Cohesive Zone Formulation Research Articles

Concurrent experimental testing and computational micromechanical modeling of a UD NCF GFRP composite ply

The current research work presents a detailed and thoroughly validated computational micromechanical modeling methodology to study the damage initiation and propagation in a uni-directional (UD) glass fiber-reinforced non-crimp fabric (NCF) composite ply. Under the applied transverse tension and compression loads, the effect of various microscale material and geometrical parameters on the ply level stress–strain behavior is studied. To this end, along with the distinctly modeled individual microscale constituents of the UD NCF composite ply (axial fibers, backing fibers, and matrix), the generated RVE (Representative Volume Element) model consists of manufacturing-induced defects and variabilities such as voids as well as the backing fiber’s out-of-plane waviness. The fiber/matrix interface failure behavior in the generated RVE model is simulated using cohesive zone formulations that follow bi-linear traction-separation law. Whereas the hydrostatic pressure-dependent non-linear stress–strain response followed by the fracture of the epoxy matrix is captured using the linear Drucker-Prager plasticity model coupled with the stress triaxiality-based failure criterion. The backing fibers stress–strain and failure behavior is modeled using the linear elastic and isotropic material model in conjunction with the maximum stress criterion.The proposed numerical methodology is thoroughly validated both qualitatively and quantitatively by conducting detailed experiments on a neat epoxy as well as at the composites laminate level. Upon comparing the experimental and computational results, the observed knee or slope change in the bi-linear stress–strain response of the UD NCF composite ply under transverse tension is attributed to the loss of the load-carrying ability of the axial fiber bundle due to fiber/matrix interface debonding followed by the ply splitting. Under the application of transverse compression load, the composite ply failure behavior is governed by the formation of a dominant matrix plastic shear band in the axial fiber bundles, which is in turn triggered by the fiber/matrix interface failure. Under the applied loads in the matrix-dominated direction, the presented research work provides a detailed insight into the microscale damage initiation and propagation in a UD NCF composite ply and its consequent influence on the macroscale stress–strain response.

Blind prediction of curved fracture surfaces in gypsum samples under three-point bending using the Discontinuous Galerkin Cohesive Zone method

A computational framework based on a Discontinuous Galerkin (DG)/Cohesive Zone formulation is utilized to simulate the experiments of the Purdue Damage Mechanics Modeling Challenge. The inelastic response of the additively-manufactured gypsum material used in the experimental tests is modeled via a dilatational plasticity model. The constitutive and fracture model parameters are calibrated using the load–displacement curves corresponding to three-point bending tests initially provided by the Challenge organizers. The test samples contained initial notches especially designed to force specific types of mixed fracture modes. The calibrated computational modeling framework is used to blindly simulate the more complex configuration of the Challenge experiments. The numerical predictions of the load–displacement curve and the shape of the curved fracture surface are compared to the experimental data provided a posteriori. It is found that the computational method is able to quantitatively describe the fracture response of the material including crack propagation, plastic wake, and the curved geometry of the fracture surface that results from the evolving fracture mode mixity with significant fidelity.

A numerical framework for simulating progressive failure in composite laminates under high-cycle fatigue loading

In this work, a recently proposed high-cycle fatigue cohesive zone model, which covers crack initiation and propagation with limited input parameters, is embedded in a robust and efficient numerical framework for simulating progressive failure in composite laminates under fatigue loading. The fatigue cohesive zone model is enhanced with an implicit time integration scheme of the fatigue damage variable which allows for larger cycle increments and more efficient analyses. The method is combined with an adaptive strategy for determining the cycle increment based on global convergence rates. Moreover, a consistent material tangent stiffness matrix has been derived by fully linearizing the underlying mixed-mode quasi-static model and the fatigue damage update. The enhanced fatigue cohesive zone model is used to describe matrix cracking and delamination in laminates. In order to allow for matrix cracks to initiate at arbitrary locations and to avoid complex and costly mesh generation, the phantom node version of the eXtended finite element method (XFEM) is employed. For the insertion of new crack segments, an XFEM fatigue crack insertion criterion is presented, which is consistent with the fatigue cohesive zone formulation. It is shown with numerical examples that the improved fatigue damage update enhances the accuracy, efficiency and robustness of the numerical simulations significantly. The numerical framework is applied to the simulation of progressive fatigue failure in an open-hole [±45]-laminate. It is demonstrated that the numerical model is capable of accurately and efficiently simulating the complete failure process from distributed damage to localized failure.

