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Journal articles on the topic 'Ductile fracture simulations'

Author: Grafiati

Published: 26 October 2024

Last updated: 19 July 2025

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Liu, HS, and MW Fu. "Prediction and analysis of ductile fracture in sheet metal forming—Part I: A modified Ayada criterion." International Journal of Damage Mechanics 23, no. 8 (2014): 1189–210. http://dx.doi.org/10.1177/1056789514541559.

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A modified ductile fracture criterion is proposed based on the traditional Ayada criterion and coded into the finite element simulation platform of VUMAT/ABAQUS for prediction and analysis of ductile fracture in metal plastic strain processes. In this modified ductile fracture criterion, stress triaxiality is taken into account, and more importantly, the exponential effect of the equivalent plastic strain on the damage behavior, which is generally ignored in other ductile fracture criteria, is also considered. The material related constants in the modified ductile fracture criterion are determined by tensile tests together with finite element simulations. The applicability and reliability of the ductile fracture criterion in ductile fracture prediction in two types of classic stress states, viz. shear stress, tensile stress in sheet metal forming, are investigated based on the deformation behavior and fracture occurrence in two case studies with two typical types of materials, i.e. Al 6061 and T10A. The materials have a wide range of plasticity. The simulation and experimental results verify the applicability and reliability of the developed ductile fracture criterion in prediction of the ductile fracture with and without necking in different stress states of plastic strain.

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Azizi, Muhammad Azim, Mohd Faiz Mohd Ridhuan, Mohd Zakiyuddin Mohd Zahari, Sharafiz Abdul Rahim, and Muhammad Amin Azman. "Peridynamic Model for Tensile Elongation and Fracture Simulations of Polymethyl Methacrylate Notched Specimens." Applied Mechanics and Materials 909 (September 28, 2022): 11–28. http://dx.doi.org/10.4028/p-2z0841.

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This paper presents the peridynamic (PD) numerical model for simulating a tensile test until total fracture for a brittle polymeric material namely polymethyl methacrylate (PMMA). U-notched and V-notched specimens were used to investigate the effect of the notches on the elongation and fracture of PMMA. The tensile elongation of PMMA exhibits nonlinearity with respect to the applied load, while the fracture occurs when the material stress has reached the ultimate tensile stress of the material. Similar elongation and fracture properties were applied on PD simulations. Two types of elongation equation are used namely brittle and ductile equations to form PD-brittle and PD-ductile models. The published experimental data of tensile fracture test on notched PMMA specimens are used as reference to validate the simulations of the PD models. The PD numerical force-extension curves have good quantitative similarity for V-notched specimen but adequate quantitative similarity for U-notched specimen. As for the quality of the fractured specimen shape, the PD simulations have good similarity for the V-notched specimen but adequate similarity for the U-notched specimen. The plot of the internal force distribution from the simulations of PD shows good qualitative similarity to the plot of the stress distribution from the published data of FEM in terms of stress concentration. From the PD results, it is observed that the PD-ductile model has better capability in producing accurate simulation of the notched specimens than the PD-brittle model.

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Dzioba, Ihor, and Sebastian Lipiec. "Fracture Mechanisms of S355 Steel—Experimental Research, FEM Simulation and SEM Observation." Materials 12, no. 23 (2019): 3959. http://dx.doi.org/10.3390/ma12233959.

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In this study, the fracture mechanisms of S355 ferritic steel were analyzed. In order to obtain different mechanisms of fracture (completely brittle, mixed brittle and ductile or completely ductile), tests were carried out over a temperature range of −120 to +20 °C. Our experimental research was supplemented with scanning electron microscopy (SEM) observations of the specimens’ fracture surfaces. Modeling and load simulations of specimens were performed using the finite element method (FEM) in the ABAQUS program, and accurate calibration of the true stress–strain material dependence was made. In addition, the development of mechanical fields before the crack tip of the cracking process in the steel was analyzed. The distributions of stresses and strains in the local area before the crack front were determined for specimens fractured according to different mechanisms. Finally, the conditions and characteristic values of stresses and strains which caused different mechanisms of fracture—fully brittle, mixed brittle and ductile or fully ductile—were determined.

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Banabic, Dorel, and Abdolvahed Kami. "Applications of the Gurson’s model in sheet metal forming." MATEC Web of Conferences 190 (2018): 01002. http://dx.doi.org/10.1051/matecconf/201819001002.

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Recent advances in the modelling of metals encompass modelling of metals structural inhomogeneity, damage, porosity, twinning/untwining and non-local and second order effects. This presentation is focused on modelling the void growth in ductile fractures. The growth and coalescence of microscopic voids are the main mechanisms in ductile fracture of bulk metallic parts. In sheet metals, ductile fracture is preceded by necking during which existing voids do not have significant growth. However, necking is highly sensitive to plastic flow direction which in turn is sensitive to the presence of voids. Also, under biaxial strain loading, the final fracture in the necking region is still controlled by void growth; hence an accurate fracture prediction is crucial for crash simulations. Finally, in super-plastic sheet forming, void growth and coalescence may precede or accompany necking. Therefore, there is as increasing interest in modelling of voids in the sheet metals. As an application, we show how the predictions of some forming limit curves (FLCs) can be affected by accurate simulation of voids growth.

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Watanabe, Atsuo, Kunio Hayakawa, and Shinichiro Fujikawa. "An Anisotropic Damage Model for Prediction of Ductile Fracture during Cold-Forging." Metals 12, no. 11 (2022): 1823. http://dx.doi.org/10.3390/met12111823.

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Researchers have formulated equations of ductile fracture to simulate and predict defects in cold-forged parts, e.g., the Cockcroft–Latham criterion. However, these equations are not applicable to certain cases of fracture in forged products. This study formulates a new equation for predicting ductile fractures with better prediction accuracy than the convention by which the cost for trial-and-error design can be reduced. The equation is expressed as a second-rank symmetric tensor, which is the inner product of the stress and strain-increment tensors. The theoretical efficacy of the equation in predicting ductile fractures is verified via a uniaxial tensile test. The practicability of the equation is confirmed by applying it to the simulations of two real cold-forged components: a cold-forged hollow shaft and a flanged shaft. For the hollow shaft, the equation predicts the position where the ductile fracture would initiate, which—to the best of the authors’ knowledge—is unique to this study. For the flanged shaft, the equation predicts the occurrence of diagonal cracks due to different lubrication conditions.

