DEPRAVACIÓN TOTAL

Current ULCF and LCF models may be broadly organized into two categories: those that couple the fracture behaviour with constitutive behaviour and those that do not. In the former, the effects of material damage are considered through the constitutive model while in the latter, material damage is an independent variable whose value has no direct effect on the constitutive model. Rather, the material damage combined with a fracture condition can predict geomet- ric changes (i.e., material separation) which, in turn, will affect the constitutive response.

Since the constitutive response of coupled models is explicitly tied to the dam- age/fracture model, coupled models can readily predict crack propagation rate and direction, whereas uncoupled models must implement a separate material separation criterion which often pre-determines the crack path. Despite this ad- vantage, coupled models are generally more difficult to calibrate because of ad- ditional parameters which may have loose physical meaning. Micromechanical- based models have been given a major emphasis to predict ULCF fracture. Among these models, the more relevant ones are:

• the Cyclic Void Growth Model (CVGM) proposed by Kanvinde and Deier-

lein [47];

• the Effective Damage Concept (EDC) developed by Ohata and Toyoda [82]; • the Leblond Model (usually called LPD model) [55] and

• the Continuum Damage Mechanics model proposed by Bonora [96].

While the first two models are uncoupled models, the latter two consider coupling between damage and constitutive equations.

2.6.1

Cyclic Void Growth Model (CVGM)

Because the mechanism of ULCF is controlled by void growth and coalescence, the CVGM proposed by Kanvinde and Deierlein [45], [47] and [46] extends upon the widely used void growth model (VGM), developed by Rice and Tracey [99], Hancock and Mackenzie [37] and others for monotonic loading. The CVGM is

2.6. ULCF and LCF models 17

defined by two equations defining the fracture demands, imposed on a material by ULCF loads, and the fracture toughness of a material, under ULCF loads. The fracture condition (crack initiation) occurs when the fracture demand exceeds the fracture toughness.

In order to account for the effects of void growth and coalescence that drive the fracture of metallic materials, the authors propose a model that calculates the void growth and compares it with a critical value to detect material failure. This parameter is obtained experimentally. The initial formulation developed for monotonic cases (Void Growth Model VGM [46]) is extended to cyclic loads by differentiating the void growth obtained in the tensile and compressive regions of the load cycle. Therefore, the void growth in the Cyclic Void Growth Model (CVGM) can be obtained as [47]: V GIcyclic= ∑ tensile cycles C1· ∫ ε2 ε1 exp(−1.5σm σe  )dεp− − ∑ compressive cycles C2· ∫ ε2 ε1 exp(−1.5σm σe  )dεp< V GIcycliccritical (2.2)

Two assumptions are inherent to the fracture demands:

• Voids grow during tensile cycles, where tensile cycles are defined to occur

whenever triaxiality is positive;

• Voids shrink during compressive cycles, similarly defined as whenever the

triaxiality is negative. In the VGM, the critical void size varies only with material; however with the CVGM the critical void size varies both with material and with the extent of material damage, induced by the reversed plasticity of the cyclic loading.

2.6.2

Effective Damage Concept (EDC)

The Effective Damage Concept (EDC), like the CVGM, is valid for arbitrary ULCF loading and controlled by the growth and coalescence of voids. The CVGM and EDC are conceptually similar. Both are based on the mechanism of void growth and coalescence with accompanying material damage induced by reversed

plasticity and both are based on extensions of the VGM. Despite these similarit- ies, there are several important differences. The EDC explicitly depends on the back stress, a second order tensor that defines the center of a materials yield sur- face in multiaxial stress space. The back stress quantifies the kinematic hardening of a material. The key assumption of the EDC is that material damage accumu- lates only when the back stress exceeds the maximum value obtained during prior load cycles.

In implementation, the EDC is summarized by two concepts:

• Applied equivalent plastic strain, during which the back stress does not

exceed the previous maximum back stress, does not contribute to material damage. Rather, only the effective equivalent plastic strain, contributes to material damage. The effective equivalent plastic strain, for a given load cycle, is defined as the portion of the total equivalent plastic strain for which the back stress equals or exceeds all prior values;

• The initiation of ductile fracture occurs when an instantaneous combination

of effective equivalent plastic strain and triaxiality equals the failure curve.

2.6.3

The Leblond model

The formulation of the Leblond model extends upon the GNT porous metal con- stitutive model. The proposed modification replaces the Cauchy stress tensor, in equation of yield surface, with the difference between the Cauchy stress tensor and the back stress tensor and, thus, accounts for the kinematic hardening asso- ciated with cyclic loads. The proposed modification advanced by Leblond allows the prediction of the constitutive response and point of ductile fracture initiation under ULCF loads. Ductile fracture initiation occurs when the void volume frac- tion exceeds a material dependent critical value and the load-carrying capacity of the material reduces to zero.

2.6.4

The Pirondi and Bonora model

Pirondi and Bonora [96], inspired on works of Lemaitre [57] and Chaboche [21] proposed a CDM model in which the constitutive behaviour under ULCF is coupled with the damage state.

2.6. ULCF and LCF models 19

The main features of the CDM model are:

• Material damage is a non-linear function of the equivalent plastic strain; • The modulus of the elasticity depends on the damage, where increases in

the material damage result in decreases in the modulus of elasticity;

• Damage accumulates and its effects are active only when the mean stress

is positive (i.e., the elastic stiffness is reduced only when the mean stress is positive). As such, any equivalent plastic strain that is accumulated when the mean stress is negative does not contribute to the damage nor does it alter the constitutive equations. Ductile fracture initiation is predicted when the material loses its load-carrying capacity.

2.6.5

The EVICD model

An interesting approach to characterize low cycle fatigue accounting for non- regular cycles is the one proposed by Jiang et al.[42], which defines an independent continuous cumulative damage function (EVICD) based on the accumulation of plastic strain energy. This formulation is based on the previous models of EVICD [42],[81] and [91] and states that the total damage can be computed as:

D =

∫

dD with dD = ζ· dWp (2.3)

Being D the fatigue damage, Wp_{the plastic strain energy density and ζa function}

determined experimentally based on the fatigue response of the material. With this approach, the authors obtain an evolution of the fatigue damage parameter as the simulation evolves, the material failure is obtained when D = 1. In [42], the model is tested for fatigue ranges between 103 _{to 10}7 _{cycles, which corresponds} to low and high cycle fatigue.

This formulation, as well as the formulation proposed by Kanvinde and Deierlein [47], are capable to account for regular and non-regular cycles, as both formula- tions are based on the addition of certain quantities while the material increases its plastic strain. However, they both have the drawback of being based on a failure criterion that is completely independent of the plastic model (uncoupled approaches): it is calculated as the simulation advances and, when it reaches a certain level, the criterion tells the code that the material has failed.

The simulation of LCF and ULCF has also been approached using non-linear constitutive laws. This is the case of Saanouni and Abdul-Latif [101] and [2], who propose the use of a representative volume element (RVE), and a non-linear law based on the slip theory, to account for the dislocation movement of metallic grains. Instead of a RVE, Naderi et al. [79] proposed simulating the progressive failure of a given structural element by applying random properties to the dif- ferent finite elements in which it is discretized. The constitutive model used to characterize LCF failure is the one defined by Lemaitre and Chaboche in [58]. The use of a stochastic approach is also the approach used by Warhadpande et al. [123], who applied random properties to a Voronoi cell. In most of these models the damage variable is also calculated independently of the non-linear constitutive law used to simulate the material performance.