Microstructural effects on ductile fracture in heterogeneous materials. Part II: Applications to cast aluminum microstructures

This is the second of a two part paper aimed at investigating the effects of microstructural morphology, material properties and loading on rate-dependent ductile fracture of heterogeneous materials. The locally enhanced Voronoi cell finite element method (LE-VCFEM) is used for micromechanical analyses of deformation and failure in complex microstructural volume elements. The first part of this paper sequence evaluates the sensitivity of strain to failure of computer simulated microstructures to loading rate, microstructural morphology and material properties. In this second part, LE-VCFEM simulations of actual microstructures of a cast aluminum alloy micrograph are used to validate a strain to failure model developed in the first part. A method for identification of critical regions within a heterogeneous microstructure is also developed and validated using in-situ observations of a two-point bending test. The influence of applied strain rates on ductile fracture of micrograph-based complex microstructures is also investigated.     

Modeling the brittle–ductile transition in ferritic steels. Part II: analysis of scatter in fracture toughness

A dislocation simulation model has been proposed to predict the brittle–ductile transition in ferritic steels in Part I. Here we extend the model to address the problem of inherent scatter in fracture toughness measurements. We carried out a series of Monte Carlo simulations using distributions of microcracks situated on the plane of a main macrocrack. Detailed statistical analysis of the simulation results showed the following: (a) fracture is initiated at one of the microcracks whose size is at the tail of the size distribution function, and (b) the inherent scatter arises from the distribution in the size of the critical microcrack that initiates the fracture and not from the variation of the location of the critical microcrack. Utilizing the weakest-link theory, Weibull analysis shows good agreement with the Weibull modulus values obtained from fracture toughness measurements.     

Assessing hurricane effects. Part 1. Sensitivity analysis

The Florida Commission on Hurricane Loss Projection Methodology (FCHLPM) performs an annual review of computer models that have been submitted by vendors for use in insurance rate filling in Florida. As part of the review process and to comply with the Sunshine Law, the FCHLPM employs a Professional Team to perform onsite (confidential) audits of these models. Members of the Professional Team represent the fields of actuarial science, computer science, meteorology, statistics and wind and structural engineering. The audit includes an assessment of modeler's compliance to a set of standards and modules established by the FCHLPM. One part of these standards requires the conduct of uncertainty and sensitivity analyses to the proprietary model. At the completion of the audit, the professional team provides a written report to the FCHLPM, who ultimately judges compliance by a vendor to the standards. To influence future such analyses, the Professional Team conducted a demonstration uncertainty and sensitivity analysis for the FCHLPM using a Rankine-vortex hurricane wind field model and surrogate damage function. This is the first of a two-part article presenting the results of those analyses. Part 1 presents sensitivity analysis results for wind speed and loss cost, while Part 2 presents the corresponding uncertainty analysis results.     

Aspects of ductile fracture and adaptive mesh refinement in damaged elasto‐plastic materials

Numerical simulation of elasto‐plastic problems involving multi‐fracturing materials requires a reliable failure prediction technique and a robust solution algorithm. This work approaches ductile fracture by means of continuum damage mechanics, from which two new failure criteria based on coupled and uncoupled damage analysis are derived. A two‐parameter stress update algorithm for damaged materials based on a Newton–Raphson iterative procedure is presented. A posteriori error estimators using ductile failure concepts are also discussed. Copyright © 2001 John Wiley & Sons, Ltd.     

Finite element analysis of non-local effects in ductile fracture

The paper introduces a weight function for the nonlocal analysis of ductile damage that has the potential to provide approximately uniform averaged field variables over a limited region. When this region is identified with the underlying microstructural length scale of the material, the method is consistent with the view that, once damage is significant, a damaging cell behaves as a uniform entity. The analyses through which these effects have been studied concerned a unit cell idealizing a periodic array of large voids, connected by material that is capable of nucleating smaller-scale porosity.     

