- Research article
- Open Access
A simple and unified implementation of phase field and gradient damage models
- E. Azinpour^{1},
- J. P. S. Ferreira^{1},
- M. P. L. Parente^{1} and
- J. Cesar de Sa^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s40323-018-0106-7
© The Author(s) 2018
- Received: 14 December 2017
- Accepted: 24 April 2018
- Published: 21 May 2018
Abstract
In this work, the analogous treatment between coupled temperature–displacement problems and material failure models is explored within the context of a commercial software (Abaqus®). The implicit gradient Lemaitre damage and phase field models are implemented utilizing the software underlying capabilities for coupled temperature–displacement problems. The heat conduction equation is made compatible with the diffusive regularization of such material models and calculations are carried out at the material point level. This bypasses the need to implement explicitly the weak form resultant from the coupling between the momentum conservation and the evolution of the diffusive field. Throughout benchmarking examples, the proposed methodology is assessed and validated by investigating typical issues risen from the considered local inelastic-based deformation models, such as mesh dependency and the difficulties to predict cracked regions.
Keywords
- Material failure
- Commercial software
- Implicit gradient Lemaitre damage model
- Phase field model
- Diffusive regularization
- Mesh dependency
Introduction
The emergence of the so-called ‘regularized’ solutions for damage and failure in engineering materials has evolved considerably in recent years. They constitute appealing simplifications of the microstructural complexity, usually resorting to non-local and gradient theories. To ensure well-posedness of the set of partial differential equations to be solved, which is affected by the presence of softening at the constitutive level, these solutions abandon the principle of local action by coupling a diffusion equation of non-local variables with the momentum balance equation. The resultant procedure acts as a spatial regularization of local constitutive relations for failure, usually with the recourse of a characteristic length associated with the size of the non-local support region definition. Herein, two types of regularization are explored, namely: the non-local implicit gradient of the Lemaitre damage model [1] and the phase field model [2] that approximate crack evolution under inelastic deformations.
Prediction of initiation and propagation of cracks defines a highly crucial task in behavior assessment of ductile metallic structures, which requires the development of accurate and robust constitutive models that trigger the formation of meso-cracks initiated by the material internal progressive degradation due to nucleation, growth, and coalescence of microvoids. Within the context of coupled models, which take into account that degradation continuously coupled with the evolving deformation, two major methodologies are commonly adopted: micromechanical or continuous damage mechanics approaches. The micromechanical-based methodologies include at the constitutive level the effect of internal degradation through material parameters calibrated with a set of experimental tests. Many were inspired on the early work of Rice and Tracey [3], that focused primarily on the microscopic evolution of a spherical void in a rigid perfectly-plastic matrix of the material, and later on the work of Gurson [4], followed by the work of Tvergaard and Needleman [5] that represent internal degradation as a volume void fraction (porosity). Another insight derived from the Continuum Damage Mechanics approach, built over the concepts defined in the seminal work of Kachanov [6], which defines the internal thermodynamically-consistent damage field phenomenologically continuously evolving to a threshold value, was firstly introduced by Lemaitre and Chaboche in [7–9]. Numerical implementations soon followed, both in the context of small strain [10] and large strain [11] analyses.
Local continuum description of damage typically may lead to an ill-posed boundary value problem as under a softening regime the governing partial differential equations may locally lose their ellipticity or change its hyperbolic character, in static and dynamical problems, respectively. This may result in mesh (geometrical discretization) sensitive solutions, where the localized zones and internal variables are highly dependent on the mesh size and discretization alignment. To avoid these inconsistencies and restore mesh objectivity, non-local approaches may be adopted by adding regularization or averaging procedures associated with a length scale effect, whether via an integral-based non-local methodology, as in Pijaudier-Cabot and Bažant [12] or gradient-type non-local methods, as in Peerlings et al. [13]. The latter has been utilized in many contexts, namely, in quasi-brittle fracture [14, 15], small strain [16] and finite strain elastoplasticity [17], finite strain elastoplasticity coupled with damage [18] or in ductile damage frameworks [19, 20], referring only to some more recent contributions.