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Tong, Ying. "The Evaluation of Ductile Fracture Criteria (DFC) of 6061-T6 Aluminum Alloy." Applied Mechanics and Materials 44-47 (December 2010): 2837–41. http://dx.doi.org/10.4028/www.scientific.net/amm.44-47.2837.

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As one of the principal failures, ductile fracturing restricts metal forming process. Cockcroft-Latham fracture criterion is suited for tenacity fracture in bulk metal-forming simulation. An innovative approach involving physical compression experiments, numerical simulations and mathematic computations provides mutual support to evaluate ductile damage cumulating process and ductile fracture criteria (DFC). The results show that the maximum cumulated damage decreases with strain rate rising, and the incremental ratios, that is damage sensitive rate, vary uniformly during the upsetting processes at different strain rates. The damage sensitive rate decreases rapidly, then it becomes stability in a constant 0.11 after true strain -0.85. The true strain -0.85 was assumed as the fracture strain, and the DFC of 6061-T6 aluminum alloy is almost a constant 0.2. According to DFC, the exact fracture moment and position during various forming processes will be predicted conveniently.

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Matsuno, Takashi, Jin Eguchi, Yasuhiro Kunii, et al. "Derivation of Point Plot-based Ductile Fracture Loci through Assimilation of Finite Element Simulation and Individual Actual Punched Surface Profile." MATEC Web of Conferences 408 (2025): 02001. https://doi.org/10.1051/matecconf/202540802001.

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This study demonstrated the inverse identification of the ductile fracture locus (DFL), successfully reproducing the actual punched surface profiles of DP980 steels. The primary aim was to visualize the plastic strain and residual stress distribution in the observed punching surface cross-section. First, cracks were incorporated into finite element (FE) half-punching simulations, on the basis of preliminary observations of the actual crack propagation behavior, to replicate individual punched surface profiles. Notably, ductile fracture modeling was not employed during the initial FE simulation. Then, the equivalent plastic strain and historical-averaged stress triaxiality were plotted for two distinct element groups: those immediately before crack propagation (fracture region) and those unaffected by the crack (non-fracture region). The DFL was inversely identified as the boundary between the fracture and non-fracture regions. Subsequently, the identified DFL was integrated with Xue's damage integration scheme for FE punching simulations. This approach successfully reproduced the punched surface profiles. The DFL exhibited robustness under various punching conditions. FE simulations with different punching clearances yielded punched surface profiles that agreed well with the experimental observations, although slight variations were noted in the fracture surface portion.

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Simkins, D. C., and S. Li. "Meshfree simulations of thermo-mechanical ductile fracture." Computational Mechanics 38, no. 3 (2005): 235–49. http://dx.doi.org/10.1007/s00466-005-0744-8.

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Sun, Dong Zhi, Michael Krawiec, and Hariaokto Hooputra. "Characterization and Modelling of the Damage Behavior of Extruded Aluminum Profiles for Crash Simulations." Materials Science Forum 877 (November 2016): 674–79. http://dx.doi.org/10.4028/www.scientific.net/msf.877.674.

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The damage behavior of aluminum profiles depends strongly on the stress state. Many investigations have shown that both ductile and shear fracture have to be taken into account in damage analysis. Since fracture strains of aluminum profiles are relatively low, damage modelling has to be included in component simulations. However, it is an open question, which kind of damage model can be used for crash simulations and which tests should be performed in order to calibrate the model. An extruded aluminum profile with double chambers of AA6060-T79 was characterized under different stress triaxialities and shear ratios. The damage criteria IDS (Instability, Ductile and Shear fracture) in ABAQUS/Explicit were used for the simulations. An explicit relationship between triaxiality and shear ratio was derived for plane stress state. The influence of the model parameter on the overlapping of both criteria (ductile and shear fracture) was systematically studied for shell element applications. The applied damage model was validated by comparing experimental and calculated results of component tests.

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Meng, Xianbo, Zixi Jiao, Haiyan Zhu, et al. "Study of the Brittle–Ductile Characteristics and Fracture Propagation Laws of Ultra-Deep Tight Sandy Conglomerate Reservoirs." Processes 13, no. 6 (2025): 1880. https://doi.org/10.3390/pr13061880.

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Ultra-deep tight sandy conglomerate reservoirs in the Junggar Basin are characterized by vertically alternating lithologies that include mudstone, sandy conglomerate, and sandstone. High in situ stresses and formation temperatures contribute to a brittle–ductile transition process in the reservoir rocks. However, the brittle behavior and ductile hydraulic fracture propagation mechanisms under in situ conditions remain inadequately understood. In this study, ultra-deep core samples were subjected to triaxial compression tests under varying confining pressures and temperatures to simulate different burial depths and evaluate their brittleness. A three-dimensional hydraulic fracture propagation model was developed in ABAQUS 2023 finite element software, incorporating a cohesive zone ductile constitutive model. Numerical simulations were conducted, considering interlayer horizontal stress differences, injection rate, and fracturing fluid viscosity, to systematically analyze the influence of geological and engineering factors on ductile fracture propagation. A fracture length–height competition diagram was constructed to illustrate the propagation mechanisms. The results reveal that high temperatures significantly accelerate the brittle–ductile transition, which occurs at confining pressures between 55 and 65 MPa. Following this transition, failure modes shift from single-shear failure to a multi-localized fracture with bulging deformation. Interlayer horizontal stress differences were found to strongly influence fracture penetration, with larger stress differences hindering vertical growth. Increasing injection rates promoted the uniform distribution of lateral fractures and fracture tip development, while medium- to high-viscosity fracturing fluids enhanced fracture width and vertical stimulation uniformity. These findings provide important insights for optimizing fracturing strategies and expanding the effective stimulation volume in the ultra-deep tight sandy conglomerate reservoirs of the Junggar Basin.

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Kacem, Ahmed, Hervé Laurent, and Sandrine Thuillier. "Prediction of forming limit curve for AA6061-T6 at room and elevated temperatures." IOP Conference Series: Materials Science and Engineering 1238, no. 1 (2022): 012044. http://dx.doi.org/10.1088/1757-899x/1238/1/012044.

