Nonlinear Cohesive Law Research Articles

On the solution of unstable fracture problems with non-linear cohesive laws

Fracture mechanics models have well-known numerical challenges when implemented within implicit quasi-static frameworks once the cumulative internal energy exceeds the capacity for dissipation through the fracture process. Although this finding has been mainly reported for linear softening traction-separation laws, this work comprehensively explores non-linear softening behaviours and proposes a more general instability criterion. The ratio of cohesive to internal power emerges as a crucial factor. As a result, even scenarios involving a single cohesive element undergoing monotonic loading may exhibit a limit point at any stage of crack propagation, not just during crack initiation. Two strategies for handling fracture problems with instabilities within an implicit solution are discussed: an arc-length technique and an extension of quasi-static formulation into a dynamic regime. A comparative assessment is performed, covering both simple single-element cases and more complex scenarios. Furthermore, the study delves into more intricate material responses, including transformation-induced plasticity effects. Notably, incorporating these dissipative phenomena in the bulk material mitigates the difficulties associated with snap-back-like behaviours.

Analytical study of the bond behavior of fiber reinforced cementitious matrix (FRCM)-substrate joints based on a two-stage nonlinear cohesive material law

Externally bonding fiber-reinforced cementitious matrix (FRCM) composite to the surface of concrete or masonry members is an effective technique to improve the performance of existing structures. In many cases, interfacial debonding between the FRCM fiber textile and embedding matrix governs the capacity of FRCM-strengthened structures. The interfacial debonding process can be studied analytically by assuming a cohesive material law (CML), which represents the relationship between interfacial shear stress and slip. In this study, a two-stage function with an exponential stage and a constant stage is proposed to describe the CML associated with the matrix-fiber interface. The latter stage is characterized by a constant shear stress to account for the friction/interlocking between the matrix and fiber observed in experimental tests. With the assumed CML, the full-range loading response was obtained. Additionally, the interfacial slip, fiber axial strain, and interfacial shear stress were analytically derived. The parameters of the two-stage CMLs for PBO, glass, and carbon FRCM were inversely determined by matching the analytical relationships of peak applied axial stresses associated with different bonded lengths with the experimental ones. Considering the inversely determined CMLs, the predicted load responses and strain profiles showed good agreement with the measurements of direct shear test specimens.

Dynamic interfacial crack propagation and kinking in sandwich panels

The dynamic interfacial crack propagation and crack kinking phenomena in sandwich panels are studied. The paper derives an analytical approach based on the Extended High-Order Sandwich Panel Theory (EHSAPT) and a cohesive interface modeling. The mathematical modeling of the panel makes a distinction between the face sheets and the core. The core is further divided into two bodies aiming to allow the nucleation of a shear crack in the core or the kinking of a core-face sheet crack into the core. The panel therefore includes four layers that are interconnected through three interfaces. The First-Order Shear Deformation Theory (FSDT) is adopted for the two face sheets and the EHSAPT is applied to the two core layers. Translational, rotational, and high order inertial effects are considered in all components. A geometrical nonlinear behavior with linear elastic laws is adopted for the face sheets while the core layers use linear kinematic relations and linear constitutive laws. The interfaces use nonlinear cohesive laws to model the dynamic nucleation, propagation, and kinking of cracks. The dynamic model is used for a numerical study of sandwich panels subjected to three point bending and end shortening loading conditions. The study explores the coupling of the crack propagation with the dynamic response. A comparison of the dynamic results with their static counterparts further explains the physical response of the sandwich panel and the dynamic characteristics of its failure mechanism.

A cohesive FE model for simulating the cracking/debonding pattern of composite NSC-HPFRC/UHPFRC members

The aim of this work is to propose to practitioners a simple cohesive Finite-Element (FE) model able to simulate the cracking/debonding pattern of retrofitted concrete elements, in particular Normal-Strength-Concrete members (slabs, bridge decks, pavements) rehabilitated by applying a layer of High-Performance or Ultra-High-Performance Fiber-Reinforced-Concrete as overlay. The interface was modeled with a proper nonlinear cohesive law which couples mode I (tension-crack) with mode II (shear-slip) behaviors. The input parameters of the FE simulation were provided by a new bond test which reproduces a realistic condition of cracking/debonding pattern. The FE simulations were accomplished by varying the overlay materials and the moisture levels of the substrate surface prior to overlay, since findings about their influence on the bond performances are still controversial. The proposed FE model proved to effectively predict the bond failure of composite NSC-HPFRC/UHPFRC members.

