Phase-field Damage Model Research Articles

The effect of crystal anisotropy and plastic response on the dynamic fracture of energetic materials

The thermomechanical behavior of solids includes dissipative processes such as plastic deformation and fracture. The relative importance of these processes on the response of energetic materials has been a subject of study for many decades due to their significance on ignition and reaction. However, a constitutive model to simulate the anisotropy of the crack patterns and the effect of plastic deformation due to slip in energetic materials is not yet available. Finite strain thermomechanical constitutive equations that couple crystal plasticity, an equation of state, and an anisotropic phase field damage model are presented. The model is implemented in a multiphysics finite element solver and used to simulate recent experiments on β-HMX (octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine) by Zaug et al. The simulations reproduce qualitatively the crack pattern and the crystal orientation dependence of the observed damage. Specifically, more damage is observed when the crystal is impacted in the (010) direction, while more plastic deformation is observed when the load is applied in the (110) direction. The present model represents a step forward to understand the interplay between plasticity and fracture in shocked β-HMX single crystals. It can be used to gain insights into temperature increase and hot-spot formation under shock.

On Coupling of Interface and Phase-field Damage Models for Quasi-brittle Fracture

Numerical simulations of crack initiation and propagation is presented. Both, interface and material cracks are modelled within the damage theory introducing two independent damage parameters. The interface cracks appearing in an adhesive layer of a contact zone between structural components consider cohesive zone models with general stress-strain relationships implemented in an energetic formulation. Accordingly, the fracture of bulk domains also includes variational consideration and leads to phase-field damage which causes a very narrow band of defected material constituting the actual crack. The proposed computational approach has a variational form, the solution being approximated by a time stepping procedure, a finite element code, and quadratic programming algorithms.

Incorporation of tension-compression asymmetry into plastic damage phase-field modeling of quasi brittle geomaterials

Both experiments and theoretical investigations have evidenced the existence of compressive-shear fracture mode in geomaterials like concrete, rocks and gypsum. Proper description and modeling of intricate fracturing pattern with consideration of strong tension-compression asymmetry in mechanical response remain an open issue in phase-field modeling and simulation. In this work, a new phase field model with plasticity-damage coupling is formulated in the framework of irreversible thermodynamics. Two essential features, tension-compression asymmetry in mechanical response as well as two distinct fracture modes, are taken into account by incorporating two scalar-valued damage variables into the classical modeling framework. By defining a specific free energy density function, the coupling between damage and plasticity is achieved by involving a phase field variable into the yield function. The proposed model is validated at two levels. In the homogeneous cases, the mechanical behaviors of typical geomaterials are investigated in the plane stress condition. Meanwhile, a stress-based crack onset criterion is utilized to capture the distinct failure behavior both in tension and in compression. In numerical simulations, a local numerical integration with implicit return mapping algorithm and plasticity-damage decoupling treatment are developed. Three numerical examples are performed to demonstrate respectively the mode I, mixed-mode and model II fracture in geomaterials. Comparisons between numerical simulations and experimental data make it possible to evaluate the predictive performance of the proposed bi-dissipative phase field damage model. In addition, the local stress analysis is carried out to explain a changing mode of fracture propagation and to demonstrate the existence of tensile-shear (hybrid) fracture mode.

Rate-dependent phase-field damage modeling of rubber and its experimental parameter identification

Phase-field models have the advantage in that no geometric descriptions of cracks are required, which means that crack coalescence and branching can be treated without additional effort. Miehe and Schänzel, 2014 introduced a rate-independent phase-field damage model for finite strains in which a viscous damage regularization was proposed. We extend the model to depend on the loading rate and time by incorporating rubber’s strain-rate dependency in the constitutive description of the bulk, as well as in the damage driving force. The parameters of the model are identified using experiments at different strain rates. Local strain fields near the crack tip, obtained with digital image correlation (DIC), are used to help identify the length scale parameter. Three different degradation functions are assessed for their accuracy to model the rubber’s rate-dependent fracture. An adaptive time-stepping approach with a corrector scheme is furthermore employed to increase the computational efficiency with a factor of six, whereas an active set method guarantees the irreversibility of damage. Results detailing the energy storage and dissipation of the different model constituents are included, as well as validation results that show promising capabilities of rate-dependent phase-field modeling.

