A rate-dependent hybrid phase field model for dynamic crack propagation
Duc Hong Doan, Tinh Quoc Bui, Thom Van Do, and Nguyen Dinh Duc
Citation: Journal of Applied Physics 122, 115102 (2017); doi: 10.1063/1.4990073 View online: http://dx.doi.org/10.1063/1.4990073
View Table of Contents: http://aip.scitation.org/toc/jap/122/11 Published by the American Institute of Physics
A rate-dependent hybrid phase field model for dynamic crack propagation
Duc HongDoan,1,a)Tinh QuocBui,2,3,a)ThomVan Do,4and Nguyen DinhDuc1,5
1Advanced Materials and Structures Lab, University of Engineering and Technology, Vietnam National University, Hanoi, Vietnam
2Institute for Research and Development, Duy Tan University, Da Nang City, Vietnam
3Department of Civil and Environmental Engineering, Tokyo Institute of Technology, 2-12-1-W8-22, Ookayama, Meguro-ku, Tokyo 152-8552, Japan
4Department of Mechanics, Le Quy Don Technical University, Hanoi, Vietnam
5Vietnam-Japan University, Vietnam National University, Hanoi, Vietnam
(Received 13 June 2017; accepted 30 August 2017; published online 18 September 2017)
Several models of variational phase field for fracture have been introduced and analyzed to different degrees of applications, and the rate-independent phase field approach has been shown to be a versatile one, but it is not able to accurately capture crack velocity and dissipated energy in dynamic crack propagation. In this paper, we introduce a novel rate-dependent regularized phase field approach to study dynamic fracture behaviors of polymethylmethacrylate materials, in which the rate coefficient is estimated through energy balance, i.e., dynamics release energy, cohesive energy and dissipated energy. The mode-I dynamics crack problem is considered, and its accuracy is validated with respect to experimental data [F. Zhou, Ph.D. dissertation (The University of Tokyo, Japan, 1996)] and other numerical methods, taking the same configuration, material prop- erty, crack location, and other relevant assumptions. The results shed light on the requirement and need for taking the rate-dependent coefficient in dynamic fracture analysis. Published by AIP Publishing.[http://dx.doi.org/10.1063/1.4990073]
I. INTRODUCTION
Prediction of the dynamic crack propagation is an impor- tant research area in materials science, solid state physics, and computational mechanics. Many striking and characteris- tic features of crack propagation have been investigated and are now well established by different approaches including experimental and numerical studies.2,3Their studies are based on the linear elastic fracture mechanics concept pioneered by Griffith4in terms of release of elastic energy and increase of surface energy. Crack growth at low speed is generally char- acterized by macroscopic linear elasticity where the traction- free boundary conditions on crack surfaces are taken into account. However, the main difficulty in the theory of high speed dynamics crack propagation is the lack of a complete theory, which is to connect between the energy flux to the crack tip and crack velocity. In experimental tests, the energy flux to the crack tip cannot directly be measured but needs to be calculated from the solution of the conservation equation of momentum.2,3
Phase field models developed since 20006for modelling of brittle fracture in elastic solids go beyond the limitations of discrete approaches in predicting the nucleation of new cracks and the direction of propagating crack. The founda- tion of phase field models for fracture is the classical Griffith criterion of linear elastic fracture mechanics. The underlying idea of the phase field model is that the crack should propa- gate along a path of least energy.10,16In the phase field the- ory, a sharp crack is smeared out in narrow band, in which the material properties are in smooth transition as a function
of the phase field parameter. Recently, the phase filed model has been successfully applied to solve complicated fracture processes such as quasi-static elastic crack growth,5 quasi- static elasto-plastic crack growth,6 and dynamic crack propagation.7
Most existing phase field models for fracture of brittle materials assume rate-independent conditions and are there- fore not able to capture dynamic effects, which take place, for instance, during the fast crack growth, consequently yielding unrealistic results. In this letter, a rate-dependent phase-field fracture model is introduced, which accounts for an important crack dynamics process namely dissipated energy dependent crack velocity. In this model, the rate coef- ficient is estimated through balance of dynamics release energy, cohesive energy, and dissipated energy. The accu- racy of the proposed rate-dependent approach is confirmed by considering the mode-I dynamic crack propagation in a polymethylmethacrylate (PMMA) rectangular plate, which has been experimentally studied by Zhou,1taking the same configuration, material property, crack location, and other relevant assumptions.