A Finite Element Based Analysis of Double Strap Bonded Joints with CFRP and Aluminium

The bonding between two different materials or between same materials is a quite popular method. Unlike fastener joints, it avoids undesirable stress concentrations and doesn't demand an intrusive application to ensure the good performance of the joint. However, depending on the configuration of the adhesively bonded joint, its performance responds differently and the choice (if possible to make) on the best configuration, i.e. the configuration that originates the highest strength and/or stiffness, may be hard to make. Within this context, several configurations of aluminium-to-aluminium bonded joints unstrengthened and strengthened with fiber reinforced polymers (FRP) were modelled using a commercial finite element code. The linearity and nonlinearity of the FRP composite and the aluminium were considered, respectively, and the adhesively bonded joints were subjected to a regular displacement that intended to simulate a tensioning load. Also, the nonlinearities of the interfaces were considered in the form of nonlinear cohesive adhesive laws. The fracture Modes I and II were defined trough a bond-slip relation with a bi-linear shape and the Mohr-Coulomb failure criterion is used for the coupling of the cohesive adhesive laws of the interface when the debonding process of the bonded joint configuration implies the interaction between both fracture modes, i.e. the joint is under a mixed-mode (Mode I+II) situation. The results are presented and discussed and the configurations of the bonded joints are all compared through bond stress distributions and load-slip responses. The study herein presented is, therefore, a contribution to the analysis of the structural integrity of bonded joints between FRP composites and aluminium substrates, helping also on the choice of the most adequate bonded joint configuration and corresponding reinforcement to be used and applied in practice.

A new generalized finite element method for two-scale simulations of propagating cohesive fractures in 3-D

Summary This paper presents a novel numerical framework based on the generalized finite element method with global–local enrichments (GFEMgl) for two-scale simulations of propagating fractures in three dimensions. A non-linear cohesive law is adopted to capture objectively the dissipated energy during the process of material degradation without the need of adaptive remeshing at the macro scale or artificial regularization parameters. The cohesive crack is capable of propagating through the interior of finite elements in virtue of the partition of unity concept provided by the generalized/extended finite element method, and thus eliminating the need of interfacial surface elements to represent the geometry of discontinuities and the requirement of finite element meshes fitting the cohesive crack surface. The proposed method employs fine-scale solutions of non-linear local boundary-value problems extracted from the original global problem in order to not only construct scale-bridging enrichment functions but also to identify damaged states in the global problem, thus enabling accurate global solutions on coarse meshes. This is in contrast with the available GFEMgl in which the local solution field contributes only to the kinematic description of global solutions. The robustness, efficiency, and accuracy of this approach are demonstrated by results obtained from representative numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.

A stable X‐FEM in cohesive transition from closed to open crack

SummaryIgnoring crack tip effects, the stability of the X‐FEM discretizations is trivial for open cracks but remains a challenge if we constrain the crack to be closed (i.e., the bi‐material problem). Here, we develop a formulation for general cohesive interactions between crack faces within the X‐FEM framework. The stability of the new formulation is demonstrated for any cohesive crack stiffness (including the closed crack) and illustrated for a nonlinear cohesive softening law. A benchmark of the new model is carried out with simpler approaches for a closed crack (i.e., Lagrange multipliers) and for a cohesive crack (i.e., penalty approach). Due to the analogies between stable cohesive X‐FEM and Nitsche's methods, the new method simplifies the implementation and is attractive in dynamic explicit codes. Copyright © 2014 John Wiley & Sons, Ltd.

