Crack Discontinuities Research Articles

A phase-field approach to model fracture of arterial walls: Theory and finite element analysis

This study uses a recently developed phase-field approach to model fracture of arterial walls with an emphasis on aortic tissues. We start by deriving the regularized crack surface to overcome complexities inherent in sharp crack discontinuities, thereby relaxing the acute crack surface topology into a diffusive one. In fact, the regularized crack surface possesses the property of Gamma-Convergence, i.e. the sharp crack topology is restored with a vanishing length-scale parameter. Next, we deal with the continuous formulation of the variational principle for the multi-field problem manifested through the deformation map and the crack phase-field at finite strains which leads to the Euler–Lagrange equations of the coupled problem. In particular, the coupled balance equations derived render the evolution of the crack phase-field and the balance of linear momentum. As an important aspect of the continuum formulation we consider an invariant-based anisotropic constitutive model which is additively decomposed into an isotropic part for the ground matrix and an exponential anisotropic part for the two families of collagen fibers embedded in the ground matrix. In addition we propose a novel energy-based anisotropic failure criterion which regulates the evolution of the crack phase-field. The coupled problem is solved using a one-pass operator-splitting algorithm composed of a mechanical predictor step (solved for the frozen crack phase-field parameter) and a crack evolution step (solved for the frozen deformation map); a history field governed by the failure criterion is successively updated. Subsequently, a conventional Galerkin procedure leads to the weak forms of the governing differential equations for the physical problem. Accordingly, we provide the discrete residual vectors and a corresponding linearization yields the element matrices for the two sub-problems. Finally, we demonstrate the numerical performance of the crack phase-field model by simulating uniaxial extension and simple shear fracture tests performed on specimens obtained from a human aneurysmatic thoracic aorta. Model parameters are obtained by fitting the set of novel experimental data to the predicted model response; the finite element results agree favorably with the experimental findings.

Phase field modeling of ductile fracture at finite strains: A variational gradient-extended plasticity-damage theory

This work outlines a rigorous variational-based framework for the phase field modeling of ductile fracture in elastic–plastic solids undergoing large strains. The phase field approach regularizes sharp crack surfaces within a pure continuum setting by a specific gradient damage modeling with geometric features rooted in fracture mechanics. It has proven immensely successful with regard to the analysis of complex crack topologies without the need for fracture-specific computational structure such as finite element design of crack discontinuities or intricate crack-tracking algorithms. Following the recent work Miehe et al. (2015), the phase field model of fracture is linked to a formulation of gradient plasticity at finite strains. The formulation includes two independent length scales which regularize both the plastic response as well as the crack discontinuities. This ensures that the damage zones of ductile fracture are inside of plastic zones, and guarantees on the computational side a mesh objectivity in post-critical ranges. The novel aspect of this work is a precise representation of this framework in a canonical format governed by variational principles. The coupling of gradient plasticity to gradient damage is realized by a constitutive work density function that includes the stored elastic energy and the dissipated work due to plasticity and fracture. The latter represents a coupled resistance to plasticity and damage, depending on the gradient-extended internal variables which enter plastic yield functions and fracture threshold functions. With this viewpoint on the generalized internal variables at hand, the thermodynamic formulation is outlined for gradient-extended dissipative solids with generalized internal variables which are passive in nature. It is specified for a conceptual model of von Mises-type elasto-plasticity at finite strains coupled with fracture. The canonical theory proposed is shown to be governed by a rate-type minimization principle, which fully determines the coupled multi-field evolution problem. This is exploited on the numerical side by a fully symmetric monolithic finite element implementation. The performance of the formulation is demonstrated by means of some representative examples.