Characterisation of damage by means of electrical measurements: Numerical predictions

AbstractUnderstanding damage mechanisms and quantifying damage is important in order to optimise structures and to increase their reliability. To achieve this goal, experimental‐ and simulation‐based techniques are to be combined. Different methods exist for the analysis of damage phenomena such as fracture mechanics, phase field models, cohesive zone formulations and continuum damage modelling. Assuming a typical ‐type damage formulation, the governing equations of continua that account for gradient‐enhanced ductile damage under mechanical and electrical loads are derived. The mechanical and electrical sub‐problems give rise to the local form of the balance equation of linear momentum, the micromorphic balance relation and the continuity equation for the electric charge, respectively. Experimental investigations indicate that changes in electrical conductivity arise due to the evolution of the underlying microstructure, for example, of cracks and dislocations. Therefore, motivated by deformation‐induced property changes, the effective electrical conductivity is assumed to be a function of the damage variable. This eventually allows the prediction of experimentally recorded changes in the electrical resistance due to mechanically‐induced damage processes. Interpreting the resistivity as a fingerprint of the material microstructure, the simulation approach proposed in the present work contributes to the development of non‐destructive electrical‐resistance‐based characterisation methods. To demonstrate the applicability of the proposed framework, different representative simulations are studied.

A thermo-electro-mechanically coupled cohesive zone formulation for predicting interfacial damage

Material interfaces can occur at different material length scales. Understanding the properties and the behaviour of interfaces is of utmost importance because interfaces can significantly influence the effective constitutive response of the material under consideration. Based on the fundamentals of electro-mechanically coupled cohesive zone formulations for electrical conductors and on the associated finite element framework proposed in Kaiser and Menzel(2021), a thermo-electro-mechanically coupled cohesive zone formulation is established in this article. To this end, the governing equations of continuum thermodynamics for materials with interfaces under combined mechanical, thermal, and electrical loads are derived. A damage variable is introduced to account for the evolution of interface damage in a thermodynamically consistent way. Motivated by deformation-induced property changes, the effective thermal and electrical conductivities are moreover assumed to be functions of the damage variable. Finally, analytical solutions are derived to validate the finite element formulation, and representative boundary value problems are studied so as to reveal key properties of the proposed cohesive-zone framework.

Computational modelling of hydrogen assisted fracture in polycrystalline materials

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 trans-granular cracking in relevant case studies. Firstly, verification calculations are conducted to show how the framework predicts the expected qualitative trends. Secondly, the model is used to simulate recent experiments on pure Ni and a Ni–Cu superalloy that have attracted particular interest. We show that the model is able to provide a good quantitative agreement with testing data and yields a mechanistic rationale for the experimental observations.

Transient creep‐fatigue crack growth in creep‐brittle materials: Application to Alloy 718

AbstractThis study presents incremental finite element computations of creep‐fatigue crack growth in Alloy 718 at in air. Alloy 718 is representative of creep‐brittle materials, in which viscoplastic deformation is restricted near the tip of a growing crack. The computations predict crack growth using a unique combination of an irreversible cohesive zone formulation and a strain gradient viscoplastic material model based on the Kocks–Mecking formulation. Cohesive zone damage parameters are estimated using sustained loading and constant‐amplitude cyclic loading experiments. Computations of crack extension under three different waveforms containing overloads all predict post‐overload retardation. The amount of retardation depends strongly on the overload ratio, consistent with experiments in the literature using similar waveforms. Analysis of the crack‐tip fields demonstrates retardation is associated with unloading in the highly deformed material near the advancing crack tip. Dynamic recovery and geometrically necessary dislocations are shown to significantly influence post‐overload crack extension.

