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A new medical device can take years to develop from early concept to product launch. The long development process can be attributed to the severe consequences for the patient if the device malfunctions. Three approaches are often combined to mitigate risks: rigorous simulation and modeling, physical test programs, and Failure Mode Effect Analysis (FMEA) — all of which are time-consuming. Physical test programs are often carried out on prototype components from the same batch and, therefore, limited in revealing the actual distribution of performance. The risk probabilities are subsequently based on educated guesses. Furthermore, simulation and modeling are usually performed on nominal geometry — not accounting for variation — and only provide a safety factor against failure. The traditional use of safety factors in structural analysis versus the probabilistic approach to risk management presents an obvious misfit. Therefore, these three approaches are not ideal for addressing the two key questions that the design engineer has: 1) How often will the design fail, and 2) How should the design be changed to improve robustness and failure rates. The present work builds upon the existing Robust and Reliability-Based Design Optimization (R2BDO) and adjusts it to address the key questions above using finite element analysis. The key feature of the new framework is the focus on minimal use of computational resources while being able to screen feasible design concepts early in the embodiment phase and subsequently optimize their probabilistic performance. A case study in collaboration with a medical design and manufacturing company demonstrates the new framework. The case study includes FEA contact modeling between two plastic molded components with twelve geometrical variables. The optimization focuses on minimizing the failure rate (and improving design robustness) concerning three constraint functions (contact pressure, strain, torque). The study finds that the new framework achieves significant improvements to the component’s performance function (failure rate) with minimal computational resources.

Assisted flipping of a slant mode I dominated tearing crack, where the crack tip flipping mechanism is made to engage by imposing a slight mode III, has recently been studied in Felter and Nielsen (2017) [Assisted crack tip flipping under Mode I thin sheet tearing, Europ. J. Mech. A/Solids, 64;2017:58–68]. In the previous study, the Gurson–Tvergaard–Needleman model was used in its original form to limit the model parameter space and facilitate a search for a set of parameter that allows flipping of the slant crack face to occur. It is well known that the adopted version of the Gurson model can predict the shear bands that travel in front of a mode I tearing crack and these are essential features to slant crack propagation — let alone the experimentally observed crack tip flipping phenomenon. In fact, assisted crack tip flipping was achieved with this numerical model set-up, but only within a very narrow parameter window. The present work adopts a phenomenological shear extended Gurson model that allows for a study of the J3 dependency in ductile fracture at engineering scale in an attempt to widen this parameter window. By running series of large-scale computations, where a ductile tearing crack propagates multiple plate thicknesses, the authors can demonstrate tearing modes for various combinations of strain hardening, initial void volume fraction, and shear parameter. The shear damage contribution is found to shift the engagement of the crack tip flipping mechanism toward lower values of the initial void volume fraction for all levels of strain hardening considered.

Size effects in a strain hardening porous solid are investigated using the Gurson-Tvergaard (GT) model enriched by a constitutive length parameter, as proposed by Niordson and Tvergaard [C.F. Niordson, V. Tvergaard, A homogenised model for size effects in porous metals, J. Mech. Phys. Solids (2019)]. The results are compared with unit cell calculations of regularly distributed voids embedded in a strain gradient enhanced matrix material. The strain gradient plasticity theory proposed by Fleck and Willis [N.A. Fleck, J.R. Willis, A mathematical basis for strain gradient plasticity theory. Part II: tensorial plastic multiplier, J. Mech. Phys. Solids 57 (2009) 1045–1057], extended to finite strains, is adopted for the cell model, consistent with the gradient enriched Gurson model. The gradient model allows for a material length parameter to enter the constitutive framework for dimensional consistency, while the enriched GT model has the same length parameter introduced through prefactors of the usual q 1 and q 2 factors. The continuum model featuring size-dependent Tvergaard-constants is used to investigate a strain hardening material with the strain gradient plasticity enriched cell model as reference. The two models are compared for three triaxialities, three initial void volume fractions, and three hardening exponents. The enriched GT model captures the effect of elevated yield point and suppressed void growth with increasing length parameter for all the cases investigated. The agreement between the models is good until severe void distortion or plastic flow localisation between neighbouring voids. The response curves and void growth curves for the enriched GT model deviate from those of the cell model at high axial strains. Void shape plots, which are only available for the cell model, show that the length parameter influences the shape of the void which in turn has impact on the material response curves and the void evolution. This is not captured by the enriched GT model as the voids are accounted for solely through a volume fraction parameter.

This paper presents a numerical basis for finite strain analysis using the higher order flow theory by Fleck and Willis [2009, A mathematical basis for strain-gradient plasticity theory - part II: tensorial plastic multiplier , J. Mech. Phys. Solids, 57, 1045–1057]. The rate-dependent (visco-plastic) formulation shares similarities with its rate-independent (generalized J2-flow) counterpart, and a combined presentation is laid out. The visco-plastic formulation [2018, A homogenized model for size-effects in porous metals , J. Mech. Phys. Solids, DOI: 10.1016/j.jmps.2018.09.004] is revisited to set the scene for the development of the rate-independent model, but the two formulations differ substantially in the numerical implementation as the rate-independent framework possesses a challenge in determining the yield of individual plastic zones. The framework developed readily accounts for repeated loading/unloading and handles the interaction of multiple plastic zones. The method presented exploits the image analysis techniques that was first combined with the Fleck-Willis theory in Nielsen and Niordson [2014, A numerical basis for strain-gradient plasticity theory: rate-independent and rate-dependent formulations, J. Mech. Phys. Solids 63, 113–127], but extends it to finite strains by adopting an updated Lagrangian framework . Besides somewhat standard extensions to deal with finite deformations , the framework largely resembles that of small strains. In fact, the adopted image analysis can be conducted in the undeformed configuration without any approximations as it relates only to the connectivity of plastic zones on the finite element mesh . The new framework is focused on the deformation and localization in both uniform and notched round bars by formulating the numerical model in an axi-symmetric setup. Comparisons between the visco-plastic and the rate-independent formulations are presented.

