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The prediction of crack initiation, propagation, and ductile fracture in a large scale metallic structure has posed a great challenge to the design and certification community. The complexity in component geometry, material heterogeneity, and the 3D stress distribution, will likely make the crack growth curvilinear. An accurate 3D stress prediction is essential to predict the ductile fracture that is controlled by void nucleation, growth and coalescence. To alleviate the computational burden associated with the use of a 3D solid finite element modeling coupled with a remeshing of the ductile failure prediction of a large scale structure, a novel approach based on the coupling of an extended finite element for shell elements and a plane strain core characterization of a cracked region is developed to efficiently simulate an arbitrary crack growth in a large scale thin-walled structure and its associated load deflection curve while retaining a sufficient level of computational efficiency. The methodology is implemented in Abaqus’ explicit solver via its VUEL where a kinematic description of a cracked shell is accomplished via the use of phantom-paired elements. A plane strain core that controls the mixture of plane stress and plane strain components is introduced for a rational representation of localized plasticity induced 3D stress state for the prediction of a ductile failure initiation. A cohesive injection is applied on the newly created crack surface to dissipate the energy during the crack propagation. The resulting XSHELL toolkit for Abaqus is applied for the ductile failure prediction of the 2012 Sandia fracture challenge problem. A blind prediction is performed first using geometry independent plane strain core parameters followed by a refined analysis based on a set of well-calibrated material and plane strain core parameters from Sandia’s tensile and compact testing data.

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... The XSHELL toolkit has been applied recently for the modeling and simulation of 2012 Sandia fracture challenge problem (SFC1) (Zhang et al. 2014). XFEM based XSHELL toolkit features the kinematic representation of a cracked shell via its phantom paired elements, crack initiation prediction using accumulated plastic strain criterion, and mesh independent crack insertion through a cracked shell. ...

... In order to characterize the 3D stress state in the vicinity of a moving crack tip, a plane strain core model in XSHELL was employed to predict the fracture pattern and load deflection curves for the SFC2 problem at the loading rate of 0.0254 and 25.4 mm/s. The detailed theory and implementation of the plane strain core model in XSHELL is described by Zhang et al. (2014). A comprehensive description of the specimen geometry and testing set up of round bars and V-notch shear specimens, and the SFC2 specimen can be found in by Boyce et al. (2016). ...

... (1) to justify the use of a 2D shell model for an accurate response prediction of the thin sheet component with multiple stress concentrators; (2) to select the plane Fig. 2 a FEA mesh of the tensile specimen; b stress strain curves at different strain rates strain modeling scheme based on either a thickness correction factor (conventional plane strain core method) or a local tip zone correction (local plane strain core method) (Zhang et al. 2014). To accomplish this, multiple elements are used in the through the thickness direction in the 3D model and the mesh of the 2D shell has the same configuration of the in-plane mesh of the 3D model. ...

A phantom paired solid shell toolkit for Abaqus explicit solver (XSHELL) coupled with accumulated plastic strain and maximum plastic strain direction is applied for the blind prediction and recalibrated analysis of the ductile rupture of a Ti–6Al–4V sheet under both quasi-static and modest-dynamic loading. The complexities of this second Sandia challenge problem include the rate dependent and anisotropic material behaviour and co-existence of multiple round notches and holes to activate a combined tensile and shear driven failure behaviour. Given the nature of this thin-walled component, both the response and kinematic description of damage initiation and propagation can be captured using a relatively coarse model with degrees of freedom in the order of 30,000. A simplified modelling strategy is employed during the blind prediction without considering the material anisotropy, shear failure data, and free rotation at its pin connections. While both the crack path and critical load at crack initiation and propagation can be predicted reasonably well especially for the quasi-static case, its initial stiffness and peak load has a deviation from the test data. In order to capture the anisotropic material behaviour, a Hill’s constitutive model along with its reduction for a 2D plane stress case is implemented in XSHELL and its model parameters are determined using the shear calibration data. By allowing a free rotation at the pin connection via the definition of a contact surface, a recalibrated analysis is performed for the ductile failure prediction of the Ti–6Al–4V sheet under both quasi-static and modest-rate dynamic loading. A much improved rupture prediction is achieved in terms of crack path, an entire load deflection curve, and peak loads associated with the crack initiation and propagation. The exercise of this Sandia challenge problem reveals that a correct data calibration and accurate representation of boundary condition is essential to perform the ductile failure prediction in the presence of limited test data and available material properties.

... The last example aims at estimating the posterior density of the unknown material properties for a specimen frequently used in the literature, namely, a Sandia fracture challenge [121]. The 2014 fracture challenge problem launched by the Sandia National Lab [121] has provided an ideal platform to assess the computational capability and limitations of each participating team [122]. Specifically, this challenge aims at evaluating the computational ability to predict crack initiation and propagation of ductile fracture with respect to experimental observations. ...

The prediction of crack initiation and propagation in ductile failure processes are challenging tasks for the design and fabrication of metallic materials and structures on a large scale. Numerical aspects of ductile failure dictate a sub-optimal calibration of plasticity- and fracture-related parameters for a large number of material properties. These parameters enter the system of partial differential equations as a forward model. Thus, an accurate estimation of the material parameters enables the precise determination of the material response in different stages, particularly for the post-yielding regime, where crack initiation and propagation take place. In this work, we develop a Bayesian inversion framework for ductile fracture to provide accurate knowledge regarding the effective mechanical parameters. To this end, synthetic and experimental observations are used to estimate the posterior density of the unknowns. To model the ductile failure behavior of solid materials, we rely on the phase-field approach to fracture, for which we present a unified formulation that allows recovering different models on a variational basis. In the variational framework, incremental minimization principles for a class of gradient-type dissipative materials are used to derive the governing equations. The overall formulation is revisited and extended to the case of anisotropic ductile fracture. Three different models are subsequently recovered by certain choices of parameters and constitutive functions, which are later assessed through Bayesian inversion techniques. A step-wise Bayesian inversion method is proposed to determine the posterior density of the material unknowns for a ductile phase-field fracture process. To estimate the posterior density function of ductile material parameters, three common Markov chain Monte Carlo (MCMC) techniques are employed: (i) the Metropolis–Hastings algorithm, (ii) delayed-rejection adaptive Metropolis, and (iii) ensemble Kalman filter combined with MCMC. To examine the computational efficiency of the MCMC methods, we employ the R^-convergence\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{R}{-}convergence$$\end{document} tool. The resulting framework is algorithmically described in detail and substantiated with numerical examples.

... The last example aims at estimating the posterior density of the unknown material properties for a specimen frequently used in the literature, namely, a Sandia fracture challenge [114]. The 2014 fracture challenge problem launched by the Sandia National Lab [114] has provided an ideal platform to assess the computational capability and limitations of each participating team [115]. Specifically, this challenge aims to evaluate the computational ability to predict crack initiation and propagation of ductile fracture with respect to the experimental observation. ...

