Fracture of materials with rate-dependent mechanical behaviour, e.g. polymers, is a highly complex process. For an adequate modelling, the coupling between rate-dependent stiffness, dissipative mechanisms present in the bulk material and crack driving force has to be accounted for in an appropriate manner. In addition, the fracture toughness, i.e. the resistance against crack propagation, can depend on rate of deformation. In this contribution, an energetic phase-field model of rate-dependent fracture at finite deformation is presented. For the deformation of the bulk material, a formulation of finite viscoelasticity is adopted with strain energy densities of Ogden type assumed. The unified formulation allows to study different expressions for the fracture driving force. Furthermore, a possibly rate-dependent toughness is incorporated. The model is calibrated using experimental results from the literature for an elastomer and predictions are qualitatively and quantitatively validated against experimental data. Predictive capabilities of the model are studied for monotonic loads as well as creep fracture. Symmetrical and asymmetrical crack patterns are discussed and the influence of a dissipative fracture driving force contribution is analysed. It is shown that, different from ductile fracture of metals, such a driving force is not required for an adequate simulation of experimentally observable crack paths and is not favourable for the description of failure in viscoelastic rubbery polymers. Furthermore, the influence of a rate-dependent toughness is discussed by means of a numerical study. From a phenomenological point of view, it is demonstrated that rate-dependency of resistance against crack propagation can be an essential ingredient for the model when specific effects such as rate-dependent brittle-to-ductile transitions shall be described.
Are the cohesive laws of interfaces sufficient for modelling fracture in polycrystals using the cohesive zone model? We examine this question by comparing a fully atomistic simulation of a silicon polycrystal to a finite element simulation with a similar overall geometry. The cohesive laws used in the finite element simulation are measured atomistically. We describe in detail how to convert the output of atomistic grain boundary fracture simulations into the piecewise linear form needed by a cohesive zone model. We discuss the effects of grain boundary microparameters (the choice of section of…
In this paper the problem of a plane strain hydraulic fracture propagating in an elasto-plastic material is analyzed. A new stress redistribution model for the proximity of the fracture tip is formulated and a resulting plasticity-dependent crack propagation condition is introduced. A modified variant of the KGD problem that accounts for the plastic deformations in the near-tip zone only is proposed. It is demonstrated that this model can be a credible substitute for the full elasto-plastic hydraulic fracture problem in the case of moderate plastic deformation. The crack-tip shielding effect i…
The mechanisms of deformation and failure of Cu/Nb nanolaminates manufactured by accumulated roll bonding were analysed using in-situ TEM mechanical tests. Dog-bone small-scale tensile specimens were prepared using a FIB-based technique with the layers parallel and perpendicular to the loading axis. Load, deformation and TEM micrographs were recorded during the in-situ TEM mechanical tests. The specimens deformed parallel to the layers presented very high strain hardening and ductility while those deformed in the perpendicular orientation were more brittle. These results were rationalized in t…
The size effect on the fracture process zone in notched and unnotched three point bending tests of concrete beams is analysed by a meso-scale approach. Concrete is modelled at the meso-scale as stiff aggregates embedded in a soft matrix separated by weak interfaces. The mechanical response of the three phases is modelled by a discrete lattice approach. The model parameters were chosen so that the global model response in the form of load-crack mouth opening displacement curves were in agreement with experimental results reported in the literature. The fracture process zone of concrete is deter…
Integral-type nonlocal damage models describe the fracture process zones by regular strain profiles insensitive to the size of finite elements, which is achieved by incorporating weighted spatial averages of certain state variables into the stress-strain equations. However, there is no consensus yet how the influence of boundaries should be taken into account by the averaging procedures. In the present study, nonlocal damage models with different averaging procedures are applied to the modelling of fracture in specimens with various boundary types. Firstly, the nonlocal models are calibrated b…
A meso-scale analysis is performed to determine the fracture process zone of concrete subjected to uniaxial tension. The meso-structure of concrete is idealised as stiff aggregates embedded in a soft matrix and separated by weak interfaces. The mechanical response of the matrix, the inclusions and the interface between the matrix and the inclusions is modelled by a discrete lattice approach. The inelastic response of the lattice elements is described by a damage approach, which corresponds to a continuous reduction of the stiffness of the springs. The fracture process in uniaxial tension is ap…