Ductile plate tearing, where the crack propagates multiple plate thicknesses, is targeted by the micro-mechanics based Gurson-Tvergaard-Needleman (GTN) model. The focus is on extracting detailed information on the fracture process that governs ductile crack initiation from a blunt pre-crack until the crack reaches steady-state propagation in order to enhance accuracy of the traditionally used cohesive zone traction-separation relations. The aim is to facilitate an accurate representation of the tearing process within the cohesive zone modeling framework as such simplistic models are largely exploited by engineers worldwide. Unfortunately, accuracy in the representation of crack propagation is often sacrificed for computation speed, but the present work allows correlating the cohesive zone modeling to a much more accurate, though computational expensive, micro-mechanics based (full 3D) model response. In the modeling of large-scale plate tearing, shell elements are typically employed to represent the engineering scale of the structure while the cohesive zone represents the micro-scale in terms of crack initiation and growth process. Thus, the cohesive zone essentially has to take over at the onset of the first localization (thinning far ahead of the crack tip). Calibration of the cohesive zone parameters has earlier been made in accordance with experimental observations such that the overall response of the system is well reproduced. But, the present work takes the calibration of the cohesive zone one step further and exploits details from a large-scale GTN model calculation. The goal is to match the response from the GTN model with the much less computation demanding cohesive zone modeling approach by incorporating knowledge of the loading history for individual cross-sections, in front of the pre-crack, through which the tearing crack propagates. The full 3D micro-mechanics based model set-up allows tracking of key parameters, such as peak traction and tearing energy, which goes into the cohesive traction-separation relation. The dependency on distance from the crack initiation site of the cohesive zone parameters is determined - from crack initiation to steady-state propagation - and followed up by a discussion on how to construct a traction-separation relation for ductile plate tearing.

A single material length parameter governs both the material size dependence and the predicted micro-structural behavior in most existing micro-structurally based continuum theories. As a consequence, smoothly varying field quantities are predicted which contrast recent years' experimental observations. In a previous work by the authors, addressing this matter, two new back stress formulations were proposed, and demonstrated to offer novel modeling capabilities in the localization behavior of geometrically necessary dislocation pile-up under monotonic loading. However, the cyclic behavior of these formulations remains to be investigated. The present work studies the new back stress formulations, within a non-work conjugate type higher order strain gradient crystal plasticity framework, and demonstrates their performance through the idealized single slip simple shear case. At high values of the material length scale parameter, a seemingly anomalous cyclic response is observed when deviating from the conventional type back stress formulation. Similar observations have recently been reported in other numerical studies, and the present work offers a discussion of the physical justification of such material behavior. It is found that the properties of the adopted formulations, in fact, open the possibility for modeling complex material behavior, tied to the presence of long range internal stresses due to dislocation pile-up. Moreover, the present study extends the discussion on micro-structure predictions as a consequence of the adopted back stress models.

The present study aims to quantify the influence of void size effect on crack initiation and growth in mode I tearing of large ductile plates. Specifically, the aim is to demonstrate how size effects influence the transition from crack initiation to steady-state tearing and reveal the evolution in both the traction-separation relation and the fracture energy dissipated in the tearing process when related to large-scale cohesive modeling. The study investigates mode I tearing by adopting the gradient enriched Gurson-Tvergaard-Needleman (GTN) model by Niordson and Tvergaard (2019), where the novelty lies in accounting for strain gradient hardening near growing micro-voids in relation to large-scale plate tearing. The steady-state crack growth conditions are compared to a 2D plane strain model setup, while the transition from crack initiation to steady-state is analyzed using full 3D plate tearing calculations. A discussion about the 2D versus 3D models’ capability, their differences, and agreement is presented. The results are related to the traction-separation relations fit for efficient large-scale simulations, e.g., using cohesive zone modeling. The present study shows that the material length parameter incorporated into the constitutive model has a negligible impact on the onset of plate thinning (or necking) but delays the onset of shear localization and, thus, increases the total cohesive fracture energy.

The two distinct tearing mechanisms observed in ductile metal plates are the void-by-void advance of the crack tip, and the simultaneous interaction of multiple voids on the plane ahead of the crack tip. Void-by-void crack advance, which leads to a cup-cup crack surface morphology, is the dominant mechanism if the plate contains a low number of small void nucleation sites (i.e., second phase particles). Conversely, a large number and/or size of nucleation sites trigger the simultaneous interaction of multiple voids resulting in a slanted crack. The present work aims to provide further insight into the parameters controlling the mechanisms and energy dissipation of plate tearing by focusing on the shape and, thereby, the orientation of the nucleation sites. The study uses a two-dimensional plane strain finite element domain to model the cross section of a plate, subject to mode I tearing, with discretely modeled, randomly distributed, finite-sized elliptic void nucleation sites. The developed finite element setup can capture the dependence of the crack surface morphology on the microstructure of the plate. The simulation results confirm that cup-cup crack propagation develops by intense plastic straining throughout the thinning region of the plate. Conversely, slanted and cup-cone cracks propagate in thin localized shear deformation bands. The energy dissipation is, therefore, greater for cup-cup cracks. The study demonstrates that the damage-related microstructure has a significant role in determining the overall hardening capacity of a plate, which in turn dictates the tearing mode and energy.

A novel numerical framework for analyzing self-similar problems in plasticity is developed and demonstrated. Self-similar problems of this kind include processes such as stationary cracks, void growth, indentation etc. The proposed technique offers a simple and efficient method for handling this class of complex problems by avoiding issues related to traditional Lagrangian procedures. Moreover, the proposed technique allows for focusing the mesh in the region of interest. In the present paper, the technique is exploited to analyze the well-known wedge indentation problem of an elastic–viscoplastic single crystal. However, the framework may be readily adapted to any constitutive law of interest. The main focus herein is the development of the self-similar framework, while the indentation study serves primarily as verification of the technique by comparing to existing numerical and analytical studies. In this study, the three most common metal crystal structures will be investigated, namely the face-centered cubic (FCC), body-centered cubic (BCC), and hexagonal close packed (HCP) crystal structures, where the stress and slip rate fields around the moving contact point singularity are presented.

Void coalescence at severe shear deformation has been studied intensively under monotonic loading conditions, and the sequence of micro-mechanisms that governs failure has been demonstrated to involve collapse, rotation, and elongation of existing voids. Under intense shearing, the voids are flattened, such that the void volume diminishes, whereafter the flattened crack-like voids rotate and elongate until interaction with neighboring micro-voids dominates the material response and coalescence sets in. Eventually, this leads to a complete loss of load carrying capacity. The severe shear loading, imposed at the far boundary, is in an early state of the deformation associated with significant stretching of parts of the void surface, while other parts remain practically un-deformed. A largely uneven distribution of the strain hardening, therefore, evolves along the void circumference and, thus, one cannot expect the void to return to its original shape in the case where the far-field loading is reversed. The present numerical work aims to investigate the evolution of micro-voids subject to constant tension and large amplitude cyclic shearing. The far-field loading, the void shape, and the void growth are monitored, and the calculations are pushed to coalescence and complete loss of load carrying capacity. The initially circular cylindrical voids are predicted to develop protrusions in the shearing plane with normal in the direction of the applied tensile load. These protrusions evolve during repeated cyclic shearing and spread towards neighboring voids - eventually being responsible for void coalescence.