Constitutive modeling of void shearing effect in ductile fracture of porous materials

Many solids, including geomaterials and commercially available metallic alloys, can be considered as a porous media. The Gurson-like model has been proposed to describe plastic deformation for such type of materials. It has attracted a great deal of attention and various modifications to this model have been proposed. The constitutive equations of Gurson-like model are governed by the first and second stress invariants and the current void volume fraction of the material. Tvergaard and Needleman included void nucleation, growth and coalescence to Gurson model in a phenomenological way [Tvergaard V, Needleman A. Analysis of the cup-cone fracture in a round tensile bar. Acta Metall 1984;32(1):157–69] – thus suggesting the so called GTN model. Meanwhile, little attention was given to the dependence of the damage evolution on the third stress invariant. McClintock et al. [McClintock FA, Kaplan SM, Berg CA. Ductile fracture by hole growth in shear bands. Int J Fract Mechan 1966;2(4):614–27] proposed damage model based on the void evolution in localized shear banding. In the present paper, a separate internal damage variable which differs from the conventional void volume fraction is introduced. The GTN model is further extended to incorporate the void shearing mechanism of damage, which depends on the third stress invariant. Numerical aspects are addressed concerning the integration of the proposed constitutive relations. A unit cell is studied to illustrate the intrinsic mechanical behavior of the modified model. Computations of the deformation in axisymmetric and transverse plane strain tension are also performed. Realistic crack modes in these simulations are achieved for the modified GTN model.     

Continuum Shape Sensitivity analysis of a mode-I fracture in functionally graded materials

This paper presents a new method for conducting a continuum shape sensitivity analysis of a crack in an isotropic, linear-elastic, functionally graded material. This method involves the material derivative concept from continuum mechanics, domain integral representation of the J-integral and direct differentiation. Unlike virtual crack extension techniques, no mesh perturbation is needed to calculate the sensitivity of stress-intensity factors. Since the governing variational equation is differentiated prior to the process of discretization, the resulting sensitivity equations are independent of approximate numerical techniques, such as the meshless method, finite element method, boundary element method, or others. In addition, since the J-integral is represented by domain integration, only the first-order sensitivity of the displacement field is needed. Several numerical examples are presented to calculate the first-order derivative of the J-integral, using the proposed method. Numerical results obtained using the proposed method are compared with the reference solutions obtained from finite-difference methods for the structural and crack geometries considered in this study.     

Geometrical relationships between side necking and plastic zone size near a crack tip in ductile metals. Part I: Modified boundary layer solutions

The plastic zone size is regarded as the measure of a material's resistance, and it also determines the fracture behaviour. Recently, stereo digital speckle photography (SDSP) has been found to be useful for measuring in-situ the side necking developed on the lateral surfaces of a specimen during a standard fracture test procedure for J_IC . Because plastic deformation occurs without any volume change, the in-plane plastic zone developed around a crack tip should be accompanied by out-of-plane deformation, that is, side necking. With the aid of the new measurement technique, side necking is expected to act as a gauge for indicating the plastic zone size. As a preliminary study, the geometrical relationships between side necking and the plastic zone size near a crack tip in ductile metals are explored by using a finite element model with modified boundary conditions. As parameters representing the geometrical similarity between side necking and the plastic zone, the shapes of each region, the distances from the crack tip to the boundaries of each region, r_p and r_s and the areas of each region, A_p and A_s are examined for their sensitivities to variables such as mode mixity, hardening exponent and so on. Among them, the areas, A_p and A_s seem to be the best for application because an excellent linearity between them is maintained in a wide range of mode mixity and load level regardless of the hardening exponent, specimen thickness and yield stress.     

Influence of ageing, inclusions and voids on ductile fracture mechanism in commercial Al-alloys

The objective of the paper is to study the effect of ageing, inclusions and voids on the mechanism of fracture and resultant toughness. It has been found that the voids are initiated at only a fraction of the larger inclusions present. The initiation of voids at small particles in the ductile fracture process appears to have little effect on fracture toughness. The strain hardening capacity has a marked effect on void size, and is an indicator of fracture toughness in the commercial Al alloy.     