Progressive damage analysis and testing of composite laminates with fiber waves

In this work, the Progressive Damage Analysis (PDA) of composite laminates with waves was developed, experimentally validated, and discussed. PDA using Continuum Damage Modeling (CDM) and Discrete Damage Modeling (DDM) was conducted. In CDM, the material continuum constitutive properties are updated to incorporate the influence of progressive damage. In DDM, the actual damage is modeled, consistent with the progressive damage model analysis and observations. A commercial finite element code ABAQUS was used for all of the analysis with specialty user subroutines for the CDM and DDM. The laminate wave parameters (wavelength and amplitude) were determined from a statistical analysis of as-manufactured laminates from failed composite wind turbine blades. Laminates with waves under tension and compression loading were considered to create a benchmark set of tests for laminates and waves, and to provide an unambiguous comparison between CDM and DDM for this type of defect. Both methods (CDM and DDM) are compared and contrasted with experimental data. It is important to note that no assumed damage (such as a crack or other discontinuity) was necessary for the analysis. The failure mode and progressive damage is a consequence of the analysis. Correlations are found with each, and the pros and cons are evaluated and discussed. Better correlations were found with DDM, but accounting for nonlinear shear in the stress–strain response using CDM in the analysis provided numerical stability and the best experimental/analytical correlations.

Using micro-planes concrete damage model in discrete least squares meshless method for predicting the crack growth

In the present study, the author’s previously developed micro-planes concrete damage model has been implemented in the Discrete Least Squares Meshless method (DLSM) for assessment of the crack performance in concrete domains. DLSM is a true meshless method needing no partition in the local approximation as well as in the assembling of the local approximations in order to obtain a global solution. Instead it uses a collection of the un-structured nodal points in place of the elements and discretizes the governing differential equations with the discrete least squares approach. This can reduce considerably the pre-processing efforts of the calculations. However, DLSM like other meshless methods loses its efficiency when it is applied for the analysis of the solid domains dealing with the growing cracks. In recent years, some researchers have utilized some techniques to overcome this problem such as visibility, transparency and diffraction for precisely detecting the crack discontinuity in the case of a static non-growing crack, but these methods require some reciprocal corrective operations within each step of the calculation process extending the time of processing. In this study, however, a new, straightforward and general applicable method has been used for accommodating this issue using the advantages of the Moving Least Squares (MLS) shape functions constructed based on the Voronoi tessellation algorithm and micro-planes concrete damage constitutive model. Voronoi based MLS shape functions have been implemented for evaluating the crack effects accurately as a discontinuity during the analysis processing and micro-planes concrete damage approach used for proper detecting the crack growth trajectory. In the proposed method, the domain of interest is divided into the Voronoi cells based on the classical Voronoi tessellation approach, a cell for each node, and then the support domain of each node located inside the domain as well as on or near the crack faces is determined based on a neighboring criterion. During each step of the analysis for all the integration points, damages are computed according to the micro-planes concrete damage model and the crack path for the next step will be defined. After the end of the iterations in each step, both the stiffness tensor and internal load vector are modified based on the final stress fields obtained on and near the crack. The accuracy and efficiency of the proposed method have been investigated by solving some benchmark examples and comparing the obtained results with the analytical or valid finite element analyses’ results.

Phase field modeling of fracture in multi-physics problems. Part III. Crack driving forces in hydro-poro-elasticity and hydraulic fracturing of fluid-saturated porous media

The prediction of fluid- and moisture-driven crack propagation in deforming porous media has achieved increasing interest in recent years, in particular with regard to the modeling of hydraulic fracturing, the so-called “fracking”. Here, the challenge is to link at least three modeling ingredients for (i) the behavior of the solid skeleton and fluid bulk phases and their interaction, (ii) the crack propagation on not a priori known paths and (iii) the extra fluid flow within developed cracks. To this end, a macroscopic framework is proposed for a continuum phase field modeling of fracture in porous media. It provides a rigorous geometric approach to a diffusive crack modeling based on the introduction of a constitutive balance equation for a regularized crack surface and its modular linkage to a Darcy–Biot-type bulk response of hydro-poro-elasticity. The approach overcomes difficulties associated with the computational realization of sharp crack discontinuities, in particular when it comes to complex crack topologies including branching. A modular concept is outlined for linking of the diffusive crack modeling with the hydro-poro-elastic response of the porous bulk material. This includes a generalization of crack driving forces from energetic definitions towards threshold-based criteria in terms of the effective stress related to the solid skeleton of a fluid-saturated porous medium. Furthermore, a Poiseuille-type constitutive continuum modeling of the extra fluid flow in developed cracks is suggested based on a deformation-dependent permeability, that is scaled by a characteristic length. This proposed modular model structure is exploited in the numerical implementation by constructing a robust finite element method, based on an algorithmic decoupling of updates for the crack phase field and the state variables of the hydro-poro-elastic bulk response. We demonstrate the performance of the phase field formulation of fracture for a spectrum of model problems of hydraulic fracture. A slight modification of the framework allows the simulation of drying-caused crack patterns in partially saturated capillar-porous media.