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Abstract Forming Limit Curve (FLC) has been widely adopted as a practical criterion for evaluating the formability of sheet metals. Predicting a reliable FLC by a virtual methodology could lead to robust process optimization before expensive tool manufacturing. In order to increase the predictive capabilities of the virtual forming tools, an accurate modeling of the forming limit curve should be considered at room and elevated temperatures. In this work, the isothermal forming limit curves of 6000 series aluminum alloy sheet metal are predicted by performing numerical simulations of Nakajima test. A stress triaxiality and Lode angle based ductile fracture criterion is used to determine the forming limit curve. Also, the ductile fracture criterion is extended to add the impact of temperature on ductile fracture prediction. The hybrid experimental-numerical approach is used to calibrate the ductile fracture criterion. The forming limit curve of AA6061-T6 sheet metal, with a thickness of 1 mm, is predicted using the calibrated ductile fracture criterion at room and elevated temperatures. Numerical simulations are performed in 3D with the finite element code Abaqus. The limit strains are determined for specimens undergoing deformation under different strain paths. Influence of temperature on the predicted forming limit curve is discussed.

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Yuan, Huang, Guoyu Lin, and Alfred Cornec. "Verification of a Cohesive Zone Model for Ductile Fracture." Journal of Engineering Materials and Technology 118, no. 2 (1996): 192–200. http://dx.doi.org/10.1115/1.2804886.

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In the present paper, ductile crack growth in an aluminium alloy is numerically simulated using a cohesive zone model under both plane stress and plane strain conditions for two different fracture types, shear and normal modes. The cohesive law for ductile fracture consists of two parts—a specific material’s separation traction and energy. Both are assumed to be constant during ductile fracture (stable crack growth). In order to verify the assumed cohesive law to be suitable for ductile fracture processes, experimental records are used as control curves for the numerical simulations. For a constant separation traction, determined experimentally from tension test data, the corresponding cohesive energy was determined by finite element calculations. It is confirmed that the cohesive zone model can be used to characterize a single ductile fracture mode and is roughly independent of stable crack extention. Both the cohesive traction and the cohesive fracture energy should be material specific parameters. The extension of the cohesive zone is restricted to a very small region near the crack tip and is in the order of the physical fracture process. Based on the present observations, the cohesive zone model is a promising criterion to characterize ductile fracture.

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Toan, Nguyen Duc, Nguyen Trong Hung, Bui Ngoc Tuyen, and Nguyen Tien Dong. "FEM STUDY TO VERIFY THE EFFECT OF EMBOSSING AND WAVE SHAPES ON FORMABILITY OF STAMPING PROCESS FOR MULTI-HOLE ETCHING METAL FOIL USING SUS316L MATERIAL." ASEAN Engineering Journal 2, no. 2 (2012): 43–50. http://dx.doi.org/10.11113/aej.v2.15348.

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In current study, to predict ductile fracture and spring-back in making embossing shape using multi-hole etching metal foil of SUS316L material, FEM simulation was adopted. Ductile fracture criterion of forming limit diagram (FLD) and stress-strain curve based on experimental data were first input to ABAQUS/Explicit finite element code to predict failure and spring-back occurrences. Several simulations were then performed with the dimension changing of etching holes in order to simulate and investigate press formability of final product after forming process. The better case was obtained by utilizing simulation results. FEM simulation results were finally confirmed by the corresponding experiment. The FEM predictions were in good agreement with experimental result.

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Fang, Yuxuan, Biao Li, Chang-Sung Seok, and Tao Shen. "The Application of Numerical Ductile Fracture Simulation in the LBB Evaluation of Nuclear Pipes." Applied Sciences 15, no. 13 (2025): 7010. https://doi.org/10.3390/app15137010.

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The leak-before-break (LBB) concept is widely used in the design and estimation of piping systems of nuclear power plants, which requires considerable test work to obtain the fracture resistance (J-R) curves of nuclear pipes. The application of numerical ductile fracture simulation can effectively limit the test work. In this study, an extended stress-modified critical strain (SMCS) model is applied to simulate the crack growth behaviors of full-scale nuclear pipes (SA312 TP304L stainless steel) with a circumferential through-wall crack under a four-point bending load. The LBB evaluation is performed based on the J-R curves of CT specimens and full-scale pipes obtained from fracture resistance tests and numerical simulations. It shows that due to the high crack-tip constraint effect, CT specimens may cause lots of conservatism in the LBB evaluation of nuclear pipes, while the application of numerical ductile fracture simulation can largely reduce the conservatism.

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Pour Mahmoud, N., and A. Zabihi. "Numerical Simulation of a Single-Phase Flow Through Fractures with Permeable, Porous and Non-Ductile Walls." Engineering, Technology & Applied Science Research 7, no. 5 (2017): 2041–46. http://dx.doi.org/10.48084/etasr.1448.

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This paper attempts to study flows within fractures through a set of numerical simulations. In addition, a special care is given to hydraulic features and characteristics of fractures. The research is performed through the application of calculative fluid dynamics and a finite volume discrete schema. The investigated flows are laminar, single-phase and stable flows of water and air through fractures with penetrable walls. The selected fracture geometry is inspired from the tomographic scan of a stone fracture. Water and air are modeled in fractures with permeable walls and different permeability levels. It has been observed that in case of permeable matrixes, the friction coefficient is lower compared to impermeable matrixes. In fact permeability reduced friction. In addition, highest pressure drops were observed in areas with smaller fracture diaphragms. Nonetheless, the surrounding area of the fracture is analyzed with the consideration of Darcy's rule.

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Pour, Mahmoud N., and A. Zabihi. "Numerical Simulation of a Single-Phase Flow Through Fractures with Permeable, Porous and Non-Ductile Walls." Engineering, Technology & Applied Science Research 7, no. 5 (2017): 2041–46. https://doi.org/10.5281/zenodo.1037234.

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Guo, Junhang, Ri-ichi Murakami, and Shengdun Zhao. "A STUDY ON EXPERIMENTS AND SIMULATIONS FOR DUCTILE FRACTURE OF ISOTROPIC MATERIAL USING ROUSSELIER'S DAMAGE MODEL." International Journal of Modern Physics: Conference Series 06 (January 2012): 257–62. http://dx.doi.org/10.1142/s2010194512003273.

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Ductile fracture has been a hot topic for a long time for its importance to mechanical design in evaluating the risk of failure. In this paper, the A5052BD-H14's ductile fracture is studied using a new constitutive equation based on the continuum damage mechanics. A novel full-implicit stress integration algorithm is developed based on Rousselier's damage model and implemented into finite element analysis (FEA) models by the ABAQUS/Explicit using the user material subroutine. The tensile tests of A5052BD-H14 with notch were taken and the load-displacement curves were recorded. By simulations, the evolutions of the void volume fraction are obtained and can be used as calibration for the critical void volume fraction. The validity of the damage model and the proposed stress integration algorithm are verified by comparing the experimental results and the simulation results. Further, by using the critical void volume fraction and element deletion, the simulation results show that this method is reliable, and can be used to predict the fracture of metals.