Study of slant fracture in ductile materials

Slant fracture is widely observed during crack growth in thin sheet specimens made of ductile materials, providing a good case for investigating three-dimensional criteria for mixed-mode ductile fracture. To gain an understanding of slant fracture events and to provide insight for establishing a slant fracture criterion, stable tearing fracture experiments on combined tension-torsion (nominal mixed-mode I/III) specimens and nominal Mode I Arcan specimens made of Al 2024-T3 are analyzed using the finite element method under three-dimensional conditions. Two types of finite element models are considered for the study of slant fracture: (a) combined tension-torsion specimens containing stationary, flat and slant cracks subject to loads corresponding to the onset of crack growth, and (b) stable tearing crack growth with slanting in a nominal Mode I Arcan specimen. Analysis results reveal that there exists a strong correlation between certain features of the crack-front effective plastic strain field and the orientation of the slant fracture surface. In particular, it is observed that (a) at the onset of crack growth in the combined tension-torsion experiments, the angular position of the maximum effective plastic strain around the crack front serves as a good indicator for the slant fracture surface orientation during subsequent crack growth; and (b) during stable tearing crack growth in the Mode I Arcan specimen, which experiences a flat-to-slant fracture surface transition, the crack growth path on each section plane through the thickness of the specimen coincides with the angular position of the maximum effective plastic strain around the crack front.     

Modelling of localized ductile fracture with volumetric locking-free tetrahedral elements

Finite element analysis of ductile fracture with tetrahedral elements faces two numerical issues: volumetric locking and mesh sensitivity. In this paper, two widely adopted remedies for volumetric locking (F-bar and mixed field) are evaluated, and the superior performance of the mixed field method is demonstrated. Building on the mixed field formulation, a gradient enhancement is further incorporated to resolve the mesh sensitivity. It is shown that a localizing gradient enhancement can avoid a spurious spreading of damage induced by the conventional gradient approach. A locking-free, regularized ductile fracture is first presented via a uniformly tapering plate example. Finally, a shear plate test on ferrite-bainite steel is considered. Numerical results obtained with the proposed approach are shown to capture the rapid strain softening and localized shear fracture phenomenon observed experimentally.     

一种韧性断裂模型的理论研究和实验验证

为研究高强钢板成形过程中的损伤破裂机理,更准确地预测高强钢的断裂失效行为,基于细观损伤力学的空穴理论,并在屈服函数就是塑性势函数的通用性假设基础上推导了各向同性的韧性断裂模型;同时引入Lode参数以反映不同应变状态下空穴形核、长大以及聚合的差异,提出了一种包含应力三轴度和Lode参数的新模型.在Hill正交各向异性屈服假设下,描述了平面应力状态下应力比值、r值与应力三轴度、等效塑性应变的关系.最后,针对DP590进行了参数确定和实验验证.结果表明:应力三轴度在高强钢韧性断裂中仍然起主导因素,在低应力三轴下,材料主要是剪切型破坏,空穴的长大及聚合方式主要受剪应力影响,高应力三轴下,空穴损伤主要受拉应力影响,断裂是韧窝形的;Lode参数决定了应力组成形式,也间接地反映了应变状态,它与应力三轴度共同决定了空穴损伤的发展.新的模型能较准确地预测DP590的成形极限.     

Micromechanics modelling of ductile fracture in tensile specimens using computational cells

This study extends the computational cell framework to model ductile fracture behaviour in tensile specimens. In the computational cell model, ductile damage occurs through void growth and coalescence (by cell extinction) within a thin layer of material located well inside the fracture process zone for the ductile process. Laboratory testing of a high strength structural steel provides the experimental stress–strain data for round bar and circumferentially notched tensile specimens to calibrate the cell model parameters for the material. Numerical simulations employing the micromechanics model reproduce the essential features of the ductile behaviour for the tensile specimens, including the development of intense necking and void growth in the centre of the specimen cross section. The resulting methodology enables the detailed study of ductile failure in small‐scale tensile specimens.     

