MATERIAL FAILURE MODELS
Why do materials actually fail? Is there a single definition of failure? Is there an approach to failure that will not fail itself from microscopic (atomic) level up to macroworld structures? In the last decade, large effort is dedicated to formulate material failure in terms of atomic dislocations, inclusions, microvoids, crack initiation and propagation, thermodynamics and macroscopic quantities such as stresses and strains. Unified approaches have emerged from these efforts, which compare well to experimental evidence. Lately, some of these approaches have been ported to the finite element method, particularly for crack propagation studies. In this chapter, a small excursion into material failure models is presented, the T failure criterion is incorporated in Abaqus/Explicit and numerical verification is conducted. Also, a novel method for calculating threshold values for the Tfailure criterion used for the simulations is presented.
In modern research and industrial applications, the estimation of an ultimate safety factor under extreme loading conditions, plays an important role in the overall safety assessment of a structure. In structural problems, where the response should be determined beyond the initiation of nonlinear material behaviour, fracture may be of profound importance for the determination of the integrity of the structure. Such problems include deep drawing, impact and crushing. Finite element methodologies, provide analysts with sophisticated tools for performing both material and geometry nonlinear calculations, which are yielding fairly accurate results. On the other hand, due to the lack of globally accepted fracture criteria, the determination of the structure’s damage, due to material failure using finite element methods, is still under intensive research. One could distinguish material failure in two broader categories depending on the scale in which the material is examined:

Failure is defined in terms of crack propagation and initiation. In this case, the microscopic behaviour of the material is examined, and respective theories are translated to finite element decoupling or even special element formulations that can reproduce crack propagation. In this case, fine meshes are required to capture the evolution of cracks. Such methodologies are useful for gaining insight in the cracking of specimens and simple structures under well defined global load distributions.

Failure is defined in terms of element removal. In this work, failure of an element is considered to occur when the element is no longer capable of carrying further loading, or equivalently, cannot store any further deformational energy. In this case, rather macroscopic failure criteria should be applied. When the conditions of such criteria are met, the element is deleted from the analysis. Such a failure mechanism is incorporated in most explicit time integration dynamic codes available today.
The second approach is most appealing for determining an upper limit to a particular loading on a structure. This work, provides a methodology for incorporating failure criteria in Abaqus/explicit, for metals following the von Mises yield conditions, as well as an application of the Tfailure criterion [2] to [5]. Abaqus/Explicit is chosen because it is the only available explicit finite element code at NTUA that allows the implementation of user subroutines that describe material failure, in the sense defined earlier. It also requires the user to fully describe plasticity calculations. This has the drawback of requiring extensive benchmarking of the user subroutine, but on the other hand provides the user with the full stress and strain tensors at each time step. This makes the implementation of any failure criterion that exists or will occur much easier.
Microscopic failure
Microscopic failure considers the initiation and propagation of a crack. Failure criteria in this case are related to microscopic fracture. Some of the most popular failure models in this area are the micromechanical failure models, which combine the advantages of continuum mechanics and classical fracture mechanics [6]. Such models are based on the concept that during plastic deformation, microvoids nucleate and grow until a local plastic neck or fracture of the intervoid matrix occurs, which causes the coalescence of neighbouring voids. Such a model that appears in the literature quite often is GTN. It was proposed by Gurson and extended by Tvergaard and Needleman.
Another approach, proposed by Rousselier and referred to in [6], is based on continuum damage mechanics (CDM) and thermodynamics. Both models form a modification of the von Mises yield potential by introducing a scalar damage quantity, which represents the void volume fraction of cavities, the porosity f [6]. The flow potential F may be written for both models in a uniform way:
where R represents the yield stress of the undamaged matrix material, p is an effective plastic strain representative of the matrix hardening, s_{0} is an effective scalar stress which is a function of both the macroscopic stress tensor and the porosity f. The latter represents the existence of cavities. The initial f_{0} is defined by the microstructure of the material and its evolution by the mass conservation modified so as to account for the stress field controlled void nucleation. Failure is triggered by the localization of strain and damage in a narrow band. Localization indicators are then developed to monitor the bifurcation at a material point. Formal characteristics of such models have been implemented in finite element codes in the form of constitutive equations. Besson et al [6], [7] have performed a series of numerical simulations using both the commercial Abaqus as well as the inhouse Zebulon. Such simulations have shown that it is possible, when using dense meshes, to predict using such models the plastic behaviour of ductile materials, yield, necking, crack initiation and propagation.
One of the most important applications of such criteria using finite element methodologies is metal forming. Chung et al [9] present a method for incorporating the CDM model for ductile damage and failure in finite element codes. In their work, crack propagation is modelled with the removal of elements when the damage indicator reaches a critical value. Based on this failure model and on numerical simulations of tensile tests with precracked specimen, they show that it is possible to predict the material behaviour and the critical load for crack initiation. They also present examples of metal forming simulations. Del Linz et al [10] present a complete model of constitutive equations combining isotropic and kinematic hardening. They also introduce a damage parameter to describe isotropic damage. This parameter depends on the damage energy release rate that corresponds to the variation of internal energy density due to damage growth at constant stress. This model compared to the classical von Mises approach, presents better fit to experimental results. The parameter is not used for element deletion from the mesh, but directly affects the evolution of plastic flow and the overall specimen behaviour. It is worth noticing that, in the results of tensile tests, the parameter maxima are located at the centre of the cross section at which necking occurs, whereas equivalent plastic strain is uniform over the whole section. A similar behaviour has also been reported in [20].
Macroscopic failure
Macroscopic failure application in finite element methods is closely related to explicit codes. This is due to the ease of element deletion that they provide. Since equilibrium equations are assembled at the element level, one may simply ignore an element. Most commercial finite element codes, allow the user to define failure in terms of a plastic strain threshold. This means that for each integration point, when plastic strain reaches a certain limit, the respective layer of the element is considered to have failed. When all layers have failed, the element is practically deleted. The maximum plastic strain has two disadvantages:

It is not well defined that it governs failure.

It depends on element dimensions (gage length).
Most failure criteria suffer from the lack of a widely accepted and applicable theoretical definition. On the other hand, it may be said that it is generally accepted that maximum plastic strain does not effectively predict failure. The dependence of plastic strain on element dimensions is also a considerable restriction. One should calculate the applicable threshold for each element in order for this criterion to converge. In order to overcome this, a dimensionless criterion should be considered.
Li in [21] presents a classification of macroscopic failure criteria in four categories:

Stress or strain failure criteria

Energy type failure criteria

Damage failure criteria

Empirical failure criteria
The relationship between material deformation and failure in reaction to applied force, with the deformation and damage mechanisms from atomic to macroscopic level is also described in a comprehensive way. Five general levels are considered, at which the meaning of deformation and failure is interpreted differently: the structural element scale, the macroscopic scale where macroscopic stress and strain are defined, the mesoscale which is represented by a typical void, the microscale and the atomic scale. The material behaviour at one level is considered as a collective of its behaviour at a sublevel. An efficient deformation and failure model should be consistent at every level. This proposition is in contrast with the maximum plastic strain failure model.
Considering the above inefficiencies introduced by the maximum plastic strain failure model and the fact that it cannot predict multiaxial failure, Lehmann and Yu [20] introduced a new method for predicting ductile failure. This method could be considered a damage failure criterion. Applications to marine structures using finite element procedures are presented. A rupture index, I_{R}, is calculated that governs failure. It is considered that since ductile rupture of ship structures is the consequence of large local strains, the use of the true stressstrain relationship is essential. For a specimen in tension, the true rupture strain is:
where A_{0} is the initial cross section area and e_{f} is the section area after failure. The magnitude of e_{f} is larger than the value of the engineering fracture elongation and is interpreted by the authors as the logarithmic engineering fracture elongation for zero gage length.
In order to account for the triaxiality of the stress field at the area of the neck, the authors use the stress triaxiality defined by:
where and are the mean and equivalent stress respectively. Three stress states may be described by the corresponding values of stress triaxiality:

uniaxial compression: ß=1

pure shear: ß=0

uniaxial tension: ß =1
Uniaxial tension is considered an ideal state since the stress triaxiality at failure is reported by the authors to be larger than unity. It is also considered that for ship structures under collision or grounding loads the range of stress triaxiality is larger than unity, and their study is restricted in this range. Large triaxiality values are calculated for prenotched specimen. Depending on the notch radius of tensile specimen, different triaxiality values are calculated. For round notched specimen and minimum diameter d at the notch area, the stress triaxiality may be efficiently calculated by Bridgman’s formula:
where R is the curvature radius of the neck.
The authors have performed a large set of tensile experiments. These experiments have shown that the fracture strain ?_{f} depends on ?.
The authors assume that the initiation of macrocracks is the result of the evolution (nucleation, growth, coalescence) of microvoids, instead of the classical fracture mechanics approach of preexisting macrocracks. The authors use the original Lemaitre definition for damage for its simple form:
Ductile plastic damage is defined as:
and its differential constitutive equation:
where and n are material constants, e _{v} is the equivalent strain and
which implies the influence of stress triaxiality. In the case of proportional loading, damage evolution is obtained as:
For uniaxial tension ? =1, the critical damage D_{c} is written in terms of the one dimensional threshold strain ?_{0} and the rupture strain ?_{R}:
During the evolution of damage, crack initiation occurs when a damage threshold, D_{c}, is reached. Considering m=2n+1 and
the effective rupture strain e _{vr} for a multiaxial stress and strain state is expressed as a function of the one dimensional rupture strain e _{R} and the corresponding stress triaxialit
While the onedimensional rupture strain e _{R} is constant the multiaxial strain e _{vr} is variable, which means that the crack will not certainly occur at the place where the maximum effective strain e _{v} exists. The authors have verified experimentally this proposition: in tensile tests, while the maximum effective plastic strain occurs at the edges of the smallest specimen’s diameter, a crack occurs at the centre of the specimen. To locate crack initiation, a rupture index I_{R} is introduced:
Fracture may be predicted as soon as:
A first crack occurs at the location of maximum I_{R}. The value of e_{R} is calculated numerically. The authors describe a method to obtain this value from a single experiment and a single finite element simulation of the experiment. For St523, this value is calculated to be equal to 0.7.
Lehmann and Yu have present two numerical simulations using finite element methods of quasistatic experiments. One of the simulations includes the indentation of a triangular sharpened plate in a stiffened cylinder. The other is the indentation of a cone in a stiffened double bottom. In both cases, agreement with experimental results is observed, while especially in the second case the agreement is much better. The results are limited up to the point where the solution can converge, since the code used is not explicit. Element failure is implemented by eliminating the stiffness of elements with maximal I_{R}. Failure is accurately determined only for the first element, since the crack shape is unrealistic due to mesh size. The rupture index I_{R} is independent of mesh dimensions, as well as of any size effects that occur when using the maximum plastic strain, and has been used effectively by the authors to predict critical rupture loads for marine structures.
Kitamura et al [18] an integral over the time domain deformational factor D(t) as a failure criterion for steel plates used in comparing different tanker designs. This factor is defined as:
where
a pressure coefficient with
, ,
the hydrostatic pressure,
s_{e} the equivalent stress
e(t) is the principal strain at time t
e_{f}(t) is the equivalent breaking strain at time t
where e_{fx} and e_{fy} are the fracture strains for gage lengths X and Y respectively, which are equal to the sides of the element. According to the authors, fracture strains are given by the following curve:
This curve, which prescribes a constant fracture strain of 15% for lengths larger than 200 mm is considered to be identical for mild steel ?315 and ?350. The fracture criterion used by Kitamura et al may be considered a strainbased semi empirical criterion. It has the disadvantage of requiring integration of equation 3.13 over the time domain for every element. Also, it involves a large number of calculations.
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