An in situ study of deformation, fracture, and fatigue behavior of silicon as a lithium-ion battery electrode material is presented. Thin films (100-200 nm) of silicon are cycled in a half-cell configuration with lithium metal foil as counter/reference electrode, with 1M lithium hexafluorophosphate in ethylene carbonate, diethylene carbonate, dimethyl carbonate solution (1:1:1, wt.%) as electrolyte. Stress evolution in the Si thin-film electrodes during electrochemical lithiation and delithiation is measured by monitoring the substrate curvature using the multi-beam optical sensing method. The…
Two-dimensional (2D) materials possess great potential for flexible devices, ascribing to their outstanding electrical, optical, and mechanical properties. However, their mechanical deformation property and fracture mechanism, which are inescapable in many applications like flexible optoelectronics, are still unclear or not thoroughly investigated due methodology limitations. In light of this, such mechanical properties and mechanisms are explored on example of gallium telluride (GaTe), a promising optoelectronic candidate with an ultrahigh photo-responsibility and a high plasticity within 2D …
We present a novel micromechanics-based phase field approach to model crack initiation and propagation in carbon nanotube (CNT) based composites. The constitutive mechanical and fracture properties of the nanocomposites are first estimated by a mean-field homogenisation approach. Inhomogeneous dispersion of CNTs is accounted for by means of equivalent inclusions representing agglomerated CNTs. Detailed parametric analyses are presented to assess the effect of the main micromechanical properties upon the fracture behaviour of CNT-based composites. The second step of the proposed approach incorp…
The width of fracture process zones in geomaterials is commonly assumed to depend on the type of heterogeneity of the material. Still, very few techniques exist, which link the type of heterogeneity to the width of the fracture process zone. Here, fracture processes in geomaterials are numerically investigated with structural network approaches, whereby the heterogeneity in the form of large aggregates and low volume fibres is modelled geometrically as poly-dispersed ellipsoids and mono-dispersed line segments, respectively. The influence of aggregates, fibres and combinations of both on fract…
In this work, a combined modelling approach for crack propagation in defective graphene is presented. Molecular dynamics (MD) simulations are used to obtain material parameters (Young's modulus and Poisson ratio) and to determine the energy contributions during the crack evolution. The elastic properties are then applied in phase-field continuum simulations which are based on the Griffith energy criterion for fracture. In particular, the influence of point defects on elastic properties and the fracture toughness are investigated. For the latter, we obtain values consistent with recent experime…
We have designed a new experimental setup able to investigate fracture of soft materials at small scales. At high crack velocity, where energy is mostly dissipated through viscoelastic processes, we observe an increasingly large high strain domain in the crack tip vicinity. Taking advantage of our ability to determine where linear elasticity breaks down, we derive a simple prediction for the evolution of the energy release rate with the crack velocity.
Dynamic fracture in a wide class of materials reveals "fracture energy" $Γ$ much larger than the expected nominal surface energy due to the formation of two fresh surfaces. Moreover, the fracture energy depends on the crack velocity, $Γ=Γ(v)$. We show that a simple dynamical theory of visco-plasticity coupled to asymptotic pure linear-elasticity provides a possible explanation to the above phenomena. The theory predicts tip blunting characterized by a dynamically determined crack tip radius of curvature. In addition, we demonstrate velocity selection for cracks in fixed-grip strip geometry acc…
The effect of hydrogen on the fracture behaviour of austenitic stainless steel has been investigated in the past [1][2]. It has been reported that fracture initiates by void formation at inclusions and regions of enhanced strain localisation [3]. There is experimental evidence that supports the fact that hydrogen influences void nucleation, growth and coalescence during material fracture [4]. This work investigates the effect of hydrogen on void growth and coalescence in austenitic stainless steel. The effect of hydrogen on void growth and coalescence for different stress triaxialities has bee…
Post mortem analysis of fracture surfaces of ductile and brittle materials on the $μ$m-mm and the nm scales respectively, reveal self affine graphs with an anomalous scaling exponent $ζ\approx 0.8$. Attempts to use elasticity theory to explain this result failed, yielding exponent $ζ\approx 0.5$ up to logarithms. We show that when the cracks propagate via plastic void formations in front of the tip, followed by void coalescence, the voids positions are positively correlated to yield exponents higher than 0.5.