Pull out of FRP reinforcement from masonry pillars: Experimental and numerical results

In this paper, debonding phenomena between carbon fiber reinforced polymer (CFRP) strips and masonry support were investigated on the basis of single-lap shear tests, considering different dimensions of the bond length. To capture the post-peak response of the CFRP–masonry joint, the slip between the support and the reinforcement strip was controlled using a clip gauge positioned at the end of the reinforcement. The tests were simulated by means of a finite element model able to capture the post-peak snap-back behavior due to the failure process. The numerical model is based on zero-thickness interface elements and on a proper non-linear cohesive law. The comparison between experimental and numerical results was performed in terms of overall response, measured by both the machine stroke and the clip gauge positioned at the free end of the reinforcement. The cases of effective bond length greater and lesser than the minimum anchorage length, suggested by the CNR Italian recommendation, were considered.

A theoretical analysis of interface debonding for coated sphere with functionally graded interphase

We present a novel nonlinear cohesive law for coated sphere with functionally graded interphase (FGI). It is derived from the van der Waals interaction accounting for the interface debonding of the coated sphere subjected to radial tensile loading. The present analytical results show that: (1) Higher values of the interphase radius rb lead to a higher interface strength and a higher debonding strain; (2) a higher Young’s modulus Ef and the Poisson’s ratio vf of the sphere lead to a lower interface strength and a smaller debonding strain; (3) large values of the index n and the inner Young’s modulus Ea of FGI as well as the Poisson’s ratio vm of the entire coat improve the interface strength and reduce the debonding strain; (4) the sphere radius ra and the outer coat radius rc can not only improve but also reduce the interface strength and the debonding strain, respectively. The established analytical solutions should be of great help for understanding the mechanical properties of the coated sphere and sphere-reinforced composites, designing microcomposites and microelectromechanical systems.

Homogenization scheme for brittle intergranular decohesion in polycrystalline aggregates

The overall fracture behaviour of polycrystalline aggregates is strongly conditioned by intergranular failure, as is the case in copper alloys subjected to dynamic embrittlement. The self-consistent scheme is extended to account for grain boundary decohesion using a nonlinear cohesive law. The effective tensile response up to failure is computed for a Cu–Ni–Si alloy based on the homogenization method. In particular, the proposed approach allows for identification of the grain boundary critical energy release rate from the macroscopic tensile curve.

Modelling of morphology evolution and macroscopic behaviour of intercalated PET–clay nanocomposites during semi-solid state processing

Morphology evolution and macroscopic stress–strain behaviour of initially intercalated PET–clay nanocomposites during semi-solid state processing were predicted using a nonlinear two-dimensional computational model. The model combined different nanocomposite length scales into a computationally-efficient, continuum and nonlinear finite element framework. The main feature of the model is its nonlinear cohesive law proposed here to describe the gallery behaviour and the phenomenon of clay platelet slippage. Our model predicted: (1) shift in the onset of macroscopic strain-stiffening caused by the increased tactoid aspect ratio and processing temperature, and (2) quantified tactoid reorientation and clay platelet slippage as a function of applied strains and processing temperatures. Our predictions were found to be in a good agreement with experimental results, and therefore the model can be used in the optimisation of the processing of PET–clay nanocomposites.

Cohesive and non‐cohesive fracture by higher‐order enrichment of XFEM

SUMMARYA comprehensive study is performed on the use of higher‐order terms of the crack tip asymptotic fields as enriching functions for the eXtended FEM (XFEM) for both cohesive and traction‐free cracks. For traction‐free cracks, the Williams asymptotic field is used to obtain highly accurate stress intensity factors (SIFs), directly from the enriched degrees of freedom without any post‐processing. The low accuracy of the results of the original research on this subject by Liu et al. [Int. J. Numer. Meth. Engng., 2004; 59:1103–1118] is remedied here by appropriate modifications of the enrichment scheme. The modifications are simple and can be easily included into an XFEM computer code. For cohesive cracks, the relevant asymptotic field is used, and two widely used criteria including the SIFs criterion and the stress criterion are examined for the crack growth simulation. Both linear and nonlinear cohesive laws are used. For the stress criterion, averaging is avoided due to the highly accurate crack tip approximation because of the higher‐order enrichment. Then, a modified stress criterion is proposed, which is shown to be applicable to a wider class of problems. Several numerical examples, including straight and curved cracks, stationary and growing cracks, single and multiple cracks, and traction‐free and cohesive cracks, are studied to investigate in detail the robustness and efficiency of the proposed enrichment scheme. Copyright © 2012 John Wiley & Sons, Ltd.