The prediction of crack initiation and propagation in ductile failure processes are challenging tasks for the design and fabrication of metallic materials and structures on a large scale. Numerical aspects of ductile failure dictate a sub-optimal calibration of plasticity- and fracture-related parameters for a large number of material properties. These parameters enter the system of partial differential equations as a forward model. In this work, we develop a step-wise Bayesian inversion framework for ductile fracture to provide accurate knowledge regarding the effective mechanical parameters. To this end, synthetic and experimental observations are used to estimate the posterior density of the unknowns. To model the ductile failure behavior of solid materials, we rely on the phase-field approach to fracture, for which we present a unified formulation that allows recovering different models on a variational basis. In the variational framework, incremental minimization principles for a class of gradient-type dissipative materials are used to derive the governing equations. The overall formulation is revisited and extended to the case of anisotropic ductile fracture. Three different models are subsequently recovered by certain choices of parameters and constitutive functions, which are later assessed through Bayesian inversion techniques. To estimate the posterior density function of ductile material parameters, three common Markov chain Monte Carlo (MCMC) techniques are employed: (i) the Metropolis-Hastings algorithm, (ii) delayed-rejection adaptive Metropolis, and (iii) ensemble Kalman filter combined with MCMC. To examine the computational efficiency of the MCMC methods, we employ the R-convergence tool.

... That said, most participants relied exclusively on the load-displacement curves of the dogbone samples for calibration of both their chosen plasticity model as well as their chosen damage model. Once the models were calibrated, the participants were asked to provide blind predictions of the load-displacement response and ductile crack path in a compact tension specimen with a blunt notch and three circular holes located at various positions in front of the notch, c.f. Pack et al. (2014); Nahshon et al. (2014); Gross and Ravi-Chandar (2014); Zhou et al. (2014); Zhang et al. (2014). A few participants' blind predictions were in remarkable agreement with the experimental measurements; however, the majority of blind predictions had significant errors with some teams predicting failure three times earlier than observed experimentally. ...

Here, we present a systematic experimental study and accompanying theoretical analysis of the dependence of ductile fracture on strain localization, strain hardening rates, deformation-induced thermal softening, and transient heat conduction under spatially uniform as well as spatially heterogeneous deformation. Spatially uniform cases are studied via standard dogbone shaped specimens subject to uniaxial tension. Spatially heterogeneous deformation is studied via the so-called Sandia Fracture Challenge (SFC) specimen, which is a standard compact tension specimen modified with three machined holes in front of a blunt notch (Boyce et al. in Int J Fract 186(1–2):5–68, 2014). We utilize the same precipitation hardened martensitic stainless steel used in the first SFC experiments, i.e. 15-5 PH with an H1075 heat treatment. We also study 15-5 PH in Condition A and with the H900 heat treatment, each of which has a different hardening behavior and ductility. We find that the ductility does not correlate with the deformation to first crack initiation in the SFC specimen. Instead, hardening rates are better correlated. Moreover, a re-examination of (Boyce et al. in Int J Fract 186(1–2):5–68, 2014) finds that an accurate calibration of the hardening rates is strongly correlated with accurate blind predictions of ductile fracture in the SFC specimens. Given that thermal softening can greatly affect hardening rates, we provided a thermographic analysis of deformation-induced heating and transient heat transfer in both the dogbone shaped samples and the SFC specimens. Transient heating in stainless steels is found to have first-order effects on the strain at the onset of necking, strain to fracture, as well as strain localization and ductile fracture in complex geometries even under so-called quasi-static loading rates.

... Thirteen participating teams were provided with a limited number of elementary test data on a typical stainless steel sheet and asked to make a blind prediction of crack initiation and propagation for a modified compact tension specimen with a round starter notch and three randomly distributed holes subjected to tensile loading (Boyce 2014). A variety of modeling approaches were taken; from porous plastic-ity (Cerrone et al. 2014 andNahshon et al. 2014) to extended finite element methods (Zhang et al. 2014) and damage indicator models uncoupled from plasticity (Gross and Ravi-Chandar 2014;Neilsen et al. 2014;Pack et al. 2014). This was an opportunity for the community to evaluate their current modeling capability and identify missing information essential to improve prediction. ...

In the context of the second Sandia Fracture Challenge, dynamic tensile experiments performed on a Ti–6Al–4V alloy with a complex fracture specimen geometry are modeled numerically. Sandia National Laboratories provided the participants with limited experimental data, comprising of uniaxial tensile test and V-notched rail shear test results. To model the material behavior up to large plastic strains, the flow stress is described with a linear combination of Swift and Voce strain hardening laws in conjunction with the inverse method. The effect of the strain rate and temperature is incorporated through the Johnson–Cook strain rate hardening and temperature softening functions. A strain rate dependent weighting function is used to compute the fraction of incremental plastic work converted to heat. The Hill’48 anisotropic yield function is adopted to capture weak deformation resistance under in-plane pure shear stress. Fracture initiation is predicted by the recently developed strain rate dependent Hosford–Coulomb fracture criterion. The calibration procedure is described in detail, and a good agreement between the blind prediction and the experiments at two different speeds is obtained for both the crack path and the force–crack opening displacement (COD) curve. A comprehensive experimental and numerical follow-up study on leftover material is conducted, and plasticity and fracture parameters are carefully re-calibrated. A more elaborate modeling approach using a non-associated flow rule is pursued, and the fracture locus of the Ti–6Al–4V is clearly identified by means of four different fracture specimens covering a wide range of stress states and strain rates. With the full characterization, a noticeable improvement in the force–COD curve is obtained. In addition, the effect of friction is studied numerically.

... XSHELL is an extended finite element based toolkit for Abaqus, which is developed for dynamic failure prediction of thin walled shell structures (Zhang et al. 2014). The use of the extended finite element methodology (XFEM) allows a mesh topology independent of any arbitrary crack surface and is proven to have great potential in automating the process as the crack grows with time with fixed mesh. ...

Ductile failure of structural metals is relevant to a wide range of engineering scenarios. Computational methods are employed to anticipate the critical conditions of failure, yet they sometimes provide inaccurate and misleading predictions. Challenge scenarios, such as the one presented in the current work, provide an opportunity to assess the blind, quantitative predictive ability of simulation methods against a previously unseen failure problem. Rather than evaluate the predictions of a single simulation approach, the Sandia Fracture Challenge relies on numerous volunteer teams with expertise in computational mechanics to apply a broad range of computational methods, numerical algorithms, and constitutive models to the challenge. This exercise is intended to evaluate the state of health of technologies available for failure prediction. In the first Sandia Fracture Challenge, a wide range of issues were raised in ductile failure modeling, including a lack of consistency in failure models, the importance of shear calibration data, and difficulties in quantifying the uncertainty of prediction [see Boyce et al. (Int J Fract 186:5–68, 2014) for details of these observations]. This second Sandia Fracture Challenge investigated the ductile rupture of a Ti–6Al–4V sheet under both quasi-static and modest-rate dynamic loading (failure in (Formula presented.) 0.1 s). Like the previous challenge, the sheet had an unusual arrangement of notches and holes that added geometric complexity and fostered a competition between tensile- and shear-dominated failure modes. The teams were asked to predict the fracture path and quantitative far-field failure metrics such as the peak force and displacement to cause crack initiation. Fourteen teams contributed blind predictions, and the experimental outcomes were quantified in three independent test labs. Additional shortcomings were revealed in this second challenge such as inconsistency in the application of appropriate boundary conditions, need for a thermomechanical treatment of the heat generation in the dynamic loading condition, and further difficulties in model calibration based on limited real-world engineering data. As with the prior challenge, this work not only documents the ‘state-of-the-art’ in computational failure prediction of ductile tearing scenarios, but also provides a detailed dataset for non-blind assessment of alternative methods.