Ductile failure with gradient plasticity coupled to the phase‐field fracture at finite strains

AbstractMost metals fail in a ductile fashion, i.e, fracture is preceded by significant plastic deformation. The modeling of failure in ductile metals must account for complex phenomena at micro‐scale, such as nucleation, growth and coalescence of micro‐voids. In this work, we start with von‐Mises plasticity model without considering void generation. The modeling of macroscopic cracks can be achieved in a convenient way by the continuum phase field approaches to fracture, which are based on the regularization of sharp crack discontinuities [1]. This avoids the use of complex discretization methods for crack discontinuities and can account for complex crack patterns. The key aspect of this work is the extension of the energetic and the stress‐based phase field driving force function in brittle fracture to account for a coupled elasto‐plastic response in line with our recent work [3]. We develop a new theoretical and computational framework for the phase field modeling of ductile fracture in elastic‐plastic solids. To account for large strains, the constitutive model is constructed in the logarithmic strain space, which simplify the model equations and results in a formulation similar to small strains. We demonstrate the modeling capabilities and algorithmic performance of the proposed formulation by representative simulations of ductile failure mechanisms in metals. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Hydraulic fracturing based on the Theory of Porous Media

AbstractHydraulic fracturing is a widely‐used technology in the field of energy exploitation. The relevant study is intricate owing to the crack‐inducing instability and the coupling between different constituents in stratum. The proposed model is based on the phase‐field theory and extended to the Theory of Porous Media. The former removes crack discontinuity by substituting sharp crack with smeared one and the later provides clear and distinct description of multi‐phasic material in macroscopic scale. A new variable, the order parameter ϕ is imported to characterise material as unbroken porous media and the crack zone, the evolution of which is governed by potential energy of solid. With the change of ϕ, the stiffness of solid is decreasing while the fluid transport is transferring from Darcy‐type to Navier‐Stokes‐type. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A phase-field model for fracture in biological tissues.

This work presents a recently developed phase-field model of fracture equipped with anisotropic crack driving force to model failure phenomena in soft biological tissues at finite deformations. The phase-field models present a promising and innovative approach to thermodynamically consistent modeling of fracture, applicable to both rate-dependent or rate-independent brittle and ductile failure modes. One key advantage of diffusive crack modeling lies in predicting the complex crack topologies where methods with sharp crack discontinuities are known to suffer. The starting point is the derivation of a regularized crack surface functional that [Formula: see text]-converges to a sharp crack topology for vanishing length-scale parameter. A constitutive balance equation of this functional governs the crack phase-field evolution in a modular format in terms of a crack driving state function. This allows flexibility to introduce alternative stress-based failure criteria, e.g., isotropic or anisotropic, whose maximum value from the deformation history drives the irreversible crack phase field. The resulting multi-field problem is solved by a robust operator split scheme that successively updates a history field, the crack phase field and finally the displacement field in a typical time step. For the representative numerical simulations, a hyperelastic anisotropic free energy, typical to incompressible soft biological tissues, is used which degrades with evolving phase field as a result of coupled constitutive setup. A quantitative comparison with experimental data is provided for verification of the proposed methodology.

X-FEM modeling of dynamic ductile fracture problems with a nonlocal damage-viscoplasticity model

In this paper, the dynamic large deformation X-FEM method is presented for modeling the full process of dynamic ductile fracture based on a nonlocal damage visco-plasticity model. The effect of inertia is modeled using an explicit central difference scheme which is enhanced through the use of mass lumping, reduced integration with hourglass control, and numerical damping. The material nonlinearity and the flow stress dependency on strain rate, hardening and temperature are modeled with the Johnson–Cook visco-plastic model. The micro-void nucleation, growth and coalescence are modeled macroscopically with an isotropic damage model. The localization phenomenon due to the damage and thermal softening is suppressed by using the visco-plastic regularization in combination with the nonlocal visco-plastic model. The large deformation and large strain formulations are implemented within the X-FEM framework to model the macro crack discontinuities using an updated Lagrangian approach. Crack propagation and crack direction criteria are presented and the issues relating to the combination of X-FEM with the damage visco-plasticity model are addressed. Finally, the robustness and accuracy of the proposed method are verified through several numerical examples.