Another type of regularization, known as the phase field method, originally departing from fracture mechanics concepts, aims to assess the evolution of macrocracks in a diffusive topology using methodologies associated with phase transition. Its implementation in the context of the finite element method allows for the possibility of tracking the cracks without any modification in the mesh grid lines which is still an utmost issue in the discrete crack approaches. The genesis of this concept was founded in the variational description of brittle fracture by Francfort and Marigo [21] and eliminates the requirement to describe a well-defined crack path. The concept was extended by the regularization approach of Bourdin et al. [22] and by the \(\Gamma \)-convergent approximations of Mumford and Shah [23].
This diffusive fracture description was further elaborated in the context of brittle fracture by, among others, Karma et al. [24], Miehe et al. [25, 26], Kuhn and Muller [27], Pham et al. [28] and Borden et al. [29, 30] (readers may refer to a concise overview of the main phase field brittle fracture methods by Ambati et al. [31]). The thermodynamically-consistent model addressed by Miehe et al. [25] was later applied to a cohesive fracture approach [32] and to crack propagation in heterogeneous microstructures [33]. Following the study of stress fields around holes in specimens via a coupled elastoplastic phase field framework by Guo et al. [34], this approach has evolved to the context of inelastic deformations, in which a considerable amount of research activities in this area can be found. To name a few, the non-local gradient damage phase field based model by Voyiadjis et al. [35], the phase field elastoplastic models by Duda et al. [36] and Ambati et al. [37], the crystal plasticity phase field coupled model of Padilla et al. [38], the phase field based anisotropic damage model of Mozaffari et al. [39] and the variational based, large deformation plasticity-phase field model by Miehe et al. [40, 41] and rate-dependent plasticity phase field model of Badnava et al. [42] mostly concentrated on approximating the post-peak material behavior via a phase field methodology. Also in [43], de Borst et al. studied several similar and different aspects of gradient enhanced damage-based approaches and the phase field framework and a discussion has made over the solution of the broadening of the damaged zone in the wake of the crack tip with these methods.
The present study addresses a methodology for solving the diffusion equation of the phase field model and gradient-enhanced non-local damage model, in the context of an existing commercial software. It takes the advantage of built-in heat equation solver in an existing software code, Abaqus/Standard in the present case, bypassing the need to establish explicitly the weak form of the governing equations containing the momentum balance and the diffusion equation to solve separately for displacement and non-local or phase field variables. In fact, this procedure could be used with other commercial softwares with similar capabilities. This work is organized as follows: “Formulation” section builds the relation between the heat conduction problem and diffusion equation of gradient problems, and explains the gradient non-local model based on Lemaitre phenomenological ductile damage model and the phase field model. Numerical implementation and results of some illustrative examples are presented in “Results and discussion” section, and finally conclusions are drawn in “Concluding remarks” section and a shortened version of routine to implement the non-local gradient damage model is provided.
Formulation
Diffusion equation forms for coupled temperature displacement, gradient-damage, and phase-field models
Name | Field variable | “Diffusion” coefficient \({\varvec{\varGamma }}\) | “Flux” term \({\varvec{\nabla \psi }}\) | “Source” term \({{\varvec{h}}}_{{\varvec{s}}}\) |
---|---|---|---|---|
Temperature | T | \(\kappa \) | \(\nabla T\) | \(\dot{q}\) |
Gradient-damage | D | \(l_n ^{2 (1)}\) | \(\nabla \bar{D}\) | \(\bar{D} -D\) |
Phase-field | d | \(\mathcal{G}_c l_d ^{(1)}\) | \(\nabla d\) | \(\left( {\mathcal{G}_c /l_d } \right) d-2\mathcal{H}\left( {1-d} \right) \) |
Lemaitre damage model coupled with plasticity
The constitutive modeling of the fully-coupled damage-plasticity model in [1] is recovered in this section, considering its extensive application in describing ductile damage in metallic materials. This model is often categorized within the Continuous Damage Mechanics material behavior description as the damage is phenomenologically introduced at the macroscopic constitutive level by internal variables whose evolution mimic the nucleation, growth and coalescence of internal micro-voids. Mostly it is based on the hypothesis of strain equivalence and the associated concept of effective stress. Herein, the simplified version of the Lemaitre thermodynamically-consistent damage model proposed by De Souza Neto et al. [45], in the context of small strains, is adopted for simplicity.