II. COMPUTATIONAL METHOD
We start with the rate-dependent hybrid (isotropic- anisotropic) phase field model as expressed in Eqs.(1)–(3), in which the usual displacement, u, and a new phase field parameter, / are defined. This phase field variable is a scalar damage indicator, which often ranges from 0 for the damage state to 1 for the intact or un-damage state, respectively
q€udivrðu;/Þ ¼0; (1)
a)Authors to whom correspondence should be addressed: doan.d.aa.eng@
gmail.com, [email protected], and [email protected]
0021-8979/2017/122(11)/115102/4/$30.00 122, 115102-1 Published by AIP Publishing.
v1/_ ¼2/HþþGc
l ð/1Þ l2D/
; (2)
Gk¼GcþFðdisÞ1 : (3) Equation(1)is the conservation of momentum in whichq is the mass density, div is the divergence operator, and the super- posed dot represents the partial differentiation with respect to time; wherer;e are the damage-dependent stress and strain tensors obeying stiffness degradation in the bulk
rðu;/Þ:¼/2@w0ðeÞ
@e : (4)
Here,w0ðeÞis the strain energy. Equation(2)represents the evolution of crack, which is driven by a strain history field of maximum positive reference energy. The decomposition of positive reference energy is based on the work reported by Amoret al.16In Eq.(2),lstands for the length scale parame- ter, an important parameter in the phase field modeling, which has been discussed in Ref.9;Hþ introduces a strain history field of maximum positive reference energy;6,7 Gc
denotes the material fracture toughness; and more impor- tantlyvis the kinetic coefficient, which controls the rate of energy dissipation and will be discussed here. During the crack propagation, the rate of energy dissipation varies as a function of not only crack velocity but also the energy release rate, which has been discussed in Karmaet al.8and Silvaet al.9In the case of quasi-static crack propagation, the kinetic coefficient, which was introduced by Mieheet al.,10 is small, which serves as a purely artificial regularization to improve numerical stability. However, when the crack propagates at high velocity, this kinetic coefficient should be considered as a dissipated energy dependent parameter.
Consequently, our present model will therefore involve one more new parameter named askinetic coefficient,v.
In order to close our equations system, a new equation for kinetic coefficient is introduced in Eq.(3), which could be explained as an equation of energy conservation at the loca- tion of the crack tip. Based on the law of crack motion reported in Refs.8and11, for mode-I dynamics crack propa- gation, the energy flow rate in the process zone (Gk: energy release rate) is balanced to the sum of the energy (Gc: Surface energy) used to create new fracture surfaces and the energy (F1(dis): dissipative energy) dissipated near the crack tip.
Based on the balance of all forces acting on crack tip in the direction parallel to the crack propagation direction,Gk, Gc
andF1(dis)are also interpreted as configuration force, cohesive force, and dissipative force, respectively.11 In accordance to Refs.12and9, the energy release rate and dissipative energy are thus expressed as follows:
Gk¼ ð
C
WþT
ð Þnktiui;k
dS þ
ð
V
q€uiui;kqu_iu_i;k
ð Þnktiui;k
½ dV (5)
whereW andTare the elastic and kinetic energies, respec- tively;Cis a close circuit around the moving crack tip, while
V is a volume close byC, andnis an outward normal toC.