Phase field modeling of fracture in multi-physics problems. Part I. Balance of crack surface and failure criteria for brittle crack propagation in thermo-elastic solids

Abstract This work presents a generalization of recently developed continuum phase field models for brittle fracture towards fully coupled thermo-mechanical and multi-physics problems at large strains. It outlines a rigorous geometric approach to the diffusive crack modeling based on the introduction of a balance of regularized crack surface, governed by a crack phase field. The regularized crack surface functional is based on a crack surface density function, that describes the macroscopic crack surface in the bulk material per unit of the reference volume. The approach overcomes difficulties associated with the computational realization of sharp crack discontinuities, in particular when it comes to complex crack topologies. The formulation proposed is essentially a gradient damage theory, however, equipped with critical ingredients rooted in fracture mechanics. A modular concept is outlined for the linking of the diffusive crack modeling with complex multi-field response of the bulk material, where focus is put on the model problem of finite thermo-elasticity. This concerns a generalization of crack driving forces from the energetic definitions towards stress-based criteria, the constitutive modeling of heat conduction across cracks and convective heat exchanges at crack faces based on additional constitutive functions. This is achieved by approximating surface load integrals of the sharp crack approach by distinct volume integrals. We demonstrate the performance of the phase field formulation of fracture at large strains by means of representative numerical examples.

Towards Phase Field Modeling of Ductile Fracture in Gradient‐Extended Elastic‐Plastic Solids

AbstractThe modeling of failure in ductile metals must account for complex phenomena at a micro‐scale as well as the final rupture at the macro‐scale. Within a top‐down viewpoint, this can be achieved by the combination of a micro‐structure‐informed elastic‐plastic model with a concept for the modeling of macroscopic crack discontinuities. In this context, it is important to account for material length scales and thermo‐mechanical coupling effects due to dissipative heating. This can be achieved by the construction of non‐standard, gradient‐enhanced models of plasticity with a full embedding into continuum thermodynamics [1,2]. The modeling of macroscopic cracks can be achieved in a convenient way by recently developed continuum phase field approaches to fracture based on regularized crack discontinuities. This avoids the use of complex discretization methods for crack discontinuities, and can account for complex crack patterns within a pure continuum formulation. Moreover, the phase field modeling of fracture is related to gradient theories of continuum damage mechanics, and fits nicely the structure of constitutive models for gradient plasticity. The main focus of this work is the extensions to gradient thermoplasticity and phase field formulation of ductile fracture, conceptually in line with the work [3]. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Variational Phase Field Modeling of Fracture Caused by Fluid Transport in Poroelastic Solids

AbstractThe numerical modeling of failure mechanisms due to fracture based on sharp crack discontinuities is extremely demanding and suffers in situations with complex crack topologies. This drawback can be overcome by recently developed diffusive crack modeling concepts, which are based on the introduction of a crack phase field. Such an approach is conceptually in line with gradient‐extended continuum damage models which include internal length scales. In this paper, we extend our recently outlined mechanical framework [1–3] towards the phase field modeling of fracture in the coupled problem of fluid transport in deforming porous media. Here, extremely complex crack patterns may occur due to drying or hydraulic induced fracture, the so called fracking. We develop new variational potentials for Biot‐type fluid transport in porous media at finite deformations coupled with phase field fracture. It is shown, that this complex coupled multi‐field problem is related to an intrinsic mixed variational principle for the evolution problem. This principle determines the rates of deformation, fracture phase field and fluid content along with the fluid potential. We develop a robust computational implementation of the coupled problem based on the potentials mentioned above and demonstrate its performance by the numerical simulation of complex fracture patterns. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Plastic pre-compression and creep damage effects on the fracture toughness behaviour of Type 316H stainless steel

The influence of inelastic damage in the form of plastic pre-strain and creep damage, on fracture toughness of Type 316H stainless steel has been examined. Creep damage has been introduced into the 8% pre-compressed material by interrupting creep crack growth tests. Comparisons have been made between the fracture toughness test results from the as-received, pre-compressed and creep damaged materials. Furthermore, the effects of creep crack discontinuities on the crack tip strain fields have been examined by digital image correlation measurements. Inelastic damage was found to reduce the fracture toughness of the material, with creep damage having more severe effects than pre-strain.

