Mixed-mode Cohesive Law Research Articles

A unified regularized variational cohesive fracture theory with directional energy decomposition

In this paper, the cohesive fracture is revisited as an energy minimization problem. The energy functional is rigorously derived based on a pair of integral transformations and a novel directional energy decomposition method. The resulting description of the regularized crack is perfectly suitable for cohesive fracture. A well-defined crack direction is associated with each material point. The resulting constitutive relation shows that a damage-induced material orthotropy within the regularized crack is introduced in this theory, which is suitable for cohesive fracture. The proposed method can implement an arbitrarily given mixed-mode cohesive law. The tensile and shear fracture modes are covered in a unified manner and controlled by the tensile-to-shear strength ratio. These theoretical results are presented with detailed proofs and verified numerically by some representative examples involving tension, shear, and compression.

A coupled mix-mode cohesive law based on a cylindrical potential function

A novel mixed-mode cohesive law derived from a potential function is presented. The potential function is formulated using physical parameters that can be extracted from any fracture mechanics test capable of providing R-curves in terms of the J-integral as a function of the normal and tangential end-openings. The proposed cohesive law is able to describe the fracture behaviour of composites with large fracture process zones, including fibre bridging. As such, it is capable of describing both the crack tip, as well as the bridging region. An important aspect of the formulation is that the shape of the mixed-mode cohesive laws are derived and not assumed. The mixed-mode cohesive law was tested using synthetic data emulating mixed-mode fracture mechanics tests. The cohesive tractions extracted from the method exhibited characteristics which were not seeded on the model such as negative normal tractions under pure mode shear loading and non-zero shear loading under pure normal mode loading.

Mixed Mode Cohesive Law of Adhesive Joints

This work presents an experimental method to determine the cohesive law of thin adhesive joints in mixed mode. The proposed method is based on the MMB (Mixed Mode Bending test). The Energy release Rate and Crack Tip Relative Displacement are determined based on a compliance variation data reduction method by processing the global load displacement curve, without monitoring the crack length and the crack tip relative displacement during the test. Representing the energy release rate versus relative displacement the cohesive law is determined by numerical differentiation. The method is validated numerically by virtual tests including the cohesive zone model. The results reveal that the proposed data reduction scheme is suitable to obtain the mixed mode cohesive law using only the load and displacement data obtained from the testing machine, without any external displacement measurement technique and without any assumption of the form of the cohesive law.

Phase-field cohesive fracture theory: A unified framework for dissipative systems based on variational inequality of virtual works

This work develops a variational framework with dissipation for deriving governing equations in the phase-field cohesive fracture theory. We show that a phase-field model consistent with cohesive zone theory can be established on only two physical foundations. One of them is the law of energy conservation and the other is a variational inequality of virtual works which serves as a stability condition. The proposed phase-field model can implement a mixed-mode cohesive law of arbitrary form. The mode-dependent fracture toughness and direction-dependent crack driving force in a general cohesive fracture can be taken into account in a unified manner. Since brittle fracture can be viewed as an extreme case of cohesive fracture, the newly proposed method is also applicable to brittle fracture. Three numerical examples are presented to demonstrate the effectiveness of the presented method, including a uniaxial tension test, a mixed-mode fracture of concrete and a compressive failure of rock with flaws. In the first example, the numerical cohesive law is extracted and compared with the target analytical cohesive law. Excellent agreement is observed. In the last two examples, the crack morphology and mechanical response predicted by the model are in excellent agreement with the experimental results.

An Experimental and Computational Study on the Low Velocity Impact-Induced Damage of a Highly Anisotropic Laminated Composite Panel

Abstract The low velocity impact (LVI)-induced damage of a highly anisotropic laminate [0/90/0/909]s has been studied experimentally and numerically. The purpose of the analyses of this laminate is that this stacking sequence resembles a sandwich composite panel, in the sense that the [0/90/0] outer layers serve as the “face sheet” while the inner 18 plies of 90 deg layers serve as the “core.” The LVI-induced damage pattern of this laminate is unique and referred to as the “kidney” shape. The “kidney” shape damage is caused by a strong interaction between matrix transverse cracking and delamination, hence is challenging to be computationally captured. The enhanced Schapery theory (EST) model has been improved with the capability to model material inelasticity, as well as a novel mixed-mode cohesive law, to tackle this problem. EST with inelasticity (EST-InELA) is shown to be able to predict load responses and damage morphology accurately and efficiently. The aim of this paper is to provide a challenging LVI case to examine and calibrate computational models.