... Others employed von Mises plasticity supplemented by a fracture model (i.e., Mohr-Coulomb [7] and Johnson-Cook [5]). The extended finite element method (XFEM) [8], material point method, peridynamic theory, and cohesive zones were also employed. Some of these methods outperformed others in determining some of the QOIs, but no single model accurately addressed all of the QOIs. ...

A simple, nonstandardized material test specimen, which fails along one of two different likely crack paths, is considered herein. The result of deviations in geometry on the order of tenths of a millimeter, this ambiguity in crack path motivates the consideration of as-manufactured component geometry in the design, assessment, and certification of structural systems. Herein, finite element models of as-manufactured specimens are generated and subsequently analyzed to resolve the crack-path ambiguity. The consequence and benefit of such a “personalized” methodology is the prediction of a crack path for each specimen based on its as-manufactured geometry, rather than a distribution of possible specimen geometries or nominal geometry. The consideration of as-manufactured characteristics is central to the Digital Twin concept. Therefore, this work is also intended to motivate its development.

... Neilsen et al. [25] have used a transient dynamic finite element code with a multi-linear elastic plastic failure (MLEPF) model developed by Wellman [26], coupled with empirical fracture criteria to describe the crack initiation and growth. Zhang et al. [27] have employed an extended finite element methodology for shell elements with a cohesive interaction model to estimate the crack propagation path. Pack et al. [28] examined the applicability of a modified MohreCoulomb (MMC) model which demonstrated superior fracture predictions than the shear modified Gurson model proposed by Nielsen and Tvergaard [29]. ...

... Others employed von Mises plasticity supplemented by a fracture model (i.e., Mohr-Coulomb [7] and Johnson-Cook [5]). The extended finite element method (XFEM) [8], material point method, peridynamic theory, and cohesive zones were also employed. Some of these methods outperformed others in determining some of the QOIs, but no single model accurately addressed all of the QOIs. ...

A simple, nonstandardized material test specimen, which fails along one of two different likely crack paths, is considered herein. The result of deviations in geometry on the order of tenths of a millimeter, this ambiguity in crack path motivates the consideration of as-manufactured component geometry in the design, assessment, and certification of structural systems. Herein, finite element models of as-manufactured specimens are generated and subsequently analyzed to resolve the crack-path ambiguity. The consequence and benefit of such a “personalized” methodology is the prediction of a crack path for each specimen based on its as-manufactured geometry, rather than a distribution of possible specimen geometries or nominal geometry. The consideration of as-manufactured characteristics is central to the Digital Twin concept. Therefore, this work is also intended to motivate its development.

Intelligent assembly of large-scale, complex structures using an intelligent manufacturing platform represents the future development direction for industrial manufacturing. During large-scale structural assembly processes, several bottleneck problems occur in the existing auxiliary assembly technology. First, the traditional LiDAR-based assembly technology is often limited by the openness of the manufacturing environment, in which there are blind spots, and continuous online assembly adjustment thus cannot be realized. Second, for assembly of large structures, a single-station LiDAR system cannot achieve complete coverage, which means that a multi-station combination method must be used to acquire the complete three-dimensional data; many more data errors are caused by the transfer between stations than by the measurement accuracy of a single station, which means that the overall system's measurement and adjustment errors are increased greatly. Third, because of the large numbers of structural components contained in a large assembly, the accumulated errors may lead to assembly interference, but the LiDAR-assisted assembly process does not have a feedback perception capability, and thus assembly component loss can easily be caused when assembly interference occurs. Therefore, this paper proposes to combine an optical fiber sensor network with digital twin technology, which will allow the test data from the assembly entity state in the real world to be applied to the “twin” model in the virtual world and thus solve the problems with test openness and data transfer. The problem of station and perception feedback is also addressed and represents the main innovation of this work. The system uses an optical fiber sensor network as a flexible sensing medium to monitor the strain field distribution within a complex area in real time, and then completes real-time parameter adjustment of the virtual assembly based on the distributed data. Complex areas include areas that are laser-unreachable, areas with complex contact surfaces, and areas with large-scale bending deformations. An assembly condition monitoring system is designed based on the optical fiber sensor network, and an assembly condition monitoring algorithm based on multiple physical quantities is proposed. The feasibility of use of the optical fiber sensor network as the real-state parameter acquisition module for the digital twin intelligent assembly system is discussed. The offset of any position in the test area is calculated using the convolutional neural network of a residual module to provide the compensation parameters required for the virtual model of the assembly structure. In the model optimization parameter module, a correction data table is obtained through iterative learning of the algorithm to realize state prediction from the test data. The experiment simulates a large-scale structure assembly process, and performs virtual and real mapping for a variety of situations with different assembly errors to enable correction of the digital twin data stream for the assembly process through the optical fiber sensor network. In the plane strain field calibration experiment, the maximum error among the test values for this system is 0.032 mm, and the average error is 0.014 mm. The results show that use of visual calibration can correct the test error to within a very small range. This result is equally applicable to gradient curvature surfaces and freeform surfaces. Statistics show that the average measurement accuracy error for regular surfaces is better than 11.2%, and the average measurement accuracy error for irregular surfaces is better than 14.8%. During simulation of large-scale structure assembly experiments, the average position deviation accuracy is 0.043 mm, which is in line with the designed accuracy.

This paper deals with ductile fracture behaviors of a typical BCC metal with the variation of the strain rate. Ductile fracture behaviors of AISI 4130 steel for BCC metals are investigated to identify their rate-dependency by tracing several stress paths based on the Lou-Huh ductile fracture criterion. Tensile tests are conducted to obtain fracture strains at a wide range of strain rates ranging from 0.001 s⁻¹ to 1000 s⁻¹ with three different shapes of specimens: dog-bone specimens for the uniaxial tension test; diagonally notched specimens for the in-plane shear test; and grooved-in-thickness specimens for the plane-strain tension test.
Equivalent strains to fracture are measured by a 2-D digital image correlation (DIC) method on the surface of the specimens just before the onset of fracture. The damage function of the Lou-Huh fracture criterion is calibrated to obtain the fracture coefficients at each strain rate so that the fracture loci of AISI 4130 steel are constructed with the variation of the strain rate. The equivalent strain to fracture decreases noticeably as the strain rate increases beyond the strain rate of 1 s⁻¹, which is due to the formation of the shear band under the in-plane shear condition. The fracture loci of OFHC copper for FCC metals are also obtained for comparison to the AISI 4130 steel for BCC metals.

