Theory and numerics for finite deformation fracture modelling using strong discontinuities

A general finite element approach for the modelling of fracture is presented for the geometrically non‐linear case. The kinematical representation is based on a strong discontinuity formulation in line with the concept of partition of unity for finite elements. Thus, the deformation map is defined in terms of one continuous and one discontinuous portion, considered as mutually independent, giving rise to a weak formulation of the equilibrium consisting of two coupled equations. In addition, two different fracture criteria are considered. Firstly, a principle stress criterion in terms of the material Mandel stress in conjunction with a material cohesive zone law, relating the cohesive Mandel traction to a material displacement ‘jump’ associated with the direct discontinuity. Secondly, a criterion of Griffith type is formulated in terms of the material‐crack‐driving force (MCDF) with the crack propagation direction determined by the direction of the force, corresponding to the direction of maximum energy release. Apart from the material modelling, the numerical treatment and aspects of computational implementation of the proposed approach is also thoroughly discussed and the paper is concluded with a few numerical examples illustrating the capabilities of the proposed approach and the connection between the two fracture criteria. Copyright © 2005 John Wiley & Sons, Ltd.     

Fatigue of spot-welded lap joints

The fatigue properties of spot-welded lap joints under a constant mean load made from 1.2 and 3 mm sheet thickness stainless steel with one, two or three-spot welds in series are reported. A log plot of cyclic load range versus fatigue life shows that for given sheet thickness and fixed load range, fatigue life increases with the number of spot welds. Oil has a beneficial effect by increasing the fatigue life of the welded joints. A fracture mechanics analysis is carried out on the data by treating the spot weld as a crack. A log plot of initial stress intensity factor range versus fatigue life is a straight line which is independent of the number of spot welds. However, increasing the sheet thickness shifts the straight line upwards in the log plot, thus indicating a longer fatigue life for the same applied initial stress intensity factor range.     

Numerical study of strengths of spot-welded joints of steel

A resistance spot-welding (RSW) joint consists of several material zones with different microstructure and properties as a result of the thermal, metallurgical and mechanical deformation process. Detailed material properties are essential to accurately simulate the mechanical behavior of a joint and its dependency on some key structural parameters (e.g. nugget size, sheet thickness etc.). The work presented in this paper utilises an inverse modelling methodology combining numerical modelling and indentation tests with a standard hardness test to characterise the detailed properties of different weld zones of spot-welded joints. The yield and strain hardening parameters of the three zones (nugget zone, HAZ: heat-affected zone and base zone) were determined and the predicted stress–strain curves for base zone were compared with standard tensile tests results. A 3-D finite element model based on the predicted constitutive material laws for different zones coupled with a fracture model was developed to predict the deformation of spot-welded joints beyond the onset of initial yield under tensile-shear loading. The deformation mode and force–displacement result showed good agreements with experimental data. The effect of nugget size and sheet thickness on the tensile-shear strength of welded joints was further systematically studied using a high performance computing system.     

Bending deformation and fracture characterisation in quenching and partitioning steel

The bending deformation recorded and fracture characterisation seen during the process of quenching and partitioning (Q&P) steel are investigated as a function of bending angle. According to experimental results, the mechanically induced martensitic transformation of the retained austenite occurred heterogeneously in the direction of thickness. Results from electron backscattered diffraction and finite element analysis confirmed that the heterogeneous phase transformation was caused by the stress difference in the direction of thickness. The results prove fracture behaviour is dominated by martensite cracking. The results provide a new insight into deformation characterisation of the Q&P steel under bending, which allows the design and optimisation of mechanical properties.     

Numerical modelling and experimental investigation on welding residual stresses in large‐scale tubular K‐joints

This paper is devoted to the experimental and numerical assessment of residual stresses created by welding in the region surrounding the weld toe of tubular K‐shaped joints (i.e. region most sensitive to fatigue cracking). Neutron‐diffraction measurements were carried out on K‐joints cut from large‐scale truss beams previously subjected to high cycle fatigue. Tri‐axial residual stresses in the transverse, longitudinal and radial direction were obtained from the weld toe as a function of the depth in the thickness of the tube wall. In addition, thermomechanical analyses were performed in three‐dimensional using ABAQUS and MORFEO finite element codes. Experimental and numerical results show that, at and near the weld‐toe surface, the highest residual stresses are critically oriented perpendicularly to the weld direction, and combined with the highest externally applied stresses. Based on a systematic study on geometric parameters, analytical residual stress distribution equations with depth are proposed.     