A mixed mode cohesive model for FRP laminates incorporating large scale bridging behaviour

A user defined cohesive material model is implemented in the LS-DYNA finite element code. The model is based on interface properties characterised from DCB specimens loaded with unequal bending moments. Different mode mixities are obtained by applying different ratios of moments to the two beams in the cracked part of the specimen. The mixed mode cohesive law is fitted for large scale bridging delamination through inverse modelling. In this way, the variations in stress and crack opening across the width of the specimen are taken into account. The J integral approach is used to find a starting point for the fitting procedure. Three properties from five moment configurations are evaluated to find a first estimate of the shape of the cohesive law: crack tip fracture energy, steady-state fracture resistance and crack end-opening at steady-state fracture resistance. The parameters of the cohesive law are then further adjusted using the optimisation tool LS-OPT. The implemented cohesive model is assessed by comparing numerical to experimental test results from the standardised ASTM double cantilever beam test and the ASTM mixed mode bending test.

Deep material network with cohesive layers: Multi-stage training and interfacial failure analysis

A fundamental issue in multiscale materials modeling and design is the consideration of traction–separation behavior at the interface. By enriching the deep material network (DMN) with cohesive layers, the paper presents a novel data-driven material model which enables accurate and efficient prediction of multiscale responses for heterogeneous materials with interfacial effect. In the newly invoked cohesive building block, the fitting parameters have physical meanings related to the length scale and orientation of the cohesive layer. It is shown that the enriched material network can be effectively optimized via a multi-stage training strategy, with training data generated only from linear elastic direct numerical simulation (DNS). The extrapolation capability of the method to unknown material and loading spaces is demonstrated through the debonding analysis of a unidirectional fiber-reinforced composite, where the interface behavior is governed by an irreversible softening mixed-mode cohesive law. Its predictive accuracy is validated against the nonlinear path-dependent DNS results, and the reduction in computational time is particularly significant.

Numerical study on notched steel beams strengthened by CFRP plates

Carbon fibre reinforced polymer (CFRP) externally bonding is an efficient and effective method to strengthen damaged steel beams, and thereby prolong their service life. However, debonding failure, which requires accurate predictions to ensure safety, can occur before the full usage of CFRP. In this study, the notched steel beams strengthened with CFRP plate were simulated by finite element method where the mixed-mode cohesive law was employed to determine the interfacial stress. The load–deflection curves and strain development at different load levels from experimental study were used to verify the validity of the numerical model. The interfacial stress distribution with increasing load was analysed, and good correlation with theoretical calculations at elastic stage was observed. In contrast to the previous elastic analytical study, the plastic behaviour of the CFRP strengthened notched steel beams was revealed. More importantly, interfacial crack initiation, propagation and debonding were accurately simulated. This simulation method can be used to predict debonding process in actual engineering applications. In addition, parametric analysis was conducted to assess the effects of notch depth, CFRP elastic modulus and CFRP thickness. The ultimate load and ductility decreased substantially with increasing notch depth. Furthermore, although increased bearing capacity was achieved by increasing the CFRP elastic modulus and thickness, ductility decreased and premature debonding failure occurred more easily.

A large deformation hybrid isogeometric-finite element method applied to cohesive interface contact/debonding

A hybrid NURBS-based isogeometric-finite element discretization technique is presented to take advantages of IGA for problems with regions demanding higher geometric accuracy or interpolation order. The validity of this coupled discretization is tested through standard patch test. Also, for demonstrative purposes, the proposed methodology is then applied to 2D interface contact/debonding of fiber–matrix as well as a 2D double cantilever beam peeling problem. Adopting a novel approach, the mortar method is used to account for both cohesive behavior and non-penetration constraint in the interface. Non-penetration constraint is enforced via Lagrange multiplier technique along with a mixed-mode cohesive law. The validity of the proposed unified contact/debonding formulation is tested against a contact/tension patch test. The stress contours after simulations show a smooth variation across IG and FE domains further demonstrating the coupling’s eligibility. Moreover, smooth contact pressure distributions are obtained along the interfaces resulting from higher-order NURBS basis functions. Having significantly less DOFs, the hybrid IG–FE discretization also behaved more robustly compared with linear FEs.

Estimation of a mixed-mode cohesive law for an interface crack between dissimilar materials

In this paper, a mixed-mode cohesive law for an interface crack between epoxy and TR (transparent thermoplastic) resin is inversely estimated by the field projection method using numerical solutions and experimentally measured displacements. Displacements in a region far away from the crack tip are measured by digital image correlation technique. An inverse analysis, the field projection method formulated from the interaction J- and M-integrals with numerical auxiliary fields, is carried out to estimate a mixed-mode cohesive law for an interface crack between dissimilar materials. In the present approach, nonlinear deformations and damage near the crack tip are converted into the relationships of tractions and separations on crack surfaces behind the crack tip. The phase angle of mixed-mode singularities of the interface crack is also obtained from measured displacements in this study.