The main objective of this paper is to exploit an efficient and a simplified extended finite element shell formulation for prediction of ductile fracture and crack branching of thin-walled aluminum structures subjected to impact loading. In order to characterize an arbitrary crack initiation, propagation and brunching, a phantom paired shell element approach is further developed and implemented into Abaqus' explicit solver via its user defined element (VUEL). The energy dissipation due to failure is captured by a cohesive force along the crack interface when its accumulative plastic strain reaches a critical value. A numerical technique for modeling out-of-plane crack branching phenomena is also developed by activating the phantom nodes and re-grouping the element connectivity. Four numerical examples are used to demonstrate the applicability and accuracy of the extended shell element approach for ductile failure prediction of an indentation test, a multi-bay stiffened panel with crack branching, an explosively loaded plate, and a cylinder subjected to impulsive loading.

Existing and emerging methods in computational mechanics are rarely validated against problems with an unknown outcome. For this reason, Sandia National Laboratories, in partnership with US National Science Foundation and Naval Surface Warfare Center Carderock Division, launched a computational challenge in mid-summer, 2012. Researchers and engineers were invited to predict crack initiation and propagation in a simple but novel geometry fabricated from a common off-the-shelf commercial engineering alloy. The goal of this international Sandia Fracture Challenge was to benchmark the capabilities for the prediction of deformation and damage evolution associated with ductile tearing in structural metals, including physics models, computational methods, and numerical implementations currently available in the computational fracture community. Thirteen teams participated, reporting blind predictions for the outcome of the Challenge. The simulations and experiments were performed independently and kept confidential. The methods for fracture prediction taken by the thirteen teams ranged from very simple engineering calculations to complicated multiscale simulations. The wide variation in modeling results showed a striking lack of consistency across research groups in addressing problems of ductile fracture. While some methods were more successful than others, it is clear that the problem of ductile fracture prediction continues to be challenging. Specific areas of deficiency have been identified through this effort. Also, the effort has underscored the need for additional blind prediction-based assessments.

Dynamic crack growth is analysed numerically for a plane strain block with an initial central crack subject to tensile loading. The continuum is characterized by a material constitutive law that relates stress and strain, and by a relation between the tractions and displacement jumps across a specified set of cohesive surfaces. The material constitutive relation is that of an isotropic hyperelastic solid. The cohesive surface constitutive relation allows for the creation of new free surface and dimensional considerations introduce a characteristic length into the formulation. Full transient analyses are carried out. Crack branching emerges as a natural outcome of the initial-boundary value problem solution, without any ad hoc assumption regarding branching criteria. Coarse mesh calculations are used to explore various qualitative features such as the effect of impact velocity on crack branching, and the effect of an inhomogeneity in strength, as in crack growth along or up to an interface. The effect of cohesive surface orientation on crack path is also explored, and for a range of orientations zigzag crack growth precedes crack branching. Finer mesh calculations are carried out where crack growth is confined to the initial crack plane. The crack accelerates and then grows at a constant speed that. for high impact velocities, can exceed the Rayleigh wave speed. This is due to the finite strength of the cohesive surfaces. A fine mesh calculation is also carried out where the path of crack growth is not constrained. The crack speed reaches about 45% of the Rayleigh wave speed, then the crack speed begins to oscillate and crack branching at an angle of about 29 from the initial crack plane occurs. The numerical results are at least qualitatively in accord with a wide variety of experimental observations on fast crack growth in brittle solids.

We present a finite element method with a finite thickness embedded weak discontinuity to analyze ductile fracture problems. The formulation is restricted to small geometry changes. The material response is characterized by a constitutive relation for a progressively cavitating elastic–plastic solid. As voids nucleate, grow and coalesce, the stiffness of the material degrades. An embedded weak discontinuity is introduced when the condition for loss of ellipticity is met. The resulting localized deformation band is given a specified thickness which introduces a length scale thus providing a regularization of the post-localization response. Also since the constitutive relation for a progressively cavitation solid is used inside the band in the post-localization regime, the traction-opening relation across the band depends on the stress triaxiality. The methodology is illustrated through several example problems including mode I crack growth and localization and failure in notched bars. Various finite element meshes and values of the thickness of the localization band are used in the calculations to illustrate the convergence with mesh refinement and the dependence on the value chosen for the localization band thickness.

lation or in the vicinity of the crack tip 14. In related work, Armero and Ehrlich 15 used embedded discontinuity elements to model hinge lines in plates. The development of a fracture criterion that is computationally efficient and is easily applied in terms of available data poses a significant difficulty. Fracture criteria for quasibrittle materials, such as aluminum, are usually expressed in terms of the critical maximum principal tensile strain. However, in low order finite element models solved by explicit time integration, the maximum principal tensile strain tends to be quite noisy, so that crack paths computed by direct application of such a criterion tend to be er- ratic and do not conform to experimentally observed crack paths. Here, we propose a nonlocal form of a strain-based fracture criterion. The nonlocal form is obtained by a kernel-weighted av- erage over a sector in front of the crack tip. In addition, we de- scribe a combination of this kernel-weighted average with an an- gular component that can be used to indicate crack branching. The methodology is applied to the fracture of shell experiments performed by Chao and Shepherd 16. Although these experi- ments are very interesting, they do not provide enough experimen- tal data for a validation of the methodology. Nevertheless, we show that the method is able to reproduce the change in failure mode that occurs for longer notches as compared with shorter notches and that the overall final configuration agrees reasonably well with that observed in the experiments.

a b s t r a c t Fracture mode of ductile solids can vary depending on the history of stress state the material experienced. For example, ductile plates under remote in-plane loading are often found to rupture in mode I or mixed mode I/III. The distinct crack patterns are observed in many different metals and alloys, but until now the underlying physical principles, though highly debated, remain unresolved. Here we show that the exist-ing theories are not capable of capturing the mixed mode I/III due to a missing ingredient in the consti-tutive equations. We introduce an azimuthal dependent fracture envelope and illustrate that two competing fracture mechanisms, governed by the pressure and the Lode angle of the stress tensor, respectively, exist ahead of the crack tip. Using the continuum damage plasticity model, we demonstrate that the distinctive features of the two crack propagation modes in ductile plates can be reproduced using three dimensional finite element simulations. The magnitude of the tunneling effect and the apparent crack growth resistance are calculated and agree with experimental observations. The finite element mesh size dependences of the fracture mode and the apparent crack growth resistance are also investigated. Published by Elsevier Ltd.

The paper discusses the effect of stress triaxiality on the onset and evolution of damage in ductile metals. A series of tests including shear tests and experiments on smooth and pre-notched tension specimens was carried out for a wide range of stress triaxialities. The underlying continuum damage model is based on kinematic definition of damage tensors. The modular structure of the approach is accomplished by the decomposition of strain rates into elastic, plastic and damage parts. Free energy functions with respect to fictitious undamaged configurations as well as damaged ones are introduced separately leading to elastic material laws which are affected by increasing damage. In addition, a macroscopic yield condition and a flow rule are used to adequately describe the plastic behavior. Numerical simulations of the experiments are performed and good correlation of tests and numerical results is achieved. Based on experimental and numerical data the damage criterion formulated in stress space is quantified. Different branches of this function are taken into account corresponding to different damage modes depending on stress triaxiality and Lode parameter. In addition, identification of material parameters is discussed in detail.

