PLASTIC ZONE OF SEMI-INFINITE CRACK IN PLANAR KAGOME AND TRIANGULAR LATTICES

The fracture investigations of the planar lattices made of ductile cell walls are currently limited to bending-dominated hexagonal honeycomb. In this paper, the plastic zones of stretching-dominated lattices, including Kagome and triangular lattices, are estimated by analyzing their effective yield loci. The normalized in-plane yield loci of these two lattices are almost identical convex curves enclosed by 4 straight lines, which is almost independent of the relative density but is highly sensitive to the principal stress directions. Therefore, the plastic zones around the crack tip of Kagome and triangular are estimated to be quite different to those of the continuum solid and also hexagonal lattice. The plastic zones predictions by convex yield surfaces of both lattices are validated by FE calculations, although the shear lag region caused by non-local bending effect in the Kagome lattice enlarges the plastic zone in cases of small ratio of rp/l.     

A cohesive plastic and damage zone model for dynamic crack growth in rate-dependent materials

Mode I steady-state dynamic crack growth in rate-dependent viscoplastic solids containing damage, under small scale yielding conditions, is analyzed based on a modified cohesive zone model. A multi-scale approach is used to describe the entire non-linear zone consisting of a plastic region and a damage region, each of which has its own constitutive law. Traction in the damage region is characterized by a softening power-law, in terms of the ultimate strength, a softening index and a rate sensitivity factor. In the plastic region, the cohesive law is assumed to be both strain hardening and rate dependent. The critical crack opening displacement at the physical crack-tip controls crack growth. The governing integral equations are derived and solved by a collocation method combined with associated boundary conditions. Numerical results are presented for the traction and opening profiles along the cohesive zone, the fracture energy and lengths of the damage and non-linear zones at different crack speeds and for different material parameters. The importance of factors, such as material softening, plastic deformation, crack speed and viscosity, is identified by parametric studies. In addition, the competition of plastic flow and material damage, and its effect on crack growth, are discussed.     

Energy dissipation mechanisms in ductile fracture

The objective is to investigate energy dissipation mechanisms that operate at different length scales during fracture in ductile materials. A dimensional analysis is performed to identify the sets of dimensionless parameters which contribute to energy dissipation via dislocation-mediated plastic deformation at a crack tip. However, rather than using phenomenological variables such as yield stress and hardening modulus in the analysis, physical variables such as dislocation density, Burgers vector and Peierls stress are used. It is then shown via elementary arguments that the resulting dimensionless parameters can be interpreted in terms of competitions between various energy dissipation mechanisms at different length scales from the crack tip; the energy dissipations mechanisms are cleavage, crack tip dislocation nucleation and also dislocation nucleation from a Frank-Read source. Therefore, the material behavior is classified into three groups. The first two groups are the well-known intrinsic brittle and intrinsic ductile behavior. The third group is designated to be extrinsic ductile behavior for which Frank-Read dislocation nucleation is the initial energy dissipation mechanism. It is shown that a material is predicted to exhibit extrinsic ductility if the dimensionless parameter bρ_disl^1/2 (b is Burgers vector, ρ_disl is dislocation density) is within a certain range defined by other dimensionless parameters, irrespective of the competition between cleavage and crack tip dislocation nucleation. The predictions compare favorably to the documented behavior of a number of different classes of materials.     

Influence of softening on fracture propagation in large-scale shell structures

In this study, fracture propagation in large shell elements is modelled with the softening law. This law is given in a general form, enabling investigations of different softening behaviours to be conducted. The final fracture is simulated by removing elements. The softening parameters are derived using the energy-based representative volume element (RVE) approach. Tracing crack propagation through the RVE defines the physically justified softening parameters for the current model. The softening model is implemented into ABAQUS using VUMAT subroutines for the shell elements. A large-scale tearing experiment is simulated with the current model and RVE-based softening parameters. In addition, the softening laws from the literature have been used. The fracture propagation is assessed in terms of plastic energy dissipation in the RVE and the whole structure, load–displacement, and crack growth. The RVE-based model is shown to have better performance compared with other models from the literature.     

Numerical simulation of fracture mode transition in ductile plates

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 existing theories are not capable of capturing the mixed mode I/III due to a missing ingredient in the constitutive 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.     