Analysis of fatigue crack behaviour based on dynamic response simulations and experiments for tensile-shear spot-welded joints

Fatigue crack initiation and propagation behaviours were studied based on the dynamic response simulation by the three‐dimensional finite‐element analysis (FEA) and dynamic response experiments for tensile‐shear spot‐welded joints. The entire fatigue propagation behaviour from the surface elliptical cracks at the initiation stage to the through thickness cracks at the final stage was taken into consideration during the three‐dimensional FEA dynamic response simulations. The results of the simulations and experiments found that the fatigue cracks of spot‐welded joint from initial detectable crack sizes to crack propagation behaviour could be described by three stages. Approximately one‐half of the total fatigue life was taken in stage I, which includes micro‐crack nucleation and the small crack growth process; 20% of the total fatigue life in stage II, in which the existing surface crack propagates through the thickness of sheet and 30% of the total fatigue life in stage III, during which the through thickness crack propagates along the direction of plate width to the final failure. According to the relationship between the crack length and depth and the dynamic response frequency during the simulated fatigue damage process, the definition of fatigue crack initiation and propagation stages was proposed. The analysis will provide some information for the fatigue life prediction of the spot‐welded structures.     

Some recent developments in computational modelling of concrete fracture

Some of the most important aspects of numerical modelling of cracking in concrete are reviewed. After a discussion of the three main lines in modelling cracking – discrete crack models, smeared representations and approaches using lattice models – a concise treatment including comparative studies is given of the various smeared crack approaches that exist to date. Next a discussion is presented of some issues pertaining to the sensitivity of numerical results on the fineness of the mesh and the direction of the mesh lines, and on size effects in concrete structures. This revised version was published online in July 2006 with corrections to the Cover Date.     

Experimental and numerical assessment of fracture toughness of dual-phase austempered ductile iron

The effects of the microstructure topology on the fracture toughness of dual-phase austempered ductile iron are studied in this paper by means of finite element modelling and experimental testing. To this end, specimens with matrix microstructures ranging from fully ferrite to fully ausferrite were studied and the preferential zones and phases for crack propagation were identified in every case. The effectiveness of the ausferrite phase as a reinforcement of the ferritic matrix via the encapsulation of the brittle and weak last-to-freeze (LTF) zones was confirmed. The toughening mechanism is consequence of the increment in the crack path longitude as it avoids the encapsulated LTF zones. Besides, the presence of small pools of allotriomorphic ferrite increase the crack propagation resistance of the ausferrite-ferrite matrices.     

A low‐order,hexahedral finite element for modelling shells

A thin, eight‐node, tri‐linear displacement, hexahedral finite element is the starting point for the derivation of a constant membrane stress resultant, constant bending stress resultant shell finite element. The derivation begins by introducing a Taylor series expansion for the stress distribution in the isoparametric co‐ordinates of the element. The effect of the Taylor series expansion for the stress distribution is to explicitly identify those strain modes of the element that are conjugate to the mean or average stress and the linear variation in stress. The constant membrane stress resultants are identified with the mean stress components, and the constant bending stress resultants are identified with the linear variation in stress through the thickness along with in‐plane linear variations of selected components of the transverse shear stress. Further, a plane‐stress constitutive assumption is introduced, and an explicit treatment of the finite element's thickness is introduced. A number of elastic simulations show the useful results that can be obtained (tip‐loaded twisted beam, point‐loaded hemisphere, point‐loaded sphere, tip‐loaded Raasch hook, and a beam bent into a ring). All of the gradient/divergence operators are evaluated in closed form providing unequivocal evaluations of membrane and bending strain rates along with the appropriate divergence calculations involving the membrane stress and bending stress resultants. The fact that a hexahedral shell finite element has two distinct surfaces aids sliding interface algorithms when a shell folds back on itself when subjected to large deformations. Published in 2004 by John Wiley & Sons, Ltd.     