Recent development of damage plasticity theory shows the critical plastic strain at fracture for ductile solids depends on the pressure and the Lode angle on the octahedral plane along the loading path. The determination of the fracture strain envelope is usually a difficult and time consuming process. This is due to the experimental difficulties in maintaining a constant pressure and Lode angle at the fracture site, which is further complicated by the coupled nature of the parameters to be calibrated and the geometrical localization of the deformation. The fracture strain envelope is one of the key ingredients of the damage plasticity theory and relates to the accuracy of predicted results. In the present paper, the Lode angle dependence and the pressure sensitivity functions for the fracture strain envelope are derived from the hardening rule of the matrix using Tresca type fracture condition and Drucker–Prager formula, respectively. Quantitative analyses of Clausing’s and Bridgman’s test data are presented. Then a pressure modified maximum shear stress condition is adopted as fracture initiation condition to examine their joint effects on the fracture strain envelope. The relationship of the strain hardening, the pressure sensitivity and the Lode angle dependence are examined and verified by existing experimental results. We show that within the moderate range of stress triaxiality, the pressure modified maximum shear condition can be used as the fracture stress envelope for ductile metals within the framework of damage plasticity. The present method reduces significantly the amount of work to calibrate the material parameters for ductile fracture.

The fracture of ductile solids has frequently been observed to result from the large growth and coalescence of microscopic voids, a process enhanced by the superposition of hydrostatic tensile stresses on a plastic deformation field. The ductile growth of voids is treated here as a problem in continuum plasticity. First, a variational principle is established to characterize the flow field in an elastically rigid and incompressible plastic material containing an internal void or voids, and subjected to a remotely uniform stress and strain rate field. Then an approximate Rayleigh-Ritz procedure is developed and applied to the enlargement of an isolated spherical void in a nonhardening material. Growth is studied in some detail for the case of a remote tensile extension field with superposed hydrostatic stresses. The volume changing contribution to void growth is found to overwhelm the shape changing part when the mean remote normal stress is large, so that growth is essentially spherical. Further, it is found that for any remote strain rate field, the void enlargement rate is amplified over the remote strain rate by a factor rising exponentially with the ratio of mean normal stress to yield stress. Some related results are discussed, including the long cylindrical void considered by F.A. McClintock (1968, J. appl. Mech. 35, 363), and an approximate relation is given to describe growth of a spherical void in a general remote field. The results suggest a rapidly decreasing fracture ductility with increasing hydrostatic tension.

Recent experimental evidence points to limitations in characterizing the critical strain in ductile fracture solely on the basis of stress triaxiality. A second measure of stress state, such as the Lode parameter, is required to discriminate between axisymmetric and shear-dominated stress states. This is brought into the sharpest relief by the fact that many structural metals have a fracture strain in shear, at zero stress triaxiality, that can be well below fracture strains under axisymmetric stressing at significantly higher triaxiality. Moreover, recent theoretical studies of void growth reveal that triaxiality alone is insufficient to characterize important growth and coalescence features. As currently formulated, the Gurson Model of metal plasticity predicts no damage change with strain under zero mean stress, except when voids are nucleated. Consequently, the model excludes shear softening due to void distortion and inter-void linking. As it stands, the model effectively excludes the possibility of shear localization and fracture under conditions of low triaxiality if void nucleation is not invoked. In this paper, an extension of the Gurson model is proposed that incorporates damage growth under low triaxiality straining for shear-dominated states. The extension retains the isotropy of the original Gurson Model by making use of the third invariant of stress to distinguish shear dominated states. The importance of the extension is illustrated by a study of shear localization over the complete range of applied stress states, clarifying recently reported experimental trends. The extension opens the possibility for computational fracture approaches based on the Gurson Model to be extended to shear-dominated failures such as projectile penetration and shear-off phenomena under impulsive loadings. (c) 2007 Elsevier Masson SAS. All rights reserved.

Two criteria of ductile fracture strain are suggested. From the theory of plasticity for porous materials, the following criterion of fracture for a triaxial state of stress is obtained: [numerical formula] where εe_qf is the equivalent fracture strain, σm the mean stress, σe_q the equivalent stress, a0 and b0 are constants. Except under certain conditions, this criterion shows reasonable agreement with experiment. To improve the accuracy of the prediction of the fracture strain, the above criterion is modified as follows: [numerical formula] where c0 is a constant. It is found that this criterion provides a greater accuracy for prediction of the fracture strain.

A finite element formulation and algorithm for the nonlinear analysis of the large deflection, materially nonlinear response of impulsively loaded shells is presented. A unique feature of this algorithm is the use of a bilinear four node quadrilateral element with single point quadrature and a simple hourglass control which is orthogonal to straining and rigid body modes on an element level. Numerous results are presented for both elastic and elastic-plastic problems with large strains.

In this paper, a method to analyse and predict crack propagation in thin‐walled structures subjected to large plastic deformations when loaded at high strain rates—such as impact and/or blast—has been proposed. To represent the crack propagation independently of the finite element discretisation, an extended finite element method based shell formulation has been employed. More precisely, an underlying 7‐parameter shell model formulation with extensible directors has been extended by locally introducing an additional displacement field, representing the displacement discontinuity independently of the mesh. Of special concern in the paper has been to find a proper balance between, level of detail and accuracy when representing the physics of the problem and, on the other hand, computational efficiency and robustness. To promote computational efficiency, an explicit time step scheme has been employed, which however has been discovered to generate unphysical oscillations in the response upon crack propagation. Therefore, special focus has been placed to investigate these oscillations as well as to find proper remedies. The paper is concluded with three numerical examples to verify and validate the proposed model.Copyright © 2013 John Wiley & Sons, Ltd.

Classical metal plasticity theory assumes that the hydrostatic pressure has no or negligible effect on the material strain hardening, and that the flow stress is independent of the third deviatoric stress invariant (or Lode angle parameter). However, recent experiments on metals have shown that both the pressure effect and the effect of the third deviatoric stress invariant should be included in the constitutive description of the material. A general form of asymmetric metal plasticity, considering both the pressure sensitivity and the Lode dependence, is postulated. The calibration method for the new metal plasticity is discussed. Experimental results on aluminum 2024-T351 are shown to validate the new material model.From the similarity between yielding surface and fracture locus, a new 3D asymmetric fracture locus, in the space of equivalent fracture strain, stress triaxiality and the Lode angle parameter, is postulated. Two methods of calibration of the fracture locus are discussed. One is based on classical round specimens and flat specimens in uniaxial tests, and the other one uses the newly designed butterfly specimen under biaxial testing. Test results of Bao (2003) [Bao, Y., 2003. Prediction of ductile crack formation in uncracked bodies. PhD Thesis, Massachusetts Institute of Technology] on aluminum 2024-T351, and test data points of A710 steel from butterfly specimens under biaxial testing validated the postulated asymmetric 3D fracture locus.