Gradient regularization of the local damage model
Integration algorithm of the non-local resolution
The integration procedure of the explained non-local gradient model is based on the conventional elastic predictor-plastic corrector, which is implemented via an operator-split scheme. The solution of the multi-field problem of Eq. (18) consists of the solution of mechanical problem based on the updated stresses and the local damage field, and the calculation of the nonlocal field from the gradient equation. The plastic correction in the mechanical problem is performed via the known iterative Newton Raphson algorithm to solve the return mapping equation, as is proved to have quadratic convergence rates.
Phase field approximation in fracture
Recently, in the context of fracture analysis of structures, the phase field approach appeared as an alternative to classical discrete crack approaches due to their intrinsic difficulty in dealing with more complex crack topologies such as kinking or branching patterns. This methodology utilizes an order parameter called phase field, which distinguishes between the intact and fully-damaged regions of the deformed body, using physical phase transformation concepts and conceptually it may be more closely related to continuous evolution of internal degradation in damage mechanics.
In its implementation, this method avoids the necessity to deal with displacement jumps due to a sharp description of cracks in discrete-based approaches, as all the calculations are performed on the original mesh without any ad hoc criteria. Adding to these the straightforward extensibility of the problem to higher dimensions in the same manner of lower dimensions, this approach has been widely used in the fracture analyses and more recently in prediction of evolution of inelastic material imperfections. As for the numerical implementation, the standard finite element shape functions can be utilized to interpolate the displacement and the diffusive field which, comparatively, bypasses the complexities of employment of enriched shape functions in methods such as XFEM. Also the treatment of the diffusion equation can be closely related to non-local gradient approach as both models use spatial gradients of the regularizing field.
Phase field model of brittle fracture
Phase field model in a ductile failure context
Integration algorithm
The integration procedure of this phase field modelling is constructed upon the elastic predictor-plastic corrector algorithm in an analogous way to the presented nonlocal damage model. The calculation of the stress field is carried out based on the von Mises rate-independent plasticity model that may be closely related to the isotropically-hardening plasticity model in [45]. The coupled plasticity model is implemented with UMAT and HETVAL subroutines in Abaqus/Standard (in the same manner that is explained earlier and in “Appendix” for the gradient damage model). In each increment, the phase field value is frozen and retrieved in routine as a temperature field. Given the flux value based on the updated history field from the previous increment in the UMAT code, the updated phase field value is computed through the HETVAL subroutine. This staggered-type solution procedure bypasses the need to calculate the coupled phase-field-displacement tangent terms as is presented in the UEL monotonic scheme in [47].
Results and discussion
The capability of the presented diffusive strategy is assessed herein, through the analysis of two common benchmarks and by observation of force-deflection diagrams, damage evolution, and crack propagation prediction.
Notched specimen tensile test
First benchmark is a notched specimen under tension, with the geometry and boundary conditions depicted in Fig. 3. This example aims to observe the propagation of the damage and assess the nonlocal gradient damage framework robustness and applicability of the proposed methodology in the post-peak regime to circumvent mesh sensitivity. Due to the symmetry, only one quarter of the specimen is modelled and discretized, assuming distinct mesh discretizations (identically adopted from [48]) and using 4-noded plane strain quadrilateral elements. The material properties used for analysis are represented in Table 2, considering that the same length scale is used and treated as a material parameter for all mesh topologies.