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Kvačkaj, Tibor, Juraj Tiža, Július Bacsó, et al. "Cockcroft-Latham Ductile Fracture Criteria for Non Ferrous Materials." Materials Science Forum 782 (April 2014): 373–78. http://dx.doi.org/10.4028/www.scientific.net/msf.782.373.

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The determination of ductile fracture criteria as well as friction coefficient, stress-strain curves, constants for Hollomon's equation and a material workability based on analytical methods as a forming limit diagram, a normalized Cockcroft-Latham criteria (nCL)) ring and compression tests for two materials based on aluminum and copper alloys were carried out. A calculation of nCL criteria on the basis of a compression test and numerical simulations was made. The critical values nCL criteria resulting from compression test were determined. Prediction of nCL criteria by numerical simulations were confirmed by laboratory compression tests. The values obtained from numerical simulations and compression tests for both materials show a good coincidence in results.

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Eom, Ji-Ho, Chul Kyu Jin, Dae-Young Ahn, JSS Babu, Jun-Young Jang, and Min Sik Lee. "Comparison of FE Simulation and Experiment on Tensile Test of TWB-HPF 22MnB5 Steel." Metals 14, no. 10 (2024): 1176. http://dx.doi.org/10.3390/met14101176.

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Finite element (FE) analysis of the tensile test of TWB-HPF 22MnB5 steel was performed and compared with the experimental results. To improve the accuracy of the simulation, the damage theory of FLD and ductile damage theory were used in 2D and 3D simulations. The tensile strength of 22MnB5 steel was determined under various welding heat inputs for FE simulation. Crack propagation of the welded region indicated that the fracture was observed in the base metal under normal welding conditions. Also, the crack propagated along the HAZ region due to higher heat input of the welding, and lead fractures have been highlighted as a potential complication.

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Keralavarma, Shyam M. "A multi-surface plasticity model for ductile fracture simulations." Journal of the Mechanics and Physics of Solids 103 (June 2017): 100–120. http://dx.doi.org/10.1016/j.jmps.2017.03.005.

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Quan, Guo-zheng, Gui-chang Luo, An Mao, Jian-ting Liang, and Dong-sen Wu. "Evaluation of Varying Ductile Fracture Criteria for 42CrMo Steel by Compressions at Different Temperatures and Strain Rates." Scientific World Journal 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/579328.

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Fracturing by ductile damage occurs quite naturally in metal forming processes, and ductile fracture of strain-softening alloy, here 42CrMo steel, cannot be evaluated through simple procedures such as tension testing. Under these circumstances, it is very significant and economical to find a way to evaluate the ductile fracture criteria (DFC) and identify the relationships between damage evolution and deformation conditions. Under the guidance of the Cockcroft-Latham fracture criteria, an innovative approach involving hot compression tests, numerical simulations, and mathematic computations provides mutual support to evaluate ductile damage cumulating process and DFC diagram along with deformation conditions, which has not been expounded by Cockcroft and Latham. The results show that the maximum damage value appears in the region of upsetting drum, while the minimal value appears in the middle region. Furthermore, DFC of 42CrMo steel at temperature range of 1123~1348 K and strain rate of 0.01~10 s-1are not constant but change in a range of 0.160~0.226; thus, they have been defined as varying ductile fracture criteria (VDFC) and characterized by a function of temperature and strain rate. In bulk forming operations, VDFC help technicians to choose suitable process parameters and avoid the occurrence of fracture.

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Ben Chabane, Nassima, Nassim Aguechari, and Mohand Ould Ouali. "Study of the slant fracture in solid and hollow cylinders: Experimental analysis and numerical prediction." Frattura ed Integrità Strutturale 17, no. 63 (2022): 169–89. http://dx.doi.org/10.3221/igf-esis.63.15.

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This paper is devoted to the numerical and experimental study of ductile fracture in bulk metal forming of the 2017A-T4 aluminum alloy. From an experimental standpoint, the ductile fracture of the 2017A-T4 aluminum alloy is investigated under compressive load. Two cross-sections of solid and hollow specimens are considered. The mechanical behavior and the microstructure of the 2017A-T4 aluminum alloy were characterized. It is found that the well-known barrel shape is obtained when a compressive load is applied. Analyses of fracture topographies show a ductile fracture with dimples under tension and coexistence of ductile fracture with dimples and slant under compression. The classical physically-based Gurson-Tvergaard-Needleman (GTN) model and its extension to incorporate shear mechanisms to predict failure at low-stress triaxiality are considered. These two models have been extended to take into account the thermal heating effect induced by the mechanical dissipation within the material during the metal forming process. The two models have been implemented into the finite element code Abaqus/Explicit using a Vectorized User MATerial (VUMAT) subroutine. Numerical simulations of the forging process made for hollow and solid cylindrical specimens show good agreement with experimental results. In contrast with the GTN model, the modified GTN model incorporating shear mechanisms can capture the final material failure.

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Rahafrooz, M., M. Sanjari, M. Moradi, and Danial Ghodsiyeh. "Prediction of Rupture in Gas Forming Process Using Continuum Damage Mechanic." Advanced Materials Research 463-464 (February 2012): 1047–51. http://dx.doi.org/10.4028/www.scientific.net/amr.463-464.1047.

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The Continuum Damage Mechanics is a branch of applied mechanics that used to predict the initiation of cracks in metal forming process. In this article, damage definition and ductile damage model are explained, and also ductile damage model is applied to predict initiation of fracture in gas metal forming process with ABAQUS/EXPLICIT simulation. In this method instead of punch, the force is applied by air pressure. In this study, first the ductile damage criterion and its relations are taken into account and, subsequently, the process of gas-aid formation process is put into consideration and ductile damage model for prediction of rupture area is simulated using ABAQUS simulation software. Eventually, the process of formation via gas on the aluminum with total thickness of 0.24 [mm] was experimentally investigated and the results acquired from experiment were compared with relating simulations. The effect of various parameters such as radius of edge matrix, gas pressure and blank temperature has been evaluated. Simulation was compared with experimental results and good agreement was observed.