Ductile failure of metals is often treated as the result of void nucleation, growth and coalescence. Various criteria have been proposed to capture this failure mechanism for various materials. In this study, ductile failure of dual phase steels is predicted in the form of plastic strain localization resulting from the incompatible deformation between the harder martensite phase and the softer ferrite matrix. Microstructure-level inhomogeneity serves as the initial imperfection triggering the instability in the form of plastic strain localization during the deformation process. Failure modes and ultimate ductility of two dual phase steels are analyzed using finite element analyses based on the actual steel microstructures. The plastic work hardening properties for the constituent phases are determined by the in-situ synchrotron-based high-energy X-ray diffraction technique. Under different loading conditions, different failure modes and ultimate ductility are predicted in the form of plastic strain localization. It is found that the local failure mode and ultimate ductility of dual phase steels are closely related to the stress state in the material. Under plane stress condition with free lateral boundary, one dominant shear band develops and leads to final failure of the material. However, if the lateral boundary is constrained, splitting failure perpendicular to the loading direction is predicted with much reduced ductility. On the other hand, under plane strain loading condition, commonly observed necking phenomenon is predicted which leads to the final failure of the material. These predictions are in reasonably good agreement with experimental observations.

We consider a problem of modeling fracture and failure preceded by large scale yielding of ductile shells from the point of view of large-scale structural analysis. We place a special emphasis on the computational efficiency of the constitutive formulation. In this context, we seek the formulation embedded in the shell mechanics framework, which is both theoretically sound and easily implementable into a large-scale explicit dynamic finite element code without precluding vectorization or parallelization. This is achieved through the elasto-plastic damage constitutive model for finite-element analysis of plates and shells. The proposed damage model is purely phenomenological with a scalar damage parameter, which has no physical interpretation, except that it represents on a global scale the micromechanical changes the material undergoes during the process of necking and fracture. The localization leading to softening and fracture is represented by the damage calibration function with exponential damage growth after the onset of necking. The proposed phenomenological damage model uses a general plasticity and shell mechanics frameworks which makes it general and easily implementable into existing finite element codes. The proposed formulation has been implemented into the explicit dynamic finite element software code EPSA (3 and 4).

From an available solution for the deformation of elliptical holes in a viscous material, a criterion is developed for fracture by the growth and coalescence of cylindrical holes under any prescribed history of applied principal components of stress and strain which do not rotate relative to the material. The criterion is extended to plastic materials by extrapolation from an analysis for the growth of circular holes under equiaxial transverse stress. Experiments on Plasticine substantiate the analysis and its extrapolation. For both plastic and viscous flow, most of the applied strain to fracture is found to occur while the holes are still small compared with their spacing. The most striking result is that in plastic materials there is a very strong inverse dependence of fracture strain on hydrostatic tension. The theory also indicates the effects of anisotropy, strain-hardening, and strain gradients on ductile fracture by the growth of holes.

Although the crack-tip-opening angle (CTOA) has been shown to be well suited for modeling stable crack growth and instability for thin-sheet aluminum alloys, its behavior for increasing thickness has not been thoroughly evaluated. This paper presents the results of two-dimensional and three-dimensional finite element based fracture analyses that were performed to characterize the critical CTOA for C(T) specimens made of 2024-T351 aluminum alloy with thicknesses of 2.3, 6.35, 12.7, and 25.4 mm. Computed CTOA, based on a center-node release methodology, was generally higher than experimentally determined surface CTOA measurements for the same thicknesses. For the C(T) specimens analyzed in this work, with the crack length and uncracked ligament generally greater than four times the specimen thickness, the generated global constraint factor data fell within those reported for M(T), DE(T), and SE(B) specimen configurations that also satisfy the above mentioned dimensional guideline. Strengthening the observation that, although critical CTOA is dependent on absolute material thickness, the CTOA characterization process is independent of specimen/loading types and specimen dimensions for cases satisfying this dimensional guideline. The CTOA values generated using 3D finite element analyses were used within a 2D finite element analysis framework to estimate plane strain core (PSC) height values for all evaluated thicknesses. The resulting PSC heights increased with increasing specimen thickness and appear to be on the order of specimen thickness.

It has been shown that the plastic response of many materials, including some metallic alloys, depends on the stress state. In this paper, we describe a plasticity model for isotropic materials, which is a function of the hydrostatic stress as well as the second and third invariants of the stress deviator, and present its finite element implementation, including integration of the constitutive equations using the backward Euler method and formulation of the consistent tangent moduli. Special attention is paid for the adoption of the non-associated flow rule. As an application, this model is calibrated and verified for a 5083 aluminum alloy. Furthermore, the Gurson–Tvergaard–Needleman porous plasticity model, which is widely used to simulate the void growth process of ductile fracture, is extended to include the effects of hydrostatic stress and the third invariant of stress deviator on the matrix material.

Through-the-thickness crack propagation in thin-walled structures is dealt with in this paper. The formulation is based on the cohesive zone concept applied to a kinematically consistent shell model enhanced with an XFEM-based discontinuous kinematical representation. The resulting formulation comprises the representation of continuous deformation, represented by midsurface placement, director and thickness inhomogeneous fields, and discontinuous deformation, represented by discontinuous placement and director fields. The shell model is implemented both for the implicit static analysis and in the context of explicit dynamic integration pertinent to impact loading, and the implementation results in a 7-parameter solid-shell element based on a 6-noded triangular element. In order to properly formulate the dynamic fracture characteristics, a rate-dependent cohesive zone model is employed with respect to, e.g. limiting crack speeds as observed experimentally. In the final example, this model has been applied to a blast loaded pressure vessel that has been experimentally tested. The results indicate that the right crack speed as well as fracture characteristics are relatively well captured. Furthermore, it appears that the discontinuous model exhibits the expected properties with respect to critical time step size in the dynamic analysis and convergence behavior towards the analytical static solution. Copyright © 2010 John Wiley & Sons, Ltd.

A new method for modelling of arbitrary dynamic crack and shear band propagation is presented. We show that by a rearrangement of the extended finite element basis and the nodal degrees of freedom, the discontinuity can be described by superposed elements and phantom nodes. Cracks are treated by adding phantom nodes and superposing elements on the original mesh. Shear bands are treated by adding phantom degrees of freedom. The proposed method simplifies the treatment of element-by-element crack and shear band propagation in explicit methods. A quadrature method for 4-node quadrilaterals is proposed based on a single quadrature point and hourglass control. The proposed method provides consistent history variables because it does not use a subdomain integration scheme for the discontinuous integrand. Numerical examples for dynamic crack and shear band propagation are provided to demonstrate the effectiveness and robustness of the proposed method. Copyright © 2006 John Wiley & Sons, Ltd.