Material parameters for the notched specimen
Name | Value |
---|---|
Young’s modulus \(\left( E \right) \) | 210 \(\hbox {GPa}\) |
Poisson’s ratio \(\left( {\upnu } \right) \) | 0.3 |
Hardening law | \(\sigma _y \left( \alpha \right) =700+300\left( \alpha \right) ^{0.3}\hbox {MPa}\) |
Lemaitre damage denominator \(\left( {r_1 } \right) \) | \(3.0\,\,\hbox {MPa}\) |
Lemaitre damage exponent \(\left( {r_2 } \right) \) | 1.0 |
Non-local model length scale \(\left( {l_n } \right) \) | 0.15 mm |
Single edged notched shear test
The proposed methodology for the phase field model is analyzed via a single edged notched test (SENT) specimen as can be found in [40, 43], among other brittle fracture based articles. The geometric setup of the 2D specimen is adopted from [37] with \(L=5\) mm and a horizontal displacement-controlled loading applied on the top edge of the specimen, as is depicted in Fig. 9a. Notice that the notch is considered as a geometrical entity with its tip located at the center of the specimen where, based on experimental evidence, the crack begins to propagate. The aim here is to observe the ability of the proposed algorithm that uses the built-in heat equation solver, by making quantitative comparisons with existing data in literature.
Material parameters for the single edge notched test
Name | Value |
---|---|
Young’s modulus \(\left( E \right) \) | \(180\,\,\hbox {GPa}\) |
Poisson’s ratio \(\left( {\upnu } \right) \) | 0.28 |
Yield stress (\(\sigma _y \)) | \(443\,\,\hbox {MPa}\) |
Hardening modulus | \(300\,\,\hbox {MPa}\) |
Phase field model length scale \(\left( {l_d } \right) \) | 0.06 |
Critical energy release rate (\(\mathcal{G}_c \)) | \(15\,\,\hbox {N}/\hbox {mm}\) |
Residual stiffness | \(1.0 \,e^{-5}\) |
\(\beta _e \) and \(\beta _p \) | 1.0 |
Plastic work threshold \((W_0 )\) | 1.0 |
Concluding remarks
In this study, the evolution of damage and failure, via non-local gradient damage and phase field models, respectively, was simulated resorting to an analogy to heat diffusion in solids, using the built-in thermo-mechanically coupled finite-element solution procedure in Abaqus. The developed procedure was justified and assessed in terms of qualitative verification of crack propagation patterns and its capability of avoiding mesh dependence pathologies.
The utilization of this approach may be viewed as a simple alternative to include the regularization procedures associated with both gradient and phase field models in commercial codes with no need of cumbersome implementations of explicit weak form derivations within required lengthy user files.
Declarations
Authors’ contributions
All authors contributed in the material modelling and simulation procedure of the article. EA carried out the user coding of the damage model and phase field method as well as the benchmark testing of the single edge notched test, and drafted the manuscript. JPSF contributed in the utilization of the heat equation solver of Abaqus in modelling the gradient regularization, and performed the numerical assessment on the notched specimen tensile test with the respective gradient damage model. JCS and MPLP supervised the study and did the corrections on the manuscript draft. All authors read and approved the final manuscript.
Acknowledgements
Authors gratefully acknowledge the funding of Project NORTE-01-0145-FEDER-000022—SciTech—Science and Technology for Competitive and Sustainable Industries, cofinanced by Programa Operacional Regional do Norte (NORTE2020), through Fundo Europeu de Desenvolvimento Regional (FEDER) and the project Grant SFRH/BD/107860/2015 of the Portuguese foundation of science.
Competing interests
The authors declare that they have no competing interests.
Availability of data and materials
Not applicable.
Consent for publication
Not applicable.
Ethics approval and consent to participate
Not applicable.
