Ductile fracture: Experiments and computations

Numerous criteria have been developed for ductile fracture (DF) prediction in metal plastic deformation. Finding a way to select these DF criteria (DFCs) and identify their applicability and reliability, however, is a non-trivial issue that still needs to be addressed in greater depth. In this study, several criteria under the categories of ‘uncoupled damage criterion’ and the ‘coupled damage criterion’, including the continuum damage mechanics (CDM)-based Lemaitre model and the Gurson-Tvergaard-Needleman (GTN) model, are investigated to determine their reliability in ductile failure prediction. To create diverse stress and strain states and fracture modes, different deformation scenarios are generated using tensile and compression tests of Al-alloy 6061 (T6) with different sample geometries and dimensions. The two categories of criteria are coded into finite element (FE) models based on the unconditional stress integration algorithm in the VUMAT/ABAQUS platform. Through physical experiments, computations and three industrial case studies, the entire correlation panorama of the DFCs, deformation modes and DF mechanisms is established and articulated. The experimental and simulation results show the following. (1) The mixed DF mode exists in every deformation of concern in this study, even in the tensile test of the round bar sample with the smallest notch radius. A decrease of stress triaxiality (η-value) leads to a reduction in the accuracy of DF prediction by the two DFC categories of DFCs, due to the interplay between the principal stress dominant fracture and the shear-stress dominant factor. (2) For deformations with a higher η-value, both categories of DFCs predict the fracture location reasonably well. For those with a lower or even negative η-value, the GTN and CDM-based criteria and some of the uncoupled criteria, including the C&L, Ayada and Oyane models, provide relatively better predictions. Only the Tresca and Freudenthal models can properly predict the shear dominant fracture. The reliability sequence of fracture moment prediction is thus the GTN model, followed by the CDM-based model and the uncoupled models. (3) The applicability of the DFCs depends on the use of suitable damage evolution rules (void nucleation/growth/coalescence and shear band) and consideration of several influential factors, including pressure stress, stress triaxiality, the Lode parameter, and the equivalent plastic strain or shear stress. These parameters determine the deformation mode (shear dominant or maximum principal stress dominant deformation) and, further, the DF mechanism (dimple fracture/shear fracture/mixed fracture).     

Ductile fracture criteria for simulating shear by node separation method

This study is concerned with the mechanics of ductile fracture in bulk metal forming processes by a finite-element analysis and experiments. Developed is a computer program using conventional finite-element method such that the behavior of crack propagation after ductile fracture can be analyzed. The phenomenon of a material separating into two pieces upon shearing and tensile tests has been simulated using the developed computer program. Special attention is focused on the effect of various ductile fracture criteria on crack initiation and propagation during shearing and tensile tests.     

Failure in notched tension bars due to high-temperature creep: Interaction between nucleation controlled cavity growth and continuum cavity growth

For axi-symmetrically notched tension bars [Dyson, B.F., Loveday, M.S., 1981, Creep Fracture in Nimonic 80A under Tri-axial Tensile Stressing, In: Ponter A.R.S., Hayhurst, D.R. (Eds.), Creep in Structures, Springer-Verlag, Berlin, pp. 406–421] show two types of damage propagation are shown: for low stress, failure propagates from the outside notch surface to the centre-line; and for high stress, failure propagates from the centre-line to the outside notch surface. The objectives of the paper are to: identify the physics of the processes controlling global failure modes; and, describe the global behaviour using physics-based constitutive equations.Two sets of constitutive equations are used to model the softening which takes place in tertiary creep of Nimonic 80A at 750 °C. Softening by multiplication of mobile dislocations is firstly combined, for low stress, with softening due to nucleation controlled creep constrained cavity growth; and secondly combined, for high stress, with softening due to continuum void growth. The Continuum Damage Mechanics, CDM, Finite Element Solver DAMAGE XX has been used to study notch creep fracture. Low stress notch behaviour is accurately predicted provided that the constitutive equations take account of the effect of stress level on creep ductility. High stress notch behaviour is accurately predicted from a normalized inverse cavity spacing d/2? = 6, and an initial normalized cavity radius r_hi/? = 3.16 × 10^?3, where 2? is the cavity spacing, and d is the grain size; however, the constants in the strain rate equation required recalibration against high stress notch data. A void nucleation mechanism is postulated for high stress behaviour which involves decohesion where slip bands intersect second phase grain boundary particles. Both equation sets accurately predict experimentally observed global failure modes.     