Thermodynamics and continuum fracture mechanics for nonlocal-elastic plastic materials

Nonlocal elasticity is used as an improved elasticity model which engenders no crack-tip stress singularities and thus makes applicable the classical stress-based failure criteria. Considering nonlocal-elastic plastic materials exposed to softening by particle decohesion in a process surface and to subsequent surface separation by fracture, fracture mechanics is addressed within the framework of irreversible internal-variable thermodynamics in the hypothesis of small strains and arbitrary (but sufficiently regular) fracture surface (crack surface plus process surface). The state equations and the energy dissipation densities are derived for the bulk material and for the process surface, for both of which thermodynamically consistent evolutive equations are also proposed. The energy consumption for the formation of the unit crack area is evaluated as the sum of two contributions, one as free energy released by the process surface microstructure, the other as mechanical work done by the surrounding bulk material. Basing on the second thermodynamics law, a crack local stability criterion is provided in terms of crack front characteristics, i.e. fracture force vector, fracture resistance vector and fracture (symmetric) stiffness matrix, all defined at the points of the crack front through the response fields and the response sensitivities to virtual crack front advancements. Stability is guaranteed if, at every point of the crack front and for any virtual advancement of the latter, the fracture resistance is greater than the fracture force, or in case of equality, the fracture stiffness matrix is positive definite. The limit case of perfectly brittle fracture is considered, also in the case of local elasticity.     

塑性动态断裂实验研究

本文利用自制的实验装置,对韧性材料在爆炸冲击载荷作用下的塑性动态断裂特性,进行了实验研究,其中包括高塑性应变速率下,塑性区裂纹扩展过程和扩展速度的测试;塑性动态断裂韧性CTOD,及其在不同裂纹扩展速度下变化规律的测试。同时,对不同裂纹扩展速度的试件断口进行微观分析。     

The size dependence of micro-toughness in ductile fracture

Micro-toughness in ductile fracture is defined as the plastic work dissipated per unit fracture surface area in the material separation processes of void growth and coalescence. A micromechanics model for the estimation of the size dependence of micro-toughness in ductile fracture is presented. Size effects are incorporated in the model using the conventional mechanism-based strain gradient plasticity (CMSG) theory. A finite element model of an axisymmetric representative unit cell with an initial spherical void is used to validate model predictions. Two characteristic length scales emerge from the model. The initial void radius sets the scale for the initial spherical void growth. For the subsequent void coalescence, the scale is set by the width of the intervoid ligament. Energy dissipation in ductile fracture is found to be dominated by the mechanisms of coalescence, and the micro-toughness in ductile fracture is found to be size dependent for dimple sizes approximately one order of magnitude larger than the material length scale.     

Modeling dynamic plasticity and spall fracture in high density polycrystalline alloys

The dynamic thermomechanical response of a tungsten heavy alloy is investigated via modeling and numerical simulation. The material of study consists of relatively stiff pure tungsten grains embedded within a more ductile matrix alloy comprised of tungsten, nickel, and iron. Constitutive models implemented for each phase account for finite deformation, heat conduction, plastic anisotropy, strain-rate dependence of flow stress, thermal softening, and thermoelastic coupling. The potentially nonlinear volumetric response in tungsten at large pressures is addressed by a pressure-dependent effective bulk modulus. Our framework also provides a quantitative prediction of the total dislocation density, associated with cumulative strain hardening in each phase, and enables calculation of the fraction of plastic dissipation converted into heat energy. Cohesive failure models are employed to represent intergranular fracture at grain and phase boundaries. Dynamic finite element simulations illustrate the response of realistic volume elements of the polycrystalline microstructure subjected to compressive impact loadings, ultimately resulting in spallation of the material. The relative effects of mixed-mode interfacial failure criteria, thermally-dependent fracture strengths, and grain shapes and orientations upon spall behavior are weighed, with interfacial properties exerting a somewhat larger influence on the average pressure supported by the volume element than grain shapes and initial lattice orientations within the bulk material. Spatially resolved profiles of particle velocities at the free surfaces of the volume elements indicate the degree to which the incident and reflected stress waves are altered by the heterogeneous microstructure.     

