to existing phase-field models where the results are sensitive to the length scale. The remainder of this paper is structured as follows. Section 2 addresses the governing equations for phasefield modeling of dynamic fracture. This is followed by Section 3 which presents the phase-field regularized CZM. Section 4 is devoted to the finite element implementation and solution algorithm of the proposed model. Numerical performances of the proposed model are investigated in Section 5, regarding several benchmark tests of dynamic fracture of brittle and quasi-brittle solids. The most relevant conclusions are drawn in Section 6. Notation. Compact tensor notation is used in the theoretical part of this paper. As general rules, scalars are denoted by italic light-face Greek or Latin letters (e.g. a or ); vectors, second- and fourth-order tensors are signified by italic boldface minuscule, majuscule and blackboard-bold majuscule characters like a, A and A, respectively. The inner products with single and double contractions are denoted by ‘’ and ‘:’, respectively. The Voigt notation of vectors and second-order tensors are denoted by boldface minuscule and majuscule letters like a and A, respectively.
5
2. Phase-field modeling of dynamic fracture As shown in Figure 1 (left), the reference configuration of a cracking solid ˝ Rndim (ndim D 1; 2; 3) with a sharp crack set S Rndim
1
is considered. The external boundary is denoted by @˝ Rndim
1
, with the outward
unit normal denoted by vector n. The material particles of the solid are labeled by their spatial coordinates x. The solid is kinematically described by the displacement field u.x; t / W ˝ Œ0; T ! Rndim in time t 2 Œ0; T 2 RC for some time interval T > 0. Upon the assumption of infinitesimal deformations, the strain field .x; t / W ˝ Œ0; T ! ŒRndim ndim sym is given by .x; t / WD r sym u.x; t /, for the symmetric gradient operator r sym ./ with respect to the spatial coordinate x.
Figure 1: A solid medium with a sharp crack (left) and its phase-field regularized counterpart (right)
2.1. Phase-field regularization of sharp cracks In the context of phase-field damage models (Bourdin et al., 2000, 2008; Miehe et al., 2010a; Pham et al., 2011; Wu, 2017, 2018a), the sharp crack S is smeared over a localization band B ˝ in which the diffuse phase-field d.x/ W B ! Œ0; 1 localizes, with the exterior domain ˝nB being intact; see Figure 1 for an illustration. Note that the damage irreversibility reads dP 0, with .P/ being the time derivative. Being a fundamental ingredient of any phase-field damage model, the geometrically regularized phase-field approximation Ad .d / of the sharp crack surface AS is expressed as Z Z dA D AS Ad .d / D
.d / dV „ ƒ‚B … „S ƒ‚ … regularized crack
(2.1)
sharp crack
for the crack surface density functional .d / Z 1p ˇ ˇ2 1 1 ˇ ˇ
.d / D ˛.d / C b rd with c0 D 4 ˛.ˇ/ dˇ c0 b 0 6
(2.2)
where b is a length scale characterizes the localization bandwidth; the geometric crack function ˛.d / 2 Œ0; 1 satisfies the properties that ˛.d / D 0 for d D 0 and ˛.d / D 1 for d D 1. Concrete forms for ˛.d / will be given later. Note that for a vanishing length scale b ! 0 the sharp crack topology AS is recovered in the context of the -convergence theorem (Braides, 1998): AS D limb!0 Ad .d /. However, it is becoming more and more popular in the literature of phase-field models that the length scale parameter is interpreted as a material property (Pham et al., 2011; Nguyen et al., 2016; Tann´e et al., 2018), disregarding the favorable
-convergence of the original phase-field
model (Bourdin et al., 2000, 2008). Nevertheless, it also introduces the issues of the length scale being too large or too small discussed in the introduction session. In this work, we adopt an alternative approach to guarantee the length scale independence of global responses while preserving the
-convergence.
2.2. Governing equations Assume the cracking solid is subjected to specified volumetric body forces (per unit mass) b .x; t / and surface boundary tractions t .x; t / for some part of the external boundary @˝ t @˝. Given displacements u .x; t / are applied to the disjointed remaining boundary @˝u @˝, i.e., @˝u \ @˝ t D ; and @˝u [ @˝ t D @˝. Upon the above setting, the dynamic equilibrium equation for the solid ˝ is expressed as r C b D uR
in ˝
(2.3)
together with the following Neumann boundary conditions n D t
on @˝ t
(2.4)
for the solid density and acceleration field uR D d2 u=dt 2 . In order for the resulting initial boundary value problem to be well-posed, the following Dirichlet boundary conditions u.x; t / D u .x; t /
on @˝u
(2.5a)
and initial conditions u.x; 0/ D u0 .x/;
P u.x; 0/ D v0 .x/
(2.5b)
P are considered, for the velocity field u.x; t / WD du.x; t /=dt . Furthermore, one needs to relate the stress to the strain and possibly, some internal variables. For the phasefield damage model interested in this work, the stress and damage energy release rate (or driving force) are expressed formally as (Wu, 2017, 2018a) WD
@ .; d /; @
Y WD
@ @d
.; d /
for the local free energy density functional
(2.6) .; d / of the cracking solid ˝ to be specified later. 7
In order to derive the evolution law for the crack phase-field d.x/, variational or thermodynamical arguments can be employed as in Bourdin et al. (2000); Miehe et al. (2010a); Wu (2017). Alternatively, motivated from the fact that the dissipation rate due to the evolution of the localization band B should be equivalent to that of the original sharp crack S, Wu (2018a) proposed a much simpler postulate of energetic equivalence, resulting in the following damage criterion g.Y; d / WD Y
G f ıd
8 ˆ 0
ˆ :< 0
dP D 0
(2.7)
and Neumann boundary condition rd nB D 0
on @B
(2.8)
where nB represents the outward unit normal vector of the boundary @B; Gf is the fracture toughness (energy) in the context of Griffith’s theory (Griffith, 1920); the variational derivative ıd crack of the crack surface density functional
.d / is defined by ıd WD @d
1 1 0 ˛ .d / r @rd D c0 b
2bd
(2.9)
for Lapalacian d WD r rd and the first derivative ˛ 0 .d / WD @˛=@d . Eqs. (2.3) and (2.7) constitute the system of governing equations for phase-field modeling of dynamic fracture, supplemented by the boundary conditions (2.4), (2.8) and (2.5a) as well as the initial conditions (2.5b). 2.3. Weak form In accordance with the weighted residual method, the above governing equations in strong form can be rewritten as the following weak form: Find u 2 Uu and d 2 Ud such that 8Z Z ˆ ˆ ˆ uR ıu dV C W r sym ıu dV D ıP 8ıu 2 Vu < ˝
Z ˆ ˆ ˆ :
˝
Y ıd C Gf ı dV 0
(2.10)
8ıd 2 Vd
B
for the virtual power ıP of external forces and the variation ı of the crack surface density function Z Z b ıu dV C t ıu dA ıP D ˝ @˝ t 1 1 0 ı D ˛ .d / ıd C 2brd rıd c0 b
(2.11a) (2.11b)
where we have introduced the following test and trial spaces n ˇ o Uu WD uˇu.x/ D u 8x 2 @˝u ; n ˇ o Ud WD d ˇd.x/ 2 Œ0; 1; dP .x/ 0 8x 2 B ;
n ˇ o Vu WD ıuˇıu.x/ D 0 8x 2 @˝u n ˇ o Vd WD ıd ˇıd.x/ 0 8x 2 B 8
(2.12a) (2.12b)
Note that the boundedness d.x/ 2 Œ0; 1 and irreversibility condition dP .x/ 0 have to be dealt with carefully in the numerical implementation; see Wu (2018b) for details. Pre-defined cracks S can be accounted for either indirectly by Dirichlet condition d.x/ D 1 for x 2 S or directly by mesh discretization. In this work, the later strategy is adopted due to its flexibility.
