Thick Level Set Research Articles

Modeling fiber bridging effects on fatigue delamination propagation in composites using the interface thick level set method

Modeling fiber bridging effects on fatigue delamination propagation in composites using the interface thick level set method

Skeleton curve and phantom node method for the Thick Level Set approach to fracture

The Thick Level Set method (TLS) is an approach for non-local damage modeling in which the damage evolution is linked to the movement of a damage front described with the level set method. More recently, a new version of the TLS, designated as the TLSV2, has been proposed as a new concept for coupling continuum damage modeling and discrete cohesive crack modeling for failure analysis in solids. The main objective of this new framework is to profit from both modeling approaches. The continuum part allows for handling crack initiation, branching and merging, whereas the cohesive part brings the capability to handle discrete cracks with large crack opening or sliding without heavily distorted elements, and with the possibility to model stiffness recovery upon contact. In this paper, a generalized framework for the TLSV2 is introduced. Two major issues with the TLSV2 method that have not been dealt with since its inception are addressed in this study, and solutions are proposed. Firstly, the method depends on the location of skeleton curve of the level set field, on which the discontinuity in the displacement field is evaluated. The problem of locating the skeleton curve can be a complicated task, mainly because topological events may emerge as the analysis progresses, such as crack branching. The skeleton curve is determined through a combination of ball-shrinking and graph-based algorithms and then mapped onto the finite element mesh. Secondly, the cohesive forces and displacement discontinuity of the TLSV2 are modeled using the phantom node method. Furthermore, a new approach to compute the averaged values of local quantities is introduced, and model calibration is discussed. The degree of stiffness recovery under compression that is still needed for the continuum part is investigated. Numerical experiments demonstrate the accuracy and ability of the proposed model to handle simulation of failure analysis presenting complex topological crack patterns.

Development of a damage viscoelastic model using the thick level set approach to fracture: 1D modeling and comparison to uniaxial tension stress tests on bituminous specimens

The present paper deals with the development of a mechanical model coupling damage and viscoelasticity for applications related to fracture in bituminous materials and structures. The behavior of these materials is highly thermo-sensitive and depends on the loading rate. Currently, there is no consensus on a modeling approach able to capture their behavior to fracture for a wide range of temperature and loading conditions. This is an objective of the model developed in this paper. Prior to damage initiation, this one is assumed purely viscoelastic and relies on the spectral decomposition of the creep function and the time–temperature superposition principle for the thermal dependency. Damage is later incorporated in the model on the basis of the concept of effective stress and a damage evolution law postulated at the local level, then regularized according to the thick level set (TLS) approach for the purpose of structural computations. This is the first modeling combining the TLS approach and viscoelasticity. This novel model is implemented in a numerical program in 1D and applied to the simulation of uniaxial tensile stress tests (UTST) carried out on a bituminous material at several temperatures and a given monotonic displacement rate. By defining the critical energy release rate as a function of temperature in the model, the numerical simulations compare favorably with the experimental results for all the tested temperatures, proving the capacity of this model to capture brittleness/ductileness of the bituminous samples depending on temperature. Based on the test simulations, other features of the model are also discussed.

Parallel Computing with the Thick Level Set Method

The thick level set (TLS) method has been proposed as a nonlocal damage model for the modeling of failure in solids being able to deal with crack initiation, branching, and merging. The nonlocality...

Modeling the R-curve effects in laminate composites using the interface thick level set method

This paper presents a new model for simulating delamination growth in composites in presence of the resistance curve (R-curve) effects. This model is extended from the interface thick level set (ITLS) method which is developed recently. In this approach the common definition of damage in the constitutive relationship of the interface elements is replaced with a new damage definition based on the thick level set (TLS) method. This new damage definition provides a direct link between fracture and damage mechanics that allows use of interface elements without a predefined cohesive law. In this paper the ITLS method is adopted to incorporate the R-curve effects caused by extended bridging, in modeling delamination. The fracture energy in the developed model is defined as a function of the crack length which allows simple use of experimentally measured R-curve as a direct input of this model. This is in contrast with the popular cohesive models that need to develop some mathematical relationships to extract the cohesive law parameters from experimental R-curve. The proposed 3D model is validated against experimental data obtained from literature to support the accuracy of obtained results.