Mechanical modelling of ductile fracture

Mechanical models of material failure by void growth to coalescence are described to give a brief overview of methods applied in the analysis of ductile fracture. Approximate constitutive relations for porous ductile materials are discussed, modelling both the nucleation and growth of voids. The application of the material models is illustrated by numerical analyses for a tensile test specimen and for dynamic, ductile crack growth. Unstable void growth is a relevant mechanism in ductile materials subject to a high level of triaxial tension. The analysis of such cavitation instabilities in elastic-perfectly plastic materials is discussed for axisymmetric stress states, and the relevance to metal/ceramic components is emphasized.General Lecture presented at the 10th Italian National Congress of Theoretical and Applied Mechanics; AIMETA, Pisa, October 1990.     

An eigenstrain formulation for the prediction of elastic moduli of defective fiber networks

A method for predicting the elastic moduli of a regular network populated by a large number of randomly located defects is presented. The prediction is based exclusively on the stiffness of individual fibers and the location of defects. The method requires a preliminary calibration step in which the eigenstrains associated with “elementary defects” of the regular network are fully characterized. Each type of defect is represented by a superposition of singular point sources in 2D elastostatics producing a field identical to the eigenstrain of the respective defect. The amplitude of the point sources is determined by probing the eigenstrain with a series of path independent integrals. This “spectral decomposition” represents the generalization that allows applying methods developed to account for crack–crack interaction in fracture mechanics to situations in which the interacting sources have eigenstrains obtained by the superposition of multiple types of singularities. Once the representation of each elementary defect is determined, any distribution of defects in the network can be mapped into a distribution of point sources in an equivalent continuum. This allows inferring the elastic behavior of a defective network of any distribution and concentration of defects. The method discussed here provides an efficient way to treat the non-affine deformation of defective regular fiber networks.     

Dynamic fracture instabilities in brittle crystals generated by thermal phonon emission: Experiments and atomistic calculations

Dynamic cleavage fracture experiments of brittle single crystal silicon revealed several length scales of surface and path instabilities: macroscale path selection, mesoscale crack deflection, and nanoscale surface ridges. These phenomena cannot be predicted or explained by any of the continuum mechanics based equations of motion of dynamic cracks, as presumably critical energy dissipation mechanisms are not fully accounted for in the theories. Experimentally measured maximum crack speed, always lower than the theoretical limit, is another phenomenon that is as yet not well understood.We suggest that these phenomena depend on velocity dependent and anisotropic material property that resists crack propagation. The basic approach is that the bond breaking mechanisms during dynamic crack propagation vibrate the atoms at the crack front to generate thermal phonon emission, or heat, which provides additional energy dissipation mechanisms. This energy dissipation mechanism is a material property that resists crack propagation. To evaluate this property, we combined the continuum based elastodynamic Freund equation of motion with molecular dynamics atomistic computer “experiments”.We analyzed the above experimental dynamic fracture instabilities in silicon with the obtained velocity dependent and anisotropic material property and show its importance in cleavage of brittle crystals.     

Crack tip blunting and cleavage under dynamic conditions

In structural materials with both brittle and ductile phases, cracks often initiate within the brittle phase and propagate dynamically towards the ductile phase. The macroscale, quasistatic toughness of the material thus depends on the outcome of this microscale, dynamic process. Indeed, dynamics has been hypothesized to suppress dislocation emission, which may explain the occurrence of brittle transgranular fracture in mild steels at low temperatures (Lin et al., 1987). Here, crack tip blunting and cleavage under dynamic conditions are explored using continuum mechanics and molecular dynamics simulations. The focus is on two questions: (1) whether dynamics can affect the energy barriers for dislocation emission and cleavage, and (2) what happens in the dynamic “overloaded” situation, in which both processes are energetically possible. In either case, dynamics may shift the balance between brittle cleavage and ductile blunting, thereby affecting the intrinsic ductility of the material. To explore these effects in simulation, a novel interatomic potential is used for which the intrinsic ductility is tunable, and a novel simulation technique is employed, termed as a “dynamic cleavage test”, in which cracks can be run dynamically at a prescribed energy release rate into a material. Both theory and simulation reveal, however, that the intrinsic ductility of a material is unaffected by dynamics. The energy barrier to dislocation emission appears to be identical in quasi-static and dynamic conditions, and, in the overloaded situation, ductile crack tip behavior ultimately prevails since a single emission event can blunt and arrest the crack, preventing further cleavage. Thus, dynamics cannot embrittle a ductile material, and the origin of brittle failure in certain alloys (e.g., mild steels) appears unrelated to dynamic effects at the crack tip.     