3. Phase-field regularized cohesive zone model (CZM) In this section the phase-field regularized cohesive zone model (CZM), developed within the unified phase-field damage theory (Wu, 2017), is recalled. Compared to the other phase-field models (Bourdin et al., 2000, 2008; Miehe et al., 2010a; Pham et al., 2011), its numerical results are independent of the incorporated length scale parameter under both brittle (Wu and Nguyen, 2018) and cohesive fracture (Wu, 2018a,b), all subjected to static loading regimes; see Wu et al. (2018) for the details. 3.1. A hybrid phase-field damage formulation For a cracking solid, it is assumed that the initial strain energy density
0
is isotropically deteriorated by the
phase-field d , i.e, .; d / D !.d /
0 ./;
0 ./
D
1 1 W E0 W D N W C0 W N 2 2
(3.1)
where N D E0 W denotes the classical effective stress tensor, with E0 and C0 being the linear elastic stiffness and compliance tensors, respectively. In accordance with the constitutive relations (2.6), the resulting stress tensor and the associated damage energy release rate (or driving force) Y are expressed as 8 @ ˆ ˆ .; d / D !.d /N with 0; a2 ; a3 to be determined. For a given traction – separation law .w/ with the failure strength f t , the initial slope k0 and ultimate crack opening wc , it follows that (Wu, 2018a) 4 Gf 2=3 lch ; a2 D 2 2k0 2 b ft 8 ˆ ˆ 2 (3.5b) pD2
for the so-called (Griffith’s or Irwin’s) internal length lch WD E0 Gf =f t2 characterizing the length of the fracture process zone (FPZ) ahead of the crack tip. Compared to the other phase-field models Bourdin et al. (2000); Miehe et al. (2010a); Pham et al. (2011) in which the length scale b is determined using the failure strength f t , herein one uses f t to determine a1 with b chosen independently as long as the latter is sufficiently small; see Remark 3.1. With the above characteristic functions, the unified phase-field damage mode particularizes into a regularized CZM with general traction – separation law .w/. That is, for a constant failure strength f t , though the parameter a1 10
and the resulting energetic function !.d / depend on the length scale b, the resulting stress versus separation relation .w/ across the localization band B case is independent of it. In particular, those softening law frequently adopted for cohesive fracture, e.g., the linear, exponential, hyperbolic and Cornelissen et al. (1986) ones, etc., can be reproduced or approximated with sufficient precision. For instance, the following softening curves are considered later in this work: 1 2
Linear softening curve: p D 2; a2 D ft .w/ D f t max 1 w; 0 2Gf
and a3 D 0 (3.6)
Cornelissen et al. (1986) softening curve for normal concrete: p D 2; a2 D 1:3868 and a3 D 0:6567 h .w/ D f t 1:0 C 31 r 3 exp
2 r
r 1:0 C 31 exp
2
i
(3.7)
for the normalized crack opening r WD w=wc as well as the parameters 1 D 3:0 and 2 D 6:93. Note that the parameter a1 is still related to the length scale parameter b through Eq. (3.5a)1 . The predicted softening curves are compared in Figure 2 against the objective ones. As can be seen, the linear softening curve is exactly reproduced, while the Cornelissen et al. (1986) one is captured quite accurately. Most importantly, the fracture energy, the initial slope and the ultimate crack opening are all precisely accounted for. σ
σ
ft
ft Analytical Approximated
Analytical Approximated
Gf Gf o
wc
o
w
(a) Linear softening law
wc
w
(b) Cornelissen’s softening law
Figure 2: General softening curves predicted from the proposed phase-field damage model
Remark 3.1 In order to guarantee the convexity of the energetic degradation function (3.4)2 and of the resulting free energy functional (3.1), the length scale b should satisfy (Wu, 2018b) a1
3 2
”
b
8 lch 0:85lch 3
(3.8)
For quasi-static fracture, length scale parameters larger than this might result in unstable numerical solutions. On the other hand, a sufficiently small length scale parameter b lch is usually selected such that the sharp crack surface can 11
be approximated with sufficient precision. For quasi-brittle materials such as concrete with E0 D 40 GPa, f t D 4:0 MPa, Gf D 80:0 J/m2 , one has lch 200 mm and b 170 mm which is too large for specimens of laboratory scales. Therefore for quasi-brittle materials, the upper bound (3.8) is virtually useless. Contrariwise, for brittle materials with these material properties E0 D 32 GPa, the failure strength f t D 12 MPa and the fracture energy Gf D 3:0 J/m2 , one has lch 0:67 mm, resulting in an upper bound for b of 0.57 mm which is usually small enough. Note again that the bound (3.8) applies only to static loading. With the addition of kinematic energy, it might be relaxed. We have found that the length scale b .1=50 1=100/L, where L is the smallest length of the structure, usually gives good results. An upper bound for the length scale parameter b under dynamic fracture needs further investigation and will be addressed elsewhere.
4. Numerical implementation In this section, the governing equations of the weak form (2.10) is numerically implemented using the so-called multi-field FEM. Precisely, multi-field elements here refer to those where nodal unknowns consist of both the displacement degrees of freedom (dofs) and the damage (phase-field) dof. 4.1. Finite element discretization The computational domain ˝ is triangulated by the mesh T h , consisting of ne elements and np element nodes. The super-index h indicates the typical mesh size of the finite element domain T h . It is necessary for the element size h within the localization band B h to be much smaller than the length scale b such that an accurate estimation of the fracture energy can be guaranteed in the discrete context (Bourdin et al., 2008). Upon the above setting, the displacement field uh and crack phase-field d h are interpolated as uh .x/ D
X
NI .x/ aI D Na;
d h .x/ D
X
I
Na NN I .x/ aN I D NN
(4.1)
I
where the interpolation matrix N WD N1 I; ; NI I; is associated with the nodal displacement vector a WD ˚ T ˚ N WD N1 ; ; NI ; with the nodal damage dofs aN WD aN 1 ; ; aN I ; T , respectively. a1 ; ; aI ; and N Note that identical interpolation functions NI .x/ are generally used in the approximation of the displacement and damage fields. The resulting strain field h and damage gradient field rd h are given by h .x/ D
X I
BI .x/ aI D Ba;
rd h .x/ D
X
Na BN I .x/ aN I D BN
(4.2)
I
where in 2-D cases, the standard displacement-strain matrix B WD B 1 ; ; B I ; and gradient vector BN WD BN 1 ; ; BN I ; have the following components 2 3 8 9 @x NI 0 6 7 dmax ; find the coordinates of the nodes I ; the crack tip is defined as the node having the maximum x-coordinate among the nodes I ;
15
tnC1 [s]
xnC1 [mm]
ynC1 [mm]
cr anC1 [mm]
cr anC1 [mm]
0.000
50
25
0.0
0.0
23.425
50
25.5
0.5
0.5
23.925
50
25.5
0.0
0.5
24.425
50
25.5
0.0
0.5
24.925
50
26.5
1.0
1.5
cr and the (total) crack length at time step Table 1: Exemplary data for the crack tip. The crack increment for time step n is denoted by anC1 P cr cr cr anC1 is computed as anC1 D mn am .