The effect of interfacial bonding on the elastic property of carbon nanotube-reinforced composite

A micromechanics model is developed to calculate the effective elastic modulus of composite with aligned straight carbon nanotubes (CNTs). The interfacial bonding between CNTs and polymer matrix gives an important effect on reinforcing specification of CNTs. The nonlinear cohesive law is used to describe the interface debonding. The effect of debonding at CNTs/polymer interface and the van der Waal's (vdw) force between carbon atoms is studied on the mechanical properties of composite reinforced by CNTs. It is also shown that small CNTs have stronger reinforcing effect than larger ones. The debonded nanotubes behave like voids in the matrix and it can significantly reduce the stiffening effect of the nanocomposites. POLYM. COMPOS., 2011. © 2011 Society of Plastics Engineers

Equivalent inclusion solution adapted to particle debonding with a non-linear cohesive law

Starting from Eshelby’s solution of the equivalent inclusion problem, an approximate solution is proposed in order to model interface debonding of a spherical inhomogeneity isolated in a uniform matrix. Both phases are linear elastic but the interface traction-separation law is non-linear. A semi-analytical incremental model is developed which is suitable for any type of loading. For computational efficiency, the model relies on two simplifying assumptions: (i) the eigenstrain is uniform inside the inhomogeneity and (ii) the interface compliance is averaged over inhomogeneity’s surface when computing the average strain within the inhomogeneity. An extensive parametric study is conducted for three loading modes and 144 combinations of non-dimensional parameters. The predictions are assessed against full-field finite element solutions based on two error measures of the mean stress field inside the inhomogeneity. The results show that the mean error value is acceptable in all cases and indicate the parameter ranges for which the model is most accurate.

Micromechanical theoretical and computational modeling of energy dissipation due to nonlinear vibration of hard ceramic coatings with microstructural recursive faults

Engine failures due to high-cycle fatigue during severe dynamic vibration have cost the US Air Force an estimated $400 million dollars per year over the past two decades. Therefore, structural materials that exhibit high damping capacities are desirable for mechanical vibration suppression and acoustic noise attenuation. Few experimental studies suggested that hard ceramic coatings, which are commonly used as thermal barrier coatings (TBCs) to protect engine components from high temperatures and corrosion, can also serve as passive dampers due to their unique microstructure which consists of several layers of splats with inter- and intra-microstructural recursive faults (micro-cracks). Therefore, the focus of this study is on the development of a fundamental understanding of the unique microstructural features and mechanisms responsible for this observed energy dissipation in ceramic coatings under nonlinear vibration through the development of a micromechanical computational framework. Inter- and intra-fatigue damage and internal friction is simulated through the development of thermodynamic-based nonlinear cohesive laws that consider interfacial degradation, debonding, plastic sliding, and Coulomb/contact friction between the interfaces of microstructural faults. Representative volume element-based micromechanical simulations are conducted in order to assess the main micromechanical mechanisms responsible for the experimentally observed nonlinear (amplitude- and frequency-dependent) damping in plasma sprayed hard ceramic coatings. It is concluded that the major part of energy dissipation is achieved through contact friction which results from sliding of the splat interfaces along the microstructural recursive faults. Energy dissipation due to progressive decohesion and evolution of new micro-cracks is not that significant as compared to energy dissipated due to increased friction from existing and new created faults. Therefore, internal friction is the main mechanism that makes TBCs effective dampers.

Macroscopic Behavior of Carbon Nanotube (CNT)-Reinforced Composite Accounting for Interface Cohesive Force

The large aspect ratio of interfacial area per unit volume at CNT/matrix interfaces may significantly influence the macroscopic behavior of CNT-reinforced composites. The property of interfaces is governed by the cohesive law, which is determined from both van der Waals forces and chemical covalent bonds. This nonlinear cohesive law is incorporated in the micromechanics model in the current work to study the mechanical behavior of CNT-reinforced composites. It is found that carbon nanotubes can improve the macroscopic behavior of the composite at small strain, but tend to weaken the composite at relatively large strain because of the interface softening behavior or debonding. The increase of interface adhesion caused by the creation of chemical covalent bonds may significantly improve the composite behavior at large strain.