This paper improves the 16 degrees-of-freedom quadrilateral shell element based on pointwise Kirchhoff–Love constraints and introduces a consistent large strain formulation for this element. The model is based on classical shell kinematics combined with continuum constitutive laws. The resulting element is valid for large rotations and displacements. The degrees-of-freedom are the displacements at the corner nodes and one rotation at each mid-side node. The formulation is free of enhancements, it is almost fully integrated and is found to be immune to locking or unstable modes. The patch test is satisfied. In addition, the formulation is simple and amenable to efficient incorporation in large-scale codes as no internal degrees-of-freedom are employed, and the overall calculations are very efficient. Results are presented for linear and non-linear problems. Copyright © 2005 John Wiley & Sons, Ltd.

The finite element method (FEM) is used to predict the applied J-integral values in highly strained tensile panels containing short center cracks. Experimental J-values are obtained by integrating strain and displacement quantities measured along an instrumented contour. FEM plane stress predictions for J-values and crack mouth opening displacements (CMOD) are much larger than experimentally measured values for short cracks (a/WJ and CMOD values. The introduction of a small stiffened zone near the crack tip using an overlay of plane strain elements brings FEM J and CMOD values into close agreement with experimental values. For longer crack lengths, conventional plane stress FEM solutions are adequate to predict J and CMOD values.On utilise la mthode des lments finis en vue de prdire les valeurs de l'intgrale J appliques des panneaux soumis un rgime de tension svre et comportant des fissures contrales courtes. Les valeurs exprimentales de l'Intgrale J sont obtenues en intgrant les dilatations et les dplacements le long d'un contour dment instrument.Il apparat que les prdictions par lments finis des valeurs de J et des dplacements d'ouverture de la portion dbouchante des fissures (CMOD), s'avrent plus leves que les valeurs mesures pour des fissures courtes (a/WOn dmontre que des modifications importantes de la gomtrie n'ont qu'une influence ngligeable sur les valeurs de J et de CMOD, dtermines par lments finis.Le fait d'introduire une zone lgrement raidie au voisinnage de l'extrmit de la fissure, en utilisant notamment un revtement d'lments oprant en tat plan de dformation, a pour consquence de rapprocher ces valeurs de J et de CMOD, des valeurs exprimentales.Dans le cas de longueurs de fissure plus importantes, il s'avre que les solutions conventionnelles par lments finis et tat plan de tension, sont appropries la prdiction des valeurs de J et de CMOD.

Necking and failure in a round tensile test specimen is analysed numerically, based on a set of elastic-plastic constitutive relations that account for the nucleation and growth of micro-voids. Final material failure by coalescence of voids, at a value of the void volume fraction in accord with experimental and computational results, is incorporated in this constitutive model via the dependence of the yield condition on the void volume fraction. In the analyses the material has no voids initially; but high voidage develops in the centre of the neck where the hydrostatic tension peaks, leading to the formation of a macroscopic crack as the material stress carrying capacity vanishes. The numerically computed crack is approximately plane in the central part of the neck, but closer to the free surface the crack propagates on a zig-zag path, finally forming the cone of the cup-cone fracture. The onset of macroscopic fracture is found to be associated with a sharp “knee” on the load deformation curve, as is also observed experimentally, and at this point the reduction in cross-sectional area stops.

A three-node, curved thin-shell triangular element with simple nodal connections is developed. The displacement and rotation components are independently interpolated by complete cubic and quadratic polynomials respectively. The Kirchhoff hypothesis is enforced at a discrete number of points in the element. The rigid-body displacement condition is satisfied by isoparametric interpolation of the shell geometry within the element. A detailed numerical evaluation through a number of standard problems is performed.

The experimental and numerical work presented in this paper reveals that stress state has strong effects on both the plastic response and the ductile fracture behavior of an aluminum 5083 alloy. As a result, the hydrostatic stress and the third invariant of the stress deviator (which is related to the Lode angle) need to be incorporated in the material modeling. These findings challenge the classical J2 plasticity theory and provide a blueprint for the establishment of the stress state dependent plasticity and ductile fracture models for aluminum structural reliability assessments. Further investigations are planned to advance, calibrate and validate the new plasticity and ductile fracture models.

A general nonlinear finite element formulation is given for two-dimensional problems. The formulation applies to the practically important cases of shells of revolution, tubes, rings, beams and frames. The approach is deduced from a corresponding three-dimensional formulation [4] and this enables a simplified implementation, especially with respect to constitutive software. Uniform reduced-integration Lagrange elements are employed and shown to be very effective for the class of problems considered.

This paper considers fracture characteristics of OFHC copper, Armco iron and 4340 steel. The materials are subjected to torsion tests over a range of strain rates, Hopkinson bar tests over a range of temperatures, and quasi-static tensile tests with various notch geometries. A cumulative-damage fracture model is introduced which expresses the strain to fracture as a function of the strain rate, temperature and pressure. The model is evaluated by comparing computed results with cylinder impact tests and biaxial (torsion-tension) tests.

This paper reviews the most important current approaches for residual strength prediction of thin-walled structures. Crack driving force parameters such the linear elastic stress intensity factor and its plastic zone corrected extension for contained yielding conditions, the crack tip opening displacement δ5, the crack tip opening angle CTOA, the cohesive zone model parameters, separation energy, critical tensile stress and critical separation and the parameters of the damage models of Gurson–Tvergaard–Needleman type are introduced and discussed with respect to their benefits and limitations for the simulation of plane and stiffened panels. In addition, specific aspects of modern non-integral and integral structures which pose a challenge are addressed. These comprise multi-site damage, crack deviation and branching, welding residual stresses, strength mismatch in material compounds and problems in bonded structures, such as delamination. A number of examples are provided to illustrate the potential of the various approaches.

A Lagrangian finite element method of fracture and fragmentation in brittle materials is developed. A cohesive-law fracture model is used to propagate multiple cracks along arbitrary paths. In axisymmetric calculations, radial cracking is accounted for through a continuum damage model. An explicit contact/friction algorithm is used to treat the multi-body dynamics which inevitably ensues after fragmentation. Rate-dependent plasticity, heat conduction and thermal coupling are also accounted for in calculations. The properties and predictive ability of the model are exhibited in two case studies: spall tests and dynamic crack propagation in a double cantilever beam specimen. As an example of application of the theory, we simulate the experiments of Field (1988) involving the impact of alumina plates by steel pellets at different velocities. The calculated conical, lateral and radial fracture histories are found to be in good agreement with experiment.

Recent experimental evidence points to limitations in characterizing the critical strain in ductile fracture solely on the basis of stress triaxiality. A second measure of stress state, such as the Lode parameter, is required to discriminate between axisymmetric and shear-dominated stress states. This is brought into the sharpest relief by the fact that many structural metals have a fracture strain in shear, at zero stress triaxiality, that can be well below fracture strains under axisymmetric stressing at significantly higher triaxiality. Moreover, recent theoretical studies of void growth reveal that triaxiality alone is insufficient to characterize important growth and coalescence features. As currently formulated, the Gurson Model of metal plasticity predicts no damage change with strain under zero mean stress, except when voids are nucleated. Consequently, the model excludes shear softening due to void distortion and inter-void linking. As it stands, the model effectively excludes the possibility of shear localization and fracture under conditions of low triaxiality if void nucleation is not invoked. In this paper, an extension of the Gurson model is proposed that incorporates damage growth under low triaxiality straining for shear-dominated states. The extension retains the isotropy of the original Gurson Model by making use of the third invariant of stress to distinguish shear dominated states. The importance of the extension is illustrated by a study of shear localization over the complete range of applied stress states, clarifying recently reported experimental trends. The extension opens the possibility for computational fracture approaches based on the Gurson Model to be extended to shear-dominated failures such as projectile penetration and shear-off phenomena under impulsive loadings.

