Modeling of crack tip high inertia zone in dynamic brittle fracture

A phenomenological modification is proposed to the existing cohesive constitutive law of Roy and Dodds to model the crack tip high inertia region proposed by Gao. The modification involves addition of a term which is attributed to fracture mechanisms that result in high energy dissipation around the crack tip. This term is assumed to be a function of external energy per unit volume input into the system. Finite element analysis is performed on PMMA with constant velocity boundary conditions and mesh discretizations based on the work of Xu and Needleman. The cohesive model with the proposed dissipative term is only applied in the high inertia zone and the traditional Roy and Dodds model is applied on cohesive elements in the rest of the domain. The results show that crack propagates in three phases with a speed of 0.35c_R before branching, confirming experimental observations. The modeling of high inertia zone is one of the key aspects to understanding brittle fracture.     

An explicit FE-model of impact fracture in an adhesive joint

Dynamic fracture of an adhesive layer in a structure is analysed. The structure represents some specific properties of an automotive structure and is simple enough to allow for closed form solutions obtained by the method of characteristics. These solutions are compared to results of explicit FE-analyses. The FE-solutions agree with the closed form solutions. Damage is included in the FE-model. Three constitutive models of the adhesive layer are used. It is shown that an amplification of the strain rate is achieved in the adhesive layer. It is also shown that an artificially increased flexibility of the adhesive in an aluminium structure gives only minor influences of the general behaviour. In some load cases, the adhesive layer will experience repeated loading/unloading. It is shown that in these cases an explicit FE-analysis with a “large” time step is more prone to give immediate rupture. Thus, the method is conservative.     

The effects of compressive and tensile prestrain on ductile fracture initiation in steels

It has been recognized that ductility of prestrained steel is inferior to that without prestrain, and the critical equivalent plastic strain of ductile fracture initiation is inversely related to stress triaxiality. In this paper, the effects of compressive and tensile prestrain on ductile fracture initiation in steels are investigated quantitatively by adopting the relationship between stress triaxiality and critical equivalent plastic strain. It is found that compressive prestrain leads to cleavage cracking and reduces ductility. In the case of the TMCP steel, compressive prestrain up to 30% does not decrease the ductility, accompanied by no evidence of cleavage cracks. However, in the case of SM490B steel, 30% compressive prestrain leads to cleavage cracking and reduces ductility significantly.     

A new cohesive zone model for mixed mode interface fracture in bimaterials

A new two-dimensional cohesive zone model which is suitable for the prediction of mixed mode interface fracture in bimaterials is presented. The model accounts for the well known fact that the interfacial fracture toughness is not a constant, but a function of the mode mixity. Within the framework of this model, the cohesive energy and the cohesive strength are not chosen to be constant, but rather functions of the mode mixity. A polynomial cohesive zone model is derived in light of analytical and experimental observations of interface cracks. The validity of the new cohesive law is examined by analyzing double cantilever beam and Brazilian disk specimens. The methodology to determine the parameters of the model is outlined and a failure criterion for a pair of ceramic clays is suggested.     

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.     

A note on pseudo-cohesive behavior in quasi-bidimensional brittle fracture

This paper addresses the study of a planar crack in a plate with a slightly curved crack front, as found in most nominally plane strain tests. Rather than trying to make a full three dimensional analysis of the elastic fields, we look for a systematic procedure to project the three-dimensional features on the plane of the plate. This is achieved by though-the-thickness averaging of the stress and displacement fields. It is shown that the resulting fields are pseudo-cohesive, which means that their expressions are formally identical to those corresponding to a traditional cohesive zone. The results for two simple cases are given to illustrate the pseudo-cohesive behavior, although general results are derived as obvious consequences of the averaging process. The general procedure to carry out the analysis for any kind of crack front shape is indicated, and the application to special cases such as fatigue-grown cracks in metals and stably growing cracks in brittle polymers is shortly discussed.     