Note that this algorithm of finding the location of the crack tip is also applicable to other continuous models and even to peridynamic models (Ha and Bobaru, 2010). The coordinates of the crack tip are written to a file, of which one example is given in Table 1. The first three columns correspond to the raw data from which one can calculate the crack length at a given time step. Having this data, the calculation of the crack velocity is proceeded following Nguyen (2014). The crack speed is computed from cr a linear fitting to three points of the .tnC1 ; anC1 / curve; that is, the crack speed at time step tnC1 is the slope of the
line that best fits, in a least square sense, .tn
cr 1 ; an 1 /,
cr cr .tnC1 ; anC1 / and .tnC1 ; anC1 /.
5. Numerical examples This section presents three numerical examples on dynamic fracture of both brittle and quasi-brittle solids. The first two include a dynamic crack branching test and the Kalthoff and Winkler (1987) experiment which are benchmarks in the computational dynamic fracture literature. The linear softening law (3.6) is employed. The third one concerns the mixed-mode fracture of a concrete beam under impact loadings, in which the Cornelissen et al. (1986) softening curve (3.7) is used. In all numerical examples presented in this section, unstructured and uniform piece-wise linear triangular elements are used to discretize the computation domain. For all cases, a plane stress state is assumed, and the explicit time integration is used to advance in time. Note that all the considered problems are concerned with brittle/quasi-brittle materials subjected to impact loadings. As the involved time scale is typically of order O.1ms/, the explicit Newmark scheme with a lumped mass matrix, which is second-order accurate and energy conservative, is very suitable. In order to circumvent the issue of conditional stability, the time increment t is determined by the Courant-Friedrichs-Lewy (CFL) condition t < tCFL D 0:8hmin =c, where hmin is the minimum mesh size among all p elements, c WD 0 = is the material sound speed, and 0.8 is the security factor. This scheme is acceptable since for problems involving extensive material nonlinearities even unconditionally stable implicit schemes need a small time increment comparable to tCFL . There are three length scales in our model: the mesh size h, the length scale parameter b and the internal length 16
lch , which should be h b lch . For brittle solids, the internal length lch is usually small, and so one needs very small mesh size (based on our experiences, h b=5 is recommended). Large problems were solved in parallel using multiple computers. Even though our model and its implementation are inherently three dimensions — 3D simulations can be carried out without any change in the code, in this manuscript only 2D simulations are reported. We anticipate that for 3D problems, the post-processing of the crack velocities would be harder since one has to deal with a diffuse crack front instead of a crack tip. Remark 5.1 As the time increment tCFL used in the simulations is related automatically to the smallest mesh size, various mesh sizes used in the simulations lead to different increment sizes. Therefore, as long as the numerical results are independent of the mesh size, they are also insensitive to the resulting time increment size. 5.1. Dynamic crack branching In this example, we consider a pre-cracked block loaded dynamically in tension. As shown in Figure 4, a uniform traction 0 is applied at the top and bottom edges of the specimen as a step function in time. This problem has been widely adopted to study dynamic crack branching; see the experimental tests reported in Ramulu and Kobayashi (1985b); Sharon and Fineberg (1996). In the literature it has been numerically addressed by many authors using different methods such as XFEM and cohesive interface elements, see e.g., Song et al. (2008); Park et al. (2012b); Rabczuk and Belytschko (2004); Nguyen (2014). In the phase-field community, this problem has been analyzed by Borden et al. (2012); Liu et al. (2016); Schl¨uter et al. (2014) using the standard phase-field model (i.e., ˛.d / D d 2 ) and by Bleyer et al. (2016); Li et al. (2016) using the non-standard one (i.e., ˛.d / D d ), all for brittle fracture with 2 !.d / D 1 d .
Figure 4: Dynamic crack branching: geometry of the edge cracked block (left) and the loading history (right). Herein, 0 D 1:0 MPa is used. The notch has a width of 0.5 mm.
This example is herein numerically analyzed using the presented phase-field regularized CZM with linear softening. The material parameters used in the simulations are taken from Park et al. (2012b): Young’s modulus E0 D 3:2 104 MPa, Poisson’s ratio of 0 D 0:2, the mass density of D 2450 kg/m3 , the failure strength f t D 12 MPa and the fracture energy Gf D 3:0 J/m2 , resulting in Griffith’s internal length lch D 0:67 mm and 17
Rayleigh’s wave speed cR = 2119 m/s. In order to investigate the influences of the length scale b and the mesh size h, the following three cases are considered: Case (a): b D 0:50 mm and h D 0:10 mm (i.e., h D b=5); Case (b): b D 0:50 mm and h D 0:05 mm (i.e., h D b=10); Case (c): b D 0:25 mm and h D 0:05 mm (i.e., h D b=5). where the length scale parameter b satisfies the upper bound given in Eq. (3.8). The crack patterns at the end of the simulation (t D 84 10
6
s) obtained with the above three cases are given in
Figure 5. The crack propagates from the notch to the right with increasing speed. At a certain point, the crack branches into two cracks which agrees well with the experiment (Ramulu and Kobayashi, 1985b). The widening of the crack right before the moment of branching is similar to other phase-field simulations reported in Borden et al. (2012); Liu et al. (2016); Li et al. (2016); Steinke et al. (2016), to non-local integral damage model Pereira et al. (2017) and to peridynamics simulations (Ha and Bobaru, 2010). This damage widening could be the signature of roughening of the crack surface experimentally observed to occur prior to branching (Ramulu and Kobayashi, 1985b). The global responses of the model are measured by the stored strain energy, dissipated surface energy and the crack tip velocity. The results are shown in Figure 6. Dynamic crack branching occurs around 36.5 s and the crack velocity never exceeds 0:6vR , which is in good agreement with the results given by other phase-field models (Borden et al., 2012; Liu et al., 2016). In all of the three cases considered above, the crack localization bandwidth is proportional to the incorporated length scale parameter b. However, the latter affects neither the crack pattern nor the global responses, and so does the mesh size: h D b=5 is sufficient to resolve the damage field. This result further consolidates our previous findings that our model is insensitive to the length scale parameter for both cohesive failure (Wu, 2017) and brittle fracture (Wu and Nguyen, 2018) under quasi-static loading. It is also the case for dynamic fracture. Contrariwise, Steinke et al. (2016) reported that the numerical predictions of the standard phase-field model depends apparently on the incorporated length scale parameter and the limiting case of b ! 0 cannot be achieved. In order to test whether the proposed phase-field regularized CZM can capture multiple crack branches, we reconsider Case (b) i.e., b D 0:25 mm and h D 0:05 mm but with 0 D 2:5 MPa. The result given in Figure 7 shows that crack branching happens earlier than 0 D 1:0 MPa, the crack angle is also smaller and there are multiple branches. All these are in good agreement with findings reported in Ha and Bobaru (2010); Bobaru and Zhang (2015); Pereira et al. (2017) using other methods. The crack patterns at various instants are shown in Figure 8. As can be seen, the evolution of multiple crack branches can be seamlessly captured by the proposed phase-field damage model. There exists damage widening prior to branching events for later branches. Similar results were presented in Ha and Bobaru (2011) with a peridynamics model.