Funding
Not applicable.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Lemaitre J. A continuous damage mechanics model for ductile fracture. J Eng Mater Technol. 1985;107:83–9. https://doi.org/10.1115/1.3225775.View ArticleGoogle Scholar
- Bourdin B, Francfort GA, Marigo JJ. The variational approach to fracture. J Elast. 2008;91(1–3):5–148.MathSciNetView ArticleMATHGoogle Scholar
- Rice JR, Tracey DM. On the ductile enlargement of voids in triaxial stress fields*. J Mech Phys Solids. 1969;17:201–17. https://doi.org/10.1016/0022-5096(69)90033-7.View ArticleGoogle Scholar
- Gurson AL. Continuum theory of ductile rupture by void nucleation and growth: part I-yield criteria and flow rules for porous ductile media. J Eng Mater Technol. 1977;99:2. https://doi.org/10.1115/1.3443401.View ArticleGoogle Scholar
- Tvergaard V, Needleman A. Analysis of the cup-cone fracture in a round tensile bar. Acta Metall. 1984;32:157–69. https://doi.org/10.1016/0001-6160(84)90213-X.View ArticleGoogle Scholar
- Kachanov L. Time of the rupture process under creep condition. Izv Akad Nauk SSSR Otd Tekhn Nauk. 1958;8:26–31.Google Scholar
- Lemaitre J. A continuous damage mechanics model for ductile fracture. J Eng Mater Technol. 1985;107:83. https://doi.org/10.1115/1.3225775.View ArticleGoogle Scholar
- Chaboche JL. Continuum damage mechanics: part 1—general concepts. J Appl Mech. 1988;55:59.View ArticleGoogle Scholar
- Chaboche JL. Continuum damage mechanics: part II—damage growth, crack initiation, and crack growth. J Mech. 1988;55:65–72. https://doi.org/10.1115/1.3173662.Google Scholar
- Benallal A, Billardon R, Doghri I. An integration algorithm and the corresponding consistent tangent operator for fully coupled elastoplastic and damage equations. Commun Appl Numer Methods. 1988;4:731–40. https://doi.org/10.1002/cnm.1630040606.View ArticleMATHGoogle Scholar
- de Souza Neto EA, Perić D, Owen DRJ. Continuum modelling and numerical simulation of material damage at finite strains. Arch Comput Methods Eng. 1998;5:311–84. https://doi.org/10.1007/BF02905910.MathSciNetView ArticleGoogle Scholar
- Pijaudier-Cabot G, Bažant ZP. Nonlocal damage theory. J Eng Mech. 1987;113(10):1512. https://doi.org/10.1061/(ASCE)0733-9399.View ArticleMATHGoogle Scholar
- Peerlings RHJ, De Borst R, Brekelmans WAM, de Vree JHP. Gradient enhanced damage for quasi-brittle materials. Int J Numer Methods Eng. 1996;39:3391–403. https://doi.org/10.1002/(SICI)1097-0207(19961015)39:19%3c3391::AID-NME7%3e3.0.CO;2-D.View ArticleMATHGoogle Scholar
- De Borst R, Pamin J, Geers MGD. On coupled gradient-dependent plasticity and damage theories with a view to localization analysis. Eur J Mech A/Solids. 1999;18:939–62. https://doi.org/10.1016/S0997-7538(99)00114-X.View ArticleMATHGoogle Scholar
- Comi C. Computational modelling of gradient-enhanced damage in quasi-brittle materials. Mech Cohesive-Frictional Mater. 1999;4:17–36. https://doi.org/10.1002/(SICI)1099-1484(199901)4:1%3c17::AID-CFM55%3e3.0.CO;2-6.View ArticleGoogle Scholar
- Engelen RAB, Geers MGD, Baaijens FPT. Nonlocal implicit gradient-enhanced elasto-plasticity for the modelling of softening behaviour. Int J Plast. 2002;19:403–33. https://doi.org/10.1016/S0749-6419(01)00042-0.View ArticleMATHGoogle Scholar
- Geers MGD, Ubachs RLJM, Engelen RAB. Strongly non-local gradient-enhanced finite strain elastoplasticity. Int J Numer Methods Eng. 2003;56:2039–68. https://doi.org/10.1002/nme.654.View ArticleMATHGoogle Scholar