Dynamic ductile fracture in aluminum round bars: experiments and simulations

The application of rate-dependent cohesive elements is validated in simulation of ductile fracture in aluminum round bars under dynamic loading conditions. Smooth and notched round bars made of AA6060-T6 are tested and simulated under quasi-static and dynamic loadings. The smooth round bar is modeled using finite elements that obey Gurson–Tvergaard–Needleman (GTN) formulation as the constitutive equation. Comparing with experimental results, corresponding GTN parameters and rate-dependent plasticity of the alloy are obtained. A single strain rate-dependent GTN element with the obtained parameters is examined under different values of stress triaxiality and loading rates. The resulting stress-elongation curves represent the traction separation law (TSL) for cohesive elements and the variations of the maximum traction and the energy absorbed are investigated. The notched round bars are modeled by axisymmetric continuum and cohesive elements. The undamaged bulk material is elastic-visco plastic and the cohesive elements obey the TSL defined from the single element calculations. The experiments are simulated by these models in which the cohesive elements are rate sensitive and automatically obtain the values of the total strain rate from their adjacent continuum elements to update the values of the cohesive strength during the analysis. The results of the analysis, including maximum load, time of failure and diameter reduction are validated with the experimental results. The effects of element size, rate-dependent plasticity of the material and stress triaxiality are also discussed.     

Modelling inter- and transgranular fracture in piezoelectric polycrystals

A cohesive zone finite element model for quasi-static fracture of piezoelectric polycrystals is proposed. Interface elements are used to model both inter- and transgranular fracture. Electromechanical constitutive relations are derived by enhancing commonly used mechanical traction-opening laws with relations for a parallel plate capacitor. Numerical simulations demonstrate that the proposed model correctly mimics several experimentally observed phenomena. Most importantly the switch from mainly intergranular to mainly transgranular fracture with increasing grain size is modelled correctly. The model also correctly mimics the influence of an electric field on the ultimate load.     

Fatigue fracture of SnAgCu solder joints by microstructural modeling

The ongoing miniaturization trend in the microelectronic industry enforces component sizes to approach the micron, or even the nano scale. At these scales, the underlying microstructural sizes and the geometrical dimensions are comparable. The increasing influence of microscopic entities on the overall mechanical properties makes conventional continuum material models more and more questionable. In this study, the thermomechanical reliability of lead-free BGA solder balls is investigated by microstructural modeling. Microstructural input is provided by orientation imaging microscopy (OIM), converted into a finite element framework. Blowholes in BGA solder balls are examined by optical microscopy and a statistical analysis on their size, position and frequency is conducted. Combining the microstructural data with the appropriate material models, three dimensional local models are created. The fatigue life of the package is determined through a critical solder ball. The thermomechanical reliability of the local models are predicted using cohesive zone based fatigue damage models. The simulation results are validated by statistical analyses provided by the industry.     

Cohesive zone modeling of interface fracture in elastic bi-materials

Some basic issues regarding the cohesive zone modeling of interface fracture between two dissimilar elastic materials are studied. The dependence of the cohesive energy density on the phase angle is first discussed under small scale cohesive zone conditions. It is then shown that in general the stress singularities in tension and shear cannot be simultaneously removed at the cohesive zone tip if a single cohesive zone length is adopted for both tensile and shear fracture modes. Finally, the energy dissipation at the tip of a prescribed cohesive zone is examined using a bilinear cohesive zone model under the uncoupled tension/shear conditions.     