Accelerating crack growth simulations through adaptive model order reduction

SummaryAccurate numerical modeling of fracture in solids is a challenging undertaking that often involves the use of computationally demanding modeling frameworks. Model order reduction techniques can be used to alleviate the computational effort associated with these models. However, the traditional offline‐online reduction approach is unsuitable for complex fracture phenomena due to their excessively large parameter spaces. In this work, we present a reduction framework for fracture simulations that leaves out the offline training phase and instead adaptively constructs reduced solutions spaces online. We apply the framework to the thick level set (TLS) method, a novel approach for modeling fracture able to model crack initiation, propagation, branching, and merging. The analysis starts with a fully‐solved load step, after which two consecutive reduction operations—the proper orthogonal decomposition and the empirical cubature method—are performed. Numerical features specific to the TLS method are used to define an adaptive domain decomposition scheme that allows for three levels of model reduction coexisting on the same finite element mesh. Special solutions are proposed that allow the framework to deal with enriched nodes and a dynamic number of integration points. We demonstrate and assess the performance of the framework with a number of numerical examples.

Combining the thick level set method with plasticity

The Thick Level Set (TLS) method has been proposed as a new approach to the modeling of damage growth in solids. The fronts of damaged zones are implicitly represented as a level set of an auxiliary field whose evolution is accomplished by the level set method. The TLS model contains a characteristic length to obtain a non-local description that prevents spurious localization in the strain field. The update of the damage is indirectly performed by integrating local values of energy release rate over this characteristic length. This model offers an automatic transition from damage to fracture, and deals with merging and branching cracks as well as crack initiation in an easy and robust manner. In this paper, the TLS is applied to simulate the formation of cusps in a polymer matrix loaded in shear. Realistic simulation of this process requires the damage model to be combined with plasticity in order to capture the behavior of the material prior to failure. To accommodate for plasticity, several changes to the TLS framework are introduced. A strength-based criterion for initiation of damage based on the ultimate yield surface of such plasticity model is proposed. A mapping operator for transferring history is included if the integration scheme in element changes. Furthermore, a new loading scheme is devised that does not rely on secant unloading. Numerical experiments demonstrate the accuracy and effectiveness of the proposed model to handle simulation of crack growth in a medium with hardening plasticity.

A cohesive XFEM model for simulating fatigue crack growth under mixed‐mode loading and overloading

SummaryStructures are subjected to cyclic loads that can vary in direction and magnitude, causing constant amplitude mode I simulations to be too simplistic. This study presents a new approach for fatigue crack propagation in ductile materials that can capture mixed‐mode loading and overloading. The extended finite element method is used to deal with arbitrary crack paths. Furthermore, adaptive meshing is applied to minimize computation time. A fracture process zone ahead of the physical crack tip is represented by means of cohesive tractions from which the energy release rate, and thus the stress intensity factor can be extracted for an elastic‐plastic material. The approach is therefore compatible with the Paris equation, which is an empirical relation to compute the fatigue crack growth rate. Two different models to compute the cohesive tractions are compared. First, a cohesive zone model with a static cohesive law is used. The second model is based on the interfacial thick level set method in which tractions follow from a given damage profile. Both models show good agreement with a mode I analytical relation and a mixed‐mode experiment. Furthermore, it is shown that the presented models can capture crack growth retardation as a result of an overload.

A new approach of graded damage modelling

To prevent the problem of spurious localisation in damage mechanics, it is necessary to control the damage gradient amplitude. The body is decomposed in three domains: the undamaged body where (), the transition zone () and the totally broken body (). For the thick level set (TLS) model, damage is a function of the signed distance to the surface . In this article, we propose to control the damage gradient using a convex internal constraint. This point of view produces a new description of graded damage. Analytical solutions on spheres and cylinders under radial loading are given and discussed. For particular internal constraints, the TLS results are recovered.

Cohesive zone and interfacial thick level set modeling of the dynamic double cantilever beam test of composite laminate

The mode-I interlaminar fracture toughness of composite laminates under different loading rates can be measured by the double cantilever beam (DCB) test. It is observed from the DCB test of a unidirectional PEEK/carbon composite laminate that as the loading rate increases from quasi-static to dynamic range: (1) delamination crack growth exhibits a transition from stable to unstable (“stick/slip”) and back to a stable type; (2) the interlaminar fracture toughness is not constant as the loading rate increases. In this paper, two numerical approaches are used to reproduce the experimental observations: a cohesive zone model (CZM) and the interfacial thick level set (ITLS) model. CZM simulations with rate-independent and rate-dependent cohesive laws are carried out. A new version of the ITLS is introduced with a phenomenological relation between crack speed and energy release rate. The simulation results of the CZM and the ITLS model are compared with the real DCB test data to evaluate the capability of these two types of models. It is found that the used CZM can reproduce rate-dependence of the fracture energy, but not the stick/slip behavior. The ITLS can capture the stick/slip behavior, but needs different parameter sets for different loading rates.