Nonlinear interface shear fracture of end notched flexure specimens

The nonlinear analytical solutions of an end notched flexure adhesive joint or fracture test specimen with identical or dissimilar adherends are investigated. In the current study, a cohesive zone model (with arbitrary nonlinear cohesive laws) based analytical solution is obtained for the interface shear fracture of an end notched flexure (ENF) specimen with sufficiently long bond length. It is found that the scatter and inconsistency in calculating Mode II toughness may be significantly reduced by this model. The present work indicates that the Mode II toughness G IIc under pure shear cracking condition is indeed very weakly dependent on the initial crack length. And this conclusion is well supported by the experimental results found in the literature. The parametric studies show that the interface shear strength is the most dominant parameter on the critical load. It is also interesting to note that with very short initial crack length and identical interface shear strength, higher Mode II toughness indeed cannot increase the critical load. Unlike the high insensitivity of critical load to the detailed shape of the cohesive law for Mode I peel fracture, the shape of the cohesive law becomes relatively important for the critical load of joints under pure Mode II fracture conditions, especially for joints with short initial crack length. The current study may help researchers deepen the understanding of interface shear fracture and clarify some previous concepts on this fracture mode.

Interfacial debonding of pipe joints under torsion loads: a model for arbitrary nonlinear cohesive laws

Adhesively bonded pipe joints are extensively used in pipelines. In the present work, Cohesive Zone Model (CZM) based analytical solutions are obtained for the bonded pipe joints under torsion. An integral form based general expression is derived which is suitable for arbitrary type of nonlinear cohesive laws. The concept of the minimum interfacial cohesive shear slip δm is introduced and used in the fundamental expression of the external torsion load. It is found that, when the bond length of the pipe joint is large enough, the torsion load capacity is indeed independent of the shape of cohesive laws and the bond length. It is interesting to note that the maximum torsion load capacity is achieved when the torsion stiffness of the pipe and coupler are identical. A good agreement with finite element analysis (FEA) result indicates that the current model works well. The formulation to develop a simple test method for determining the τ–δ constitutive relationship in pipe joints under torsional loads is suggested. Parametric studies of various cohesive laws are conducted. This model deepens the understanding of the interfacial debonding problem of bonded joints. The fracture energy based formulas of the torsion load capacity derived in the present work can be directly used in the design of adhesively bonded pipe joints.

Experimental validation of large-scale simulations of dynamic fracture along weak planes

A well-controlled and minimal experimental scheme for dynamic fracture along weak planes is specifically designed for the validation of large-scale simulations using cohesive finite elements. The role of the experiments in the integrated approach is two-fold. On the one hand, careful measurements provide accurate boundary conditions and material parameters for a complete setup of the simulations without free parameters. On the other hand, quantitative performance metrics are provided by the experiments, which are compared a posteriori with the results of the simulations. A modified Hopkinson bar setup in association with notch-face loading is used to obtain controlled loading of the fracture specimens. An inverse problem of cohesive zone modeling is performed to obtain accurate mode-I cohesive zone laws from experimentally measured deformation fields. The speckle interferometry technique is employed to obtain the experimentally measured deformation field. Dynamic photoelasticity in conjunction with high-speed photography is used to capture experimental records of crack propagation. The comparison shows that both the experiments and the numerical simulations result in very similar crack initiation times and produce crack tip velocities which differ by less than 6%. The results also confirm that the detailed shape of the non-linear cohesive zone law has no significant influence on the numerical results.

The viscoelastic composite with interface debonding

The effect of particle/matrix interface debonding on composites consisting of elastic particles and viscoelastic matrix is studied. The interface debonding is represented by a nonlinear cohesive law for the high explosive PBX 9501. The Mori–Tanaka method gives the constitutive relation of the composite in terms of the properties of elastic particles, viscoelastic matrix, and nonlinear cohesive law for interfaces. For the example of a composite with spherical particles subject to hydrostatic tension, simple analytical expressions of the composite stress–strain relation are obtained. The strain rate and temperature have strong effects on the composite behavior. High strain rate and low temperature give high strength of the composite. However, the rate and temperature effects decrease as the particle volume fraction increases.