18
Damage 0.000e+00
0.25
0.5
0.75
1.000e+00
(a) b D 0:5 mm and h D 0:10 mm Damage 0.000e+00
0.25
0.5
0.75
1.000e+00
(b) b D 0:5 mm and h D 0:05 mm Damage 0.000e+00
0.25
0.5
0.75
1.000e+00
(c) b D 0:25 mm and h D 0:05 mm Figure 5: Dynamic crack branching: Numerical crack patterns at time 84 s.
19
Surface energy [J]
Stored energy [J]
0.18 b = 0.50 mm, h = 0.10 mm b = 0.50 mm, h = 0.05 mm b = 0.25 mm, h = 0.05 mm
0.16 0.14 0.12 0.10 0.08
0.4 b = 0.50 mm, h = 0.10 mm b = 0.50 mm, h = 0.05 mm b = 0.25 mm, h = 0.05 mm
0.3
0.2
0.06 0.1
0.04 0.02 0.00
0
10
20
30
40
50
60
0.0
70 80 Time [µs]
0
10
Velocity [m/s]
(a) Stored strain energy
20
30
40
50
60
70 80 Time [µs]
(b) Dissipated surface energy
2000 b = 0.50 mm, h = 0.10 mm b = 0.50 mm, h = 0.05 mm b = 0.25 mm, h = 0.05 mm
1750 1500
0.6vR
1250 1000 750 500 250 0
0
10
20
30
40
50
60
70 80 Time [µs]
(c) Crack tip velocity Figure 6: Dynamic crack branching: Strain energy, surface energy and crack tip velocities in time.
5.2. Edge-cracked plate under impulsive loading This section studies a doubly notched specimen under an impact load, which is often referred to as the KalthoffWinkler experiment. The geometry of the specimen is shown in Figure 9, and the impact loading is applied by a projectile. In the experiment (Kalthoff and Winkler, 1987), two different failure modes were observed by modifying the projectile speed, v0 ; at high impact velocities (50 m/s and 100 m/s, respectively, were reported in Hofacker and Miehe (2012b) and Li et al. (2016) in which different phase-field models were considered), a shear band is observed to emanate from the notch at an angle of 10o with respect to the initial notch and at lower strain rates (v0 D 16:54 m/s), brittle failure with a crack propagation angle of about 70o is observed. We are interested only in the velocity range that resulted in a brittle failure mode. If the transition of brittle-ductile failure modes is of interest, one may need to consider the plastic behavior in the phase field model as suggested in Miehe et al. (2015a), which is, however, beyond the scope of this work. The simulations of this problem have been reported by several researchers, see e.g., Klein et al. (2001); Zhang and Paulino (2005); Park et al. (2012b); Song and Belytschko (2009); Nguyen (2014) using different methods such as XFEM and adaptive cohesive interface elements. In the phase-field community this problem 20
Figure 7: Dynamic crack branching: influence of the load intensity. The higher the load amplitude the more branches are present. Crack branching also happens sooner and the crack angle (of the first branch) is smaller.
(a) t D 25:1s
(b) t D 40:2s
(c) t D 47:7s
(d) t D 61:9s
(e) t D 84:0s Figure 8: Dynamic crack branching: Crack patterns at various instants for the traction 0 D 2:5 MPa.
21
has been analyzed by Borden et al. (2012); Li et al. (2016); Liu et al. (2016); Hofacker and Miehe (2012a, 2013). The material parameters of Maraging steel 18Ni(300) are taken from Park et al. (2012b): Young’s modulus E0 D 1:9 105 MPa, Poisson’s ratio 0 D 0:3, the mass density D 8000 kg/m3 , the failure strength f t D 2812:25 MPa and the facture energy Gf D 22:2 N/mm, resulting in Griffith’s internal length lch D 0:53 mm. The Rayleigh wave speed is cR D 2745 m/s. Two cases are considered to study the sensitivity of the results with respect to b: b D 0:50 mm and h D 0:10 mm; b D 0:25 mm and h D 0:05 mm. Due to symmetry only the upper half was modeled at the impact velocity v0 D 16:54 m/s.
Figure 9: Edge-cracked plate under impulsive loading.
Damage 1.000e+00
0.75
0.5
68o
0.25
68o
0.000e+00
Figure 10: Edge-cracked plate under impulsive loading: Numerical crack patterns at time 90 s. Left (b D 0:50 mm and h D 0:10 mm) and right (b D 0:25 mm and h D 0:05 mm).
The notch is introduced in the mesh a a geometry discontinuity following Li et al. (2016) as phase-field induced crack did not result in crack initiation correctly. The crack pattern is shown in Figure 10 which agrees with available 22
findings Borden et al. (2012); Li et al. (2016); Liu et al. (2016). Initially the crack starts to grow at a larger angle (about 70o ) then the angle decreases as the crack grows. The average angle from the initial crack tip to the point where
6
Surface energy [kJ]
Stored energy [kJ]
the crack intersects the top boundary is about 68o and in fairly good agreement with the experimental result.
b = 0.50 mm, h = 0.10 mm b = 0.25 mm, h = 0.05 mm
5 4 3
3.0
2.0 1.5
2
1.0
1
0.5
0
0
10
20
30
40
50
60
70
0.0
80 90 Time [µs]
b = 0.50 mm, h = 0.10 mm b = 0.25 mm, h = 0.05 mm
2.5
0
10
3000
vR
2500 b = 0.50 mm, h = 0.10 mm b = 0.25 mm, h = 0.05 mm
2000
30
40
50
60
70
80 90 Time [µs]
70
80 90 Time [µs]
(b) Dissipated surface energy Crack length [mm]
Velocity [m/s]
(a) Stored strain energy
20
1500
120 b = 0.50 mm, h = 0.10 mm b = 0.25 mm, h = 0.05 mm
100 80 60
0.4vR 1000
40
500
20
0
0
10
20
30
40
50
60
70
0
80 90 Time [µs]
(c) Crack tip velocity
0
10
20
30
40
50
60
(d) Crack length
Figure 11: Edge-cracked plate under impulsive loading: Strain energy, surface energy, crack tip velocity and crack length.