We present a novel computational framework to simulate the electromechanical response of self-sensing carbon nanotube (CNT)-based composites experiencing fracture. The computational framework combines electrical-deformation-fracture finite element modelling with a mixed micromechanics formulation. The latter is used to estimate the constitutive properties of CNT-based composites, including the elastic tensor, fracture energy, electrical conductivity, and linear piezoresistive coefficients. These properties are inputted into a coupled electro-structural finite element model, which simulates the…
A theoretical-computational framework is proposed for predicting the failure behavior of two anisotropic brittle materials, namely, single crystal magnesium and boron carbide. Constitutive equations are derived, in both small and large deformations, by using thermodynamics in order to establish a fully coupled and transient twin and crack system. To study the common deformation mechanisms (e.g., twinning and fracture), which can be caused by extreme mechanical loading, a monolithically-solved Ginzburg--Landau-based phase-field theory coupled with the mechanical equilibrium equation is implemen…
We propose a one-dimensional, nonconvex elastic constitutive model with higher gradients that can predict spontaneous fracture at a critical load via a bifurcation analysis. It overcomes the problem of discontinuous deformations without additional field variables, such as damage or phase-field variables, and without a priori specified surface energy. Our main tool is the use of the inverse deformation, which can be extended to be a piecewise smooth mapping even when the original deformation has discontinuities describing cracks opening. We exploit this via the inverse-deformation formulation o…
A 2D lattice approach to describe hydraulic fracturing is presented. The interaction of fluid pressure and mechanical response is described by Biot's theory. The lattice model is applied to the analysis of a thick-walled cylinder, for which an analytical solution for the elastic response is derived. The numerical results obtained with the lattice model agree well with the analytical solution. Furthermore, the coupled lattice approach is applied to the fracture analysis of the thick-walled cylinder. It is shown that the proposed lattice approach provides results that are independent of the mesh…
We present a combined phase field and cohesive zone formulation for hydrogen embrittlement that resolves the polycrystalline microstructure of metals. Unlike previous studies, our deformation-diffusion-fracture modelling framework accounts for hydrogen-microstructure interactions and explicitly captures the interplay between bulk (transgranular) fracture and intergranular fracture, with the latter being facilitated by hydrogen through mechanisms such as grain boundary decohesion. We demonstrate the potential of the theoretical and computational formulation presented by simulating inter- and tr…
For the fatigue life of thin-walled components, not only fatigue crack initiation, but also crack growth is decisive. The phase-field method for fracture is a powerful tool to simulate arbitrary crack phenomena. Recently, it has been applied to fatigue fracture. Those models pose an alternative to classical fracture-mechanical approaches for fatigue life estimation. In the first part of this paper, the parameters of a phase-field fatigue model are calibrated and its predictions are compared to results of fatigue crack growth experiments of aluminium sheet material. In the second part, compress…
Despite numerous experimental and theoretical investigations of the mechanical behavior of high-capacity Si and Ge Li-ion battery anodes, our basic understanding of swelling-driven fracture in these materials remains limited. Existing theoretical studies have provided insights into elasto-plastic deformations caused by large volume change phase transformations, but have not modeled fracture explicitly beyond Griffith's criterion. Here we use a multi-physics phase-field approach to model self-consistently anisotropic phase transformation, elasto-plastic deformation, and crack initiation and pro…
The aim of the paper is to propose a paradigm shift for the variational approach of brittle fracture. Both dynamics and the limit case of statics are treated in a same framework. By contrast with the usual incremental approach, we use a space-time principle covering the whole loading and crack evolution. The emphasis is given on the modelling of the crack extension by the internal variable formalism and a dissipation potential as in plasticity, rather than Griffith's original approach based on the surface area. The new formulation appears to be more fruitful for generalization than the standar…
A decrease of fracture toughness of irradiated materials is usually observed, as reported for austenitic stainless steels in Light Water Reactors (LWRs) or copper alloys for fusion applications. For a wide range of applications (e.g. structural steels irradiated at low homologous temperature), void growth and coalescence fracture mechanism has been shown to be still predominant. As a consequence, a comprehensive study of the effects of irradiation-induced hardening mechanisms on void growth and coalescence in irradiated materials is required. The effects of irradiation on ductile fracture mech…