A unified phase field damage model for modeling the brittle-ductile dynamic failure mode transition in metals

Dynamic brittle fracture and shear banding are typical failure modes in metals under high strain rate loading. In this paper, a unified phase field damage model is developed to capture the above two failure modes’ transition naturally by allowing the critical energy release rate to vary with the stress triaxiality to distinguish the material failure properties of tensile fracture and shear banding. Besides this, compared with the existing elastic-plastic phase field fracture models, the plastic dissipation energy before the damage evolution is excluded in the failure energy to model ductile failure more physically, and the degradation function of the yield stress is introduced which not only provides a damage softening mechanism for the ductile failure but also ensures a proper simulation of the brittle fracture. The unified phase field damage model is applied to study the brittle-ductile failure mode transition in Kalthoff test. The results show that stress wave propagation plays an important role in the brittle-ductile failure mode transition. The shear banding requires more energy for initiation and will be activated straightly as the plastic wave passes the notch tip for high impact velocity. While for low impact velocity, the energy will be released by selecting the brittle fracture mode which requires less energy when the reflected tensile wave superimposes at the notch tip. For our simulation, the damage softening contributes considerably more to the shear banding than the thermal softening. This model can be used to simulate the complex dynamic failure of metal under impact loading.

Higher regularity for solutions to elliptic systems in divergence form subject to mixed boundary conditions

This work combines results from operator and interpolation theory to show that elliptic systems in divergence form admit maximal elliptic regularity on the Bessel potential scale $$ H ^{s-1}_D(\varOmega )$$ for $$s>0$$ sufficiently small, if the coefficient in the main part satisfies a certain multiplier property on the spaces $$ H ^{s}(\varOmega )$$ . Ellipticity is enforced by assuming a Garding inequality, and the result is established for spaces incorporating mixed boundary conditions with very low regularity requirements for the underlying spatial set. To illustrate the applicability of our results, two examples are provided. Firstly, a phase-field damage model is given as a practical application where higher differentiability results are obtained as a consequence to our findings. These are necessary to show an improved numerical approximation rate. Secondly, it is shown how the maximal elliptic regularity result can be used in the context of quasilinear parabolic equations incorporating quadratic gradient terms.

Particle-induced damage in Fe–TiB2 high stiffness metal matrix composite steels

Fe–TiB2 metal matrix composites, termed high modulus steels, have great potential for lightweight design applications due to their high stiffness/density ratio. However, the observed embrittlement, caused by the TiB2 particles, critically limits application of these steels. Experimental studies to identify the influence of particle microstructure on ductility and toughness are complex in view of the multitude of parameters affecting microstructural damage. We therefore pursue instead an integrated computational materials engineering approach to gain understanding and derive guidelines for optimizing the particle microstructure and thus improve the mechanical properties, particularly the damage tolerance of these high modulus steels. Key microstructural parameters such as particle clustering degree, size and volume fraction were investigated. Model geometries were statistically and systematically generated with varied particle configurations from random to clustered distributions. Simulations were performed using a crystal plasticity Fast Fourier transformation method coupled with a novel phase field damage model. The influence of particle configuration on damage initiation and evolution was evaluated from the simulation results, and it was observed that microstructures with homogeneous particle distributions of 7 to 15 vol% TiB2, devoid of large TiB2 particles stemming from primary solidification, appear most favorable for obtaining high modulus steels with optimized mechanical properties.

A multiphysics model that can capture crack patterns in Si thin films based on their microstructure

Fracture in silicon anodes has fascinated the electrochemistry community for two decades, as it can result in a 80% capacity loss over the first few electrochemical cycles and is the limiting factor in commercializing such high capacity anodes. Although numerous experimental data exist illustrating severe fracture patterns and their dependence on the scale of the microstructure, no theoretical model has been able to re-produce and capture such behaviour. In this article, a multi-physics phase-field damage model is presented that can accurately capture the long standing problem of dry bed-lake crack patterns observed for Si thin film anodes. A promising aspect of the model is that, in addition to accounting for Li-ion diffusion, it can explicitly capture the microstructure, and therefore when applied to a Si film with a thickness below 100 nm no fracture was observed, which is consistent with experiments. As fracture in continuous thin films is random, micron-hole patterned Si films were also fabricated and cycled, resulting in ordered crack patterns. The proposed model was able to capture these elaborate, yet ordered, crack patterns, further validating its efficiency in predicting damage during lithiation of Si. This paves the way to using multiscale modeling for predicting the dimensions that limit and control fracture during lithiation, prolonging hence the electrode lifetime.