The global responses of the model are measured by the stored strain energy, dissipated surface energy and the crack tip velocity. The results are shown in Figure 11. Again, the responses are independent with respect to the length scale and the mesh sizes. The final crack length, Figure 11(d), is about 102 mm (corresponding to an energy dissipation of 22:2 102 D 2264:4 J for a sharp crack) and the calculated surface energy is 2460 J which is 8.6% over-estimated due to the discretization error. If we assume that the crack is straight, then with the angle of 68ı , its length would be 81 mm and its surface energy would be 1796 J. Our value is about 37% higher, but is significantly better than the value reported in Borden et al. (2012) (90% over-estimated) who used a phase-field model without an elastic regime. Next we consider a larger impact velocity v0 D 100 m/s (Li et al., 2016) using b D 0:25 mm and h D 0:05 mm. The mesh consists of about two million nodes and four million elements. Successive crack branching can be observed 23
from Figure 12. Again, there exists damage widening prior to all branching events. Note that large elements are used at the lower right corner (to reduce computational cost) and thus large damage bands occur at this region. Similar findings are reported in Li et al. (2016); Hofacker and Miehe (2012a, 2013). So, with sufficiently refined meshes, our model can capture multiple dynamic crack branchings quite straightforwardly. This result is promising for studying dynamic crack instability problem where there are many very small cracks along a major dominant one (Zhang et al., 2007). Note that the resulting crack patterns are sensitive to the tension-compression split model (Li et al., 2016). We recall that in the experiment a failure-mode transition from mode-I to mode-II was observed when the impact velocity increases. This cannot be captured using an elastic-damage model (Li et al., 2016).
(a) t D 40 s
(b) t D 53 s
Figure 12: Edge-cracked plate under impulsive loading with a large impact velocity of 100 m/s: Numerical crack patterns
5.3. John-Shah test We finally consider the three-point bending concrete beams subjected to impact loading (John and Shah, 1990). The problem configuration is given in Figure 13, where an offset notch from the midspan (with varying locations depending on the dimensionless parameter ) is made to study mixed-mode fracture. John and Shah (1990) observed different crack patterns for various and interestingly there exists a transition value t that defines a change in failure mode – crack grows from the notch for < t and from the midspan for > t (Figure 14). The experimental value of t is 0.77. This test has been numerically studied by many researchers, for example Belytschko et al. (2000) with the Element Free Galerkin method, Ruiz et al. (2001); Sam et al. (2005) with extrinsic cohesive interface elements, Zi et al. (2005) with XFEM among others. Different authors reported slightly different values of t : t D 0:6 in Ruiz et al. (2001), t D 0:635 in Zi et al. (2005), t 2 f0:75; 0:8g in Sam et al. (2005) and t D 0:734 in Belytschko et al. (2000). This is not surprising as they have used different material properties, particularly the tensile strength and the fracture energy. Herein we study this example using the presented phase-field regularized CZM with a plane stress condition. Material properties are taken from the aforementioned references: Young’s modulus E0 D 31:37 24
GPa, Poisson’s ratio 0 D 0:2, the mass density D 2400 kg/m3 , the tensile strength f t D 3:0 MPa and the fracture energy Gf D 31:1 J/m2 .
Figure 13: John-Shah test: Geometry (Unit of length: mm) and loading. The notch width is 1.5 mm and t denotes the beam thickness.
Figure 14: John-Shah test: Notch-location dependent fracture mode (John and Shah, 1990).
The imposed velocity is given by (Belytschko et al., 2000; Zi et al., 2005) 8 v ˆ < 1t if t t0 v.t / D t0 ˆ :v otherwise 1 for t0 D 196 s and v1 D 60 mm/s. These values approximately produce the measured strain rate of 0.3 s
(5.1)
1
in a
linear elastic analysis. As mesh convergence and length scale sensitivity have been extensively for the previous two problems, herein we ˚ consider various notch locations t D 0:50; 0:60; 0:65; 0:72; 0:78 using a single set of b D 1:5 mm ( h=50) and a mesh size h D b=5. Figure 15 shows the corresponding numerical crack patterns. It is obvious that the presented phase-field regularized CZM can capture the transition of the failure mode. Our estimate for t is somewhere in between 0.72 and 0.78 which is in good agreement with the experimental value. The obtained crack patterns are similar to EFG results reported in Belytschko et al. (2000), and to cohesive elements in Sam et al. (2005). As crack nucleation can be dealt with in the presented model, it is not necessary to introduce a small notch at the beam mid-span as in Belytschko et al. (2000); Zi et al. (2005).
25
Figure 15: John-Shah test: Numerical crack patterns for various notch locations.
6. Conclusions We have extended the unified phase-field damage model for quasi-static brittle/cohesive fracture presented in Wu (2017); Wu and Nguyen (2018) to the dynamic case. As our model is equivalent to the well-known cohesive zone model (CZM), at least in one dimension, we have named it as a phase-field regularized CZM. Based on a number of benchmark tests, it is confirmed that the phase-field regularized CZM gives length scale parameter independent numerical results for both dynamic brittle and cohesive fracture when the mesh is sufficiently fine (i.e., h b=5). Similar to existing phase-field models, ours was able to capture qualitatively the characteristics of dynamic fracture such as multiple crack branching, damage widening prior to branching, crack velocities well below the Rayleigh wave speed. Compared to existing phase-field models and CZM based discontinuous approaches to fracture, the presented phase-field regularized CZM is of several advantages. Firstly, it is equivalent to the CZM with general softening laws (at least in the 1-D cases), but it does not need the cumbersome crack tracking and the elastic penalty (dummy) stiffness necessary for discrete approaches (to e.g., handle crack lips interpenetration). Secondly, it gives length scale independent global responses while preserving the favored
-convergence property of phase-field models.
The biggest complaint about phase-field models is their high computational cost. Our view to this point is twofold. First, it is obvious that problems that were intractable decades ago are now solved routinely. Second, phasefield models seem to be the only continuous approach that can analyse complex 3D crack problems. Advancements in computer hardware and computing technologies (anisotropic mesh adaptivity, global/local methods, multi-scale methods, model order reduction methods) will definitely increase the efficiency of phase-field simulations.