We investigate the potential of quasi-Newton methods in facilitating convergence of monolithic solution schemes for phase field fracture modelling. Several paradigmatic boundary value problems are addressed, spanning the fields of quasi-static fracture, fatigue damage and dynamic cracking. The finite element results obtained reveal the robustness of quasi-Newton monolithic schemes, with convergence readily attained under both stable and unstable cracking conditions. Moreover, since the solution method is unconditionally stable, very significant computational gains are observed relative to the …
In this paper the problem of a plane strain hydraulic fracture in elasto-plastic material is analyzed. The analysis is based on Finite Element Method computations of crack propagation and closure to simulate the Mini-Frac calibration test. The basic trends of fracture evolution are identified for both, the propagation and the closure stage in elasto-plastic materials and compared with the results obtained for purely elastic deformation of rock. Pressure decline analysis is conducted to identify the closure stress and leak-off coefficient. The results show that the standard techniques of pressu…
A new multiscale modeling approach to unravel the influence of interlayer sp3 bonds on the nonlinear large-deformation and fracture behaviors of 2D carbon nanostructures under tension
The effects of shear on energy release rate and mode mixity in a symmetric sandwich beam with isotropic layers and a debond crack at the face sheet/core interface are investigated through a semi-analytic approach based on two-dimensional elasticity and linear elastic fracture mechanics. Semi-analytic expressions are derived for the shear components of energy release rate and mode mixity phase angle which depend on four numerical coefficients derived through accurate finite element analyses. The expressions are combined with earlier results for three-layer configurations subjected to bending-mo…
The phase field fracture method has emerged as a promising computational tool for modelling a variety of problems including, since recently, hydrogen embrittlement and stress corrosion cracking. In this work, we demonstrate the potential of phase field-based multi-physics models in transforming the engineering assessment and design of structural components in hydrogen-containing environments. First, we present a theoretical and numerical framework coupling deformation, diffusion and fracture, which accounts for inertia effects.Several constitutive choices are considered for the crack density f…
Soft materials including elastomers and gels are pervasive in biological systems and technological applications. Whereas it is known that intrinsic fracture energies of soft materials are relatively low, how the intrinsic fracture energy cooperates with mechanical dissipation in process zone to give high fracture toughness of soft materials is not well understood. In addition, it is still challenging to predict fracture energies and crack-tip strain fields of soft tough materials. Here, we report a scaling theory that accounts for synergistic effects of intrinsic fracture energies and dissipat…
Fracture of lithium-ion battery electrodes is found to contribute to capacity fade and reduce the lifespan of a battery. Traditional fracture models for batteries are restricted to consideration of a single, idealised particle; here, advanced X-ray computed tomography (CT) imaging, an electro-chemo-mechanical model and a phase field fracture framework are combined to predict the void-driven fracture in the electrode particles of a realistic battery electrode microstructure. The electrode is shown to exhibit a highly heterogeneous electrochemical and fracture response that depends on the partic…
Taper-free InP twinning superlattice (TSL) nanowires with an average twin spacing of ~ 13 nm were grown along the zinc-blende close-packed [111] direction using metalorganic vapor phase epitaxy. The mechanical properties and fracture mechanisms of individual InP TSL nanowires in tension were ascertained by means of in situ uniaxial tensile tests in a transmission electron microscope. The elastic modulus, failure strain and tensile strength along the [111] direction were determined. No evidence of inelastic deformation mechanisms was found before fracture, which took place in a brittle manner a…
Phase-field models are a leading approach for realistic fracture problems. They treat the crack as a second phase and use gradient terms to smear out the crack faces, enabling the use of standard numerical methods for simulations. This regularization causes cracks to occupy a finite volume in the reference, and leads to the inability to appropriately model the closing or contacting -- without healing -- of crack faces. Specifically, the classical idealized crack face tractions are that the shear component is zero, and that the normal component is zero when the crack opens and identical to the …
Modeling dynamics fracture in materials involves usage of hydrodynamic codes which solve basic conservation laws of mass, energy and momentum in space and time. This requires appropriate models to handle elastic-plastic deformation, equation of state, material strength, and fracture. Nucleation and Growth (NAG) damage model is a micro-physical model which computes amount of damage in the material by accounting for phenomena like nucleation, growth and coalescence of voids or cracks. The NAG model involves several material model parameters, such as nucleation threshold, growth threshold, etc. T…
An improved peridynamic framework to eliminate unphysical stress and fictitious yield at geometry surface for geomaterials elastoplastic deformation and fracture analysis