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Goijaerts, A. M., L. E. Govaert, and F. P. T. Baaijens. "Prediction of Ductile Fracture in Metal Blanking." Journal of Manufacturing Science and Engineering 122, no. 3 (1999): 476–83. http://dx.doi.org/10.1115/1.1285909.

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This study is focused on the description of ductile fracture initiation, which is needed to predict product shapes in the blanking process. Two approaches are elaborated using a local ductile fracture model. According to literature, characterization of such a model should take place under loading conditions, comparable to the application. Therefore, the first approach incorporates the characterization of a ductile fracture model in a blanking experiment. The second approach is more favorable for industry. In this approach a tensile test is used to characterize the fracture model, instead of a complex and elaborate blanking experiment. Finite element simulations and blanking experiments are performed for five different clearances to validate both approaches. In conclusion it can be stated that for the investigated material, the first approach gives very good results within the experimental error. The second approach, the more favorable one for industry, yields results within 6 percent of the experiments over a wide, industrial range of clearances, when a newly proposed criterion is used. [S1087-1357(00)02202-4]

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Cai, Wei, Zhihui Zhou, Xudong Qian, et al. "Numerical Study on Ductile Failure Behaviours of Steel Structures under Quasi-Static Punch Loading." Journal of Marine Science and Engineering 11, no. 6 (2023): 1197. http://dx.doi.org/10.3390/jmse11061197.

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A reliable finite element procedure to simulate shear-dominated ductile fractures in large-scale, thin-walled steel structures is still evolving primarily due to the challenges in determining the failure criterion of metal materials under complex stress states. This paper aims to examine the accuracy of the modified Gurson–Tvergaard–Needleman (GTN) model considering the shear failure in simulating the ductile fracture of steel plate structures under quasi-static punch loading. The modified GTN damage models are performed by the ABAQUS user-defined material subroutine (VUMAT). The void-related parameters and shear damage parameters of Xue’s and the N-H modified GTN models are calibrated from test specimens with various geometries corresponding to different stress triaxiality and shearing conditions. The damage evolution associated with shearing of voids in the modified GTN models has strong influences on the stress triaxiality versus plastic strain under complex stress states, especially for the shear-dominated loading conditions. Based on the original GTN model, Xue’s and the N-H modified GTN model with calibrated material parameters, a numerical comparative study examines the ductile fracture of steel non-stiffened plates and stiffened plates under punch loading. Benchmarked against the experimental studies, the numerical simulations demonstrate that the shear-driven void evolution in the modified GTN model imposes significant effects on the load–displacement responses as well as the onset and extension of ductile fractures in steel plates under punch actions. The N-H modified model with calibrated shear damage parameters shows a better correlation with the ductile fractures in steel plates observed in the experiment than the original GTN model and Xue’s modified GTN model. As a result of this study, the modified GTN model considering shear action can be applied for practical applications in the crashworthiness assessment of ship collision and grounding.

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Lu, Yue, Kun Liu, Zili Wang, Wenyong Tang, and Jørgen Amdahl. "Development of ductile fracture modelling approach in ship impact simulations." Ocean Engineering 252 (May 2022): 111173. http://dx.doi.org/10.1016/j.oceaneng.2022.111173.

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Kang, K., and W. Cai. "Brittle and ductile fracture of semiconductor nanowires – molecular dynamics simulations." Philosophical Magazine 87, no. 14-15 (2007): 2169–89. http://dx.doi.org/10.1080/14786430701222739.

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SUGIYAMA, Hirofumi, Kazumi MATSUI, and Takahiro YAMADA. "Ductile fracture simulations by damage model and finite cover method." Proceedings of The Computational Mechanics Conference 2016.29 (2016): 4_210. http://dx.doi.org/10.1299/jsmecmd.2016.29.4_210.

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Anvari, Majid, Jun Liu, and Christian Thaulow. "Dynamic ductile fracture in aluminum round bars: experiments and simulations." International Journal of Fracture 143, no. 4 (2007): 317–32. http://dx.doi.org/10.1007/s10704-007-9062-9.

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Guo, Jun Hang, Ri Ichi Murakami, and Sheng Dun Zhao. "Simulation of Ductile Fracture in an Aluminum Alloy Using Various Criteria." Advanced Materials Research 560-561 (August 2012): 973–78. http://dx.doi.org/10.4028/www.scientific.net/amr.560-561.973.

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There are many models and failure criteria have been developed to predict the ductile fracture (DF) in metal plastic deformation. But usually, it is difficult to select a suitable model and the corresponding criterion from them. So, finding a way to identity their applicability and reliability is useful for selecting these DF criteria. In this paper, ductile fracture of aluminum alloy A5052P-H34 is studied by experiments and finite element simulations. In experiments, the mode I crack was obtained by uniaxial tension of plate with a circular hole in the center. The von Mises yield model and continuum damage mechanics based Rousselier model and modified Rousselier model are chosen to describe the material behavior. Three failure criteria, including the Cockcroft-Latham integral, maximum shear stress theory and critical void volume fraction criterion are investigated to determine their reliability in ductile failure prediction. These constitutive models and DF criteria are implemented by user material subroutine in ABAQUS/Explicit to predict the crack. And the crack initiation and propagation is implemented by element erosion method. By comparing the experiments and simulations, the modified Rousselier’s model with the corresponding criterion shows agreement with the experiments.

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Oh, Chang Kyun, Yun Jae Kim, Jong Hyun Baek, Young Pyo Kim, and Woo Sik Kim. "A Micromechanical Model for Ductile Fracture of API X65." Key Engineering Materials 321-323 (October 2006): 43–47. http://dx.doi.org/10.4028/www.scientific.net/kem.321-323.43.

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This paper presents a micro-mechanical model of ductile fracture for the API X65 steel, using the Gurson-Tvergaard-Needleman (GTN) model. Experimental tests and FE damage simulations using the GTN model are performed for smooth and notched tensile bars with three different notch radii, from which micromechanical parameters in the GTN model are calibrated. The calibrated micro-mechanical model is applied to quantify pre-strain effects on plastic deformation and fracture of the API X65 steel. Good agreements of the FE damage results with experimental data suggest confidence in the use of the proposed micro-mechanical model to simulate ductile failure of pipelines made of API X65 steels.

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Qian, L., Hiroyuki Toda, Kentaro Uesugi, Masakazu Kobayashi, and Toshiro Kobayashi. "3D Image-Based Modeling of Ductile Fracture in an Aluminum Alloy Using Synchrotron X-Ray CT Images." Materials Science Forum 561-565 (October 2007): 263–66. http://dx.doi.org/10.4028/www.scientific.net/msf.561-565.263.