Effect of initial damage variability on hot-spot nucleation in energetic materials

Mechanical insult may be able to produce chemical transformations in solids when the energy is released in highly localized regions. This phenomenon is responsible for the nucleation of hot-spots that are responsible for ignition of energetic materials. The concentration of energy at microstructural defects leads to the probabilistic nature of ignition. The effect of the microstructure of the energetic particles, specifically the influence of the initial crack distribution on the sensitivity to ignition, is studied for a particle embedded in a polymeric matrix at impact velocities 100 m/s and 400 m/s with finite element simulations that couple fracture dynamics and heat transport. A phase field damage model that includes heat sources due to frictional heating at the crack surfaces and heat dissipation during crack propagation is developed and verified. These heat sources are compared and, in the range of impact velocities studied, heat generation due to friction is more important than dissipation due to crack propagation. Hot-spots nucleated at 100 m/s do not reach the critical temperature while conditions consistent with the Lee-Tarver criterion for ignition are observed at 400 m/s impact velocity. The variability observed due to the stochasticity of the initial crack distribution is studied and it increases with a higher impact velocity. In particular, regions of high temperature develop close to cracks intersecting the particle polymer interface. Therefore, controlling the surface quality of the energetic particles may lead to a reduction on the sensitivity uncertainty in polymer-bonded explosives.

Robust numerical implementation of non-standard phase-field damage models for failure in solids

Recently several non-standard phase-field models different from the standard one for brittle fracture (Bourdin et al. 2000, 2008; Miehe et al. 2010a) have been proposed for the modeling of damage and fracture in solids. On the one hand, linear elastic behavior before the onset of damage and even cohesive cracking induced quasi-brittle failure can be considered by such non-standard phase-field models. On the other hand, they present great challenges to the numerical implementation since the damage boundedness is no longer automatically fulfilled. In this work, the numerical implementation of the unified phase-field damage model (Wu 2017, 2018) is addressed, though it also applies to the standard and other non-standard ones. In particular, an iterative alternate minimization (AM) algorithm enhanced with path-following strategies is presented. The phase-field or damage sub-problem is solved by the bound-constrained optimization solver in which the boundedness and irreversibility conditions of the crack phase-field are exactly enforced. Moreover, material softening induced snap-backs typically for localized failure in solids can also be effectively dealt with. The equilibrium paths in terms of the fracture surface and of the indirect displacement (e.g., the crack mouth opening or sliding displacement) are discussed in the context of the AM algorithm. The AM algorithm with the indirect displacement control is advocated, since both prescribed external forces and displacements can be naturally dealt with, which is more versatile than the one with the fracture surface control applicable only for prescribed displacements. Representative examples of benchmark tests show that the AM algorithm enhanced with the indirect displacement control is extremely robust even for large increment sizes. The insensitivity of the unified phase-field damage model to the incorporated length scale and mesh size is also confirmed.

A length scale insensitive phase-field damage model for brittle fracture

Being able to model complex nucleation, propagation, branching and merging of cracks in solids within a unified framework, the classical phase-field models for brittle fracture fail in predicting length scale independent global responses for a solid lacking elastic singularities (e.g., corners, notches, etc.). Motivated from Barenblatt’s approximation of Griffith’s brittle fracture with a vanishing Irwin’s internal length, this paper extends our recent work in quasi-brittle failure (Wu, 2017, 2018a) and presents for the first time a length scale insensitive phase-field damage model for brittle fracture. More specifically, with a set of optimal characteristic functions, a phase-field regularized cohesive zone model (CZM) with linear softening law is addressed and applied to brittle fracture. Both the failure strength and the traction – separation law are independent of the incorporated length scale parameter. Compared to other phase-field models and CZM based discontinuous approaches for brittle fracture, the proposed phase-field regularized CZM is of several merits. On the one hand, being theoretically equivalent to Barenblatt’s CZM (at least in the 1-D case), it needs neither the explicit crack representation/tracking nor the elastic penalty stiffness which both are necessary but cumbersome for discontinuous approaches. On the other hand, it gives length scale independent global responses for problems with or without elastic singularities while preserving the expected Γ-convergence property of phase-field models. Representative numerical examples of several well-known benchmark tests support the above conclusions, validating its capability of modeling both mode-I and mixed-mode brittle fracture.

A simple and unified implementation of phase field and gradient damage models

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.