A cohesive finite element formulation for modelling fracture and delamination in solids

In recent years, cohesive zone models have been employed to simulate fracture and delamination in solids. This paper presents in detail the formulation for incorporating cohesive zone models within the framework of a large deformation finite element procedure. A special Ritz-finite element technique is employed to control nodal instabilities that may arise when the cohesive elements experience material softening and lose their stress carrying capacity. A few simple problems are presented to validate the implementation of the cohesive element formulation and to demonstrate the robustness of the Ritz solution method. Finally, quasi-static crack growth along the interface in an adhesively bonded system is simulated employing the cohesive zone model. The crack growth resistance curves obtained from the simulations show trends similar to those observed in experimental studies     

Determination of fracture energy and tensile cohesive strength in Mode I delamination of angle-ply laminated composites

The energy release rate in delamination of angle-ply laminated double cantilever composite beam specimens was calculated using the compliance equation, and interlaminar cohesive strengths were obtained. Instead of the traditional approach of a beam on an elastic foundation, a second-order shear-thickness deformation beam theory (SSTDBT) was considered. The equilibrium equations were obtained using the principle of minimum total potential energy and the system of ordinary differential equations were solved analytically. The problem was solved for [0°]₆ , [±30°]₅, and [±45°]₅ laminates with mid-plane delaminations and the results were verified using experimental evidence available in the literature.     

Comparison of cohesive zone model and linear elastic fracture mechanics for a mode I crack near a compliant/stiff interface

Cohesive zone model has been widely applied to simulate crack growth along interfaces, but its application to crack growth perpendicularly across the interface is rare. In this paper, the cohesive zone model is applied to a crack perpendicularly approaching a compliant/stiff interface in a layered material model. One aim is to understand the differences between the cohesive zone model and linear elastic fracture mechanics in simulating mode I crack growth near a compliant/stiff interface. Another aim is to understand the effects of elastic modulus mismatch and cohesive strength of the stiff layer on the crack behavior near the interface. To simulate crack growth approaching an interface, the cohesive zone model which incorporates both the energy criterion and the strength criterion is an effective method.     

Multiscale dynamic fracture analysis of composite materials using adaptive microstructure modeling

In this study, a computational framework is proposed to investigate multiscale dynamic fracture phenomena in materials with microstructures. The micro- and macro-scales of a composite material are integrated by introducing an adaptive microstructure representation. Then, the far and local fields are simultaneously computed using the equation of motion, which satisfies the boundary conditions between the two fields. Cohesive surface elements are dynamically inserted where and when needed, and the Park-Paulino-Roesler cohesive model is employed to approximate nonlinear fracture processes in a local field. A topology-based data structure is utilized to efficiently handle adjacency information during mesh modification events. The efficiency and validity of the proposed computational framework are demonstrated by checking the energy balances and comparing the results of the proposed computation with direct computations. Furthermore, the effects of microstructural properties, such as interfacial bonding strength and unit cell arrangement, on the dynamic fracture behavior are investigated. The computational results demonstrate that local crack patterns depend on the combination of microstructural properties such as unit cell arrangement and interfacial bonding strength; therefore, the microstructure of a material should be carefully considered for dynamic cohesive fracture investigations.     

Analysis of a compressive shear test for adhesion between elastomeric polymers and rigid substrates

A compressive shear test for investigating adhesion between an elastomeric polymer and a rigid substrate has been studied. The test consists of loading a specimen comprising of a 3-ply laminate: substrate/polymer/ substrate, in compression and shear at a specified angle to the loading direction. Under displacement control and when adhesion is sufficiently low, an interfacial crack nucleates at one interface early during loading and propagates stably up to a critical load at which unstable propagation with an associated load drop ensues. The case of an isothennal hyperelastic material has been analyzed by computing the energy release rate for an interfacial crack as a function of crack length. The analysis shows that for a range of initial crack size interfacial crack propagation is stable until crack length reaches a critical size at which unstable propagation ensues. The energy release rate at this instability is relatively insensitive to angle of loading, strain, and hyperelastic parameters, which allows one to extract an interfacial toughness, ₀, from overall measurement of stress and strain. The analysis has been extended to consider combined hyperelasticity and viscoelasticity by using a cohesive zone model for crack propagation implemented as a cohesive finite element. The energy release rate and cohesive zone analyses give identical results for an hyperelastic material. For a viscoelastic-hyperelastic material, the cohesive zone approach allows the viscous losses in the bulk polymer to be estimated separately from the value of interfacial fracture toughness. Both analyses have been applied to experiments on glass/polyvinyl butyral (Butacite®)/glass laminate specimens. The intrinsic interfacial toughness, consisting of contributions from bond rupture and a near-tip process zone, is found to be rate-dependent and lies in the range 50–200 J m^–2.     