Crack tip flipping, where the fracture surface alternates from side to side in roughly 45° shear bands, seems to be an overlooked propagation mode in Mode I thin sheet tearing. In fact, observations of crack tip flipping is rarely found in the literature. Unlike the already established modes such as slanting, cup-cone (rooftop), or cup-cup (bathtub) the flipping crack never settles in a steady-state as the near tip stress/strain field continuously change when the flip successively initiates and develops shear-lips. A recent experimental investigation has revealed new insight by exploiting 3D X-ray tomography scanning of a developing crack tip flip. But, it remains to be understood what makes the crack flip systematically, what sets the flipping frequency, and under which material conditions this mode occurs. The present study aims at investigating the idea that a slight out-of-plane action (Mode III type loading) on the tip of a slant Mode I crack can provoke it to flip to the opposite side. Both experiments and micro-mechanics based modeling support this hypothesis.

The present work investigates a new approach to formulating a rate-independent strain gradient theory for crystal plasticity. The approach takes as offset recent discussions published in the literature for isotropic plasticity, and a key ingredient of the present work is the manner in which a gradient enhanced effective slip measure governs hardening evolution. The effect of both plastic strains and plastic strain gradients are combined into this scalar effective slip quantity, the energy associated with plastic strain is dissipative (unrecoverable), while the energy from plastic strain gradients is recoverable (free). The framework developed forms the basis of a finite element implementation and is demonstrated on benchmark problems designed to bring out effects such as strengthening and hardening. Monotonic loading and plane strain deformation is assumed throughout, but despite this, non-proportional straining is predicted in the plastic regime even under pure shear conditions. Results of single slip and symmetric double slip reveal that strengthening and hardening are governed by the slip system orientation and the material length parameter only.

Simulation techniques such as Finite Element Analysis (FEA) have substantially contributed to reducing the time and cost of developing new mechanical products over the last decades. However, FEA is primarily carried out on the nominal geometry, i.e. not considering the variation in geometrical and material parameters that originates from either the production or use-cases. This can lead to either overly conservative or overly optimistic designs, which in turn leads to unnecessary cost and/or unforeseen product failures. Despite the introduction of commercial FEA tool packages (such as 3DX/Abaqus/ANSYS etc.) that enables designers and engineers to combine FEA and Robust Design (RD) methods, performing sensitivity studies to identify robustness issues is rarely conducted in a systematic manner. The objective of this study is to identify the barriers for why the combined use of FEA and robust design techniques are not a standard part of the design process. The study concludes on identified barriers and that combining FEA and RD will require a substantial effort, however create significant value and enhance the design process.

The ductile fracture process in porous metals due to growth and coalescence of micron scale voids is affected not only by the imposed stress state but also by the distribution of the voids and the material size effect. The objective of this study is to understand the interaction of the inter-void spacing (or ligaments) and the resultant gradient-induced material size effect on void coalescence for a range of imposed stress states. To this end, three-dimensional finite element calculations of unit cell models with a discrete void embedded in a strain gradient-enhanced material matrix are performed. The calculations are carried out for a range of initial inter-void ligament sizes and imposed stress states characterized by fixed values of the stress triaxiality and the Lode parameter. Our results show that in the absence of strain gradient effects on the material response, decreasing the inter-void ligament size results in an increase in the propensity for void coalescence. However, in a strain gradient-enhanced material matrix, the strain gradients harden the material in the inter-void ligament and decrease the effect of inter-void ligament size on the propensity for void coalescence.

The objective of this paper is to investigate the ductile fracture behavior of thin metal plates under nominally plane strain conditions, with an emphasis on the differences between far-field plane strain tension and plane strain bending. Recent experimental results show that the fracture strain obtained using far-field plane strain tension tests is significantly lower than that obtained under plane strain bending conditions. Investigating the source of these differences is of critical importance for multiple industries since it affects formability, crashworthiness and ductile failure of thin metal plates. The present work provides critical insights into this issue based on the results of a detailed numerical investigation of both tests to failure, using the micro-mechanics based Gurson material model. The numerical simulations reveal many similarities between plane strain bending and far-field plane strain tension until the onset of localization. In the far-field plane strain tension case, after Considere thinning, a through-thickness shear localization occurs, followed by a micro-crack formation in the center of the plate. This is accompanied by an increase of the stress triaxiality well above the nominal plane strain tension level (i.e., ~ 0.57 for thin plates). In contrast, the absence of thinning in plane strain bending yields a stress triaxiality that equals nominal plane strain tension, which results in a significantly lower void growth rate (in reference to far-field plane strain tension). A surface instability (undulations) eventually forms on the tension side of the plate in plane strain bending, followed by multiple shear bands. The crack initiates inside the dominant shear band and propagates from the surface into the plate. These differences in the localization mechanisms and local stress states are responsible for the difference in the macroscopic fracture strain obtained under far-field plane strain tension and plane strain bending.

In metal plates, the crack propagation mechanism sets the amount of the plastic deformation before failure: a slanted or a cup–cone crack typically yields limited plate thinning within the fracture process zone, while large deformation precedes cup–cup crack propagation. The present work investigates the effect of intermetallic particles on the propagation mechanisms and the associated fracture surface morphologies when tearing Al 1050 plates under far-field mode I loading. Both single edge notched and double edge notched tension specimens, with thicknesses ranging from 0.5 to 5 mm, were tested. The chemical compositions of intermetallic particles were determined by performing energy dispersive X-ray measurements, and their morphological features were characterized by Scanning Electron Microscopy (SEM). Likewise, SEM images were taken to display the fracture surfaces, and the details of the surface morphology were visualized in three dimensions by using X-ray Tomography scanning. The experimental results indicate that an increase in the volume fraction, size, and aspect ratio of the intermetallic particles all promote slanted/cup–cone cracks, while a low amount of small, circular particles leads to cup–cup cracks. Furthermore, two-dimensional finite element simulations for mode I crack propagation support the experimental findings.

Size effects in metal plasticity are widely accepted, and different theoretical approaches to handle the phenomenon are developing in the literature. The present work considers the Fleck and Willis (2009b) framework, and creates a new gradient enhanced rate-independent crystal plasticity FE-implementation. The study considers both energetic and dissipative gradient hardening and strengthening, and adopts a general form of the gradient enhanced effective slip rate. Monotonic and cyclic shearing of an infinite crystal slab containing a single slip system at an angle of 90° to the loading direction is a first benchmark case. A second case considers combined shear and tension in 2D of a constrained HCP single crystal. The HCP crystal is loaded in its basal plane by a so-called butterfly deformation path that inflicts repeated loading and unloading of the three crystallographic slip systems. Finally, the evolution and interaction of multiple plastic zones are demonstrated by considering a notched tensile sample. A direct comparison to visco-plastic (rate-dependent) simulations confirms that the proposed crystal plasticity framework forms a rate-independent limit for the gradient enhanced Fleck–Willis theory. The model response also reduces to that of conventional crystal plasticity in the limit of zero length parameters.