Dynamic fracture and hot-spot modeling in energetic composites

Defects such as cracks, pores, and particle-matrix interface debonding affect the sensitivity of energetic materials by reducing the time-to-ignition and the threshold pressure to initiate an explosion. Frictional sliding of preexisting cracks is considered to be one of the most important causes of localized heating. Therefore, understanding the dynamic fracture of crystalline energetic materials is of extreme importance to assess the reliability and safety of polymer-bonded explosives. Phase field damage model simulations, based on the regularization of the crack surface as a diffuse delta function, are used to describe crack propagation in cyclotetramethylene-tetranitramine crystals embedded in a Sylgard matrix. A thermal transport model that includes heat generation by friction at crack interfaces is coupled to the solution of crack propagation. 2D and 3D dynamic compression simulations are performed with different boundary velocities and initial distributions of cracks and interface defects to understand their effect on crack propagation and heat generation. It is found that, at an impact velocity of 400 m/s, localized damage at the particle-binder interface is of key importance and that the sample reaches temperatures high enough to create a hot-spot that will lead to ignition. At an impact velocity of 10 m/s, preexisting cracks advanced inside the particle, but the increase of temperature will not cause ignition.

A Phase-Field Damage Model for Orthotropic Materials and Delamination in Composites

A phase-field damage model for orthotropic materials is proposed and used to simulate delamination of orthotropic laminated composites. Using the deviatoric and hydrostatic tensile components of the stress tensor for elastic orthotropic materials, a degraded elastic free energy that can accommodate damage is derived. The governing equations follow from the principle of virtual power and the resulting damage model, by its construction, conforms with the physical relevant condition of no matter interpenetration along the crack faces. The model also dispenses with the traction separation law, an extraneous hypothesis conventionally brought in to model the interlaminar zones. The model is assessed through numerical simulations on delaminations in mode I, mode II, and another such problem with multiple initial notches. The present method is able to reproduce nearly all the features of the experimental load displacement curves, allowing only for small deviations in the softening regime. Numerical results also show forth a superior performance of the proposed method over existing approaches based on a cohesive law.

A phase-field damage model based on evolving microstructure

A phase-field damage model based on evolving microstructure

Numerical simulations of inter-laminar fracture in particle-toughened carbon fiber reinforced composites

The most predominant failure mechanism in carbon fiber reinforced composites is delamination. Dispersed second phase particles have been added to the interlayer region to reduce this damaging effect. In this work a phase field damage model is used to study Mode I crack propagation in particle-toughened interlayers. The simulations show several damage mechanisms observed in experiments and the processes that produce crack growth. In particular, failure involves multiple micro-cracks forming ahead of the main crack tip that coalesce and lead to crack branching, deflection and the formation of ligaments. Additionally, increasing the particle stiffness and the surface energy of the matrix-particle interface do not lead to an improvement of the interlaminate toughness as the fiber-matrix interface becomes a preferential path for the crack through the nucleation of extended delamination zones ahead of the crack tip.

Comparison of a phase-field model and of a thick level set model for brittle and quasi-brittle fracture

Summary This paper provides a comparison between one particular phase-field damage model and a thick level set (TLS) damage model for the simulation of brittle and quasi-brittle fractures. The TLS model is recasted in a variational framework, which allows comparison with the phase-field model. Using this framework, both the equilibrium equations and the damage evolution laws are guided by the initial choice of the potential energy. The potentials of the phase-field model and of the TLS model are quite different. TLS potential enforces a priori a bound on damage gradient whereas the phase-field potential does not. The TLS damage model is defined such that the damage profile fits to the one of the phase-field model for a beam of infinite length. The model parameters are calibrated to obtain the same surface fracture energy. Numerical results are provided for unidimensional and bidimensional tests for both models. Qualitatively, similar results are observed, although TLS model is observed to be less sensible to boundary conditions. Copyright © 2015 John Wiley & Sons, Ltd.

Local versus average field failure criterion in amorphous polymers

There is extensive work developing laws that predict yielding in amorphous polymers, ranging from the pioneer experimental work of Sternstein et al (1968 Appl. Polym. Symp. 7 175–99) to the novel molecular dynamics simulations of Jaramillo et al (2012 Phys. Rev. B 85 024114). While atomistic models render damage criteria in terms of local values of the stress and strain fields, experiments provide yield conditions in terms of the average values of these fields. Unfortunately, it is not possible to compare these results due to the differences in time and length scales. Here, we use a micromechanical phase-field damage model with parameters calculated from atomistic simulations to connect atomistic and macroscopic scale experiments. The phase-field damage model is used to study failure in composite materials. We find that the yield criterion should be described in terms of local stress and strains fields and cannot be extended directly from applied stress field values to determine yield conditions.