Crack tip is defined as the point of/¼0:1 contour with the largest value of x, and the crack tip velocity,v, is defined as a temporal differentiation of the crack tip, which can explic- itly be calculated from the evolution of the phase field profile. The dissipated energy for mode-I dynamics crack propagation is derived as in Eq. (6), relied on considering configuration forces and asymptotic match.9,11
FðdisÞ1 ¼vv1 ð
1
1
ð
1
1
@x/
ð Þ2dx: (6)
From Eqs. (3), (5), and (6), the kinetic coefficient v can finally be expressed as
v¼ v
ð
1
1
ð
1
1
@x/ ð Þ2dx FðdisÞ1 ¼
v ð
1
1
ð
1
1
@x/ ð Þ2dx GkGc
(7) Equation(7)implies that the energy flow rate in the process zone in excess of the surface energy,Gk– Gc, must be dissi- pated in the process zone.8 The rate of energy dissipation, which is represented by the kinetic coefficient,v, is inversely proportional to the excess energy and proportional to the crack propagation velocity. As the dimensionless integral term in Eq.(7)is evaluated in a coordinate system translating with the crack tip, this term is governed by the change of the shape of the crack tip. In the case of low speed crack propa- gation without crack tip blunting and splitting, this term however is almost constant.
In our work, a staggered iterative scheme is adopted to solve the resulting uncoupled equations. First, we calculate the kinetic coefficient by Eq. (5), using the known value of phase field profile, crack tip velocity, and energy release rate at previous time step. Then we solve Eq. (1)for dynamics displacement field by using an in-house adaptive finite ele- ment method using 6-node quadratic triangular elements.
The time-derivative term in Eq. (1) is discretized with the center difference scheme. Then we solve Eq.(2) for evolu- tion of phase field using the kinetic coefficient calculated at the first step. The time-derivative term in Eq. (2) is discre- tized with the backward scheme. Equations (1) and (2) are solved interactively until the phase field is convergent.
III. RESULTS AND DISCUSSION
To show the advance of this new rate-dependent model, we study the dynamic crack propagation in a polymethylme- thacrylate (PMMA) rectangular place. Reference results derived from the experiment,1 and other numerical methods like the cohesive element,13 the nonlocal integral damage model,14 and the conventional phase field model15 are used for comparison purpose. The geometry and boundary condi- tions of the plate are depicted in Fig. 1. The plate of 32 mm wide and 16 mm height is pre-strained by applying displace- ment in the vertical direction at both bottom and top surfaces.
The pre-strain increases linearly from 0 to a defined value dur- ing the first 1ls, then keeps as a constant value of 0.06 mm,
115102-2 Doanet al. J. Appl. Phys.122, 115102 (2017)
which is shown in Fig. 1(b). The mechanical properties of PMMA are E¼309 GPa, ¼0:35, q¼1180 kg/m3, Gc
¼300 J/m2, andl¼0:1 mm.15 In the experimental work,1 Zhou reported that the crack was accelerated until reaching a steady-state propagation velocity of 338 m/s.
Figure2shows the crack velocityversuscrack propaga- tion time. In order to mimic the rate-independent mode, the kinetic coefficient is set by a value of 106m2/N s,10and the results are then shown in Fig.2(a). Without considering the energy dissipated rate, both the present phase-field model and nonlocal integral damage model yield almost the same steady velocity, which is higher than the experimental value reported by Zhou1 at the same loading. This discrepancy was also stated by Wolffet al.,14 and the original source of this dis- crepancy, as previously mentioned, is due to the lack of rate- dependent constitute relation. In Ref. 14, a rate-dependent model with some fitting parameters to reduce their calculated steady velocity of crack is used in order to fit the experimen- tal results. It is important to emphasize here that in Wolff’s model14all of shape parameters were fitting parameters and needed to be connected with the dissipated energy. In Fig.
2(b), the numerical results of the crack tip velocity calculated by the present rate-dependent model is in good agreement with a steady velocity, 338 m/s, measured by Zhou.1
This is the first time a pure and close physical rate- dependent model has been developed to successfully capture dissipated energy dependent behavior of high velocity crack where the conventional rate-independent model is not. In con- trast to crack velocity histories calculated by rate-independent model which crack velocity increased rapidly than oscillated around a constant velocity. Numerical results calculated using the rate-dependent model however show more complicated
behavior. A strong correlation between crack velocity and rate coefficient is illustrated in Fig. 3, representing the rate coefficient varies as the crack tip position. The crack tip velocity decreases with increasing dissipated energy, which is clearly expressed in Eq.(7).