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Traditional computational models always assume idealized crack geometry. However, actual crack geometry is very complex in real materials and thus, those simulations do not realistically represent the actual loading conditions of a real crack. In this paper, three-dimensional (3D) image-based simulation was performed to investigate the fracture behavior of an aluminum alloy, and the model takes into account the real crack geometry based on the 3D images of the crack. Accordingly, many essential features of fracture can be identified and interpreted, and some new insight into fracture behavior in real materials can be offered.

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Zhuang, Xincun, Yehui Meng, and Zhen Zhao. "Evaluation of prediction error resulting from using average state variables in the calibration of ductile fracture criterion." International Journal of Damage Mechanics 27, no. 8 (2017): 1231–51. http://dx.doi.org/10.1177/1056789517728563.

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In order to evaluate the prediction error resulting from using average state variables in the calibration procedure of the ductile fracture criterion, a series of experiments and corresponding simulations were performed to extract the evolution of fracture-related state variables such as stress triaxiality (η), Lode parameter, and equivalent strain to fracture at the fracture initiation points. The average stress triaxiality, average Lode parameter, and equivalent strain to fracture were used to calibrate the Lou-Huh (L-H) ductile fracture criterion. The average induced prediction error was evaluated by comparing the accumulated damage value, which was computed with the calibrated L-H ductile fracture criterion at the fracture initiation point, with the critical threshold value. Comparisons based on a series of experiments covering a wide range of values for stress triaxiality indicated the existence of an average induced prediction error for the compression tests, and demonstrated that different values of embedded-constants C1 and C2 of L-H ductile fracture criterion resulted in entirely different average induced prediction errors. Thus, a parameter study was performed to investigate the influences of C1, C2, the relationship of η and equivalent plastic strain ([Formula: see text]), and the internal function of the integral formula on the average induced relative error. The influence of the relationship of [Formula: see text] could be represented by the influence of the exponent a, the intercept for the stress triaxiality, and the allocation of equivalent strain for the segmented function. Among these influence factors, the value of C2, the value of the exponent a, and the value of the negative intercept for stress triaxiality contributed significantly to an increase in relative error.

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Haskul, Mehmet, and Eray Arslan. "Fracture Initiation in Aluminum Alloys Under Multiaxial Loading at Various Low Strain Rates." Metals 15, no. 7 (2025): 785. https://doi.org/10.3390/met15070785.

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The initiation of ductile fractures in medium-strength AW5754 and high-strength AW6082 aluminum alloys at different quasi-static strain rates and under multiaxial stress states was investigated through a series of tensile tests using various specimen geometries. The sensitivity of the stress triaxiality locus to variations in the loading rate was examined for these two aluminum alloy families. Fractographic and elemental analyses were also conducted via SEM and EDS. Numerical simulations based on the finite element method (FEM) were performed using ABAQUS/Standard to determine the actual stress triaxialities and the equivalent plastic strains at fracture. The numerical approach was validated by comparing the simulation results with the experimental findings. These simulations facilitated the generation of a stress triaxiality locus through a curve-fitting process. Among the considered fitting functions, an exponential function was selected as it provided the most accurate relation between the equivalent plastic strain at fracture and the corresponding stress state across different strain rates. The results reveal different strain rate dependencies for the two alloys within a very low strain rate range. The resulting stress triaxiality loci provide a valuable tool for predicting fracture strains and for more accurately evaluating stress states. Overall, the findings of this study significantly advance the understanding of the fracture initiation behavior of aluminum alloys under multiaxial loading conditions and their sensitivity to various quasi-static loading rates.

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Zhao, Lianxing, Xiaotao Fei, Chaifeng Sun, Peng Liu, and Di Li. "The Study and Application on Ductile Fracture Criterion of Dual Phase Steels During Forming." Metals 14, no. 11 (2024): 1301. http://dx.doi.org/10.3390/met14111301.

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High-strength steel exhibits complex fracture behavior due to the interplay between shear and necking mechanisms during stamping and forming processes, posing challenges to achieving the dimensional accuracy and reliability demanded for automotive body panels. Existing prediction methods often fail to simultaneously account for both tensile and shear fracture characteristics, thereby limiting their predictive accuracy under diverse stress conditions. To address this limitation, we propose a ductile fracture criterion that integrates both tensile and shear mechanisms, calibrated using a single tensile–shear test to facilitate practical engineering applications. This study investigates the fracture characteristics of DP780 dual-phase steel through numerical analysis and tensile–shear experiments. The findings establish a relationship between stress triaxiality and ultimate fracture strain across varying stress states, represented by the B–W curve. Simulations reveal distinct stress triaxiality behaviors under different loading conditions: under uniaxial tensile loading, triaxiality ranges from 0.33 to 0.6, with fracture strain decreasing monotonically as triaxiality increases. Under shear loading, triaxiality ranges from 0 to 0.33, with fracture strain increasing monotonically as triaxiality rises. Additional bending simulations validate that this criterion, along with the B–W curve, reliably predicts the fracture behavior of DP780, offering an effective tool for predicting fracture in dual-phase steels during stamping and forming processes.

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Hentati, Hamdi, Ilyes Ben Naceur, Wassila Bouzid, and Aref Maalej. "Numerical Analysis of Damage Thermo-Mechanical Models." Advances in Applied Mathematics and Mechanics 7, no. 5 (2015): 625–43. http://dx.doi.org/10.4208/aamm.2014.m517.

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AbstractIn this paper, we present numerical computational methods for solving the fracture problem in brittle and ductile materials with no prior knowledge of the topology of crack path. Moreover, these methods are capable of modeling the crack initiation. We perform numerical simulations of pieces of brittle material based on global approach and taken into account the thermal effect in crack propagation. On the other hand, we propose also a numerical method for solving the fracture problem in a ductile material based on elements deletion method and also using thermo-mechanical behavior and damage laws. In order to achieve the last purpose, we simulate the orthogonal cutting process.

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Chuzhoy, L., R. E. DeVor, S. G. Kapoor, A. J. Beaudoin, and D. J. Bammann. "Machining Simulation of Ductile Iron and Its Constituents, Part 1: Estimation of Material Model Parameters and Their Validation." Journal of Manufacturing Science and Engineering 125, no. 2 (2003): 181–91. http://dx.doi.org/10.1115/1.1557294.