Ductile fracture of materials with high void volumefraction

In the present paper, axisymmetric cell models containing one or two voids and athree-dimensional cell model containing two voids have been used to investigate void size andspacing effect on the ductile fracture in materials with high initial void volume fraction. They areperformed for round smooth and round notched specimens under uniaxial tension. The examplematerial used for comparison is a nodular cast iron material GGG-40 with initial void volumefraction of 7.7%. The parameters considered in this paper are void size and shape foraxisymmetric cell models containing a single void, and void distribution pattern foraxisymmetric and 3D cell models containing two voids of different sizes. The results obtainedfrom these cell models by using FEM calculations are compared with the Gurson model, theGurson–Tvergaard–Needleman model, the Rice–Tracey model and the modified Rice–Traceymodel. It can be stated that the influence of void size and void spacing on the growth in volumeof voids is very large, and it is dependent on the distribution of voids. Using non-uniform voiddistribution, the results of axisymmetric cell models can explain how a void can grow in anunstable state under very low stress triaxiality at very small strain as observed in experiments.Calculations using cell models containing two voids give very different results about the stableand unstable growth of voids which are strongly dependent on the configuration of cell model.     

Simulation of voids effect on ductile fracture of metallic alloys

Most commerical alloys have inclusions which reduce their ductility and toughness. Ductile fracture of perforated mild steel sheet was investigated in order to simulate the fracture of metallic alloys containing a second phase or inclusions. The growth of the holes during tensile straining has been studied. The results indicate that fracture of the specimens did not result from the coalescence of growing holes, but from the initiation of a shear crack at a side of the hole after the strain of the matrix close to that hole reached a critical value.Theories of ductile fracture^1–4 were examined using experimental data obtained using mild steel sheet and ductile cast iron. It was found that these theories can be used to predict the experimental results within some limitations.     

Ductile reinforcements for enhancing fracture resistance in composite materials

Ductile reinforcements can supply fracture toughness to a polymer matrix by pulling out and by plastically deforming. In the case of metal reinforcements that are not in a toughened condition, there may be more toughening to be gained when the fibers remain in the matrix and plastically deform rather than pulling out. These fibers can be said to have an unused plastic potential. When these fibers bridge a crack, their plastic deformation causes a rise in the force which is trying to pull out the fiber. Because of this, the shape of the fiber must be adjusted along its length if it is to remain anchored and contribute its plastic work. The use of anchored, ductile fibers provides a new design axis that brings new possibilities not achievable by the current research focus on the fiber–matrix interface. This paper describes the experimental pullout of aligned ductile fibers from a polymer matrix, and indicates the effect of the shape and embedded length of the fiber on the toughness increase of the composite. Anchored, plastically deforming fibers are shown to provide a major improvement to the toughening. Even for unoptimized ductile fibers, the calculated toughening improvement equals or exceeds the toughening available from current short glass or graphite fibers. In addition, pullout values are obtained for fibers that are embedded at an angle, simulating fiber bridging from fibers not perpendicular to the crack surface. These results further demonstrate the toughening efficiency of ductile fibers.     

A damage-mechanics model for fracture nucleation and propagation

In this paper, a composite model for earthquake rupture initiation and propagation is proposed. The model includes aspects of damage mechanics, fiber-bundle models, and slider-block models. An array of elements is introduced in analogy to the fibers of a fiber bundle. Time to failure for each element is specified from a Poisson distribution. The hazard rate is assumed to have a power-law dependence on stress. When an element fails it is removed, the stress on a failed element is redistributed uniformly to a specified number of neighboring elements in a given range of interaction. Damage is defined to be the fraction of elements that have failed. Time to failure and modes of rupture propagation are determined as a function of the hazard-rate exponent and the range of interaction.     