This paper describes atomistic simulations of deformation and fracture of Al reinforced with carbon nanotubes (CNTs). We use density functional theory (DFT) to understand the energetics of Al-graphene interfaces and gain reference data for the parameterization of Al-C empirical potentials. We then investigate the load transfer between CNTs and Al and its effect on composite strengthening. To this end, we perform uniaxial tensile simulations of an Al crystal reinforced with CNTs of various volume fractions. We also study the interaction of the embedded CNTs with a crack. We show that the intera…
In this work, we extend the recently proposed adaptive phase field method to model fracture in orthotropic functionally graded materials (FGMs). A recovery type error indicator combined with quadtree decomposition is employed for adaptive mesh refinement. The proposed approach is capable of capturing the fracture process with a localized mesh refinement that provides notable gains in computational efficiency. The implementation is validated against experimental data and other numerical experiments on orthotropic materials with different material orientations. The results reveal an increase in …
The mechanical properties of the stoichiometric B2 $β$-phase of NiAl are well established, however the effect of off-stoichiometric composition on the fracture toughness has not yet been systematically studied over the entire composition range of 40-50% Al. Here we use microbending tests on notched cantilever beams FIB-milled from NiAl single crystals with an aluminized as well as an oxidation-induced composition gradient to determine the influence of the Al concentration on the mechanical properties. The fracture toughness is maximal for the stoichiometric composition. It decreases with incre…
Discrete element modelling (DEM) is one of the most efficient computational approaches to the fracture processes of heterogeneous materials on mesoscopic scales. From the dynamics of single crack propagation through the statistics of crack ensembles to the rapid fragmentation of materials DEM had a substantial contribution to our understanding over the past decades. Recently, the combination of DEM with other simulation techniques like Finite Element Modelling further extended the field of applicability. In this paper we briefly review the motivations and basic idea behind the DEM approach to …
We study a planar thin brittle beam subject to elastic deformations and cracks described in terms of a nonlinear Griffith energy functional acting on $SBV$ deformations of the beam. In particular we consider the case in which elastic bulk contributions due to finite bending of the beam are comparable to the surface energy which is necessary to completely break the beam into several large pieces. In the limit of vanishing aspect ratio we rigorously derive an effective Griffith-Euler-Bernoulli functional which acts on piecewise $W^{2,2}$ regular curves representing the midline of the beam. The e…
We combine molecular dynamics simulations of deformations at the submicron scale with a continuum fracture mechanics calculation for the onset of crack propagation to predict the macroscopic fracture toughness of amorphous glassy polymers. Key ingredients in this multiscale approach are the elastic properties of polymer crazes and the stress at which craze fibrils fail through chain pullout or scission. Our results are in quantitative agreement with dimensionless ratios that describe experimental polymers and their variation with temperature, polymer length and polymer rigidity.
Fracture growth in a material is strongly influenced by the presence of inhomogeneities, which deviate crack trajectories from rectilinearity and deeply affect failure. Increasing crack tortuosity is connected to enhancement of fracture toughness, while often a crack may even be stopped when it impinges a void, which releases the stress concentration. Therefore, the determination of crack trajectories is important in the design against failure of materials and mechanical pieces. The recently developed phase-field approach (AT1 and AT2 models), based on a variational approach to damage localiza…
We study a lattice model for mode III crack propagation in brittle materials in a stripe geometry at constant applied stretching. Stiffening of the material at large deformation produces supersonic crack propagation. For large stretching the propagation is guided by well developed soliton waves. For low stretching, the crack-tip velocity has a universal dependence on stretching that can be obtained using a simple geometrical argument.
One of the most important scaling laws of time dependent fracture is Basquin's law of fatigue, namely, that the lifetime of the system increases as a power law with decreasing external load amplitude, $t_f\sim σ_0^{-α}$, where the exponent $α$ has a strong material dependence. We show that in spite of the broad scatter of the Basquin exponent $α$, the fatigue fracture of heterogeneous materials exhibits intriguing universal features. Based on stochastic fracture models we propose a generic scaling form for the macroscopic deformation and show that at the fatigue limit the system undergoes a co…
Predicting the Fracture Propensity of Amorphous Silica Using Molecular Dynamics Simulations and Machine Learning
Sibley School of Mechanical and Aerospace Engineering, Cornell University（康奈尔大学机械与航空航天工程学院）；Department of Engineering Mechanics, Xi’an Jiaotong University（西安交通大学工程力学系）