Analysis of crack growth and crack-tip plasticity in ductile materials using cohesive zone models

Crack initiation and crack growth resistance in elastic plastic materials, dominated by crack-tip plasticity are analyzed with the crack modeled as a cohesive zone. Two different types (exponential and bilinear) of cohesive zone models (CZMs) have been used to represent the mechanical behavior of the cohesive zones. In this work, it is suggested that different forms of CZMs (e.g., exponential, bilinear) are the manifestations of different micromechanisms-based inelastic processes that participate in dissipating energy during the fracture process and each form is specific to each material system. It is postulated that the total energy release rate comprises the plastic dissipation rate in the bounding material and the separation energy rate within the fracture process zone, the latter is determined by CZMs. The total energy release rate then becomes a function of the material properties (e.g., yield strength, strain hardening exponent) and cohesive properties of the fracture process zone (e.g., cohesive strength and cohesive energy), and the form of cohesive zone model (CZM) that determines the rate of energy dissipation in the forward and wake regions of the crack. The effects of material parameters, cohesive zone parameters as well as the form/shape of CZMs in predicting the crack growth resistance and the size of plastic zone (SPZ) surrounding the crack tip are systematically examined. It is found that in addition to the cohesive strength and cohesive energy, the form (shape) of the traction–separation law of CZM plays a very critical role in determining the crack growth resistance (R-curve) of a given material. It is further observed that the shape of the CZM corresponds to inelastic processes active in the forward and wake regions of the crack, and has a profound influence on the R-curve and SPZ.     

Multiscale modeling of crack initiation and propagation at the nanoscale

Fracture occurs on multiple interacting length scales; atoms separate on the atomic scale while plasticity develops on the microscale. A dynamic multiscale approach (CADD: coupled atomistics and discrete dislocations) is employed to investigate an edge-cracked specimen of single-crystal nickel, Ni, (brittle failure) and aluminum, Al, (ductile failure) subjected to mode-I loading. The dynamic model couples continuum finite elements to a fully atomistic region, with key advantages such as the ability to accommodate discrete dislocations in the continuum region and an algorithm for automatically detecting dislocations as they move from the atomistic region to the continuum region and then correctly “converting” the atomistic dislocations into discrete dislocations, or vice-versa. An ad hoc computational technique is also applied to dissipate localized waves formed during crack advance in the atomistic zone, whereby an embedded damping zone at the atomistic/continuum interface effectively eliminates the spurious reflection of high-frequency phonons, while allowing low-frequency phonons to pass into the continuum region.The simulations accurately capture the essential physics of the crack propagation in a Ni specimen at different temperatures, including the formation of nano-voids and the sudden acceleration of the crack tip to a velocity close to the material Rayleigh wave speed. The nanoscale brittle fracture happens through the crack growth in the form of nano-void nucleation, growth and coalescence ahead of the crack tip, and as such resembles fracture at the microscale. When the crack tip behaves in a ductile manner, the crack does not advance rapidly after the pre-opening process but is blunted by dislocation generation from its tip. The effect of temperature on crack speed is found to be perceptible in both ductile and brittle specimens.     