Ductile and brittle fracture analysis of surface flaws using CTOD

A reference stress method is used to analyze both brittle and ductile fracture in structures containing surface flaws. Crack-tip opening displacement (CTOD) is used as the fracture-toughness input, althoughJ-based reference stress analyses are also possible. Both detailed and simplified analyses for brittle and ductile fracture are described. A brittle fracture analysis which takes account of stress concentrations, secondary stresses and stress gradients is presented, together with a complete ductile tearing analysis which utilizes a single CTOD value measured at maximum load. In addition, two simplified approaches are proposed: a yield-before-break criterion for brittle fracture and a critical learing modulus for ductile fracture.     

A physical model for nucleation and early growth of voids in ductile materials under dynamic loading

Spall fracture and other rapid tensile failures in ductile materials are often dominated by the rapid growth of voids. Recent research on the mechanics of void growth clearly shows that void nucleation may be represented as a bifurcation phenomenon, wherein a void forms spontaneously followed by highly localized plastic flow around the new void. Although thermal, viscoplastic, and work hardening effects all play an essential role in the earliest stages of nucleation and growth, the flow becomes dominated by spherical radial inertia, which soon causes all voids to grow asymptotically at the same rate, regardless of differences in initial conditions or constitutive details, provided only that there is the same density of matrix material and the same excess loading history beyond the cavitation stress.These two facts, initiation by bifurcation at a cavitation stress, at which a void first appears, and rapid domination by inertia, are used to postulate a simple, but physically realistic, model for nucleation and early growth of voids in a ductile material under rapid tensile loading. A reasonable statistical distribution for the cavitation stress at various nucleation sites and a simple similarity solution for inertially dominated void growth permit a simple calculation of the initiation and early growth of porosity in the material.Parametric analyses are presented to show the effect that loading rate, peak loading stress, density of nucleation sites, physical properties of the material, etc. have on the applied pressure and distribution of void sizes when a critical porosity is reached.     

Modelling temperature-dependent fracture nucleation of SWCNTs using atomistic-based continuum theory

The fracture behaviour of carbon nanotubes depends largely on temperature, defect distribution, and geometric features. In this paper, the effect of temperature upon fracture nucleation of single-walled carbon nanotubes (SWCNTs) is investigated using an atomistic-based continuum theory. The temperature effects are described in terms of a modified Cauchy–Born rule based on the assumptions that the deformation is sufficiently small and locally homogeneous. Furthermore, it is assumed that the atoms have the same local vibration mode at a given temperature. The first derivative of the free energy density, which is a function of both the deformation gradient and the temperature, enables the determination of the second Piola–Kirchhoff stress. In the present study, the fracture nucleation is modelled as a bifurcation of a homogeneously deformed nanotube at a critical strain. The model predictions show that the fracture strain decreases with increasing temperature, while the elastic stiffness remains largely unchanged.     

Influence of void nucleation on ductile shear fracture at a free surface

An approximate continuum model of a ductile, porous material is used to study the influence of the nucleation and growth of micro-voids on the formation of shear bands and the occurrence of surface shear fracture in a solid subject to plane strain tension. Bifurcation into diffuse modes is analysed for a plane strain tensile specimen described by these constitutive relations, which account for a considerable plastic dilatancy due to void growth and for the possibility of non-normality of the plastic flow law. In particular, bifurcation into surface wave modes and the possible influence of such modes triggering shear bands is investigated. For solids with initial imperfactions such as a surface undulation, a local material inhomogeneity on an inclusion colony, the inception and growth of plastic flow localization is analysed numerically. Both the formation of void-sheets and the final growth of cracks in the shear bands is described numerically. Some special features of shear band development in the solid obeying non-normality are studied by a simple model problem.     