A numerical framework for analyzing steady-state elastic–plastic material deformation at finite strains is developed and demonstrated in the present work. The framework is an extension of the original method by Dean and Hutchinson (1980) develop to analyze steady-state crack propagation, under a small strain assumption, in which the history-dependent material response is captured through streamline integration. Steady-state problems are encountered in numerous engineering processes, and the original studies of crack growth have already been extended to include rolling and drawing, though within a small strain framework. However, the investigation of such manufacturing processes, where strains greater than 10% easily develop, requires a finite strain formulation to provide accurate results. The framework proposed in the present work offers an efficient and accurate method to extract the steady-state solution at finite strains without encountering the numerical issues related to traditional Lagrangian procedures. Furthermore, the framework also accounts for elastic unloading compared to many existing numerical steady-state schemes as they are often restricted to rigid plasticity. The new numerical framework employs a hyperelastic material model, in terms of a Neo-Hookean material, combined with an isotropic viscoplastic material behavior. However, the framework is not limited to any specific hyperelastic material model, nor any specific model for the plastic behavior. The finite strain steady-state framework has been verified through comparison to a traditional Lagrangian analysis conducted in ANSYS. The benchmark case constitutes a plane strain drawing process where the thickness of the specimen is reduced by drawing it between two circular cylindrical tools.

Wire drawing processes at the micron scale have received increased interest as micro wires are increasingly required in electrical components. It is well-established that size effects due to large strain gradient effects play an important role at this scale and the present study aims to quantify these effects for the wire drawing process. Focus will be on investigating the impact of size effects on the most favourable tool geometry (in terms of minimizing the drawing force) for various conditions between the wire/tool interface. The numerical analysis is based on a steady-state framework that enables convergence without dealing with the transient regime, but still fully accounts for the history dependence as-well as the elastic unloading. Thus, it forms the basis for a comprehensive parameter study. During the deformation process in wire drawing, large plastic strain gradients evolve in the contact region. This creates a need for a higher order plasticity theory to accurately predict the material behaviour across the multiple scales involved. The present study reveals that the contribution from an energetic (recoverable) length parameter is limited, while the corresponding dissipative contribution dominates and tends to shift the drawing force to a higher level. As a direct consequence, the strain gradient hardening effect reduces the most favourable tool angle of a sharp tool with up to 50% (in terms of the required drawing force), whereas a circular shaped tool is proven less sensitive to scaling effects. By considering the contact force profile between tool and material it becomes clear that the strain gradients have a smoothing effect and both the magnitude and position of the peak pressure are affected significantly. A round tool is found to reduce the peak force, while the location of the peak is found to move from outlet to inlet depending on the tool geometry.

The fracture toughness for a mode I/II crack propagating in a ductile material has been subject to numerous investigations. However, the influence of the material hardening law has received very limited attention, with isotropic hardening being the default choice if cyclic loads are absent. The present work extends the existing studies of monotonic mode I/II steady-state crack propagation with the goal to compare the predictions from an isotropic hardening model with that of a kinematic hardening model. The work is conducted through a purpose-built steady-state framework that directly delivers the steady-state solution. In order to provide a fracture criterion, a cohesive zone model is adopted and embedded at the crack tip in the steady-state framework, while a control algorithm for the far-field, that significantly reduces the number of equilibrium iterations is employed to couple the far-field loading to the correct crack tip opening. Results show that the steady-state fracture toughness (shielding ratio) obtained for a kinematic hardening material is larger than for the corresponding isotropic hardening case. The difference between the isotropic and kinematic model is tied to the non-proportional loading conditions and reverse plasticity. This also explains the vanishing difference in the shielding ratio when considering mode II crack propagation as the non-proportional loading is less pronounced and the reverse plasticity is absent.

A new modeling framework for analyzing steady-state elastic–viscoplastic single crystal problems at micron scale is presented. The model takes as offset the higher-order gradient plasticity theory by Kuroda and Tvergaard (J Mech Phys Solids 54:1789–1810 2006), and it is strongly inspired by an existing steady-state framework for conventional elastic–viscoplastic materials. Details on the modeling framework are laid out, and the numerical challenges and stability issues are discussed. Subsequently, the modeling framework is demonstrated on Mode I steady-state crack growth in HCP single crystals, where the impact of size effects on the active plastic zone that surrounds the crack tip is investigated. The focus is on the plastic zone shape and size, as well as its influence on the macroscopic fracture toughness. In this way, the chosen benchmark problem for the new modeling framework serves as an extension to the conventional study of the corresponding problem without size effects presented in Juul et al. (J Mech Phys Solids 101:209–222 2017). The tendency is that the plastic zone is smeared out (seen as less distinct features) when increasing the material rate sensitivity. This is in-line with already published work in the literature. It is shown that the size effect has a limited effect on the qualitative features of the plastic zone, whereas it clearly suppresses the magnitude of the features due to the added gradient hardening effect. Moreover, the hardening effect tends to lower the shielding ratio, and hence a key result is that single crystal materials appear less crack resistant than conventional studies suggest at the micron scale.

The tearing of ductile metal plates can take place in three distinctly different modes: cup-cup, cup-cone, and crack slanting, but they are often observed in combination. It is well established that all tearing modes are governed by nucleation, growth, and coalescence of voids at the micron scale. What controls the shift between different tearing modes is, however, unclear and a micro-mechanics-based investigation is launched here to shed light on the issue. The present work talces as a starting point the hypothesis that the volume fraction, (average) size, and distribution of second phase particles, which act as void nucleation sites, are the key microstructural parameters that determine the tearing mode. In accordance with this hypothesis, the plates are modeled here by embedding randomly distributed void nucleation sites in a homogeneous matrix material. A parameter study is performed, and by adjusting the number, size and distribution of the nucleation sites in the fracture process zone, a shift in the tearing mode is achieved: a low number of small (relative to the plate thicicness) randomly distributed particles link up in a void-by-void-type failure, whereas bigger particles, or a large number of small particles, can facilitate multiple void interactions. The present work also demonstrates that, for plates with intermediate or low volume fraction of nucleation sites, the localization of deformation in a macroscopic band precedes the microscopic localization that eventually links the voids. However, the two modes of plastic flow localization occur simultaneously for large volume fractions., VILLUM FOUNDATION [VKR023451]

A numerical investigation of wedge indentation, with a nearly flat indenter, into a monazite (LaPO4) single crystal is carried out to obtain the asymptotic field solution associated with the moving contact point singularities. The crystal orientation is such that plane strain conditions prevail, under the assumption of small scale yielding, as out-of-plane deformations are eliminated due to out-of-plane mirror symmetry of the crystal, specimen and loading state. The plastic deformation in such a 2D study can be described in terms of effective in-plane slip systems comprised of crystallographic slip systems with equal and opposite out-of-plane deformation and rotation. The numerical simulations are conducted within a framework specialized for self-similar problems and adopts a visco-plastic single crystal material model. The detailed numerical investigation of the monazite single crystal reveals that the effective slip systems lead to a non-symmetric in-plane deformation field, which is consistent with the absence of in-plane mirror symmetries of the crystal. Interestingly, the non-symmetric deformation field results in one contact point singularity travelling at a greater speed than the other. The deformation near the moving contact point singularities are found to be divided into two angular sectors separated by a boundary of glide shear type. The slip rates on the individual systems reveal that one slip system dominates at both contact points, whereas the other slip system shows negligible activity. Thus, only one slip system gives rise to a discontinuity in the slip rate field.