Damage tolerance reliability analysis of automotive spot-welded joints

This paper develops a damage tolerance reliability analysis methodology for automotive spot-welded joints under multi-axial and variable amplitude loading history. The total fatigue life of a spot weld is divided into two parts, crack initiation and crack propagation. The multi-axial loading history is obtained from transient response finite element analysis of a vehicle model. A three-dimensional finite element model of a simplified joint with four spot welds is developed for static stress/strain analysis. A probabilistic Miner's rule is combined with a randomized strain-life curve family and the stress/strain analysis result to develop a strain-based probabilistic fatigue crack initiation life prediction for spot welds. Afterwards, the fatigue crack inside the base material sheet is modeled as a surface crack. Then a probabilistic crack growth model is combined with the stress analysis result to develop a probabilistic fatigue crack growth life prediction for spot welds. Both methods are implemented with MSC/NASTRAN and MSC/FATIGUE software, and are useful for reliability assessment of automotive spot-welded joints against fatigue and fracture.     

Cohesive zone modelling of interface fracture near flaws in adhesive joints

A cohesive zone model is suggested for modelling of interface fracture near flaws in adhesive joints. A shear-loaded adhesive joint bonded with a planar circular bond region is modelled using both the cohesive zone model and a fracture mechanical model. Results from the models show good agreement of crack propagation on the location and shape of the crack front and on the initial joint strength. Subsequently, the cohesive zone model is used to model interface fracture through a planar adhesive layer containing a periodic array of elliptical flaws. The effects of flaw shape are investigated, as well as the significance of fracture process parameters. The results from simulations of fracture in a bond containing circular flaws show that localization of crack propagation in the vicinity of a flaw has significant effect on the joint strength and crack front shape. The localization effects are highly dependent on the fracture process zone width relative to the flaw dimensions. It is also seen that with increasing fracture process zone width, the strength variation with the flaw shape decreases, however, the strength is effected over a wider range of propagation.     

The use of a cohesive zone model to study the fracture of fibre composites and adhesively-bonded joints

Analytical solutions for beam specimens used in fracture-mechanics testing of composites and adhesively-bonded joints typically use a beam on an elastic foundation model which assumes that a non-infinite, linear-elastic stiffness exists for the beam on the elastic foundation in the region ahead of the crack tip. Such an approach therefore assumes an elastic-stiffness model but without the need to assume a critical, limiting value of the stress, _max, for the crack tip region. Hence, they yield a single fracture parameter, namely the fracture energy, G _c. However, the corresponding value of _max that results can, of course, be calculated from knowledge of the value of G _c. On the other hand, fracture models and criteria have been developed which are based on the approach that two parameters exist to describe the fracture process: namely G _c and _max. Here _max is assumed to be a critical, limiting maximum value of the stress in the damage zone ahead of the crack and is often assumed to have some physical significance. A general representation of the two-parameter failure criteria approach is that of the cohesive zone model (CZM). In the present paper, the two-parameter CZM approach has been coupled mainly with finite-element analysis (FEA) methods. The main aims of the present work are to explore whether the value of _max has a unique value for a given problem and whether any physical significance can be ascribed to this parameter. In some instances, both FEA and analytical methods are used to provide a useful crosscheck of the two different approaches and the two different analysis methods.     

Fracture mechanics‐based progressive damage modelling of adhesively bonded fibre‐reinforced polymer joints

A quasi‐static progressive damage model for prediction of the fracture behaviour and strength of adhesively bonded fibre‐reinforced polymer joints is introduced in this paper. The model is based on the development of a mixed‐mode failure criterion as a function of a master R‐curve derived from the experimental results obtained from standard fracture mechanics joints. Consequently, the developed failure criterion is crack‐length and mode‐mixity dependent, and it takes into account the contribution of the fibre‐bridging effect. Energy release rate values for adhesively bonded double‐lap joints are obtained by using the virtual crack closure technique method in a finite element model, and the numerically obtained strain energy release rate is compared to the critical strain energy release rate given by the mixed‐mode failure criterion. The entire procedure is implemented in a numerical algorithm, which was successfully used for predicting the strength and R‐curve response of adhesively bonded double‐lap structural joints made of pultruded glass fibre‐reinforced polymers and epoxy adhesives.     