26
Acknowledgments The corresponding author (J.Y. Wu) acknowledges the support from the National Key R&D Program of China (2017YFC0803300) and the National Natural Science Foundation of China (51678246). The first author (V.P. Nguyen) thanks the funding support from the Australian Research Council via DECRA project DE160100577. Partial support from the State Key Laboratory of Subtropical Building Science (2018ZC04) and the Scientific/Technological Project of Guangzhou (201607020005) to J.Y. Wu is also acknowledged. References Abedi, R., Haber, R. B., Clarke, P. L., Dec 2017. Effect of random defects on dynamic fracture in quasi-brittle materials. International Journal of Fracture 208 (1), 241–268. Abedi, R., Hawker, M. A., Haber, R. B., Matous, K., 2006. An adaptive spacetime discontinuous galerkin method for cohesive models of elastodynamic fracture. International Journal for Numerical Methods in Engineering 81 (10), 1207–1241. Ambati, M., Gerasimov, T., de Lorenzis, L., 2015. A review on phase-field models for brittle fracture and a new fast hybrid formulation. Comput. Mech. 55, 383–405. Amor, H., 2008. Approche variationnelle des lois de griffith et de paris via des modeles non-locaux d’endommagement: Etude theorique et mise en oeuvre num´erique. Ph.D. thesis, Universit´e Paris 13, Paris, France. Amor, H., Marigo, J., Maurini, C., 2009. Regularized formulation of the variational brittle fracture with unilateral contact: numerical experiments. J. Mech. Phys. Solids 57, 1209–1229. Aranson, I. S., Kalatsky, V. A., Vinokur, V. M., 2000. Continuum field description of crack propagation. Physic. Review Letters 85, 118–121. Areias, P., Msekh, M. A., Rabczuk, T., 2016a. Damage and fracture algorithm using the screened poisson equation and local remeshing. Eng. Fract. Mech. 158, 116–143. Areias, P., Rabczuk, T., Msekh, M. A., 2016b. Phase-field analysis of finite-strain plates and shells including element subdivision. Comput. Methods Appl. Mech. Engrg. 312, 322–350. Areias, P., Reinoso, J., Camanho, P. P., C´esar de S´a, J., Rabczuk, T., 2018. Effective 2d and 3d crack propagation with local mesh refinement and the screened poisson equation. Eng. Fract. Mech. 189, 339–360. Armero, F., Linder, C., 2009. Numerical simulation of dynamic fracture using finite elements with embedded discontinuities. International Journal of Fracture 160 (2), 119–141. Balay, S., Abhyankar, S., Adams, M. F., Brown, J., Brune, P., Buschelman, K., Dalcin, L., Eijkhout, V., Gropp, W. D., Kaushik, D., Knepley, M. G., McInnes, L. C., Rupp, K., Smith, B. F., Zampini, S., Zhang, H., Zhang, H., 2016. PETSc users manual. Tech. Rep. ANL-95/11 - Revision 3.7, Argonne National Laboratory. Barenblatt, G. I., 1959. The formation of equilibrium cracks during brittle fracture. general ideas and hypotheses. axially-symmetric cracks. Journal of Applied Mathematics and Mechanics 23, 622–636. Baˇzant, Z. P., Oh, B. H., 1983. Crack band theory for fracture of concrete. Mater. Struct. 16, 155–177. Belytschko, T., Chen, H., Xu, J., Zi, G., 2003. Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment. International Journal for Numerical Methods in Engineering 58 (12), 1873–1905. Belytschko, T., Krongauz, Y., Organ, D., Fleming, M., Krysl, P., 1996. Meshless methods: An overview and recent developments. Computer Methods in Applied Mechanics and Engineering 139 (1), 3 – 47. Belytschko, T., Organ, D., Gerlach, C., 2000. Element-free galerkin methods for dynamic fracture in concrete. Computer Methods in Applied Mechanics and Engineering 187 (3-4), 385 – 399.
27
Bleyer, J., Molinari, J.-F., 2017. Microbranching instability in phase-field modelling of dynamic brittle fracture. Appl. Phys. Lett. 110, 151903. Bleyer, J., Roux-Langlois, C., Molinari, J.-F., 2016. Dynamic crack propagation with a variational phase-field model: limiting speed, crack branching and velocity-toughening mechanisms. International Journal of Fracture, 1–22. Bobaru, F., Zhang, G., 2015. Why do cracks branch? A peridynamic investigation of dynamic brittle fracture. International Journal of Fracture 196 (1), 59–98. Borden, M. J., Verhoosel, C. V., Scott, M. A., Hughes, T. J., Landis, C. M., 2012. A phase-field description of dynamic brittle fracture. Computer Methods in Applied Mechanics and Engineering 217-220, 77 – 95. Bourdin, B., Francfort, G., Marigo, J.-J., 2000. Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48 (4), 797–826. Bourdin, B., Francfort, G., Marigo, J.-J., 2008. The variational approach to fracture. Springer, Berlin. Bourdin, B., Larsen, C. J., Richardson, C. L., 2011. A time-discrete model for dynamic fracture based on crack regularization. International Journal of Fracture 168 (2), 133–143. Braides, A., 1998. Approximation of Free-Discontinuity Problems. Springer science & Business Media, Berlin. Carollo, V., Reinoso, J., Paggi, M., 2017. A 3d finite strain model for intralayer and interlayer crack simulation coupling the phase field approach and cohesive zone model. Composite Structures 182, 636–651. Combescure, A., Gravouil, A., Grgoire, D., Rthor, J., 2008. X-FEM a good candidate for energy conservation in simulation of brittle dynamic crack propagation. Computer Methods in Applied Mechanics and Engineering 197 (5), 309 – 318. Conti, S., Focardi, M., Iurlano, F., 2016. Phase field approximation of cohesive fracture models. Annales de l’Institut Henri Poincar´e Nonlinear Analysis 33 (4), 1033 – 1067. Cornelissen, H., Hordijk, D., Reinhardt, H., 1986. Experimental determination of crack softening characteristics of normalweight and lightweight concrete. Heron 31 (2), 45–56. Cox, B. N., Gao, H., Gross, D., Rittel, D., 2005. Modern topics and challenges in dynamic fracture. Journal of the Mechanics and Physics of Solids 53 (3), 565 – 596. Dugdale, D., 1960. Yielding of steel sheets containing slits. Journal of Mechanics and Physics of Solids 8, 100–109. Facchinei, F., Pang, J.-S., 2003. Finite Dimensional Variational Inequalities and Complementarity Problems, Vol. 1 and Vol. 2. Springer-Verlag, New York. Farrell, P., Maurini, C., 2017. Linear and nonlinear solvers for variational phase-field models of brittle fracture. Int. J. Numer. Meth. Engng. 109 (5), 648667. Feng, D. C., Wu, J. Y., 2018. Phase-field regularized cohesize zone model (czm) and size effect of concrete. Eng. Fract. Mech. 197, 66–79. Fineberg, J., Bouchbinder, E., Nov 2015. Recent developments in dynamic fracture: some perspectives. International Journal of Fracture 196 (1), 33–57. Francfort, G., Marigo, J.-J., 1998. Revisiting brittle fracture as an energy minimization problem. Journal of the Mechanics and Physics of Solids 46 (8), 1319 – 1342. Freddi, F., Iurlano, F., 2017. Numerical insight of a variational smeared approach to cohesive fracture. Journal of the Mechanics and Physics of Solids 98, 156 – 171. Freund, L., 1998. Dynamic Fracture Mechanics. Cambridge University Press, Cambridge. Ganzenmuller, G. C., Hiermaier, S., May, M., 2015. On the similarity of meshless discretizations of Peridynamics and Smooth-Particle Hydrodynamics. Computers & Structures 150, 71 – 78. Griffith, A. A., 1920. The phenomena of rupture and flow in solids. Philosophical Transactions of the Royal Society of Londres 221, 163–198. Ha, Y. D., Bobaru, F., 2010. Studies of dynamic crack propagation and crack branching with peridynamics. International Journal of Fracture 162 (1), 229–244.