The continuing demand on device miniaturization has motivated the interest in micro metal forming technologies for manufacturing metal parts with sub-mm characteristic dimensions. In this work, axisymmetric reverse extrusion experiments were conducted on Cu 110 and Al 1100 alloy rod specimens with widely varying initial grain sizes, achieved through equal channel angular pressing (ECAP) and post annealing. Accompanying crystal plasticity finite element (CPFE) simulations were carried out. At a characteristic extrusion dimension (CED) of ∼100 µm, defined by the sidewall thickness of the extruded cup-shaped structures, the deformation characteristics of the reverse extrusion process were investigated. Specifically, the characteristic plastic strain for extrusion and the influence of initial grain size on the extrusion mechanical response and shape of extruded parts were examined in detail through scanning electron microscopy (SEM) and electron backscatter diffraction (EBSD), in combination with CPFE simulations. The mechanical response of extrusion showed deviations from continuum scaling behavior as the CED became small as compared to the initial grain size. The present study serves as a baseline for further studies of metal forming at the meso to micro scales.

The size effect in wedge indentation of an FCC single crystal is investigated. Conventional plasticity fails to describe the mechanical response of crystalline materials at the micron level due to the accumulation of Geometrically Necessary Dislocations (GNDs) which experiments have shown to introduce a size-dependent increase in the apparent yield stress and subsequent hardening. GND-densities scale with the gradient of plastic deformation and their effect on the mechanical response is here modelled by adopting a dissipative strain gradient single-crystal plasticity theory. Numerical solutions to the classical wedge indentation problem are obtained from a purpose-built Finite Element model accounting for finite strains. A special 2D plane strain set-up, with three effective in-plane slip systems, is adopted to comply with state-of-the-art experimental results. The indentation process is modelled for a nearly flat wedge as well as a wedge with an included angle of 90 ∘ . The distribution of slip and the GND-densities are investigated and compared to conventional plasticity predictions.

Ductile crack initiation, governed by void growth to coalescence, is controlled by two distinct mechanisms, namely: (i) void-by-void growth, where substantial growth of the void closest to the crack tip precedes growth of other voids, and (ii) multiple void interaction, where the voids grow simultaneously and interact in the vicinity of the crack tip prior to crack advance. Tvergaard and Hutchinson (2002) [Int. J. Solids and Struct, 39: 3581-3597] studied the two mechanisms in detail under far-field mode I loading conditions and demonstrated their interference by changing the size and spacing of discretely modeled voids. The present numerical work focuses on mixed mode loading conditions rather than limiting the study to mode I and takes up the question; how will a change in the far-field loading conditions affect the shift between the two mechanisms? A pre-existing straight crack with nearby discretely modeled voids in an elastic-plastic material is considered by a specialized 2D plane strain setting capable of out-of-plane displacements to allow for combinations of mode I, mode II, and mode III. Details of the void-by-void growth versus the multiple void interaction mechanisms are laid out for a range of load cases and geometrical void configurations, and increasing the mode mixity is found to favor void-by-void growth accompanied by a higher load intensity required on the far-field boundary to initiate the crack. The change between the two mechanisms is tied to a change in the deformation field that surrounds the voids, and the study reveals significant deformation that stretches above (and below) the row of discrete voids for increasing shear mode contributions yielding an increase in the void rotation.

Published experimental measurements on deformed metal crystals show distinct pattern formation, in which dislocations are arranged in wall and cell structures. The distribution of dislocations is highly non-uniform, which produces discontinuities in the lattice rotations. Modeling the experimentally observed micro-structural behavior, within a framework based on continuous field quantities, poses obvious challenges, since the evolution of dislocation structures is inherently a discrete and discontinuous process. This challenge, in particular, motivates the present study, and the aim is to improve the micro-structural response predicted using strain gradient crystal plasticity within a continuum mechanics framework. One approach to modeling the dislocation structures observed is through a back stress formulation, which can be related directly to the strain gradient energy. The present work offers an investigation of constitutive equations for the back stress based on both considerations of the gradient energy, but also includes results obtained from a purely phenomenological starting point. The influence of model parameters is brought out in a parametric study, and it is demonstrated how a proper treatment of the back stress enables dislocation wall and cell structure type response in the adopted framework.

Two rate-independent strain gradient crystal plasticity models, one new and one previously published, are compared and a numerical framework that encompasses both is developed. The model previously published is briefly outlined, while an in-depth description is given for the new, yet somewhat related, model. The difference between the two models is found in the definitions of the plastic work expended in the material and their relation to spatial gradients of plastic strains. The model predictions are highly relevant to the ongoing discussion in the literature, concerning 1) what governs the increase in the apparent yield stress due to strain gradients (also referred to as strengthening)? And 2), what is the implication of such strengthening in relation to crystalline material behavior at the micron scale? The present work characterizes material behavior, and the corresponding plastic slip evolution, by use of the finite element method. The pure shear problem of an infinite material slab is investigated, with the previously published model displaying strengthening, while the new model does not. In addition to the numerical approach an exact closed form solution, to the pure shear problem, is obtained for the new model, and it is demonstrated that the model predicts proportional straining in the entire plastic regime. Somewhat surprising it is found that the predictions for strain gradient hardening coincide for the two models.

Experiments on soft polycrystalline aluminum have yielded evidence that, besides the required punch load, both the size and shape of imprinted features are affected by the scale of the set-up, e.g. substantial details are lost when the characteristic length is on the order of 10 μ m . The objective of this work is to clarify the role played by strain gradients on this issue, and to shed light on the underlying mechanisms. For this, indentation by a periodic array of flat punch indenters is considered, and a gradient enhanced material model that allows for a numerical investigation of the fundamentals are employed. During a largely non-homogeneous deformation, the material is forced up in between the indenters so that an array of identical imprinted features is formed once the tool is retreated. It is confirmed that the additional hardening owing to plastic strain gradients severely affects both the size and shape of these imprinted features. In particular, this is tied to a large increase in the mean contact pressure underneath the punch, which gives rise to significant elastic spring-back effects during unloading.