Modeling of ductile fracture: Application of the mechanism-based concepts

This paper summarizes our recent studies on modeling ductile fracture in structural materials using the mechanism-based concepts. We describe two numerical approaches to model the material failure process by void growth and coalescence. In the first approach, voids are considered explicitly and modeled using refined finite elements. In order to predict crack initiation and propagation, a void coalescence criterion is established by conducting a series of systematic finite element analyses of the void-containing, representative material volume (RMV) subjected to different macroscopic stress states and expressed as a function of the stress triaxiality ratio and the Lode angle. The discrete void approach provides a straightforward way for studying the effects of microstructure on fracture toughness. In the second approach, the void-containing material is considered as a homogenized continuum governed by porous plasticity models. This makes it possible to simulate large amount of crack extension because only one element is needed for a representative material volume. As an example, a numerical approach is proposed to predict ductile crack growth in thin panels of a 2024-T3 aluminum alloy, where a modified Gologanu–Leblond–Devaux model [Gologanu, M., Leblond, J.B., Devaux, J., 1993. Approximate models for ductile metals containing nonspherical voids – Case of axisymmetric prolate ellipsoidal cavities. J. Mech. Phys. Solids 41, 1723–1754; Gologanu, M., Leblond, J.B., Devaux, J., 1994. Approximate models for ductile metals containing nonspherical voids – Case of axisymmetric oblate ellipsoidal cavities. J. Eng. Mater. Tech. 116, 290–297; Gologanu, M., Leblond, J.B., Perrin, G., Devaux, J., 1995. Recent extensions of Gurson’s model for porous ductile metals. In: Suquet, P. (Ed.) Continuum Micromechanics. Springer-Verlag, pp. 61–130] is used to describe the evolution of void shape and void volume fraction and the associated material softening, and the material failure criterion is calibrated using experimental data. The calibrated computational model successfully predicts crack extension in various fracture specimens, including the compact tension specimen, middle crack tension specimens, multi-site damage specimens and the pressurized cylindrical shell specimen.     

Numerical implementation and assessment of the GLPD micromorphic model of ductile rupture

Just like all constitutive models involving softening, Gurson's classical model for porous ductile solids predicts unrealistic, unlimited localization of strain and damage. An improved variant of this model aimed at solving this problem has been proposed by Gologanu, Leblond, Perrin and Devaux (GLPD) on the basis of some refinement of Gurson's original homogenization procedure. The GLPD model is of “micromorphic” nature since it involves the second gradient of the macroscopic velocity and generalized macroscopic stresses of “moment” type, together with some characteristic “microstructural distance”. This work is devoted to its numerical implementation and the assessment of its practical relevance. This assessment is based on two criteria: absence of mesh size effects in finite element computations and agreement of numerical and experimental results for some typical experiments of ductile fracture. The GLPD model is found to pass both tests. It is therefore concluded that it represents a viable, although admittedly complex solution to the problem of unlimited localization in Gurson's model of ductile rupture.     

Mode-3 spontaneous crack propagation in symmetric functionally graded materials

The effects of spatially varying the material properties on the mode-3 planar crack propagation characteristics are numerically investigated. The spectral scheme that is available for homogeneous materials is modified to account for the symmetrically varying material properties. Crack propagation in hardening, softening and unsymmetric type of functionally graded have been simulated. A parametric study was performed by systematically varying the material inhomogeneity length scale. Our study indicated that softening and unsymmetric graded materials reduce the resistance to fracture, while a hardening material offers higher fracture resistance with increase in inhomogeneity. Only the transient phase of crack propagation speed was affected by the material property variation, irrespective of whether the material was hardening, softening or an unsymmetric type. The crack always reached a quasi-steady-state velocity, which remained unaffected by the material property inhomogeneity.     

Elastic-plastic mechanics of steady crack growth under anti-plane shear

For a crack with steady growth under anti-plane shear, analysis shows a primary plastic zone included in an angle of ±19.7° ahead of the crack tip, and two very thin secondary (reverse) plastic zones along the crack flanks, each included in an angle of 0.37°. Numerical solutions give the shape of the plastic zones which determine the active and residual plastic strains, and give the crack tip displacement, which is approximately 0.07 of that for monotonic loading without growth. The length of the primary plastic zone is almost the same as that without growth, but the thickness is about 3/5 as great. Coupled with ductile fracture criteria, the present results predict initially stable crack growth, whereas analyses based on the simplification of yielding on just one plane predict unstable fracture immediately following initiation.     

混凝土拉伸软化曲线折线近似的逆解方法

研究基于Hillerborg的虚拟裂纹模型，利用有限元分析方法，求得折线近似的拉伸软化曲线的逆解方法。对弹性模量，初始开裂应力的决定方法进行了研究。以双直线模型的计算结果为算例进行了逆推分析，算例符合得很好。也较好地从实验得到的荷载位移曲线再现了拉伸软化曲线。这对于研究混凝土的断裂能，尺寸效应等问题很具意义。