Edge Detection in Pipe Images Using Classification of Haar Wavelet Transforms

Automatic image interpretation for pipe inspection is a relatively recent area of research, which has great potential benefit. An important component of such systems is crack detection, or, more generally, edge or discontinuity detection. This paper describes a new approach to edge detection and applies it to pipe images. The method labels each pixel in an image as an edge pixel or a nonedge pixel by processing the Haar wavelet transform of the image in a window about the pixel using a support vector machine. As a pixel classifier, to within a moderate morphological tolerance, the detector has an accuracy of 99% on the images on which it has been tested and compares favorably with the commonly used Canny edge detector.

Design method for flexural strengthening with fiber reinforced polymer (FRP) laminates avoiding its premature debonding

According to the available experimental research related to structures flexurally strengthened by bonding fiber reinforced polymer (FRP) laminates, the laminate debonds in a brittle manner in most tests. This debonding failure usually initiates in the most loaded regions due the effect of intermediate cracks (intermediate crack debonding). From the beginning of this technique, several models have been developed to avoid this premature failure. However, some of them present a significant scatter in relation to the experimental data. This paper presents a comparative analysis of the existing formulations through a database of tests compiled by the authors. Afterwards, a design procedure based on a shear-bending moment interaction diagram related to intermediate crack debonding is presented. This interaction diagram depends on the maximum shear force between crack discontinuities. Finally, the proposed method is compared to the rest of the models obtaining satisfactory results when predicting the failure load.

Phase field modeling of fracture in rubbery polymers. Part I: Finite elasticity coupled with brittle failure

This work presents a new phase field model for rate-independent crack propagation in rubbery polymers at large strains and considers details of its numerical implementation. The approach accounts for micro-mechanically based features of both the elastic bulk response as well as the crack toughness of idealized polymer networks. The proposed diffusive crack modeling based on the introduction of a crack phase field overcomes difficulties associated with the computational realization of sharp crack discontinuities, in particular when it comes to complex crack topologies. The crack phase field governs a crack density function, which describes the macroscopic crack surface in the polymer per unit of the reference volume. It provides the basis for the constitutive modeling of a degrading free energy storage and a crack threshold function with a Griffith-type critical energy release rate, that governs the crack propagation in the polymer. Both the energy storage as well as the critical energy release due to fracture can be related to classical statistical network theories of polymers. The proposed framework of diffusive fracture in polymers is formulated in terms of a rate-type variational principle that determines the evolution of the coupled primary variable fields, i.e. the deformation field and the crack phase field. On the computational side, we outline a staggered solution procedure based on a one-pass operator split of the coupled equations, that successively updates in a typical time step the crack phase field and the displacement field. Such a solution algorithm is extremely robust, easy to implement and ideally suited for engineering problems. We finally demonstrate the performance of the phase field formulation of fracture at large strains by means of representative numerical examples.

EXTENDED FINITE ELEMENT METHOD WITH NONLINEAR GEOMETRY AND ITS APPLICATION IN FRACTURE MECHANICS

A extended finite element (X-FEM) method and corresponding Fortran code are developed in the modeling and simulation for nonlinear geometry and fracture mechanics problems. X-FEM can model a domain without explicitly meshing the crack surface. This method can treat an arbitrary crack independent of the mesh and crack growth. The X-FEM formulas with nonlinear geometry are deduced. In order to model the crack discontinuity, a Heaviside step function and a two-dimensional asymptotic crack-tip displacement field are added to the traditional finite element approximation for the local enrichment based on the theory of partition of unity. The crack is described by two level set functions. The X-FEM computational algorithm is presented in the framework of Lagrangian description in order to model the arbitrary discontinuities in large deformations. The stress intensifying the factors of a crack are calculated by using the multi-point displacement extrapolation method and least square fitting. Finally, a numerical example is presented to demonstrate the accuracy and efficiency of the X-FEM and the FORTRAN code in large deformation crack problems. It is found that X-FEM is superior to the traditional FEM in the modeling and simulation of crack being and crack growth program.