A numerical model formulation of the higher order flow theory (rate-independent) by Fleck and Willis [2009. A mathematical basis for strain-gradient plasticity theory – part II: tensorial plastic multiplier. Journal of the Mechanics and Physics of Solids 57, 1045-1057.], that allows for elastic–plastic loading/unloading and the interaction of multiple plastic zones, is proposed. The predicted model response is compared to the corresponding rate-dependent version of visco-plastic origin, and coinciding results are obtained in the limit of small strain-rate sensitivity. First, (i) the evolution of a single plastic zone is analyzed to illustrate the agreement with earlier published results, whereafter examples of (ii) multiple plastic zone interaction, and (iii) elastic–plastic loading/unloading are presented. Here, the simple shear problem of an infinite slab constrained between rigid plates is considered, and the effect of strain gradients, strain hardening and rate sensitivity is brought out. For clarity of results, a 1D model is constructed following a procedure suitable for generalization to 2D and 3D.

Cracked ductile sheet metals, subject to Mode I tearing, have been observed to display a variety of fracture surface morphologies depending on the material properties, and a range of studies on the fracture surface appearance have been published in the literature. Whereas classical fractures such as cup-cone, cup–cup and slanting are widely observed, the phenomenon of a slanted crack which systematically “flips” back and forth in a roughly 45° orientation has only recently been reported. The present study aims to add details and understanding to this crack growth phenomenon – through experimental testing and comparison with published results. A series of crack propagation tests have been carried out, where cracks are driven many plate thicknesses under global Mode I loading. The current study employs both the edge crack specimen (ECS) loaded in combined in-plane bending and extension, and the double edge notched tension (DENT) specimen loaded in extension. Clear evidence of the alternating slant failure is observed for a normal strength steel and details on the phenomenon are brought out by grinding and polishing cross-sections in successive steps along the crack growth direction. Moreover, these results are compared to a soft aluminum (1050A) which predominantly shows cup–cup or slant failure. In general, both materials fail by the mechanism of void growth to coalescence (fractographs are included), and evidence points toward cup–cup being favored by materials that experience extensive amounts of void growth (thus severe thinning in the process zone), whereas slanting is typical in high strength materials that rapidly nucleate large populations of smaller voids.

The lay-out of a numerical solution procedure for the strain gradient flow (rate-independent) theory by Fleck and Willis [A mathematical basis for strain-gradient theory – Part II: Tensorial plastic multiplier, 57:1045–1057; 2009, JMPS] has been an open issue, and its finite element implementation is yet to be completed. Only recently, a sound solution procedure that allows for elastic–plastic loading/unloading and the interaction of multiple plastic zones has been put forward, and demonstrated within a 1D model set-up. The aim of the present work is to extend this procedure to form a basis for 2D and 3D calculations, and thereby to broaden its use within engineering applications. Focus is on the numerical implementation that adopts image analysis techniques to identify individual plastic zones and to treat none-regular mesh configurations. The developed finite element model is demonstrated through; i) tensile loading of a finite homogeneous material slab that offers a simple interpretation of results, ii) tensile loading of a double edge notch tension specimen involving symmetry considerations, and iii) shearing of a periodically voided structure that displays the ability of the procedure to treat periodic boundary conditions. A comparison between the implemented flow theory model and the corresponding rate-dependent visco-plastic version of the model shows coinciding results in the limit of zero rate-sensitivity.

Conclusive insight to the mechanics that govern so-called “crack tip flipping”remains to be revealed, but details continue to fall into place as researcher dig deeper. The work presents an overview of the latest findings and the next steps to be made.

Elastic and elastic-plastic results are obtained for a semi-infinite slanted through-crack propagating in a symmetrically loaded plate strip with the aim of providing theoretical background to commonly observed plate tearing behavior. Were it is not for the slant of the crack through the thickness of the plate, the problem would be mode I, but due to the slant the local conditions along the crack front are a combination of mode I and mode III. A three-dimensional formulation for steady-state crack propagation is employed to generate distributions of effective stress, stress triaxiality and Lode parameter through the plate in the plastic zone at the crack tip. The distribution of the mode I and mode III stress intensity factors along the crack front are obtained for the elastic problem. The out-of-plane bending constraint imposed on the plate significantly influences the mixed mode behavior along the crack front. The solution is examined for clues as to why propagating slant cracks sometimes undergo a transition and flip about 90° to propagate with the opposite slant orientation.

The active plastic zone that surrounds the tip of a sharp crack growing under plane strain Mode I loading conditions at a constant velocity in a single crystal is studied. Both the characteristics of the plastic zone and its effect on the macroscopic toughness is investigated in terms of crack tip shielding due to plasticity (quantified by employing the Suo, Shih, and Varias set-up). Three single crystals (FCC, BCC, HCP) are modelled in a steady-state elastic visco-plastic framework, with emphasis on the influence of rate-sensitivity and crystal structures. Distinct velocity discontinuities at the crack tip predicted by Rice [Rice J.R., 1987. Tensile crack tip fields in elastic-ideally plastic crystals. Mech. Mater. 6, pp. 317–335] for quasi-static crack growth are confirmed through the numerical simulations and highly refined details are revealed. Through a detailed study, it is demonstrated that the largest shielding effect develops in HCP crystals, while the lowest shielding exists for FCC crystals. Rate-sensitivity is found to affect the plastic zone size, but the characteristics overall remain similar for each individual crystal structure. An increasing rate-sensitivity at low crack velocities monotonically increases the crack tip shielding, whereas the opposite behaviour is observed at high velocities. This observation leads to the existence of a characteristic velocity at which the crack tip shielding becomes independent of the rate-sensitivity.

A new numerical framework specialized for analyzing self-similar problems in plasticity is developed. Self-similarity in plasticity is encountered in a number of different problems such as stationary cracks, void growth, indentation etc. To date, such problems are handled by traditional Lagrangian procedures that may be associated with severe numerical difficulties relating to sufficient discretization, moving contact points, etc. In the present work, self-similarity is exploited to construct the numerical framework that offers a simple and efficient method to handle self-similar problems in history dependent materials. The procedure allows for focusing the mesh only in regions of interest giving highly detailed results in fractions of the time compared to traditional frameworks. The framework is not limited to a specific constitutive law and may be applied to a wide range of material models. The technique is here applied to wedge indentation in elastic-viscoplastic single crystals.The three most common metal structures are investigated, namely the FCC, BCC, and HCP crystal structures, where the slip rate fields and stress fields will be compared to analytical predictions [1][2] and traditional numerical simulations [3] when possible. To mimic the condition for the analytical predictions, the wedge indenter is considered nearly flat and the material is perfectly plastic with a very low yield strain. Under these conditions, [1][2] proved analytically the existence of discontinuities in the slip rate field. The numerical simulations reveal a striking match to the analytical prediction showing the expected discontinuities in the slip rate field. In addition, the current results provide much more detailed views of the stress and slip rate fields than previously obtained. The results are all obtained without encountering many of the issues related to the traditional procedures and guarantees that it is indeed the self-similar solution that has been found.