Shear testing and failure modelling of calcium phosphate coated AZ31 magnesium alloys for orthopaedic applications

Magnesium orthopaedic fracture fixation devices can potentially provide significant clinical benefits, such as the elimination of secondary surgeries for device removal due to in-vivo resorption and reduced stress shielding due to reduced device stiffness. However, development, approval, and clinical adoption of magnesium devices has been hindered by the excessively high rates of in-vivo corrosion such that the structural integrity of the device can be catastrophically reduced before fracture healing occurs. Coating of devices with calcium phosphate coatings has been shown to significantly reduce corrosion rates, while enhancing osseointegration. However, the adhesion strength between the CaP coatings and magnesium substrates has not been previously investigated. Clinical insertion of fracture fixation devices such as intramedullary nails and k-wires will impose significant shear loading on the coated surface of the implant. If the effective shear strength of the coating-device interface is not sufficiently high, the coating will be damaged and removed during device insertion. In the current study a bespoke experimental-computational approach is developed to provide a new understanding of the relationship between coating thickness, surface roughness, and effective shear strength of the CaP coating- Mg substrate interface. Nine test cases were created by adjusting either the deposition time (3 thickness values) or the surface treatment of the Mg alloy using SiC paper (3 roughness values) and double-lap shear testing was performed for these coating configurations. Strain development in the Mg substrates was monitored using strain gauges, and failure stress was determined for each configuration. Test results revealed that the effective shear strength of the coating-substrate interface is significantly higher for coatings on the rougher substrate surfaces when compared to those on smoother surfaces. Coating thickness was not found to significantly influence the effective shear strength over the range considered in this study (0.37–1.34 μm). Micro-scale finite element models of lap-shear tests were constructed using experimental profilometry data. Simulations of rough coating-substrate interfaces reveal that significant localised compression occurs at the coating-substrate interface in regions of large asperities. A novel cohesive zone formulation has been developed to simulate compression induced shear hardening, and the resultant simulations are found to accurately predict the significantly higher effective shear strength measured experimentally for rougher coatings compared to smoother Mg substrate surfaces.

Investigation of concrete cracking phenomena by using cohesive fracture-based techniques: A comparison between an embedded crack model and a refined diffuse interface model

Cracking simulation in concrete-like structures is the subject of numerous investigations of computational kind, in which different modeling techniques based on either smeared or discrete fracture approaches have been proposed. In this work, two different finite element-based cohesive fracture models, proposed by some of the authors and belonging to the discrete approaches, have been compared: (i) a novel refined diffuse interface model and (ii) a well-established embedded crack model, which are based on an inter- and intra- element fracture approach, respectively. In particular, the first one is based on an intrinsic cohesive zone formulation, according to which nonlinear interface elements, equipped with a softening constitutive law, are inserted among all the finite elements of the computational domain. The second one, which has been successfully used to simulate concrete cracking along nonprescribed paths and is here used a reference model, combines the cohesive crack concept with the embedded strong discontinuity approach, allowing the cracks to be inserted across finite elements as discontinuities in the corresponding displacement field once a certain stress criterion is satisfied. These two models have been compared by analyzing the cracking behavior in concrete specimens subjected to general loading conditions. A critical discussion of the obtained computational outcomes, in terms of computational efficiency and numerical accuracy for both predicted loading curve and crack pattern, shows the strengths and limits of the investigated discrete fracture models.