Modelling ductile fracture behaviour from deformation parameters in HSLA steels

In this work, an attempt is made to model the ductile fracture behaviour of two Cu‐strengthened high strength low alloy (HSLA) steels through the understanding of their deformation behaviour. The variations in deformation behaviour are imparted by prior deformation of steels to various predetermined strains. The variations in parameters such as yield strength and true uniform elongation with prior deformation is studied and was found to be analogous to that of initiation fracture toughness determined by independent method. A unique method is used to measure the crack tip deformation characterized by stretch zone depth that also depicted a similar trend. Fracture toughness values derived from the stretch zone depth measurements were found to vary in the same fashion as the experimental values. A semiempirical relationship for obtaining ductile fracture toughness from basic deformation parameters is derived and model is demonstrated to estimate initiation ductile fracture toughness accurately.     

An implicit discontinuous Galerkin finite element framework for modeling fracture failure of ductile materials undergoing finite plastic deformation

It is a challenge to achieve a complete simulation of fracture failure in ductile materials undergoing large plastic deformation within implicit finite element frameworks due to instability issues. Currently, traditional nodal force or crack surface traction release methods target the direct release of tractions on cracked surfaces within the current time/load step. An abrupt change from a system without cracks to another system with cracks may contribute to the instability issues. Specifically, because of broken meshes, discontinuous Galerkin (DG) methods have an advantage over traditional continuous elements in naturally accommodating crack openings along DG interfaces across elements. To improve the convergence in nonlinear iterations during crack openings, we propose a relaxation scheme for DG formulations to gradually recover the traction‐free condition on cracked surfaces. Furthermore, this DG‐based relaxation scheme for crack openings in finite plastic media has been consistently formulated within the incomplete interior penalty DG framework. Finally, we have demonstrated a good performance of the proposed implicit DG formulation along with the DG relaxation scheme by successfully solving a nuclear fuel rod structural failure problem with multiple hydride crack openings and the Sandia Fracture Challenge benchmark.     

板料成形中韧性断裂准则应用研究进展

对板料成形中的成形极限应力图、最大变薄率、成形极限图以及韧性断裂准则等预测成形极限的方法,进行了综述和分析。指出利用韧性断裂准则,不但能够较好地预测塑性差的板料成形极限,而且还能考虑应变路径的变化。利用有限元方法模拟时,韧性断裂准则既可以应用到完全耦合的弹塑性损伤模型的增量方法中,也可以应用到一步有限元逆算法中。为了准确地预测成形极限,除了提高有限元模拟精度外,应找到一种本质地反映材料性能的韧性断裂准则。     

A simple damage-accumulation model for constraint effects on ductile fracture

A simple model is presented to account for the effects of void-type damage on crack initiation and propagation in ductile steels under plane strain conditions by virtue of elementary fracture mechanics solutions. Multiple primary voids from large inclusions are uniformly distributed ahead of the crack tip. The growth of these primary voids is followed by nucleation of a large population of secondary voids from second-phase particles. A critical accumulative damage based on the length ratio of the damage zone to the spacing of primary voids, is employed as a failure criterion, including contributions from two populations of voids. Damage accumulation depends much on the strain and stress states such as stress triaxilities, which are extracted from existing results instead of detailed computation. Results show the dependence of fracture toughness on the size of damage zones associated with constraints. Initiation of crack growth is insensitive to the constraints since nucleation of fine voids is determined by local deformation. The model captures the transition in mechanisms from void-by-void growth to multiple void interactions in terms of a decreasing trend in the slopes of fracture resistance curves. At high constraints and large damage zone, a steady-state crack advance is identified with constant toughness. Damage accumulation from the growth of primary voids determines subsequent crack growth resistance and the study demonstrates its dependences on the crack-tip constraints.     