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A microstructure-level simulation model was recently developed to characterize machining behavior of heterogeneous materials. During machining of heterogeneous materials such as cast iron, the material around the machining-affected zone undergoes reverse loading, which manifests itself in permanent material softening. In addition, cracks are formed below and ahead of the tool. To accurately simulate machining of heterogeneous materials the microstructure-level model has to reproduce the effect of material softening on reverse loading (MSRL effect) and material damage. This paper describes procedures used to calculate the material behavior parameters for the aforementioned phenomena. To calculate the parameters associated with the MSRL effect, uniaxial reverse loading experiments and simulations were conducted using individual constituents of ductile iron. The material model was validated with reverse loading experiments of ductile iron specimens. To determine the parameters associated with fracture of each constituent, experiments and simulation of notched specimens are performed. During the validation stage, response of simulated ductile iron was in good agreement with the experimental data.

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Wciślik, Wiktor, and Robert Pała. "Some Microstructural Aspects of Ductile Fracture of Metals." Materials 14, no. 15 (2021): 4321. http://dx.doi.org/10.3390/ma14154321.

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The paper discusses the basic issues of the local approach to ductile fracture of structural metals, with particular emphasis on the failure due to microvoid development. The mechanisms of nucleation of voids around inclusions and precipitates are characterized. The criteria for the nucleation of voids resulting from cracking of the existing particles or their separation from the material matrix are presented. Selected results of experimental studies and Finite Element Method (FEM) simulations on nucleation of voids are discussed. The analytical and numerical models of growth and coalescence of voids are described, indicating the effect of the stress state components on the morphology of voids and the course of the cracking on a microscopic scale.

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Grolleau, Vincent, Vincent Lafilé, Christian C. Roth, Bertrand Galpin, Laurent Mahéo, and Dirk Mohr. "Rate-dependent ductile fracture under plane strain tension: experiments and simulations." EPJ Web of Conferences 183 (2018): 02022. http://dx.doi.org/10.1051/epjconf/201818302022.

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Among all other stress states achievable under plane stress conditions, the lowest ductility is consistently observed for plane strain tension. For static loading conditions, V-bending of small sheet coupons is the most reliable way of characterising the strain to fracture for plane strain tension. Different from conventional notched tension specimens, necking is suppressed during V-bending which results in a remarkably constant stress state all the way until fracture initiation. The present DYMAT talk is concerned with the extension of the V-bending technique from low to high strain rate experiments. A new technique is designed with the help of finite element simulations. It makes use of modified Nakazima specimens that are subjected to V-bending. Irrespective of the loading velocity, plane strain tension conditions are maintained throughout the entire loading history up to fracture initiation. Experiments are performed on specimens extracted from aluminum 2024-T3 and dual phase DP450 steel sheets. The experimental program includes quasi static loading conditions which are achieved on a universal testing machine. In addition, high strain rate experiments are performed using a specially-designed drop tower system. In all experiments, images are acquired with two cameras to determine the surface strain history through stereo Digital Image Correlation (DIC). The experimental observations are discussed in detail and also compared with the numerical simulations to validate the proposed experimental technique

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Vladimirov, Ivaylo N., Michael P. Pietryga, Yalin Kiliclar, Vivian Tini, and Stefanie Reese. "Failure modelling in metal forming by means of an anisotropic hyperelastic-plasticity model with damage." International Journal of Damage Mechanics 23, no. 8 (2014): 1096–132. http://dx.doi.org/10.1177/1056789513518953.

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In metal forming, formability is limited by the evolution of ductile damage in the work piece. The accurate prediction of material failure requires, in addition to the description of anisotropic plasticity, the inclusion of damage in the finite element simulation. This paper discusses the application of an anisotropic hyperelastic-plasticity model with isotropic damage to the numerical simulation of fracture limits in metal forming. The model incorporates plastic anisotropy, nonlinear kinematic and isotropic hardening and ductile damage. The constitutive equations of the proposed model are numerically integrated both implicitly and explicitly, and the model is implemented as a user material subroutine UMAT in the commercial solvers ABAQUS/Standard and LS-DYNA, respectively. The numerical examples investigate the potential of the constitutive framework regarding the prediction of failure in metal forming processes such as, e.g. cross-die deep drawing. In particular, simulations of the Nakazima stretching test with varying specimen geometry are utilized to simulate the forming limit diagram at fracture and the numerical results are compared to experimental data for aluminium alloy sheets.

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Amri, A. El, M. H. El Yakhloufi, and A. Khamlichi. "Identification Damage Model for Thermomechanical Degradation of Ductile Heterogeneous Materials." International Journal of Applied Mechanics and Engineering 22, no. 2 (2017): 475–81. http://dx.doi.org/10.1515/ijame-2017-0031.

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AbstractThe failure of ductile materials subject to high thermal and mechanical loading rates is notably affected by material inertia. The mechanisms of fatigue-crack propagation are examined with particular emphasis on the similarities and differences between cyclic crack growth in ductile materials, such as metals, and corresponding behavior in brittle materials, such as intermetallic and ceramics. Numerical simulations of crack propagation in a cylindrical specimen demonstrate that the proposed method provides an effective means to simulate ductile fracture in large scale cylindrical structures with engineering accuracy. The influence of damage on the intensity of the destruction of materials is studied as well.

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Charoensuk, Kritchanan, and Viton Uthaisangsuk. "Determination of 3D Ductile Failure Criteria for Advanced High Strength Steel Sheet." Key Engineering Materials 658 (July 2015): 53–58. http://dx.doi.org/10.4028/www.scientific.net/kem.658.53.

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In this work, 3D ductile fracture locus was determined for the advanced high strength (AHS) steel sheet grade DP780 using a hybrid approach between experiment and FE simulation. Tensile tests of different sample geometries were performed for the investigated dual phase steel, by which varying stress triaxiality (η) and lode angle (θ) values developed in the material during loading were introduced. During the tests, the direct current potential drop (DCPD) method and digital image correlation (DIC) technique were applied for identifying crack initiation on the micro-scale and fracture of the specimens due to local plastic deformation. Obtained force and displacement curves were correlated with the electric potential curves. Then, the moments of crack onset were determined for various states of stress. In parallel, the most critical areas of deformed samples before fracture were observed by the DIC method. Subsequently, FE simulations of the tensile tests were carried out and calculated local stresses and strains were gathered. The stress triaxialities, equivalent plastic strains and lode angles were evaluated for the corresponding detected areas. These threshold variables obtained from different specimens were plotted as the 3D failure locus for defining crack initiation and fracture occurrence in the DP steel.