The fracture surface morphology that results from mode I tearing of ductile plate metals depends heavily on both the elastic-plastic material properties and the microstructure. Severe tunneling of the advancing crack tip (resulting in cup-cup, or bath-tub like fracture surfaces) can take place in a range of materials, often of low strength, while tearing of high strength metals typically progress by the shear band failure mechanism (slanting). In reality, however, most fracture surfaces display a mixture of morphologies. For example, slant crack propagation can be accompanied by large shear lips near the outer free plate surface or a complete shear band switch - seemingly distributed randomly on the fracture surface. The occasionally observed shear band switch of mode I slant cracks, related to ductile plate tearing, is far from random as the crack can flip systematically from one side to the other in roughly 45-degree shear bands. This "flipping" action of a slanted crack remains to be fully understood, and the present study serves to share details on the phenomenon by exploiting X-ray tomography scanning to access the plate interior and the very crack tip. Throughout, the focus is on a crack tip where the flip is underway. Extensive growth of single edge cracks under mode I loading is achieved in a purpose build test set-up. Here, considering a 4 mm plate of normal strength / high strain hardening steel which has been found to display successive flipping of the slant crack face. While undergoing a shear band switch, such that the flipping mechanism is active, the plate tearing test is interrupted and the crack tip extracted for further investigation. The conducted X-ray tomography scans reveal the failure process ahead of the advancing crack tip to resemble the ductile slant crack growth governed by local thinning and moderate crack tip tunneling. However, small shear lips form at the outer free plate surface, well behind the 45-degree slant (tunneling) crack tip, as the flipping action engages. Upon further loading, the shear lips subsequently grow to form a set of secondary crack fronts at an angle to the primary tunneling slant crack. Eventually, these secondary crack fronts catch up on the primary slant crack front and overtake the growth to complete the shear band switch. Once the crack slants, an out-of-plane action occurs due to the loss of symmetry in the system. It is this out-of-plane action which is believed to set-off the flipping mechanism.

Steady state crack propagation produce substantial plastic strain gradients near the tip, which are accompanied by a high density of geometrically necessary dislocations and additional local strain hardening. Here, the objective is to study these gradient effects on Mode I toughness of a homogeneous rate-sensitive metal, using a higher order plasticity theory. Throughout, emphasis is on the toughness rate-sensitivity, as a recent numerical study of a conventional material (no gradient effects) has indicated a significant influence of both strain rate hardening and crack tip velocity. Moreover, a characteristic velocity, at which the toughness becomes independent of the rate-sensitivity, has been observed. It is the aim to bring forward a similar characteristic velocity for the current strain gradient visco-plastic model, as-well as to signify its use in future visco-plastic material modeling.

The failure process ahead of a mode I crack advancing in a ductile thin metal plate or sheet produces plastic dissipation through a sequence of deformation steps that include necking well ahead of the crack tip and shear localization followed by a slant fracture in the necked region somewhat closer to the tip. The objective of this paper is to analyze this sequential process to characterize the traction–separation behavior and the associated effective cohesive fracture energy of the entire failure process. The emphasis is on what is often described as plane stress behavior taking place after the crack tip has advanced a distance of one or two plate thicknesses. Traction–separation laws are an essential component of finite element methods currently under development for analyzing fracture of large scale plate or shell structures. The present study resolves the sequence of failure details using the Gurson constitutive law based on the micromechanics of the ductile fracture process, including a recent extension that accounts for damage growth in shear. The fracture process in front of an advancing crack, subject to overall mode I loading, is approximated by a 2D plane strain finite element model, which allows for an intensive study of the parameters influencing local necking, shear localization and the final slant failure. The deformation history relevant to a cohesive zone for a large scale model is identified and the traction–separation relation is determined, including the dissipated energy. For ductile structural materials, the dissipation generated during necking prior to the onset of shear localization is the dominant contribution; it scales with the plate thickness and is mesh-independent in the present numerical model. The energy associated with the shear localization and fracture is secondary; it scales with the width of the shear band, and inherits the finite element mesh dependency of the Gurson model. The cohesive traction–separation laws have been characterized for various material conditions.

The post-welding stress state, strain history and material conditions of friction stir welded joints are often strongly idealized when used in subsequent modeling analyses, typically by neglecting one or more of the features above. But, it is obvious that the conditions after welding do influence the weld performance. The objective of this paper is to discuss some of the main conflicts that arise when taking both the post-welding material conditions and stress–strain state into account in a subsequent structural analysis. The discussion is here based on a preliminary numerical study of the possible effect of the post-welding conditions when subjecting a friction stir weld to loading transverse to the weld line. The numerical model of the friction stir welded joint, employs a step-wise modeling approach to combine an in-situ weld simulation with a post-welding failure analysis. Using the commercial software ANSYS, a thermo-mechanical model is employed to predict the thermally induced stresses and strains during welding, while an in-house finite element code is used to study the plastic flow localization and failure in a subsequent structural analysis. The coupling between the two models is made by mapping the post-welding stress–strain conditions predicted in ANSYS to the in-house code by using re-meshing techniques. The study indicates a noticeable influence of the post-welding stress–strain condition when subjecting the welded plate to tension. E.g. an increase of the tensile curve prior to plastic flow localization was observed, with a substantial influence on the specimen elongation at the onset of localization and thereby failure. This influence is, however, shown to be strongly affected by the applied boundary conditions. Specimens cut from the welded plate, transverse to the weld line, showed the largest influence of the post-welding conditions, even though significant relaxation of the residual stress state was predicted.

Crack propagation in metallic materials produces plastic dissipation when material in front for the crack tip enters the active plastic zone traveling with the tip, and later ends up being part of the residual plastic strain wake. Thus, the macroscopic work required to advance the crack is typically much larger than the work needed in the near tip fracture process. For rate sensitive materials, the amount of plastic dissipation typically depends on the rate at which the material is deformed. A dependency on the crack velocity should therefore be expected. The objective of this paper is to study the macroscopic toughness of crack advance along an interface joining two dissimilar rate dependent materials, characterized by an elastic-viscoplastic material model that approaches the response of a J2-flow material in the rate independent limit. The emphasis here is on the rate sensitivity of the macroscopic fracture toughness under mixed Mode I/II loading. Moreover, special cases of joined similar rate dependent materials, as well as dissimilar materials where one substrate remains either elastic or approaches the rate independent limit is also included. The numerical analysis is carried out using the SSV model [Suo, Z., Shih, C., Varias, A., 1993. A theory for cleavage cracking in the presence of plastic flow. Acta Metall. Mater. 41, 1551–1557] embedded in a steady state finite element formulation, here assuming plane strain conditions and small-scale yielding. Results are presented for a wide range of material parameters, including noteworthy observations of a characteristic crack velocity at which the macroscopic toughness becomes independent of the material rate sensitivity. The potential of this phenomenon is elaborated on from a modeling point of view.