- Areias PMA, de Sá JC, António CC. A gradient model for finite strain elastoplasticity coupled with damage. Finite Elem Anal Des. 2003;39:1191–235. https://doi.org/10.1016/S0168-874X(02)00164-6.View ArticleGoogle Scholar
- Almansba M. Isotropic elastoplasticity fully coupled with non-local damage. Engineering. 2010;2:420–31. https://doi.org/10.4236/eng.2010.26055.View ArticleGoogle Scholar
- Javani HR, Peerlings RHJ, Geers MGD. Consistent remeshing and transfer for a three dimensional enriched mixed formulation of plasticity and non-local damage. Comput Mech. 2014;53:625–39. https://doi.org/10.1007/s00466-013-0922-z.MathSciNetView ArticleMATHGoogle Scholar
- Francfort GA, Marigo JJ. Revisiting brittle fracture as an energy minimization problem. J Mech Phys Solids. 1998;46:1319–42. https://doi.org/10.1016/S0022-5096(98)00034-9.MathSciNetView ArticleMATHGoogle Scholar
- Bourdin B, Francfort GA, Marigo JJ. Numerical experiments in revisited brittle fracture. J Mech Phys Solids. 2000;48:797–826. https://doi.org/10.1016/S0022-5096(99)00028-9.MathSciNetView ArticleMATHGoogle Scholar
- Mumford D, Shah J. Optimal approximations by piecewise smooth functions and associated variational problems. Commun Pure Appl Math. 1989;42:577–685. https://doi.org/10.1002/cpa.3160420503.MathSciNetView ArticleMATHGoogle Scholar
- Karma A, Kessler DA, Levine H. Phase-field model of mode III dynamic fracture. Phys Rev Lett. 2001;87(4):045501. https://doi.org/10.1103/PhysRevLett.87.045501.View ArticleGoogle Scholar
- Miehe C, Welschinger F, Hofacker M. Thermodynamically-consistent phase field models of fracture: variational principles and multi-field FE implementations. J Numer Methods Eng. 2009;83(10):1273–311.MathSciNetView ArticleMATHGoogle Scholar
- Miehe C, Hofacker M, Welschinger F. A phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits. Comput Methods Appl Mech Eng. 2010;199:2765–78. https://doi.org/10.1016/j.cma.2010.04.011.MathSciNetView ArticleMATHGoogle Scholar
- Kuhn C, Müller R. A continuum phase field model for fracture. Eng Fract Mech. 2010;77:3625–34. https://doi.org/10.1016/j.engfracmech.2010.08.009.View ArticleGoogle Scholar
- Pham K, Amor H, Marigo JJ, Maurini C. Gradient damage models and their use to approximate brittle fracture. Int J Damage Mech. 2011;20:618–52. https://doi.org/10.1177/1056789510386852.View ArticleGoogle Scholar
- Borden MJ, Verhoosel CV, Scott MA, et al. A phase-field description of dynamic brittle fracture. Comput Methods Appl Mech Eng. 2012;217–220:77–95. https://doi.org/10.1016/j.cma.2012.01.008.MathSciNetView ArticleMATHGoogle Scholar
- Borden MJ, Hughes TJR, Landis CM, Verhoosel CV. A higher-order phase-field model for brittle fracture: formulation and analysis within the isogeometric analysis framework. Comput Methods Appl Mech Eng. 2014;273:100–18. https://doi.org/10.1016/j.cma.2014.01.016.MathSciNetView ArticleMATHGoogle Scholar
- Ambati M, Gerasimov T, De Lorenzis L. A review on phase-field models of brittle fracture and a new fast hybrid formulation. Comput Mech. 2014;55:383–405. https://doi.org/10.1007/s00466-014-1109-y.MathSciNetView ArticleMATHGoogle Scholar
- Verhoosel CV, de Borst R. A phase-field model for cohesive fracture. Int J Numer Methods Eng. 2013;96:43–62. https://doi.org/10.1002/nme.4553.MathSciNetView ArticleMATHGoogle Scholar
- Nguyen TT, Yvonnet J, Zhu QZ, et al. A phase field method to simulate crack nucleation and propagation in strongly heterogeneous materials from direct imaging of their microstructure. Eng Fract Mech. 2015;139:18–39. https://doi.org/10.1016/j.engfracmech.2015.03.045.View ArticleGoogle Scholar
