2.5
Continuum Damage Mechanics method
The CDM is an alternative to the Fracture Mechanics Approach, with important advantages, namely that it handles, in a unified way, both crack initiation and crack propagation.
Failure in metals and alloys is a multiscale phenomenon, in general. Macroscopic rupture is the result of the irreversible processes that occur at smaller material length-scales. Both pre-existing and load generated micro/mesoscale flaws may grow reducing the nominal material resistance. Consequently, the conditions un- der which failure can occur should be evaluated based on meso/micromechanics considerations. In metals, all failure modes can be ascribed only to five micro- mechanisms and their combinations: cleavage, ductile fracture, creep, fatigue, and corrosion. For all of them, the reduction of the material load carrying capability is always associated to the appearance of irreversible strain, which may be either highly localized in the microstructure, as for cleavage, or spread across the entire geometry volume, as for ductile rupture or creep.
In addition to global theories, such as fracture mechanics concepts, in the last two decades, micromechanical modelling has been proved to be a powerful ap- proach to understand and predict failure in materials. The real advantage of micromechanics relies on the assumption that the model parameters are only material characteristics and do not depend on the geometry. Damage resulting from plastic deformation is mainly due to the formation of micro voids which initiate either as a result of fracturing or debonding from the ductile matrix, of inclusions such as carbides and sulphides. In pure metals, micro voids are nuc- leated at the grain triple points and along the grain boundary as a result of the incapacity of the microstructure to accommodate, in a congruent manner, the imposed displacement field. The growth of micro voids, under increasing strain level, progressively reduces the material capability to carry loads up to complete failure. A proper modelling of this micro mechanism at the mesoscale is the basis for the prediction of ductile failure in real life components and structures (i.e., the macroscale).
Damage models can be grouped in three main categories:
• Abrupt failure criteria • Porous solid plasticity
In abrupt failure criteria, failure is predicted to occur when one external variable, that is uncoupled from other internal variables, reaches its critical value (e.g. Rice and Tracy critical cavity growth criterion).
In porous solid plasticity, the effect of ductile damage (Gurson [35], Needleman and Tvergaard [80], hereafter GTN) is accounted for in the yield condition by a porosity term that progressively shrinks the yield surface. The GTN model, although extensively used is known to suffer from a number of limitations:
• A large number of material parameters which makes difficult to evaluate
possible mutual influence;
• The material parameters are not physically based and cannot be directly
measured for a material;
• A poor geometry transferability of damage parameters which often requires
a posteriori adjustments;
• Damage softening introduces a length scale dependency, (mesh effect) [3].
In the last category, damage is assumed to be one of the internal constitutive vari- ables that accounts for the effects on the material constitutive response induced by the irreversible processes that occurs in the material microstructure.
Starting from the early work of Kachanov [43], the CDM framework for ductile damage was later developed by Lemaitre [57] and Chaboche [21]. In the last two decades, a number of CDM based formulations have been proposed. Also, these models show a number of limitations:
• The proposed choice for the damage dissipation potential is, in many cases,
specific of the particular material;
• Damage evolution laws are often validated only with experimental data
obtained under uniaxial stress. Therefore, the transferability of parameters to multiaxial stresses is not always demonstrated;
• Similarly to the porosity models, also the CDM formulations are affected
by mesh size effect due to damage softening.
More recently CDM formulations have been proposed [17] which try to over- come the above limitations: the damage variable is uncoupled from plasticity, so
2.5. Continuum Damage Mechanics method 15
avoiding mesh size influence, and the damage evolution law take into account for nonlinear accumulation effects. All the cited approaches are based on the void growth and coalescence descriptions which, as already stated, strongly depend on stress triaxiality. Anyway, in the final phase of the fracture process two differ- ent mechanisms may compete, the internal necking of ligaments between voids, which is mainly influenced by medium-high values of the stress triaxiality, and the shear failure which is evident at low stress triaxiality, as described by Bao and Wierzbicki [7].
Experimental evidences of a different fracture behaviour at similar triaxiality level, but obtained from different geometrical conditions were early found by Clausing [23]. McClintock [76] also found that for many materials the equival- ent plastic strain at failure is lower in torsion than in tension, even if the stress triaxiality in torsion is zero, which is not consistent with any of the hydrostatic pressure dependence (i.e. triaxiality) models described. A step forward with re- spect to the classical damage theories was proposed by Wierzbicki et al. [124] by introducing another normalized parameter - the deviatoric parameter, based on the third invariant of the stress tensor, beside the stress triaxiality, to capture the strain to fracture dependence from the stress state, thus covering both the hydrostatic and the shear type failure modes. The sensitivity of a material to the third invariant has as consequence the non-uniqueness of the relation between triaxiality and fracture strain, which is bound between two distinct limits (upper and lower). These evidences have also been confirmed in recent works by Barsoum [13] and Coppola [26].
In contrast to monotonic ductile failures and low/high-cycle fatigue, models are less well developed for ULCF. The fundamental physical processes responsible for this type of fracture cannot be modelled using traditional fracture mechanics and fatigue models. ULCF is often accompanied by large-scale yielding, which may invalidate stress intensity-based K- or J-type approaches. It is well known that Coffin-Manson approach used in Low-cycle fatigue tends to over-predict the cyclic life under extremely low cycle fatigue conditions [125]. Like monotonic ductile fracture, ULCF is ultimately controlled by the growth and coalescence of microscopic voids. However, the reversed plasticity induced by the severe cyclic loading degrades the fracture resistance, or toughness, due to the accumulation of material damage.
This degradation mechanism is similar, in concept, to that of low cycle fatigue. Thus, ULCF is more accurately conceptualized as an interaction between ductile fracture and fatigue.