Analysis of the failure at notches and cavities in quasi-brittle media using the Thick Level Set damage model and comparison with the coupled criterion

The failure of quasi-brittle specimen weakened by sharp or blunted notches and cavities is analyzed under quasi-static loading. The load at failure is obtained with the Thick Level Set (TLS) damage modeling. In this model, the damage gradient is bounded implying that the minimal distance between a point where damage 0 (sound material) to 1 (fully damaged) is an imposed characteristic length in the model. This length plays an important role on the damage evolution and on the failure load. The paper shows that the TLS predictions are relevant. A comparison with the coupled criterion (CC) of Leguillon (2002) is given. A good agreement is obtained for cavities and V-notches provided that the characteristic length of Irwin is small compared to the notch depth (condition for the applicability of the CC criterion). A comparison with failure loads obtained experimentally is also given. In the numerical simulations, uniform stresses are imposed at infinity using a new finite element mapping technique (Cloirec 2005).

Modeling Restrained Shrinkage Induced Cracking in Concrete Rings Using the Thick Level Set Approach

Modeling restrained shrinkage-induced damage and cracking in concrete is addressed herein. The novel Thick Level Set (TLS) damage growth and crack propagation model is used and adapted by introducing shrinkage contribution into the formulation. The TLS capacity to predict damage evolution, crack initiation and growth triggered by restrained shrinkage in absence of external loads is evaluated. A study dealing with shrinkage-induced cracking in elliptical concrete rings is presented herein. Key results such as the effect of rings oblateness on stress distribution and critical shrinkage strain needed to initiate damage are highlighted. In addition, crack positions are compared to those observed in experiments and are found satisfactory.

Coupling damage and cohesive zone models with the Thick Level Set approach to fracture

This paper couples bulk damage modeling and cohesive zone modeling to get the benefits of both. Damage brings the directionality for the crack propagation as well as the possibility of crack branching while cohesive zone modeling allows for an explicit discrete crack modeling. The coupling is made easy through the Thick Level Set approach. The originality is that the coupling induces concurrent development of bulk and interface degradation.

Concurrent development of local and non-local damage with the Thick Level Set approach: Implementation aspects and application to quasi-brittle failure

The present paper focuses on the discretization and implementation of the Thick Level Set (TLS) approach for quasi-brittle failure when dealing with both local and non-local development of damage.The TLS damage model introduces a length scale and nonlocality in the Fracture Process Zone (FPZ) and it handles the continuous to discontinuous failure transition. The added nonlocality is limited to the FPZ and therefore the model is local away from the FPZ or before its emergence. It leads to concurrent development of local and nonlocal damage during simulations. This paper presents a space–time discretization that handles concurrent computations in a unified manner. It is based on an explicit time discretization written as a predictor–corrector scheme and a space discretization that uses specific approximation functions (modes) that embed nonlocality. It also uses an alternative implementation of the eXtended Finite Element Method (X-FEM) enrichment used in the TLS which is based on virtual nodes.

On the capability of the Thick Level Set (TLS) damage model to fit experimental data of size and shape effects

Size and shape effects are important issues in predicting the global response of concrete structures. Small-scale tests performed in laboratory to determine the material properties are not enough to simulate large-scale structures. Many models are used to extrapolate small scale results to large scale simulations, but only few are able to recover size and shape effects. Recently a model of graded damage (TLS) has been proposed and comparison with cohesive zone models shows that this new model contains a new degree of freedom, the length of transition between totally damaged material and undamaged zone (ie the process zone size). In this paper, the capability of the model (TLS) to represent size and shape effects for two recently published experimental campaigns is studied.

Fatigue modeling in composites with the thick level set interface method

This paper presents a new discontinuous damage model for modeling fatigue crack growth in composites. This new fatigue model is formulated based on the thick level set interface approach which we developed recently. In this approach the thick level set (TLS) method is combined with interface elements for modeling delamination growth. Crack growth under cyclic loading is described with the Paris relation. In contrast with popular cohesive zone methods, this new approach provides an accurate non-local evaluation of the energy release rate as well as a framework in which the crack growth rate can be directly imposed. The proposed 3D mixed-mode model is validated against experimental and theoretical data.