- Guo XH, Shi SQ, Ma XQ. Elastoplastic phase field model for microstructure evolution. Appl Phys Lett. 2005;87:1–3. https://doi.org/10.1063/1.2138358.Google Scholar
- Voyiadjis GZ, Mozaffari N. Nonlocal damage model using the phase field method: theory and applications. Int J Solids Struct. 2013;50:3136–51. https://doi.org/10.1016/j.ijsolstr.2013.05.015.View ArticleGoogle Scholar
- Duda FP, Ciarbonetti A, Sánchez PJ, Huespe AE. A phase-field/gradient damage model for brittle fracture in elastic-plastic solids. Int J Plast. 2014;65:269–96. https://doi.org/10.1016/j.ijplas.2014.09.005.View ArticleGoogle Scholar
- Ambati M, Gerasimov T, De Lorenzis L. Phase-field modeling of ductile fracture. Comput Mech. 2015;55:1017–40. https://doi.org/10.1007/s00466-015-1151-4.MathSciNetView ArticleMATHGoogle Scholar
- Hernandez Padilla CA, Markert B. A coupled phase-field model for ductile fracture in crystal plasticity. PAMM. 2014;14:441–2. https://doi.org/10.1002/pamm.201410208.View ArticleMATHGoogle Scholar
- Mozaffari N, Voyiadjis GZ. Phase field based nonlocal anisotropic damage mechanics model. Phys D Nonlinear Phenom. 2015;308:11–25. https://doi.org/10.1016/j.physd.2015.06.003.MathSciNetView ArticleMATHGoogle Scholar
- Miehe C, Teichtmeister S, Aldakheel F. Phase-field modelling of ductile fracture: a variational gradient-extended plasticity-damage theory and its micromorphic regularization. Philos Trans A Math Phys Eng Sci. 2016;374:20150170. https://doi.org/10.1098/rsta.2015.0170.MathSciNetView ArticleMATHGoogle Scholar
- Miehe C, Hofacker M, Schaenzel LMM, Aldakheel F. Phase field modeling of fracture in multi-physics problems. Part II. Coupled brittle-to-ductile failure criteria and crack propagation in thermo-elastic-plastic solids. Comput Methods Appl Mech Eng. 2015;294:1–37. https://doi.org/10.1016/j.cma.2014.11.017.MathSciNetView ArticleGoogle Scholar
- Badnava H, Etemadi E, Msekh M. A phase field model for rate-dependent ductile fracture. Metals (Basel). 2017;7:180. https://doi.org/10.3390/met7050180.View ArticleGoogle Scholar
- de Borst R, Verhoosel CV. Gradient damage vs phase-field approaches for fracture: similarities and differences. Comput Methods Appl Mech Eng. 2016;312:78–94. https://doi.org/10.1016/j.cma.2016.05.015.MathSciNetView ArticleGoogle Scholar
- Geers MGD, de Borst R, Brekelmans WAM, Peerlings RHJ. Strain-based transient-gradient damage model for failure analyses. Comput Methods Appl Mech Eng. 1998;160:133–53. https://doi.org/10.1016/S0045-7825(98)80011-X.View ArticleMATHGoogle Scholar
- de Souza Neto EA, Peric D, Owen RJ. Computational methods for plasticity. New York: Wiley; 2008.View ArticleGoogle Scholar
- Borden MJ, Hughes TJR, Landis CM, et al. A phase-field formulation for fracture in ductile materials: finite deformation balance law derivation, plastic degradation, and stress triaxiality effects. Comput Methods Appl Mech Eng. 2016;312:130–66. https://doi.org/10.1016/j.cma.2016.09.005.MathSciNetView ArticleGoogle Scholar
- Msekh MA, Sargado JM, Jamshidian M, et al. Abaqus implementation of phase-field model for brittle fracture. Comput Mater Sci. 2015;96:472–84. https://doi.org/10.1016/j.commatsci.2014.05.071.View ArticleGoogle Scholar
- Andrade FX, de Sá JC, Pires FA. Assessment and comparison of non-local integral models for ductile damage. Int J Damage Mech. 2014;23:261–96. https://doi.org/10.1177/1056789513493103.View ArticleGoogle Scholar