An atomistic-based continuum theory for carbon nanotubes: analysis of fracture nucleation

Carbon nanotubes (CNTs) display unique properties and have many potential applications. Prior theoretical studies on CNTs are based on atomistic models such as empirical potential molecular dynamics (MD), tight-binding methods, or first-principles calculations. Here we develop an atomistic-based continuum theory for CNTs. The interatomic potential is directly incorporated into the continuum analysis through constitutive models. Such an approach involves no additional parameter fitting beyond those introduced in the interatomic potential. The atomistic-based continuum theory is then applied to study fracture nucleation in CNTs by modelling it as a bifurcation problem. The results agree well with the MD simulations.     

Microvoid multistage nucleation model and its application in analysis of micro damage and fracture

In this paper, the microvoid multistage nucleation model [14,15] suggested by the authors of this paper has been studied on the micro ductile damage and fracture of metallic material under large elastic-plastic deformation.Using this model, the analyses of micro damage and fracture for various axisymmetric tensile specimens and for TPB and CCP cracked specimens have been carried out. And the results from these analyses on damage development and fracture are in good agreement with the experimental ones for axisymmetric specimens and reasonable for cracked specimens from the microscopic point of view.The project surported by National Science Foundations of China.     

Mesoscopic natural element analysis of elastic moduli, yield stress and fracture of solids containing a number of voids

A two-dimensional mesoscale simulation method based on the natural element method, which is a kind of meshless method, is developed and applied to the analysis of overall elastic moduli, macro yield stress and void-linking fracture. The calculated results are compared with the theoretical solutions for overall elastic moduli, the experimental macro yield stress for aluminum and brass as well as the improved Gurson’s yield function, and the experimental void linking fracture to show the validity of the proposed method.     

Homogenization of geomaterials containing voids by random fields and finite elements

The paper describes the use of random fields and finite elements to assess the influence of porosity and void size on the effective elastic stiffness of geomaterials. A finite element model is developed involving “tied freedoms” that allows analysis of an ideal block of materials leading to direct evaluation of the effective Young’s modulus and Poisson’s ratio. The influence of block size and representative volume elements (RVE) are discussed. The use of random fields and Monte-Carlo simulations deliver a mean and standard deviation of the elastic parameters that lead naturally to a probabilistic interpretation. The methodology is extended to a foundation problem involving a footing on an elastic foundation containing voids. The approach enables estimates to be made of the probability of excessive settlement.     

Optimal Scaling in Solids Undergoing Ductile Fracture by Crazing

We derive optimal scaling laws for the macroscopic fracture energy of polymers failing by crazing. We assume that the effective deformation-theoretical free-energy density is additive in the first and fractional deformation-gradients, with zero growth in the former and linear growth in the latter. The specific problem considered concerns a material sample in the form of an infinite slab of finite thickness subjected to prescribed opening displacements on its two surfaces. For this particular geometry, we derive optimal scaling laws for the dependence of the specific fracture energy on cross-sectional area, micromechanical parameters, opening displacement and intrinsic length of the material. In particular, the upper bound is obtained by means of a construction of the crazing type.     

Coalescence of voids by internal necking: Theoretical estimates and numerical results

Upper-bound estimates and supposedly exact numerical results are obtained for the limit loads associated with cylindrical cells containing voids and subjected to boundary conditions that are consistent with post-localization kinematics in porous plastic solids. When supplemented with evolution equations for the microstructural variables, the results can be used in the modeling of void coalescence by internal necking in ductile materials.     

The mechanical behaviour of ductile materials damaged by two generations of voids

Large strain finite element method is employed to investigate the damaging effect of two generations of voids in ductile materials. An axisymmetric cylinder embedding an initially spherical void is chosen as the model cell. Secondary voids will initiate around the initial void when the local stress/strain in the matrix increases to certain critical conditions. This event is numerically simulated through an empty element technique. The interaction between these two generations of voids has been proved to be favourable to the voiding condition, thus accelerating the material damage, characterized by the value of the overall elastic modulus which may undergo drastic drop when nearing final fracture.