On phase field modeling of ductile fracture

Kuhn, Charlotte; Noll, Timo; Müller, Ralf

doi:10.1002/gamm.201610003

Cited by 118 publications

(69 citation statements)

References 30 publications

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“…The sample has been discretized in 80x80x1 brick elements and mounted, such that plane strain conditions apply. For the simulations below, the model has been transformed in a dimensionless form as shown in [5]. …”

Section: Numerical Examplesmentioning

confidence: 99%

A Monolithic Solution Scheme for a Phase Field Model of Ductile Fracture

Noll

Kuhn

Müller

2017

Proc Appl Math and Mech

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A phase field model for elastic-plastic fracture is presented, which is based on an energy functional composed of an elastic energy contribution, a plastic dissipation potential and a fracture energy. The coupling of the mechanical fields with the fracture field is modeled by a degradation function. Due to the proposed coupling it is possible to solve the global system of differential equations in a monolithic iterative solution scheme. Numerical simulations are presented, where the choice of the degradation function is investigated and a staggered is compared to monolithic iteration scheme. The proposed ductile phase field fracture model with linear isotropic hardening is an extension of an elastic phase field fracture model [1], which can be regarded as a regularized approximation of the variational formulation of brittle fracture by [2]. The crack field variable s ranging from 1 for undamaged to 0 for fully damaged material, together with the displacement vector represent the set of independent variables in the energy density formulation Ψ that consists of three parts: the Griffith-type fracture energy Ψ fr along with the contributions of elastic Ψ el and plastic deformation Ψ pl . Compared to the linear elastic model a dissipation potential Π pl accounting for the accumulated plastic strain, represented by the internal variable α, is added to the elastic energy contribution W el . The elastic strain is the difference between the total and the plastic strain ε p , the second internal variable. A small strain scenario is assumed. The fourth order stiffness tensor C is assumed to be isotropic. The plastic material behavior is characterized by the plastic variables, the linear hardening modulus H and the initial yield stress σ Y . The degradation function g(s) models the decrease in stiffness of broken material as well as the coupling between plastic deformation and the evolution of fracture. The dimensionless parameter η is introduced for numerical reasons. Parameter G c describes the fracture resistance and can be related to the fracture toughness in the elastic material. The width of the transition zone between broken and undamaged material is controlled by the length parameter .withandEquilibrium of stresses has to hold, wherein the stresses are computed from the energy density formulationThe Ginzburg-Landau equation describes the evolution of the fracture fieldThe parameter M in this viscous approximation has to be chosen high enough in order to approximate quasi-static conditions, where δΨ δs = 0. The hardening force q, which is driving the plasticity is given as

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Section: Numerical Examplesmentioning

confidence: 99%

A Monolithic Solution Scheme for a Phase Field Model of Ductile Fracture

Noll

Kuhn

Müller

2017

Proc Appl Math and Mech

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“…An analysis of the spatially homogeneous solution of the model in one dimension leads to the insight that the plastic model parametersσ Y andH do not coincide with the effective constitutive phase field parametersσ pf Y andH pf observed in the phase field fracture model [2]. Crack nucleation in this scenario is observed when the spatially homogeneous solution becomes unstable.…”

Section: Effective Elastic Materials Propertiesmentioning

confidence: 99%

“…Crack nucleation in this scenario is observed when the spatially homogeneous solution becomes unstable. Since the fracture field is smaller than one at this stage, the effective yield stressσ pf y is already lower than the nominal yield stressσ Y due to the coupling between crack field and stress, see (2). This can lead to an effective softening of the material.…”