Methodology to Calibrate the Dissection Properties of Aorta Layers from Two Sets of Experimental Measurements

Aortic dissection is a prevalent cardiovascular pathology that can have a fatal outcome. However, the mechanisms that trigger this disease and the mechanics of its progression are not fully understood. Computational models can help understand these issues, but they need a proper characterisation of the tissues. Therefore, we propose a methodology to obtain the dissection parameters of all layers in aortic tissue via the computational modelling of two different delamination tests: the peel and mixed tests. Both experimental tests have been performed in specimens of porcine aorta, where the intima-media and media-adventitia interfaces, as well as the medial layer, were dissected. These two tests have been modelled using a cohesive zone formulation for the separating interface and a hyperelastic anisotropic material model via an implicit static analysis. The dissection properties of each interface have been calibrated by reproducing the force-displacement curves obtained in the experimental tests. The values of peak and mean force of the experiments were fitted with an error below 10%. With this methodology, we intend to contribute to the development of reliable numerical tools for simulating aortic dissection and aortic aneurysm rupture.

Fundamentals of electro-mechanically coupled cohesive zone formulations for electrical conductors

Motivated by the influence of (micro-)cracks on the effective electrical properties of material systems and components, this contribution deals with fundamental developments on electro-mechanically coupled cohesive zone formulations for electrical conductors. For the quasi-stationary problems considered, Maxwell’s equations of electromagnetism reduce to the continuity equation for the electric current and to Faraday’s law of induction, for which non-standard jump conditions at the interface are derived. In addition, electrical interface contributions to the balance equation of energy are discussed and the restrictions posed by the dissipation inequality are studied. Together with well-established cohesive zone formulations for purely mechanical problems, the present developments provide the basis to study the influence of mechanically-induced interface damage processes on effective electrical properties of conductors. This is further illustrated by a study of representative boundary value problems based on a multi-field finite element implementation.

Interfacial transition zones in concrete meso-scale models – Balancing physical realism and computational efficiency

Using meso-structural representations of concrete, with aggregates dispersed in mortar, has become a standard approach for damage and fracture analysis. However, there is no full agreement on appropriate modelling of different phases and on the inclusion of interfacial transition zones (ITZ). This work explores different mortar and ITZ formulations and by comparison with own experiments demonstrates that the optimal strategy balancing physical realism and computational efficiency requires: (1) damage-plasticity formulation for mortar, calibrated with mortar tension and compression experiments; (2) cohesive-zone formulation for ITZ with zero-thickness cohesive elements, calibrated with concrete tension and compression experiments. Models omitting ITZ are shown to be in poor agreement with experiments, both qualitatively and quantitatively. Models with finite thickness ITZ are also in poorer agreement with experiments compared to those with zero-thickness, despite higher computational complexity. It is recommended that concrete analyses follow the proposed strategy for meso-structure modelling and constituents’ calibration.

A mechanistic analysis of delamination of elastic coatings from the surface of plastically deformed stents

Several medical papers have reported delamination of the coating from the stent-substrate following intravascular deployment leading to adverse outcomes for patients. However, the mechanisms of delamination of such polymer coatings from the surface of a stent due to large deformations during device deployment have not been studied. In this paper, a novel and in-depth investigation of the mechanisms and parameters that govern stent-coating delamination is performed, using a cohesive zone formulation to simulate the evolution of traction at the stent-coating interface. The study firstly analyses the behaviour of elastic coatings on idealised elastic stent substrates. Simulations reveal that the mode mixity of delamination is strongly dependent on the level of stent deployment at initiation. In general, peak normal tractions exceed peak shear tractions at low levels of stent deployment whereas the reverse trend is computed at high levels of stent deployment. Interface tractions increase with both increasing stent thickness and coating thickness suggesting that thinner stents and thinner coatings should be utilised for the delivery of antiproliferative drugs to reduce the risk of coating delamination. Next, the influence of stent plasticity on interface tractions and coating delamination is investigated. Even at low levels of deployment, plastic yielding occurs in the stent hinge region and the patterns of normal and shear tractions are found to be significantly more complex than those computed for the elastic stents, with both tensile and compressive regions of normal traction occurring in the stent arch. At a high level of stent deployment shear tractions at the stent-coating interface are computed to increase with decreasing strain hardening modulus. The findings of this paper provide a new insight into the stress-state at the stent-coating interface as a function of the stent design parameters and large deformation elasticity and plasticity during deployment, allowing for a more reliable assessment of the limits relating to safe implantation of coated stents.