In Fig.4, we further show the correlation between dissi- pated energy, which is normalized by fracture toughness Gc, and the phase field distribution. Basically, the evolution of crack can be interpreted though the representation of the nor- malized dissipated energy computed by the developed rate- dependent phase field model. The crack propagation path is shown in Fig. 4 (upper) through the phase field parameter denoted by isoline of s¼0.1. Figure4(lower) sheds light on the variation of normalized dissipated energy at different loca- tions of the crack-tip. As denoted in Fig.4(upper), the zone A indicates the initial crack, or crack has not propagated yet, cor- responding to a value of the rate coefficient 106 m2/N s as shown in Fig.3. When loaded, the crack starts propagating at which the energy dissipated rapidly, see zone B. In this zone, the crack smears out with the band width increasing to the maximum value and then decreases to the constant band width as the initial state. The changing of the crack band width has a strong correlation with the dissipated energy. Both the crack band width and dissipated energy reached the maximum value in zone B at the same position of the crack tip, see Fig.4. The characteristics between the crack band width and dissipated energy could be explained as follows: as the energy flux at the crack tip increases over the energy that needs to form the new crack surface, which is relevant to the crack velocity, (i) the dissipated energy increases, thus the rate coefficient and crack velocity in the x-direction decreases and (ii) the released energy induced the crack, which can also be seen the
(a) (b)
FIG. 1. Specimen geometry and bound- ary condition (a) and loading condition (b) with t0¼1ls.
FIG. 2. Crack velocity histories calculated by: (a) rate-independent model: the solid line is calculated result by the rate-independent model; the dash-line repre- sents a steady velocity, 554 m/s, calculated by the nonlocal integral damage with the rate-independent model by Wolffet al.;14(b) rate-dependent model: the solid line is calculated result by the present rate-dependent model; the dash-line represents a steady velocity, 338 m/s, measured by Zhou.1
in y-direction, decreases, which could be thought as sub- surface micro-branches, which was observed in the experiment by Sharon and Fineberg.17,18A strong oscillation in the crack velocity in zone B can also be found in Fig.2(b). This trend however was also experimentally observed in Refs.17and18.
Here, under given load, the crack does not show to be branching since the energy dissipated at crack-tip is not strong enough. The variation of the dissipated energy and crack propa- gation in the zones C is similar to zone B but with smaller varia- tion of the crack band width. At zone D, the crack reaches the saturated velocity without the oscillation in the crack band width.
It is obvious that the propagation of crack can be interpreted through a rigorous relation among the crack velocity, dissipated energy, and rate-coefficient as described in Eqs.(1)–(3).
IV. CONCLUSIONS
In this study, we introduce a rate-dependent hybrid phase- field fracture model for dynamic crack propagation, exploring the dissipated energy dependent crack velocity, one of the important aspects of dynamic fracture. This new rate- dependent model, in contrast to the rate-independent one, is able to accurately capture crack velocity and dissipated energy under dynamic loading, substantially illustrated through the mode-I dynamic crack problem. Based on our results, we con- firm that the rate-dependent coefficient plays an important role and that must be taken into account in dynamic fracture analy- sis. Numerical results also shed light on the dominance of the developed rate-dependent phase-field model over the rate- independent one in terms of dynamic crack propagation.
ACKNOWLEDGMENTS
D.H.D. and N.D.D. gratefully acknowledge the support of Vietnam National University (VNU), Hanoi, under Project No. QG.17.45. The authors would like to thank all reviewers for their valuable comments and suggestions.
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FIG. 4. Correlation between phase field distribution and dissipated energy.
Upper picture: snapshot of phase field distribution which varies continuously from 0 (red region: damaged) to 1 (blue region: intact) and note thats<0.1 is indicated in this study as its bandwidth represents the area inside the white curves. Lower graph: plot of dissipated energy,Gk–Gc, as a function of nor- malized by surface energy,Gc, as a function of the crack tip position.
FIG. 3. Rate coefficient varies as a function of the crack tip position.
115102-4 Doanet al. J. Appl. Phys.122, 115102 (2017)