Numerical studies on CFRP strengthened steel circular members under marine environment

The durability of carbon fibre reinforced polymer (CFRP) strengthened steel circular hollow section (CHS) members has now become a real challenge to researchers. In addition, various parameters that may affect the durability of such members have not been revealed yet. This paper presents brief experimental results and the first finite element (FE) approach of CFRP strengthened steel CHS beams conditioned in simulated sea water, along with an accelerated corrosion environment at ambient (24 ± 4 °C) and 50 °C temperatures. The beams were loaded to failure under four-point bending. It was found that the strength and stiffness reduced significantly after conditioning in an accelerated corrosion environment. Numerical simulation was implemented using the ABAQUS static general approach. A cohesive element was utilised to model the interface element and an 8-node quadrilateral in-plane general-purpose continuum shell was used to model CFRP elements. A mixed mode cohesive law was deployed for all the three components of stresses in the proposed FE approach, which were one normal component and two shear components. The validity of the FE models was ascertained by comparing the ultimate load and load vs deflection response from experimental results. A range of parametric studies were conducted to investigate the effects of bond length, adhesive types, thickness and diameter of tubes. The results of parametric studies indicated that the adhesive with high tensile modulus performed better and durability design factors varied from section to section.

Cohesive zone modelling and the fracture process of structural tape

Structural tapes provide comparable toughness as structural adhesives at orders of magnitude lower stresses. This is potentially useful to minimize the effects of differences in thermal expansion in the joining of mixed materials. The strength properties are modelled using the cohesive zone model. Thus, a cohesive zone represents the tape, i.e. stresses in the tape are transmitted to the substrates through tractions determined by the separations of the surfaces of substrates. This simplification allows for structural analysis of large complex structures. The relation between the traction and the separation is measured experimentally using methods based on the path independence of the J-integral. Repeated experiments are performed at quasi-static loading. A mixed mode cohesive law is adapted to the experimental data. The law is implemented as a UMAT in Abaqus. Simulations show minor thermal distortions due to thermal loading and substantial structural strength in mechanical loading of a mixed material structure.

Finite element modelling of debonding failures in steel beams flexurally strengthened with CFRP laminates

A steel beam may be strengthened in flexure by bonding a carbon fibre-reinforced polymer (CFRP) plate to the tension face. Such a beam may fail by debonding of the CFRP plate that initiates at one of the plate ends (i.e. plate end debonding) or by debonding that initiates at a local damage (e.g. a crack or concentrated yielding) away from the plate ends (intermediate debonding). This paper presents the first finite element (FE) approach that is capable of accurate predictions of such debonding failures, with particular attention to plate-end debonding. In the proposed FE approach, a mixed-mode cohesive law is employed to depict interfacial behaviour under a combination of normal stresses (i.e. mode-I loading) and shear stresses (i.e. mode-II loading); the interfacial behaviour under pure mode-I loading or pure mode-II loading is represented by bi-linear traction–separation models. Damage initiation is defined using a quadratic strength criterion, and damage evolution is defined using a linear fracture energy-based criterion. Detailed FE models of steel beams tested by previous researchers are presented, and their predictions are shown to be in close agreement with the test results. Using the proposed FE approach, the behaviour of CFRP-strengthened steel beams is examined, indicating that: (1) if the failure is governed by plate end debonding, the use of a CFRP plate with a higher elastic modulus and/or a larger thickness may lead to a lower ultimate load because plate end debonding may then occur earlier; (2) plate end debonding is more likely to occur when a short CFRP plate is used, as is commonly expected; and (3) the failure mode may change to intermediate debonding or other failure modes such as compression flange buckling if a longer plate is used.

Mixed Mode cohesive law with interface dilatation

Experimental investigations of adhesive joints and fibre composites have shown that under Mode II cracking, the fracture process induces a displacement normal to the fracture plane. This effect can be attributed e.g. to roughness of the fracture surface under dominating tangential crack face displacements. As the crack faces displace relatively to each other, the roughness asperities ride on top of each other and result in an opening (dilatation) in the normal direction. Furthermore, the interaction of the crack surfaces in the contact zone gives rise to compressive normal stresses and frictional shear stresses opposing the crack face displacements. A phenomenological Mixed Mode cohesive zone law, derived from a potential function, is developed to describe the above mentioned fracture behaviour under monotonic opening. The interface dilatation introduces two new lengths. The cohesive law is implemented in the commercial finite element program Abaqus. The model is validated and tested against experimental results under various mode mixities.