Wave diffraction by a line of finite crack in a saturated two-phase medium

An application of the Biot’s theory to the diffraction problem of plane harmonic dilatational waves (P-waves) of the first kind and the second kind by a line crack or geometric discontinuity of finite width embedded in a saturated two-phase medium is presented in this paper. The crack surfaces are assumed impermeable, and the integral transform method is utilized to reduce the mixed boundary-value problem to a single Fredholm integral equation. The magnitudes of the intensity of the stress fields near the crack tips measured by Mode I dynamic stress-intensity factor (dimensionless) are computed and displayed graphically against dimensionless circular frequency (ω) for several dimensionless material property values, namely, viscosity-to-permeability and mass density ratios. In the case of the normally incident P-waves of the first kind, the results in terms of stress-intensity factor are also compared with the corresponding values of dry elastic material. All the stress-intensity factor curves are shown to exhibit a similar character in that they rise to the peaks at certain frequency values and then decay with increasing frequencies. At certain frequency ranges and material property values, amplification in the dynamic stress-intensity factor can be substantially larger than those encountered in dry elastic materials. The stress-intensity factor is found to be more affected by the changes in the ratio of viscosity-to-permeability at lower mass density ratio. With fluid mass density 10% of the bulk mass density, the viscosity-to-permeability ratio of 0.01 gives the highest increase of about 32% in the magnitude of stress-intensity factor compared to the dry material counterpart value, while a decrease of about 9% is observed for the viscosity-to-permeability ratio of 100. It is also found that change in mass density ratio has significant effect upon the magnitude of stress-intensity factor at lower ratio of viscosity-to-permeability. As for the normally incident P-waves of the second kind, the presence of the pore fluid affects both the magnitude and character of the stress-intensity factor. Large variations in the magnitude of stress-intensity factor are observed as viscosity-to-permeability ratio changes from 1 to 100. At the ratio of viscosity-to-permeability of 1.0, the stress-intensity factor curves increase gradually with frequency and exhibit the peaks in curves for mass density ratio of 0.3 and higher. As the viscosity-to-permeability ratio is raised to 100, the stress-intensity factor curves increase monotonically with frequency at a much faster rate throughout the frequency range of interest (ω=0–2), and the change in mass density ratio is shown to have little effect on the stress-intensity factor, especially within the low frequency ranges. The results obtained in this study are useful in the mechanics of fracture initiation of saturated porous materials under the fluctuating mechanical and/or pore fluid loadings that are periodic with time.

Phase Field Modeling of Fracture in Plates and Shells

AbstractThe numerical modeling of failure mechanisms in plates and shells due to fracture based on sharp crack discontinuities is extremely demanding and suffers in situations with complex crack topologies. This drawback can be overcome by a diffusive crack modeling, which is based on the introduction of a crack phase field. In this paper, we extend ideas recently outlined in [1, 2] towards the phase field modeling of fracture in dimension‐reduced continua with application to Kirchhoff plates and shells. The introduction of history fields, containing the maximum reference energy obtained in history, provides a very transparent representation of the coupled balance equations and allows the construction of an extremely robust operator split technique. The performance of the proposed models is demonstrated by means of representative numerical examples. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A Phase Field Model for Ductile to Brittle Failure Mode Transition

AbstractThe computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in dynamic problems with complex crack topologies. This can be overcome by a diffusive crack modeling based on the introduction of a crack phase field. We outline a conceptual framework for phase field models of crack propagation in brittle elastic and ductile elastic‐plastic solids under dynamic loading and investigate the ductile to brittle failure mode transition observed in the experiment performed by Kalthoff and Winkeler [3]. We develop incremental variational principles and consider their numerical implementations by multi‐field finite element methods. To this end, we define energy storage and dissipation functions for the plastic flow including the fracture phase field. The introduction of local history fields that drive the evolution of the crack phase field inspires the construction of robust operator split schemes. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)