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Bao, Zeying, та Fulin Shang. "New Insights on the Tensile Strength and Fracture Mechanism of c-ZrO2/α-Al2O3 Interfaces". Applied Sciences 13, № 6 (2023): 3742. http://dx.doi.org/10.3390/app13063742.

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The tensile strength and fracture properties of the c-ZrO2(001)/α-Al2O3(11¯02) interfaces were investigated by first-principle tensile simulations. Models with different stacking sequences of c-ZrO2(001) were examined. The theoretical tensile strength and work of adhesion were present. It was found that the adhesive strength of the interface was strongly influenced by the termination of c-ZrO2(001), and the c-ZrO2(001)/α-Al2O3(11¯02) interfaces adhered weakly. Then, variations of the atomic bonds were observed to clarify the fracture characteristics of the interfaces. Our study indicates that the fracture modes of the O- and Zr-model tend to be ductile fractures, while the fracture mode of the 2O-model is a brittle fracture. Furthermore, all three models were completely separated along the intermediate layer between the initial ZrO2 and Al2O3 slabs. Finally, we compared our results with those available in the published literature, and the potential application of the first-principle results will be further discussed.

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Spielmannová, Alena, Anna Machová, and Petr Hora. "Crack Orientation versus Ductile-Brittle Behavior in 3D Atomistic Simulations." Materials Science Forum 567-568 (December 2007): 61–64. http://dx.doi.org/10.4028/www.scientific.net/msf.567-568.61.

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The paper presents results of molecular dynamic (MD) simulations in 3D bcc iron crystals with edge pre-existing cracks (001)[110] and (110) [110] (crack plane/crack front) loaded uni-axially in tension mode I at temperature of 300 K. The iron crystals in MD have the same orientation and similar geometry as in our recent fracture tests performed at room temperature on iron (3wt.%Si) single crystals [1].

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Bolli, Eleonora, Alessandra Fava, Paolo Ferro, et al. "Cr Segregation and Impact Fracture in a Martensitic Stainless Steel." Coatings 10, no. 9 (2020): 843. http://dx.doi.org/10.3390/coatings10090843.

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The fracture surfaces of a 10.5 wt.% Cr martensitic stainless steel broken in Charpy tests have been investigated through X-ray photoelectron spectroscopy (XPS). The specimens have been examined in two different conditions: as-quenched and heat treated for 10 h at 700 °C. The trends of Fe/Cr ratio vs. test temperature are similar to the sigmoidal curves of absorbed energy and, after both ductile and quasi-cleavage brittle fractures, such ratio is always significantly lower than the nominal value of the steel chemical composition. Cr segregation does not occur on a macroscopic scale but takes place in microscopic zones which represent weaker spots in the steel matrix and a preferred path for moving cracks. Small area (diameter 300 µm) XPS measurements evidenced a higher density of such microscopic zones in the inner part of probes; this is explained by the different diffusion length of Cr atoms in the external and inner parts during quenching from austenitic field which has been calculated through FEM simulations. No significant differences of Cr concentration were observed in fracture surfaces of probes with and without heat treatment. The results highlight how Cr segregation plays a role not only in the intergranular mode of fracture but also in the quasi-cleavage and ductile ones.

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Zhang, Hua, Hong Zhang, Fuguo Li, and Jun Cao. "A Novel Damage Model to Predict Ductile Fracture Behavior for Anisotropic Sheet Metal." Metals 9, no. 5 (2019): 595. http://dx.doi.org/10.3390/met9050595.

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The purpose of the present work is to investigate the fracture behavior of anisotropic sheet metal under various stress states. Notched tension and flat-grooved tension tests at 0°, 45°, and 90° directions with respect to rolling direction were carried out by a hybrid experimental–numerical approach, and then a novel damage model was proposed by coupling Hill48’s criterion. Based on this, finite element method (FEM) analysis models were established. The force–displacement responses of experiments and simulations are in good agreement, which verify the FEM models. The predictability of the damage model established for the fracture behavior of anisotropic materials was studied by comparing the fracture displacements between experiments and simulations. It is found that the predictability of novel damage model is basically consistent with predictive results. The difference of damage locations and local strain evolutions at a 45° direction is greater than the other directions. In addition, stress triaxiality does not play a predominant role in the fracture process for notched tension specimens, while it does play a predominant role for flat-grooved tension specimens.

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Cerik, Burak Can, and Joonmo Choung. "Fracture Prediction of Steel-Plated Structures under Low-Velocity Impact." Journal of Marine Science and Engineering 11, no. 4 (2023): 699. http://dx.doi.org/10.3390/jmse11040699.

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In this paper, a validation study of a recently proposed rate-dependent shell element fracture model using quasi-static and dynamic impact tests on square hollow sections (SHS) made from offshore high-tensile strength steel was presented. A rate-dependent forming limit curve was used to predict the membrane loading-dominated failure, while a rate-dependent ductile fracture locus was applied for predicting failure governed by bend loading. The predicted peak force and fracture initiation using the adopted material and fracture model agreed well with the experimental results. The fracture mode was also captured accurately. Further simulations were performed to discuss the importance of the inclusion of dynamic effects and the separate treatment of failure modes. Finally, the shortcomings of the common practice of treatment of rate-effects in low-velocity impact simulations involving fracture were highlighted.

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SUGIYAMA, Hirofumi, Kazumi MATSUI, Takahiro YAMADA, and Shigenobu OKAZAWA. "Ductile fracture simulations by damage model based on continuous damage mechanics." Proceedings of The Computational Mechanics Conference 2017.30 (2017): 307. http://dx.doi.org/10.1299/jsmecmd.2017.30.307.

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WANG, Fangyi, Hiroki YASUFUKU, and Takehiro FUJIMOTO. "Numerical simulations for 3 point ductile metal fracture under projectile collision." Proceedings of The Computational Mechanics Conference 2019.32 (2019): 137. http://dx.doi.org/10.1299/jsmecmd.2019.32.137.

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Mediavilla, J., R. H. J. Peerlings, and M. G. D. Geers. "A robust and consistent remeshing-transfer operator for ductile fracture simulations." Computers & Structures 84, no. 8-9 (2006): 604–23. http://dx.doi.org/10.1016/j.compstruc.2005.10.007.

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