A comparative study of dynamic, ductile fracture initiation in two specimen configurations

In this work a single edge notched plate (SEN(T)) subjected to a tensile stress pulse is analysed, using a 2D plane strain dynamic finite element procedure. The interaction of the notch with a pre-nucleated hole ahead of it is examined. The background material is modelled by the Gurson constitutive law and ductile failure by microvoid coalescence in the ligament connecting the notch and the hole is simulated. Both rate independent and rate dependent material behaviour is considered. The notch tip region is subjected to a range of loading rates J by varying the peak value and the rise time of the applied stress pulse. The results obtained from these simulations are compared with a three point bend (TPB) specimen subjected to impact loading analysed in an earlier work [3]. The variation of J at fracture initiation, J _c, with average loading rate J is obtained from the finite element simulations. It is found that the functional relationship between J _c and J is fairly independent of the specimen geometry and is only dependent on material behaviour.     

Constraint effects on probabilistic analysis of cracks in ductile solids

This paper presents a method for evaluating constraint effects on probabilistic elastic–plastic analysis of cracks in ductile solids. It is based on fracture parameters J and Q , correlation between Q and J– resistance curve of the material, and J -tearing theory for predicting fracture initiation and instability in cracked structures. Based on experimental data from small-scale fracture specimens, correlation equations were developed for fracture toughness at crack initiation and the slope of the J– resistance curve as a function of constraint condition. The random parameters may involve crack geometry, tensile and fracture toughness properties of the material, and applied loads. Standard reliability methods were applied to predict probabilistic fracture response and reliability of cracked structures. The results suggest that crack-tip constraints have little effect on the probability of crack initiation. However, the probability of fracture instability can be significantly reduced when constraint effects are taken into account. Hence, for a structure where some amount of stable crack-growth can be tolerated, crack-tip constraints should be considered for probabilistic fracture-mechanics analysis.     

Mixed-mode fracture of ductile thin-sheet materials under combined in-plane and out-of-plane loading

Cracks in thin structures often are subjected to combined in-plane and out-of-plane loading conditions leading to complex mixed mode conditions in the crack tip region. When applied to ductile materials, large out-of-plane displacements make both experimentation and modeling difficult. In this work, the mixed-mode behavior of thin, ductile materials containing cracks undergoing combined in-plane tension (mode I) and out-of-plane shear (mode III) deformation is investigated experimentally. Mixed-mode fracture experiments are performed and full, three-dimensional (3D) surface deformations of thin-sheet specimens from aluminum alloy and steel are acquired using 3D digital image correlation. General characteristics of the fracture process are described and quantitative results are presented, including (a) the fracture surface, (b) crack path, (c) load-displacement response, (d) 3D full-field surface displacement and strain fields prior to crack growth, (e) radial and angular distributions of the crack-tip strain fields prior to crack growth and (f) singularity analysis of the crack-tip strains prior to crack growth. Results indicate that the introduction of a mode III component to the loading process (a) alters the crack tip fields relative to those measured during nominally mode I loading and (b) significantly increases the initial and stable critical crack-opening-displacement. The data on strain fields in both AL6061-T6 aluminum and GM6208 steel consistently show that for a given strain component, the normalized angular and radial strains at all load levels can be reasonably represented by a single functional form over the range of loading considered, confirming that the strain fields in highly ductile, thin-sheet material undergoing combined in-plane tension and out-of-plane shear loading can be expressed in terms of separable angular and radial functions. For both materials, the displacement and strain fields are (a) similar for both mixed-mode loading angles Φ = 30° and Φ = 60° and (b) different from the fields measured for Mode I loading angle Φ = 0°. Relative to the radial distribution, results indicate that the in-plane strain components do not uniformly exhibit the singularity trends implicit in the HRR theory.