Section: Effective Elastic Materials Propertiesmentioning

confidence: 99%

Investigation of a Phase Field Model for Elasto‐plastic Fracture

Noll

Kuhn

Müller

2016

Proc Appl Math and Mech

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Phase field modelling of brittle fracture is very well understood today. However, the attempts of investigation of elastoplastic fracture by the phase field approach are limited. This contribution deals with the investigation of a phase field model for elasto-plastic fracture. Based on a free energy density comprising elastic, fracture and plastic contributions, the model describes an extension of the linear elastic model towards von Mises plasticity. In this work it is analyzed numerically to which extend analytical findings concerning the interpretation of the model parameters in 1D are transferable to 2D scenarios. The main ingredient of the potential Ψ of the ductile fracture model is the potential for brittle fracture from [1]. The ductile potentialΨ, in which the infinitesimal total strain tensorε is replaced by the elastic strain tensorε e =ε −ε p , and an additional term accounting for plastic deformation is added compared to the brittle potential is given bȳwhere (·) indicates the dimensionless form of the respective quantity. The crack field is represented by s. While s = 1 indicates undamaged material, a crack is represented by s = 0. The plastic strain tensorε p andᾱ, accounting for accumulated plastic deformation are internal variables. The regularization parameter is represented by¯ . For numerical reasons the residual stiffness η is introduced. The initial yield stress and the linear isotropic hardening coefficient are represented byσ Y andH, respectively. The Cauchy stress tensorσ has to satisfy the mechanical equilibrium conditionwhereC denotes the fourth order stiffness tensor. The evolution of the fracture field is determined by the time dependent Ginzburg-Landau evolution equatioṅwhereM is a positive kinetic coefficient. The driving force for plasticity and the von Mises type yield criterion are given bywheres denotes the deviatoric part of the stress tensor. Effective elastic material propertiesAn analysis of the spatially homogeneous solution of the model in one dimension leads to the insight that the plastic model parametersσ Y andH do not coincide with the effective constitutive phase field parametersσ pf Y andH pf observed in the phase field fracture model [2]. Crack nucleation in this scenario is observed when the spatially homogeneous solution becomes unstable. Since the fracture field is smaller than one at this stage, the effective yield stressσ pf y is already lower than the nominal yield stressσ Y due to the coupling between crack field and stress, see (2). This can lead to an effective softening of the material. The relations between the nominal plastic parameters and the effective parameters at the onset of yielding are obtained from the one dimensional analysis as where s Y is the spatially homogeneous solution of the crack field at the onset of yielding in the 1D scenario.

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“…These works have been subsequently extended to include geometric nonlinearities in [10,51,19]. Further insights on the role of the constitutive parameters on the crack path are given in [44]. Imposing an intrinsic link between damage and plastic dissipation, [11,39] proposed a phase-field model of plastic slip-lines.…”

Section: Introductionmentioning

confidence: 99%

Coupling damage and plasticity for a phase-field regularisation of brittle, cohesive and ductile fracture: One-dimensional examples

Alessi

Marigo

Maurini

et al. 2018

International Journal of Mechanical Sciences

113

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Plasticity and damage are two fundamental phenomena in nonlinear solid mechanics associated to the development of inelastic deformations and the reduction of the material stiffness. Alessi et al. [4] have recently shown, through a variational framework, that coupling a gradient-damage model with plasticity can lead to macroscopic behaviours assimilable to ductile and cohesive fracture. Here, we further expand this approach considering specific constitutive functions frequently used in phase-field models of brittle fracture. A numerical solution technique of the coupled elastodamage-plasticity problem, based on an alternate minimisation algorithm, is proposed and tested against semi-analytical results. Considering a one-dimensional traction test, we illustrate the properties of four different regimes obtained by a suitable tuning of few key constitutive parameters. Namely, depending on the relative yield stresses and softening behaviours of the plasticity and the damage criteria, we obtain macroscopic responses assimilable to (i) brittle fracturè a la Griffith, (ii) cohesive fractures of the Barenblatt or Dugdale type, and (iii) a sort of cohesive fracture including a depinning energy contribution. The comparisons between numerical and analytical results prove the accuracy of the proposed numerical approaches in the considered quasi-static time-discrete setting, but they also emphasise some subtle issues occurring during time-discontinuous evolutions.

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On phase field modeling of ductile fracture

Cited by 118 publications

References 30 publications

A Monolithic Solution Scheme for a Phase Field Model of Ductile Fracture

A Monolithic Solution Scheme for a Phase Field Model of Ductile Fracture

Investigation of a Phase Field Model for Elasto‐plastic Fracture

Coupling damage and plasticity for a phase-field regularisation of brittle, cohesive and ductile fracture: One-dimensional examples

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