A new method for fatigue life prediction based on the Thick Level Set approach

The last decade has seen a growing interest in cohesive zone models for fatigue applications. These cohesive zone models often suffer from a lack of generality and applying them typically requires calibrating a large number of model-specific parameters. To improve on these issues a new method has been proposed in this paper based on the Thick Level Set approach. In this concept, material degradation due to cyclic loading is the result of interaction between damage evolution and fracture mechanics. The Thick Level Set formulation has been extended to interface elements, in order to allow for separation of strain energy in the bulk and energy required for surface creation. Global fracture parameters, derived from a free energy description governing the interface elements, are used as input for the empirical crack growth rate relation (Paris’ equation). It must be emphasized that in contrast to existing fatigue models, the Thick Level Set approach does not require the definition of a damage evolution law. Instead, damage is updated automatically by a continuously moving damage front. It is shown that applicability is not limited to fatigue behavior of linear elastic materials; elastic-plastic materials such as steels can be analysed as well. The sensitivity of model parameters is investigated and discussed and the practical relevance is explored for standard test configurations.

An interface thick level set model for simulating delamination in composites

SummaryThis paper presents a discontinuous thick level set model for modeling delamination initiation and propagation in composites. Interface elements are widely applied in delamination models to define a discontinuity between layers of a laminate. In this paper, the common damage definition in the constitutive relation of interface elements is replaced with a new definition developed using the thick level set approach. Following this approach, a band of damage with predefined length is considered where the damage is defined as a function of distance to the damage front. The specifications of suitable damage functions for the developed method are investigated, and an efficient damage function is introduced. A sensitivity analysis of numerical input parameters is performed, which proves that the model is not sensitive to the length of the damage band. Because the required element size is linked to the length of the damage band, the insensitivity to this length provides freedom to use coarser meshes. Furthermore, the model provides a direct link between fracture mechanics and damage mechanics, which enables further development of the model for fatigue analysis. Validation of this model is presented by conducting three‐dimensional mode I, mode II, and mixed‐mode simulations and comparing the results with analytical solutions. Copyright © 2016 John Wiley & Sons, Ltd.

The Thick Level-Set model for dynamic fragmentation

The Thick Level-Set (TLS) model is implemented to simulate brittle media undergoing dynamic fragmentation. This non-local model is discretized by the finite element method with damage represented as a continuous field over the domain. A level-set function defines the extent and severity of damage, and a length scale is introduced to limit the damage gradient. Numerical studies of one-dimensional problems demonstrate that the proposed method reproduces the rate-dependent energy dissipation and fragment length observations from analytical, numerical, and experimental approaches. Additional studies emphasize the importance of appropriate bulk constitutive models and sufficient spatial resolution of the length scale.

On use of the thick level set method in 3D quasi-static crack simulation of quasi-brittle material

This work demonstrates the 3D capability of the thick level set (TLS) method, first introduced by Moes et al. (Int J Numer Methods Eng 86:358–380, 2011. doi:10.1002/nme.3069) and later in Stolz and Moes (Int J Fract 174(1):49–60, 2012. doi:10.1007/s10704-012-9693-3). The thick level set approach is a non-local damage method embedding fracture mechanics discontinuity. Enhanced numerical implementation for elastic quasi-brittle materials in 2D under quasi-static loading conditions was presented in Bernard et al. (Comput Methods Appl Mech Eng 233–236:11–27, 2012. doi:10.1016/j.cma.2012.02.020). The present work focuses on using this enhanced numerical implementation in a 3D context. This work adds a new way to construct the crack faces by use of a “double cut algorithm”. The regularization computation, part of the non-local feature of the model, is also reviewed to improve its accuracy. As 3D models are computationally intensive, CPU aspects are discussed. Five test cases are presented. The first one illustrates the capability of the method to deal with crack coalescence, which is quite unique for this kind of simulation. Three other cases point out a comparison with literature examples (numerical and experimental) and good agreement is observed. One is a more complex example, which deals with an engineering oriented application. This work confirms good performance of the thick level set method in 3D context. The use of the new “double cut” algorithm is giving well discretized crack path and allows for discontinuous displacement.