28
Ha, Y. D., Bobaru, F., 2011. Characteristics of dynamic brittle fracture captured with peridynamics. Engineering Fracture Mechanics 78, 1156– 1168. Hilber, H., Hughes, T., Tayler, R., 1977. Improved numerical dissipation for time integration algorithms in structural dynamics. Earthquake Engrg. Struct. Dyn. 5, 283–292. Hillerborg, A., Mod´eer, M., Petersson, P., 1976. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement Concr. Res. 6, 773–781. Hofacker, M., Miehe, C., 2012a. Continuum phase field modeling of dynamic fracture: variational principles and staggered FE implementation. International Journal of Fracture 178 (1), 113–129. Hofacker, M., Miehe, C., 2012b. Continuum phase field modeling of dynamic fracture: variational principles and staggered FE implementation. International Journal of Fracture 178 (1), 113–129. Hofacker, M., Miehe, C., 2013. A phase field model of dynamic fracture: Robust field updates for the analysis of complex crack patterns. International Journal for Numerical Methods in Engineering 93 (3), 276–301. Hughes, T., 1987. The finite element method. Prentice-Hall: Englewood Cliffs, NJ. Irwin, G. R., 1957. Analysis of stresses and strains near the end of a crack traversing a plate. Journal of Applied Mechanics 24, 361–364. John, R., Shah, S. P., 1990. Mixed mode fracture of concrete subjected to i mpact loading. ASCE Journal of Structural Engineering 116, 585–602. Kachanov, L., 1958. Time rupture process under creep conditions. Izv. A Rad. Nauk. SSSR otd Tekh. Nauk 8, 26–31. Kalthoff, J. F., Winkler, S., 1987. Failure mode transition at high rates of shear loading. International Conference on Impact Loading and Dynamic Behavior of Materials 1, 185–195. Karamnejad, A., Nguyen, V. P., Sluys, L. J., 2013. A multi-scale rate dependent crack model for quasi-brittle heterogeneous materials. Engineering Fracture Mechanics 104, 96 – 113. Klein, P., Foulk, J., Chen, E., Wimmer, S., Gao, H., 2001. Physics-based modeling of brittle fracture: cohesive formulations and the application of meshfree methods. Theoretical and Applied Fracture Mechanics 37 (1-3), 99 – 166. Krajcinovic, D., 2003. Damage Mechanics. Elsevier B.V., the Netherlands. Larsen, C. J., Ortner, C., Sali, E., 2010. Existence of solutions to a regularized model of dynamic fracture. Mathematical Models and Methods in Applied Sciences 20 (07), 1021–1048. Li, T., Marigo, J.-J., Guilbaud, D., Potapov, S., 2016. Gradient damage modeling of brittle fracture in an explicit dynamics context. Int. J. Numer. Meth. Engng. 108 (11), 1381–1405. Liu, G., Li, Q., Msekh, M. A., Zuo, Z., 2016. Abaqus implementation of monolithic and staggered schemes for quasi-static and dynamic fracture phase-field model. Computational Materials Science 121, 35 – 47. Malvern, L. E., 1969. Introduction to the Mechanics of a Continuous Medium. Prentice-Hall International, Englewood Cliffs, New Jersey. May, S., Vignollet, J., de Borst, R., 2015. A numerical assessment of phase-field models for brittle and cohesive fracture: -convergence and stress oscillations. European Journal of Mechanics - A/Solids 52, 72 – 84. Miehe, C., Hofacker, M., Sch¨anzel, L.-M., Aldakheel, F., 2015a. Phase field modeling of fracture in multi-physics problems. part ii. coupled brittleto-ductile failure criteria and crack propagation in thermo-elastic-plastic solids. Computer Methods in Applied Mechanics and Engineering 294, 486 – 522. Miehe, C., Hofacker, M., Welschinger, F., 2010a. A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits. Computer Methods in Applied Mechanics and Engineering 199 (45-48), 2765 – 2778. Miehe, C., Sch¨anzel, L., Ulmer, H., 2015b. Phase field modeling of fracture in multi-physics problems. part i. balance of crack surface and failure criteria for brittle crack propagation in thermo-elastic solids. Comput. Methods Appl. Mech. Engrg. 294, 449–485. Miehe, C., Welschinger, F., Hofacker, M., 2010b. Thermodynamically consistent phase-field models of fracture: Variational principles and multi-
29
field fe implementations. Int. J. Numer. Meth. Engng. 83, 1273–1311. Msekh, M. A., Cuong, N. H., Zi, G., Areias, P., Zhuang, X., Rabczuk, T., 2018. Fracture properties prediction of clay/epoxy nanocomposites with interphase zones using a phase field model. Eng. Fract. Mech 188, 287–299. Newmark, N., 1956. A method of computation for structural dynamics. Journal of Engineering Mechanics 85, 67–94. Nguyen, T. T., Yvonnet, J., Bornert, M., Chateau, C., Sab, K., Romani, R., Roy, R. L., 2016. On the choice of parameters in the phase field method for simulating crack initiation with experimental validation. Int. J. Fract. 197, 213–226. Nguyen, V. P., 2014. Discontinuous Galerkin/Extrinsic cohesive zone modeling: implementation caveats and applications in computational fracture mechanics. Engineering Fracture Mechanics 128, 37–68. Nguyen, V. P., Rabczuk, T., Bordas, S., Duflot, M., 2008. Meshless methods: A review and computer implementation aspects. Mathematics and Computers in Simulation 79 (3), 763–813. Oˇzbolt, J., Boˇsnjak, J., Sola, E., 2013. Dynamic fracture of concrete compact tension specimen: Experimental and numerical study. International Journal of Solids and Structures 50 (25), 4270 – 4278. Park, K., Paulino, G. H., Celes, W., Espinha, R., 2012a. Adaptive mesh refinement and coarsening for cohesive zone modeling of dynamic fracture. International Journal for Numerical Methods in Engineering 92 (1), 1–35. Park, K., Paulino, G. H., Celes, W., Espinha, R., 2012b. Adaptive mesh refinement and coarsening for cohesive zone modeling of dynamic fracture. International Journal for Numerical Methods in Engineering 92 (1), 1–35. Peerlings, R. H. J., De Borst, R., Brekelmans, W. A. M., De Vree, J. H. P., 1996. Gradient enhanced damage for quasi-brittle materials. International Journal for Numerical Methods in Engineering 39 (19), 3391–3403. Pereira, L., Weerheijm, J., Sluys, L., 2017. A numerical study on crack branching in quasi-brittle materials with a new effective rate-dependent nonlocal damage model. Engineering Fracture Mechanics, –. Pham, K., Amor, H., Marigo, J.-J., Maurini, C., 2011. Gradient damage models and their use to approximate brittle fracture. International Journal of Damage Mechanics 20, 618–652. Pham, K. H., Ravi-Chandar, K., Landis, C. M., 2017. Experimental validation of a phase-field model for fracture. Int. J. Fract. 