The effects of inter-surface cohesive tractions on linear and penny-shaped cracks

A mesoscopic fracture model of equilibrium slit cracks in brittle solids, including inter-surface cohesive tractions acting near the crack tip, is analyzed and the effects of the cohesive tractions on the in-plane stress fields, crack-opening displacement profiles, and crack driving forces examined quantitatively for linear and penny-shaped cracks. The (numerical) analysis method is described in detail, along with results for four different cohesive forces. The assumed distribution of cohesive tractions were found to suppress the in-plane stress field adjacent to cracks in a homogeneous, isotropic medium when uniformly loaded in mode-I, and the suppression was a function of crack length. The crack-opening displacement profile was also perturbed and a new regime identified between the near-field Barenblatt zone and the far-field continuum zone. The extent of this `cohesive zone' was quantified by use of an interpolating function fit to the calculated profiles and found to be independent of crack size for a given cohesive tractions distribution. Furthermore, the crack-opening displacement at the edge of the cohesive zone was found to be independent of crack size, implying that despite significant perturbations to the stress field, the crack driving force at unstable equilibrium remains unchanged with crack size.     

Mixed-mode I/II wood fracture characterization using the mixed-mode bending test

In this work fracture characterization of wood under mixed-mode I/II loading is addressed. The mixed-mode bending test is used owing to its aptitude for easier alteration of mode ratio. Experimental tests were performed covering a wide range of mode ratios in order to obtain a mixed-mode fracture criterion for the maritime pine (Pinus pinaster Ait.) in the RL crack propagation system. A data reduction scheme based on beam theory and crack equivalent concept was used to overcome some difficulties inherent to the test. The method does not require crack length monitoring during propagation and provide an entire resistance curve allowing easier identification of the fracture energy. A numerical analysis using cohesive elements was also performed to validate the method. The linear energetic fracture criterion was proved to be the most adequate to describe the failure envelop of this wood species.     

Mixed-mode cohesive-zone models for fracture of an adhesively bonded polymer-matrix composite

As a direct extension of previous mode-I work on the adhesion of composite joints, this paper uses a cohesive-zone approach to model the mixed-mode fracture of adhesive joints made from a polymer-matrix composite. Mode-II cohesive-zone parameters were obtained using sandwich end-notch flexure specimens. These parameters were used directly with the previously determined mode-I parameters to predict the fracture and deformation of mixed-mode geometries. It was shown that numerical simulations provided quantitative predictions for these geometries, including predictions for both the strengths of the joints and for the failure mechanisms. In conjunction with the earlier work, these results demonstrate the use of cohesive-zone approaches for the design of adhesively bonded composite joints, and indicate approaches for determining the relevant material properties to describe mixed-mode fracture.     

Constitutive models for cohesive zones in mixed-mode fracture of plain concrete

Discrete mixed-mode fracture (modes I and II) of plain concrete is investigated using a coupled and an uncoupled cohesive zone constitutive model in a finite element context. Fracture surfaces are confined to inter-element boundaries that are not necessarily coincident with the actual fracture surfaces. For this reason, traction components on the cohesive zone do not correspond to actual values either. In this work is demonstrated that only the coupled model is able to cope with these spurious traction components, that must decrease with crack opening. It is shown also that, in this regard, the key variable is the plastic potential adopted in the integration of tractions. Three mixed-mode fracture examples were tested in this work: a three-point single-edge notched beam, double-edge notched plates under variable lateral and normal deformation and four-point double-edge notched beams. A good fitting with experiments was obtained only for the coupled model. Mode II parameters can change in a large range without noticeable change in results, at least in the tested examples.