A phantom node formulation with mixed mode cohesive law for splitting in laminates

A phantom node method with mixed mode cohesive law is proposed for the simulation of splitting in laminates. With this method, a discontinuity in the displacement field can be modeled at arbitrary locations. The micromechanical phenomenon that splitting cracks grow parallel to the fiber, is incorporated on the mesolevel, i.e., in the homogenized ply, by setting the direction of the crack propagation equal to the fiber direction. A new mixed mode cohesive law is introduced for increased robustness of the incremental-iterative solution procedure. The model is validated with mixed mode bending tests, and its utility is illustrated with examples for a single ply and for a laminate.

Identification of adhesive properties in GLARE assemblies using digital image correlation

An experimental–numerical methodology is introduced to identify the parameters of a cohesive law of an adhesive layer within a joined assembly on the basis of kinematic data provided by digital image correlation. Non-conventional experiments on joined samples were designed to generate within the assembly and the adhesive film complex strain and stress states close to those expected in-service and up to complete debonding. The modeling is developed with reference to the observed sub-domain in which the experimental boundary conditions are prescribed. The nonlinear behavior of the adhesive layer is described as a finite-thickness interface endowed with a mixed-mode cohesive law whose parameters are identified so as to match at best the measured displacement field.

Numerical investigation on the dynamic behavior of advanced ceramics

This paper studies the dynamic behavior of advanced ceramics by a finite element model. It implements the direct simulation of fracture and fragmentation together with a mixed-mode cohesive law to describe the fracture process. In particular, we simulate dynamic Brazilian tests performed with a Hopkinson bar, at strain rates ranging from 65 to 89 s −1, on six different materials: three kinds of alumina with different average grain sizes and degrees of purity, a blend of alumina and zirconia, silicon carbide and boron carbide. The rate dependence of the results emerges explicitly from the calculations, thanks both to the inertia attendant to the fracture process, and to the time effect provided by the cohesive law. Indeed, the simulations give accurate values for the dynamic strength of the six ceramics under study. The simulations also predict the main features of the crack pattern.

Modeling dynamic crack propagation in fiber reinforced composites including frictional effects

Dynamic crack propagation in a unidirectional carbon/epoxy composite is studied through finite element analyses of asymmetric impact (shear loading) of a rod against a rectangular plate. A finite deformation anisotropic visco-plastic model is used to describe the constitutive response of the composite. Crack propagation is simulated by embedding zero thickness interface element along the crack path. An irreversible mixed-mode cohesive law is used to describe the evolution of interface tractions as a function of displacement jumps. Contact and friction behind the crack tip are accounted for in the simulations. The failure of the first interface element at the pre-notch tip models onset of crack extension. Crack propagation is modeled through consecutive failure of interface elements. The dynamic crack propagation phenomenon is studied in terms of crack initiation time, crack speed, mode I and mode II displacement jumps and tractions associated with the failure of interface elements, effective plastic strain at the crack tip and path independent integral J ′. Analyses are carried out at impact velocities of 5, 10, 20, 30 and 40 m/s, assuming the crack wake is frictionless. Moreover, analyses at impact velocities of 30 and 40 m/s are also carried out with a friction coefficient of 0.5, 1, 5 and 10 along the crack surfaces. The analyses show that steady-state intersonic crack propagation in fiber reinforced composite materials occurs when the impact velocity exceeds a given threshold. A steady-state crack speed of 3.9 times the shear wave speed and 83% of the longitudinal wave speed is predicted in the cases in which the impact velocity is above 10 m/s. Detailed discussion is given on the features of sub-sonic and intersonic crack propagation. It is shown that friction effects, behind the crack tip, do not have a significant effect on maximum crack speed; however, they do on characteristics of the shock wave trailing the crack tip. The analyses also show that the contour integral J ′, computed at contours near the crack tip, is indeed path independent and can serve as a parameter for characterizing intersonic crack propagation.

Specific strain work as both failure criterion and material damage measure: Zuchowski, R. Res. Mech. 1989 27, (4), 309–322

Experimental investigations of adhesive joints and fibre composites have shown that under Mode II cracking, the fracture process induces a displacement normal to the fracture plane. This effect can be attributed e.g. to roughness of the fracture surface under dominating tangential crack face displacements. As the crack faces displace relatively to each other, the roughness asperities ride on top of each other and result in an opening (dilatation) in the normal direction. Furthermore, the interaction of the crack surfaces in the contact zone gives rise to compressive normal stresses and frictional shear stresses opposing the crack face displacements. A phenomenological Mixed Mode cohesive zone law, derived from a potential function, is developed to describe the above mentioned fracture behaviour under monotonic opening. The interface dilatation introduces two new lengths. The cohesive law is implemented in the commercial finite element program Abaqus. The model is validated and tested against experimental results under various mode mixities.