A duplex oxide cohesive zone model to simulate intergranular stress corrosion cracking

A finite element model with slip-oxidation is proposed for solving intergranular stress corrosion cracking (IGSCC) with duplex oxides replicating the cyclic physics of the slip oxidation. The purpose is to investigate the crack growth effect due to different rate, compositions and kinetics of the duplex oxide. The finite element model is based on a coupling between cohesive zone formulation, slip-oxidation model and a diffusion model. The cohesive zone formulation includes a degradation formulation which is linked to the slip-oxidation formulation. The environmental properties in the slip-oxidation were obtained from the diffusion modeled with Fick's second law in one-dimension. This was then coupled to the structural model by a segregated solution scheme. The mesh of the cohesive zone adapts to the oxide thickness of the duplex oxide during the crack growth. The duplex oxide has the mathematical form of a power law or a logarithmic form. The model showed matching results for all duplex oxide combinations in varying stress, but the inner logarithmical oxide gave higher crack growth rates than the power law. The power law with the thicker inner oxide showed good results for the change of stress intensity factor and gave the best results when the yield stress was varied. Grain misorientation effect was higher for the duplex oxides with thicker outer oxides.

Investigation of over-nonlocal damage and interface cohesive models for simulating structural behaviors of composite UHPC-NC members

In this paper, a finite element (FE) modeling method is proposed to predict the structural behaviors of composite members of ultrahigh-performance concrete (UHPC) and normal concrete (NC). This FE modeling method integrates: (i) over-nonlocal gradient-enhanced damage model for UHPC and NC, capable of circumventing strong mesh dependency of the classical continuum damage model and spurious damage growth of the standard gradient-enhanced damage model; and (ii) the irreversible viscosity-regularized cohesive zone formulation to describe UHPC-NC interfacial behaviors. Linearization of relative governing equations is presented and the framework is realized on general FE platform ABAQUS through user subroutines UEL and UMAT. Three UHPC-NC composite specimens were prepared to validate the versatility of the numerical method. Improved slant shear test was accompanied to realistically quantify the UHPC-NC bond strength. The height of UHPC substrate governed the overall behaviors and leaded to complete different failure patterns. It is validated that incorporating the material properties and bond behaviors in this paper predicts the composite specimens’ behaviors in an accurate and efficient way.

A stable extended/generalized finite element method with Lagrange multipliers and explicit damage update for distributed cracking in cohesive materials

A flexible, general and stable mixed formulation is developed to model distributed cracking in cohesive grain-based materials in the framework of the extended/generalized finite element method. The displacement field is discretized on each grain separately, and the continuity of the displacement and traction fields across the interfaces between grains is enforced by Lagrange multipliers. The design of the discrete Lagrange multiplier space is detailed for bilinear quadrangular elements with the potential presence of multiple interfaces/discontinuities within an element. We give numerical evidence that the designed Lagrange multiplier space is stable and provide examples demonstrating the robustness of the method. Relying on the stable discretization, a cohesive zone formulation equipped with a damage constitutive formulation expressed in terms of the traction is used to model propagation of multiple cracks at the interfaces between grains. The damage formulation makes use of an explicit solution procedure, couples the normal and tangential failure modes, accounts for different tension and compression behaviours and takes into account a compression-dependent fracture energy in mixed mode. The framework is applied to complex 2D problems inspired by indirect tension tests of heterogeneous rock-like materials.