This paper presents a part of an on-going study of light-weight shelters made of aluminium alloy AA5083–H116. A necessary prerequisite for numerical simulations of impact behaviour of such shelters is a calibrated numerical model. Slightly modified versions of the two Johnson–Cook models describing flow stress and fracture strain are applied. In addition to ordinary quasistatic tests with smooth specimens, these models demand tests at elevated temperatures and strain rates, and different triaxiality ratios. All tests to be presented in this paper are uniaxial tension tests performed in servohydraulic test machines or a Split–Hopkinson bar. The test programme involves approximately 100 specimens, and includes coupons from three different directions of the plate material, thereby taking account for the anisotropy. The AA5083 alloy is susceptible to dynamic strain ageing, and this phenomenon is shown to result in serrated stress–strain curves and negative strain rate sensitivity for a rather wide range of strain rates. In particular, the specimen geometry seems to have a strong influence on the serrated curves. The calibration shows that the Johnson–Cook–type models represent most of the observed material behaviour reasonably well, although the negative strain rate sensitivity is not adequately described.

A nonlinear finite element formulation is presented for the three-dimensional quasistatic analysis of shells which accounts for large strain and rotation effects, and accommodates a fairly general class of nonlinear, finite-deformation constitutive equations. Several features of the developments are noteworthy, namely: the extension of the selective integration procedure to the general nonlinear case which, in particular, facilitates the development of a ‘heterosis-type’ nonlinear shell element; the presentation of a nonlinear constitutive algorithm which is ‘incrementally objective’ for large rotation increments, and maintains the zero normal-stress condition in the rotating stress coordinate system; and a simple treatment of finite-rotational nodal degrees-of-freedom which precludes the appearance of zero-energy in-plane rotational modes. Numerical results indicate the good behavior of the elements studied.

A finite element formulation and algorithm for the nonlinear analysis of the large deflection, materially nonlinear response of impulsively loaded shells is presented. A unique feature of this algorithm is the use of a bilinear four-node quadrilateral element with single-point quadrature and a simple hourglass control which is orthogonal to rigid body modes on an element level and does not compromise the consistency of the equations. The geometric nonlinearities are treated by using a corotational description wherein a coordinate system that rotates with the material is embedded at the integration point; thus the algorithm is directly applicable to anisotropic materials without any corrections for frame invariance of material property tensors. This algorithm can treat about 200 element-time-steps per CPU second on a CYBER 170/730 computer in the explicit time integration mode. Numerous results are presented for both elastic and elastic-plastic problems with large strains that show that the method in most cases is comparable in accuracy with an earlier version of this algorithm employing a cubic triangular plate-shell element, but substantially faster.

In this paper we introduce and analyze a finite element method for elasticity problems with interfaces. The method allows for discontinuities, internal to the elements, in the approximation across the interface. We propose a general approach that can handle both perfectly and imperfectly bonded interfaces without modifications of the code. For the case of linear elasticity, we show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh. We present numerical examples for the linear case as well as for contact and crack propagation model problems.

Widely used constitutive laws for engineering materials assume plastic incompressibility, and no effect on yield of the hydrostatic component of stress. However, void nucleation and growth (and thus bulk dilatancy) are commonly observed is some processes which are characterized by large local plastic flow, such as ductile fracture. The purpose of this work is to develop approximate yield criteria and flow rules for porous (dilatant) ductile materials, showing the role of hydrostatic stress in plastic yield and void growth. Other elements of a constitutive theory for porous ductile materials, such as void nucleation, plastic flow and hardening behavior, and a criterion for ductile fracture will be discussed in Part II of this series. The yield criteria are approximated through an upper bound approach. Simplified physical models for ductile porous materials (aggregates of voids and ductile matrix) are employed, with the matrix material idealized as rigid-perfectly plastic and obeying the von Mises yield criterion. Velocity fields are developed for the matrix which conform to the macroscopic flow behavior of the bulk material. Using a distribution of macroscopic flow fields and working through a dissipation integral, upper bounds to the macroscopic stress fields required for yield are calculated. Their locus in stress space forms the yield locus. It is shown that normality holds for this yield locus, so a flow rule results. Approximate functional forms for the yield loci are developed.

NASA has developed a comprehensive analytical methodology for predicting the onset of widespread fatigue damage in fuselage structure. The determination of the number of flights and operational hours of aircraft service life that are related to the onset of widespread fatigue damage includes analyses for crack initiation, fatigue crack growth, and residual strength. Therefore, the computational capability required to predict analytically the onset of widespread fatigue damage must be able to represent a wide range of crack sizes from the material (microscale) level to the global structural-scale level. NASA studies indicate that the fatigue crack behavior in aircraft structure can be represented conveniently by the following three analysis scales: small three-dimensional cracks at the microscale level, through-the-thickness two-dimensional cracks at the local structural level, and long cracks at the global structural level. The computational requirements for each of these three analysis scales are described in this paper.

Dynamic fracture modeling in shell structures based on XFEM A criterion for ductile fracture by the growth of holes A triangular thin-shell finite element based on discrete Kirchhoff theory

- R Larsson
- J Mediavilla
- Fagerström
- Murthy
- Ss
- Gallagher

Larsson R, Mediavilla J, Fagerström M (2011) Dynamic fracture modeling in shell structures based on XFEM. Int J Numer Meth Eng 86:499–527 McClintock FA (1968) A criterion for ductile fracture by the growth of holes. J Appl Mech 35(2):363–371 Murthy SS, Gallagher RH (1986) A triangular thin-shell finite element based on discrete Kirchhoff theory. Comput Methods Appl Mech Eng 54:197–222

On the ductile enlargement of voids in triaxial stress fields A method for dynamic crack and shear band propagation with phantom nodes Dynamic fracture of shells sub-jected to impulsive loads

- Jr
- Tracey
- Jh Song
- Areias
- Jh Song
- Belytschko

JR, Tracey DM (1969) On the ductile enlargement of voids in triaxial stress fields. J Mech Phys Solids 17(3):201–217 Song JH, Areias PMA (2006) A method for dynamic crack and shear band propagation with phantom nodes. Int J Numer Meth Eng 2006(67):868–893 Song JH, Belytschko T (2009) Dynamic fracture of shells sub-jected to impulsive loads. J Appl Mech 76(5):051301

Crack tip dtrain—a comparison of finite element method calculations and Moiré measurements. Cracks and fracture, ASTM STP 601

- W Hu
- H Liu