Interface fracture of polymer films: blister test experiments and modelling

The interface fracture between a rigid substrate and polymer film is investigated in this work using pressurised blister test experiments and modelling. The interface crack growth is studied for two different types of polymer films: stiff and compliant ones. The pressurised blister test is used to provide critical pressure-crack length curves for different loading media (water and electrolyte solutions) and loading rates. Two different analytical approaches and a numerical modelling concept are used to determine the critical total energy release rate as a function of the crack length (crack resistance curve or R-curve). A relatively flat R-curve is observed for the system with the stiff polymer film, whilst R-curve for the compliant film system exhibits an increasing tendency. The mixed-mode fracture behaviour occurs for both investigated polymer film systems, as shown by the value of the mixed-mode angle that is constant for all investigated crack lengths. R-curves are nearly unaffected by different loading media, whereas the loading rate has a strong influence on the interface fracture of the compliant file system. Finite element method-based prediction of the total energy release rate is in good agreement with that obtained from analytical expressions.     

Mixed-mode fracture characterization of adhesive joints

A novel load jig is presented which allows mixed-mode fracture testing of adhesive joints and composite laminates over the entire range from mode I to mode II, by using a single equal adherend double-cantilever-beam specimen. Experiments performed with the load jig showed that G_IIC was approximately three times higher than G_IC for the tested adhesive system consisting of FPL-etched 7075-T6 aluminium adherends bonded with Cybond 4523GB (American Cyanamid) epoxy adhesive. Experimental data showed that G_C was independent of crack length and that there was no dependence of G_IC on adherend thickness. Comparison of G_IIC values obtained by using the load jig to test conventional end notch flexure (ENF) specimens indicated that there are relatively small friction effects between crack faces in mode II testing of ENF specimens. The experimental data were also used to evaluate three different analytical techniques for the mode partitioning of unequal adherend specimens.     

Extrinsic cohesive modelling of dynamic fracture and microbranching instability in brittle materials

Dynamic crack microbranching processes in brittle materials are investigated by means of a computational fracture mechanics approach using the finite element method with special interface elements and a topological data structure representation. Experiments indicate presence of a limiting crack speed for dynamic crack in brittle materials as well as increasing fracture resistance with crack speed. These phenomena are numerically investigated by means of a cohesive zone model (CZM) to characterize the fracture process. A critical evaluation of intrinsic versus extrinsic CZMs is briefly presented, which highlights the necessity of adopting an extrinsic approach in the current analysis. A novel topology‐based data structure is employed to enable fast and robust manipulation of evolving mesh information when extrinsic cohesive elements are inserted adaptively. Compared to intrinsic CZMs, which include an initial hardening segment in the traction–separation curve, extrinsic CZMs involve additional issues both in implementing the procedure and in interpreting simulation results. These include time discontinuity in stress history, fracture pattern dependence on time step control, and numerical energy balance. These issues are investigated in detail through a ‘quasi‐steady‐state’ crack propagation problem in polymethylmethacrylate. The simulation results compare reasonably well with experimental observations both globally and locally, and demonstrate certain advantageous features of the extrinsic CZM with respect to the intrinsic CZM. Copyright © 2007 John Wiley & Sons, Ltd.     

On the fracture toughness anisotropy of mechanically poled ferroelectric ceramics

In this paper an incremental constitutive theory for the deformation due to switching in ferroelectrics is applied to predict the fracture toughness anisotropy in these materials after mechanical poling. Mechanical poling of an initially unpoled specimen differs from electrical poling in that only mechanical stresses are applied to the material. Therefore, no electrical polarization can develop. After mechanical poling, for example by a uniaxial applied stress, the fracture toughness of a ferroelectric ceramic for cracks running parallel or orthogonal to the poling direction will differ. Finite element computations of the steady crack growth process have been carried out to quantify these differences. Results are generated for a range of constitutive properties for three crack growth directions with respect to the initial mechanical poling direction. The results are discussed in relation to available experimental data and to the toughness anisotropy due to electrical poling.     

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     

On the characteristic distance and minimum fracture toughness for cleavage fracture in a C-Mn steel

This study makes a further investigation on the characteristic distance, minimum fracture toughness and its temperature dependence for cleavage fracture in a C-Mn steel by the detailed finite element analysis combined with experimental observation and measurement. Results show that there is a minimum active zone for cleavage initiation, and the minimum fracture toughness of steel results from the minimum active zone necessary. Corresponding to the minimum fracture toughness, the cleavage fracture ahead of a crack tip can only initiate in a distance range from the minimum distance X_fmin determined by the lower boundary of the active zone to the maximum distance X_fmax determined by its upper boundary. The reason for the occurrence of the minimum active zone and the factors influencing it are analyzed. The temperature dependence of the characteristic distance and minimum fracture toughness and its mechanism are also discussed.