Failure under intense shearing at close to zero stress triaxiality is widely observed for ductile metallic materials, and is identified in experiments as smeared-out dimples on the fracture surface. Numerical cell-model studies of equal sized voids have revealed that the mechanism governing this shear failure mode boils down to the interaction between primary voids which rotate and elongate until coalescence occurs under severe plastic deformation of the internal ligaments. The objective of this paper is to analyze this failure mechanism of primary voids and to study the effect of smaller secondary damage that co-exists with or nucleation in the ligaments between larger voids that coalesce during intense shearing. A numerical cell-model study is carried out to gain a parametric understanding of the overall material response for different initial conditions of the two void populations, subject to shear dominated loading. To account for both length scales involved in this study, a continuum model that includes the softening effect of damage evolution in shear is used to represent the matrix material surrounding the primary voids. Here, a recently extended Gurson-type model is used, which represents the effect of the small secondary voids under the low triaxiality loading conditions considered. This work suggests a failure mechanism for materials that contain voids on two different length scales, subject to intense shearing, in terms of; (i) the interaction of the primary voids, and (ii) the material softening of the ligaments due to the evolution of secondary damage. It is found that coalescence of primary voids under shear loading is severely affected by the presence of smaller secondary voids or defects in the ligaments. The change in overall ductility is presented for a wide range of initial material conditions, and an empirical correlation with the peak load is reported.

The strain hardening and damage behaviour of isothermally heat treated 6005A aluminium is investigated in order to link the thermal treatment conditions, microstructure and fracture strain. The need for a plastic flow rule involving a stage IV hardening at large strain was found essential to generate quantitative predictions when using an enhanced Gurson type damage model for the material behaviour. This model relies on an explicit description of the three stages of nucleation, growth and coalescence of voids, involving void shape effects. An implementation within a 3D finite element code allows for the simulation of the full tensile response curves up to cracking initiation and final failure. Reasonable agreement is found when comparing the predicted and experimentally measured fracture strains for a wide range of heat treatment conditions using real microstructure based parameters. (C) 2010 Elsevier Ltd. All rights reserved.

Gurson type constitutive models that account for void growth to coalescence are not able to describe ductile fracture in simple shear, where there is no hydrostatic tension in the material. But recent micro-mechanical studies have shown that in shear the voids are flattened out to micro-cracks, which rotate and elongate until interaction with neighbouring micro-cracks gives coalescence. Thus, the failure mechanism is very different from that under tensile loading. Also, the Gurson model has recently been extended to describe failure in shear, by adding a damage term to the expression for the growth of the void volume fraction, and it has been shown that this extended model can represent experimental observations. Here, numerical studies are carried out to compare predictions of the shear-extended Gurson model with the shear failures predicted by the micro-mechanical cell model. Both models show a strong dependence on the level of hydrostatic tension. Even though the reason for this pressure dependence is different in the two models, as the shear-extended Gurson model does not describe voids flattening out and the associated failure mechanism by micro-cracks interacting with neighbouring micro-cracks, it is shown that the trends of the predictions are in good agreement.

For resistance spot welded shear-lab specimens, interfacial failure under ductile shearing or ductile plug failure are analyzed numerically, using a shear modified Gurson model. The interfacial shear failure occurs under very low stress triaxiality, where the original Gurson model would predict void nucleation and very limited void growth. Void coalescence would therefore be largely postponed. However, using the shear modification of the Gurson model, recently introduced by Nahshon and Hutchinson (2008) [1] , failure prediction is possible at zero or even negative mean stress. Since, this shear modification has too large effect in some cases where the stress triaxiality is rather high, an extension is proposed in the present study to better represent the damage development at moderate to high stress triaxiality, which is known to be well described by the Gurson model. Failure prediction and tensile response curves for an interfacial shear failure or a ductile plug failure, are here compared when using either the original Gurson model, the shear modified model, or the extension to the shear modified model. The suggested extension makes it possible to use the shear modified model as a simple way of accounting for damage development under low triaxiality shearing, without further increasing the damage rate in regions of moderate to high stress triaxiality.

The plug failure modes of resistance spot welded shear-lab and cross-tension test specimens are studied, using recent extensions to the Gurson model. A comparison of the predicted mechanical response is presented when using either: (i) the Gurson–Tvergaard–Needleman model (GTN-model), (ii) the shear-modified GTN-model by Nahshon and Hutchinson that also describes damage development at low triaxiality (NH-model) or (iii) the Gologanu–Leblond–Devaux model (GLD-model) accounting for non-spherical void growth. The failure responses predicted by the various models are discussed in relation to their approximate description of the nucleation, growth and coalescence of microvoids. Using the void shape factor of the GLD-model, a simple approach for approximating void nucleation by either particle fracture or particle–matrix decohesion is applied and a study of the subsequent void shape evolution is presented. The models are applied to predict failure of specimens containing a fully intact weld nugget as well as a partly removed weld nugget to address the problems of shrinkage voids or larger weld defects. All analysis are carried out by full 3D finite element modelling.

A friction stir weld in 6005A-T6 aluminum alloy has been prepared and analyzed by micro-hardness measurements, tensile testing and scanning electron microscopy (SEM). The locations of the various weld zones were determined by micro-hardness indentation measurements. The flow behavior of the various zones of the weld was extracted using micro-tensile specimens cut out parallel to the welding direction. The measured material properties and weld topology were then introduced in a fully coupled micro-mechanical finite element model, accounting for nucleation and growth of voids as well as void shape evolution. The model shows satisfactory preliminary results in predicting the tensile behaviour of the weld and the true strain at fracture.

For a friction stir welded aluminum plate the resistance to ductile failure is studied by analyzing tensile test specimens cut out across the weldline. As the stress triaxiality is rather low in these tests, the Gurson material model is not expected to give a very accurate description of the void growth to coalescence. A recently proposed modified version of the Gurson model is used, in which an extra term in the damage evolution law allows for the prediction of failure even at zero or negative values of the mean stress. This modification of the Gurson model is purely phenomenological, such that the damage parameter does not really represent the void volume fraction. Various amounts of the additional damage evolution are compared with predictions of the original Gurson model. The analyses are carried out for different yield stress profiles transverse to the weld and for different specimen widths. It is found that the modification does provide additional damage development in the friction stir weld, which may help to fit experimental data. But the suggested modification depends strongly on the overall stress state, and may have a too strong effect in some cases where the stress triaxiality is rather high.