A unified formulation for fatigue crack onset and growth via cohesive zone modelling

This paper presents a unified cohesive zone formulation for describing the fatigue-driven crack onset and propagation in quasi-brittle materials. This approach is based on formulating a modified traction-displacement law involving linear softening, which accounts for the onset of fatigue failure described by materials S-N curves. The fatigue crack propagation in a steady-state regime is described as a sequence of initiation stages that progressively take place within the process zone ahead of the “zero-stress” fracture tip. It is demonstrated that the damage distribution within the process zone under cyclic load is governed by a non-linear Fredholm integral equation of the second kind, which is solved numerically using an iterative scheme. It is also shown that the Paris-Erdogan law for fatigue crack growth can be directly obtained from engineering S-N curves via the linear traction displacement model introduced here. The Paris-Erdogan propagation regime is attained when the crack propagation rate is much smaller that the length of the process zone. The exponent of the Paris-Erdogan law is equal to half the inverse of that characterising the material S-N curve. The main advantage of the cohesive zone model introduced here is that it does require neither ad-hoc additional parameters nor calibration constant to obtain the Paris-Erdogan law from material S-N data. It is also proved that the cohesive zone length in fatigue depends on the exponent of the S-N curve and it always shorter than its static counterpart. The unified cohesive zone model is validated against experimental data for the specific cases of mode I and mode II fatigue delamination growth in the carbon/epoxy composite material IM7/8552. Finally, an analysis of the role played by size effects is presented, with emphasis on the influence of the laminate thickness on the fatigue damage accumulation within the process zone and the ensuing propagation rates of cohesive cracks.

Nanoindentation characterization of Glass/Epoxy composite for viscoelastic damage interlaminar modeling

Polymer matrix composites exhibit rate dependent behavior due to viscoelastic properties of polymers. It has been assumed that the strain rate dependent mode-II delaminations in End Notched Flexure (ENF) tests are due to the viscoelastic properties of resin rich interface. In this paper a characterization procedure has been proposed within nanoindentation technique and viscoelastic properties of resin rich interface of woven E-glass/epoxy composite laminate have been characterized. Afterwards the ENF tests have been simulated by finite element method using characterized properties and developed viscoelastic cohesive zone formulation. The obtained results validate the characterization procedure for the estimation of time-dependent delamination.

Crack Growth Resistance in Metallic Alloys: The Role of Isotropic Versus Kinematic Hardening

The sensitivity of crack growth resistance to the choice of isotropic or kinematic hardening is investigated. Monotonic mode I crack advance under small scale yielding conditions is modeled via a cohesive zone formulation endowed with a traction–separation law. R-curves are computed for materials that exhibit linear or power law hardening. Kinematic hardening leads to an enhanced crack growth resistance relative to isotropic hardening. Moreover, kinematic hardening requires greater crack extension to achieve the steady-state. These differences are traced to the nonproportional loading of material elements near the crack tip as the crack advances. The sensitivity of the R-curve to the cohesive zone properties and to the level of material strain hardening is explored for both isotropic and kinematic hardening.

Non-local plasticity effects on notch fracture mechanics

We investigate the influence of gradient-enhanced dislocation hardening on the mechanics of notch-induced failure. The role of geometrically necessary dislocations (GNDs) in enhancing cracking is assessed by means of a mechanism-based strain gradient plasticity theory. Both stationary and propagating cracks from notch-like defects are investigated through the finite element method. A cohesive zone formulation incorporating monotonic and cyclic damage contributions is employed to address both loading conditions. Computations are performed for a very wide range of length scale parameters and numerous geometries are addressed, covering the main types of notches. Results reveal a strong influence of the plastic strain gradients in all the scenarios considered. Transitional combinations of notch angle, radius and length scale parameter are identified that establish the regimes of GNDs-relevance, laying the foundations for the rational application of gradient plasticity models in damage assessment of notched components.