205, 83–101. Pijaudier-Cabot, G., Baˇzant, Z. P., 1987. Nonlocal damage theory. Journal of Engineering Mechanics 113, 1512–1533. Rabczuk, T., 2013. Computational methods for fracture in brittle and quasi-brittle solids: State-of-the-art review and future perspectives. ISRN Applied Mathematics. Rabczuk, T., Belytschko, T., 2004. Cracking particles: a simplified meshfree method for arbitrary evolving cracks. International Journal for Numerical Methods in Engineering 61 (13), 2316–2343. Ramulu, M., Kobayashi, A. S., 1985a. Mechanics of crack curving and branching – a dynamic fracture analysis. International Journal of Fracture 27 (3), 187–201. Ramulu, M., Kobayashi, A. S., 1985b. Mechanics of crack curving and branching – a dynamic fracture analysis. International Journal of Fracture 27 (3), 187–201. Ravi-Chandar, K., 2004. Dynamic Fracture. Elsevier, Amsterdam. Reinoso, J., Paggi, M., 2017. Revisiting the problem of a crack impinging on an interface: A modeling framework for the interaction between the phase field approach for brittle fracture and the interface cohesive zone model. Comput. Methods Appl. Mech. Engng. 321, 145–172. Remmers, J. J. C., Borst, R. d., Needleman, A., 2003. A cohesive segments method for the simulation of crack growth. Computational Mechanics 31 (1), 69–77. Ren, H. L., Zhuang, X. Y., Cai, Y. C., Rabczuk, T., 2016. Dual-horizon peridynamics. International Journal for Numerical Methods in Engineering 108 (12), 1451–1476. Ren, H. L., Zhuang, X. Y., Rabczuk, T., 2018. Dual-horizon peridynamics: A stable solution to varying horizons. Computer Methods in Applied
30
Mechanics and Engineering 318, 762–782. R´ethor´e, J., Gravouil, A., Combescure, A., 2004. A stable numerical scheme for the finite element simulation of dynamic crack propagation with remeshing. Computer Methods in Applied Mechanics and Engineering 193 (42), 4493 – 4510. Ruiz, G., Pandolfi, A., Ortiz, M., 2001. Three-dimensional cohesive modeling of dynamic mixed-mode fracture. International Journal for Numerical Methods in Engineering 52 (1-2), 97–120. Sam, C.-H., Papoulia, K. D., Vavasis, S. A., 2005. Obtaining initially rigid cohesive finite element models that are temporally convergent. Engineering Fracture Mechanics 72 (14), 2247 – 2267. Schl¨uter, A., Willenb¨ucher, A., Kuhn, C., M¨uller, R., 2014. Phase field approximation of dynamic brittle fracture. Computational Mechanics 54 (5), 1141–1161. Sharon, E., Fineberg, J., 1996. Microbranching instability and the dynamic fracture of brittle materials. Physical Review B 54, 7128–7139. Silling, S. A., 2000. Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48, 175–209. Silling, S. A., Askari, E., 2005. A meshfree method based on the peridynamic model of solid mechanics. Computers & Structures 83 (17-18), 1526 – 1535. Silling, S. A., Epton, M., Weckner, O., Xu, J., Askari, A., 2007. Peridynamics states and constitutive modeling,. J. Elasticity 88, 151–184. Song, J., Belytschko, T., 2009. Cracking node method for dynamic fracture with finite elements. International Journal for Numerical Methods in Engineering 77 (3), 360–385. Song, J., Wang, H., Belytschko, T., 2008. A comparative study on finite element methods for dynamic fracture. Computational Mechanics 42 (2), 239–250. Spatschek, R., Brener, E., Karma, A., 2011. Phase field modeling of crack propagation. Philosophical Magazine 91, 75 – 95. Spring, D. W., Paulino, G. H., Mar 2018. Achieving pervasive fracture and fragmentation in three-dimensions: an unstructuring-based approach. International Journal of Fracture 210 (1), 113–136. ¨ Steinke, C., Ozenc ¸ , K., Chinaryan, G., Kaliske, M., 2016. A comparative study of the r-adaptive material force approach and the phase-field method in dynamic fracture. International Journal of Fracture 201 (1), 97–118. Tann´e, E., Li, T., Bourdin, B., Marigo, J.-J., Maurini, C., 2018. Crack nucleation in variational phase-field models of brittle fracture. Journal of the Mechanics and Physics of Solids 110, 80–99. Verhoosel, C. V., de Borst, R., 2013. A phase-field model for cohesive fracture. International Journal for Numerical Methods in Engineering 96 (1), 43–62. Vignollet, J., May, S., de Borst, R., Verhoosel, C. V., 2014. Phase-field models for brittle and cohesive fracture. Meccanica 49 (11), 2587–2601. Wolff, C., Richart, N., Molinari, J.-F., 2015. A non-local continuum damage approach to model dynamic crack branching. International Journal for Numerical Methods in Engineering 101 (12). Wu, J. Y., 2017. A unified phase-field theory for the mechanics of damage and quasi-brittle failure in solids. Journal of the Mechanics and Physics of Solids 103, 72–99. Wu, J. Y., 2018a. A geometrically regularized gradient-damage model with energetic equivalence. Comput. Methods Appl. Mech. Engrg. 328, 612–637. Wu, J. Y., 2018b. Numerical implementation of non-standard phase-field damage models. Computer Methods in Applied Mechanics and Engineering in press. Wu, J. Y., Nguyen, V. P., 2018. A length scale insensitive phase-field damage model for brittle fracture. Journal of the Mechanics and Physics of Solids in press. Wu, J. Y., Nguyen, V. P., Nguyen, C. T., Sutula, D., Bordas, S., Sinaie, S., 2018. Phase field modeling of fracture. Advances in Applied Mechancis: Multi-scale Theory and Computation 52, submitted.
31
Xu, X., Needleman, A., 1994. Numerical simulations of fast crack growth in brittle solids. Journal of the Mechanics and Physics of Solids 42 (9). Zhang, X., Sloan, S. W., Vignes, C., Sheng, D., 2017a. A modification of the phase-field model for mixed mode crack propagation in rock-like materials. Computer Methods in Applied Mechanics and Engineering 322, 123 – 136. Zhang, X., Vignes, C., Sloan, S. W., Sheng, D., 2017b. Numerical evaluation of the phase-field model for brittle fracture with emphasis on the length scale. Computational Mechanics, 1–16. Zhang, Z., Paulino, G. H., 2005. Cohesive zone modeling of dynamic failure in homogeneous and functionally graded materials. International Journal of Plasticity 21 (6), 1195 – 1254. Zhang, Z., Paulino, G. H., Celes, W., 2007. Extrinsic cohesive modelling of dynamic fracture and microbranching instability in brittle materials. International Journal for Numerical Methods in Engineering 72 (8), 893–923. Zhou, F., Molinari, J.-F., Shioya, T., 2005. A rate-dependent cohesive model for simulating dynamic crack propagation in brittle materials. Engineering Fracture Mechanics 72 (9), 1383 – 1410. Zi, G., Chen, H., Xu, J., Belytschko, T., 2005. The extended finite element method for dynamic fractures. Journal of Sound and Vibration 12, 9–23. Ziaei-Rad, V., Shen, L., Jiang, J., Shen, Y., 2016. Identifying the crack path for the phase field approach to fracture with non-maximum suppression. Computer Methods in Applied Mechanics and Engineering 312, 304 – 321. Zienkiewicz, O. C., Taylor, R. L., 2006. The Finite Element Method for Solid and Structural Mechanics, sixth Edition. Butterworth-Heinemann, Oxford, UK.
32
