Applied strain-rate triaxiality

The applied in-plane load is such that the only non-zero components of the strain-rate tensor are D116= 0, D226= 0. (5.20) In the following, it is convenient to assume without any loss of generality that|D22|>|D11|. Then, making use of the definitions for the mean and equivalent macroscopic strain-rate,DmandDeq, and the strain-rate triaxiality,XE, which are recalled here for completeness

Deq= |D11−D22|

√3 , Dm=D11+D22

2 , XE= Dm

Deq

, (5.21)

we can rewrite the components of the strain-rate tensor as D11= In the last expression, Deq andXE are externally applied in the problem andρ=±1 such that the condition|D22|>|D11|is always true. In the sequel, we discuss the evolution of microstructure and the possible instabilities in the porous medium for given values of the strain-rate triaxiality, XE, as predicted by the “second-order” method (SOM), the “variational” method (V AR) and the Gurson model (GU R).

0 0.2 0.4 0.6 0.8 1

Figure 5.15: Results are shown for the evolution of the relevant microstructural and macroscopic variables for a porous material consisted of cylindrical pores with an initially, circular cross-section and porosityfo= 10%.

The matrix phase exhibits an ideally-plastic behavior, while the composite is subjected to biaxial tension and compression loading conditions with fixed strain-rate triaxiality,XE =−0.05,0.1. SOM,V AR and GU R (Gurson, 1977) estimates are shown for the evolution of the (a) porosity f, (b) the aspect ratiow, (c) the hardening rateH and (d) the normalized, macroscopic equivalent stressσeq/σo (σo denotes the flow stress of the matrix phase) of the composite as a function of the macroscopic, equivalent strain εeq. In (c) the symbols◦and¤denote the loss of stability for the porous medium as predicted by theSOM and theV AR, respectively.

Fig. 5.15 shows evolution curves for the porosity f, the aspect ratio w, the hardening rate H and the macroscopic equivalent stress σeq normalized by the flow stress of the matrix phase σo as a function of the strain-rate triaxiality XE and the total equivalent strain-rate εeq for a porous material consisting of cylindrical voids with circular cross-section and initial porosityfo= 10%. For completeness, the SOM is compared with corresponding estimates obtained by the V AR and the

GU R models. Two values of the strain-rate XE = −0.05,0.1 are chosen to study the instability conditions described above. More specifically, Fig. 5.15a shows evolution curves for the porosity f as predicted by the three models considered here. In this case that the loading is strain-controlled and the matrix phase is incompressible, all the methods deliver exactly the same evolution for the porosity. On the other hand, in Fig.5.15b, the corresponding evolution of the aspect ratio is different for the various methods. Of course, the GU R model does not include any information about the shape of the voids and therefore, no curve is shown here. In contrast, theSOM predicts a higher rate of change in the aspect ratiow, when compared with the correspondingV ARestimate in both cases of XE considered. In particular, for XE = 0.1 the ellipsoidal void elongates in the direction of the maximum principal loading, i.e., forXE= 0.1,D22> D11 the aspect ratio grows in the 2−direction implying thatw >1. Consequently, the evolution of the aspect ratiowacts as a hardening mechanism during the deformation process in both methods, whereas in the case of the SOM this hardening effect is stronger than in theV ARmethod. On the other hand, the porosityf also grows inducing a softening in the overall behavior of the composite. These two mechanisms, i.e., the hardening effect induced by the growth of the aspect ratiowand the softening caused by the growth of the porosity f are in competition. Looking at Fig. 5.15c the curves forXE= 0.1, we observe that for the SOM estimate the hardening effect caused by the growth of the aspect ratiowdominates initially over the softening mechanism of the porosity growth until a total strain of ∼25%, where the hardening rate H crosses zero. This point, denoted with a circle on the graph, corresponds to a possible instability of the material. On the other hand, in the V AR method and certainly in the GU R model, the softening effect induced by the growth of the porosity f is dominant and the corresponding curves forH never cross zero. This has as a consequence that both theV ARand theGU Rmodels do not predict instability for strain-rate triaxiality XE = 0.1. Fig. 5.15d in turn shows the corresponding stress curves where forXE = 0.1 theSOM predicts a lower value forσeq than both theV AR and theGU Rresults. Of course,GU Rmodel is the stiffest of the three models in this case and predicts higher stress values.

Similar observations can be made in the context of Fig. 5.15 forXE = −0.05. In this case the material is subjected to compression in the 2−direction and, therefore, the void elongates in the 1−direction (i.e.,w <1). While the material softens in the 2−direction, (i.e., in the direction of the maximum (absolute) principal loading) due to the change of the aspect ratiow, the overall porosity f is decreasing which induces hardening in the material. In this case, both theSOM and theV AR exhibit an initial softening due to the elongation ofw in the 1−direction, which is observed by the initial negative hardening rateH, in Fig.5.15c. Nonetheless, the correspondingSOM and theV AR estimates for H cross zero and become positive (i.e., the medium hardens) at a critical total strain εeq∼10%, which indicates a possible point of instability (theSOM becomes unstable earlier than the V AR). In contrast, theGU Rmodel predicts hardening as a consequence of the decreasing porosity and never loses stability.

This procedure, described above, can be repeated for the entire range of strain-rate triaxialities, i.e.,XE∈(−∞,∞). In this regard, Fig.5.16summarizes results for the critical equivalent strain-rate ε^cr_eq for loss of stability as a function of the strain-rate triaxiality in the case of a porous medium with

-0.2 -0.1 0 0.1 0.2 0

0.2 0.4 0.6 0.8 1

LOE, f =10%, m=0

SOM VAR

cr

ε

X

Figure 5.16: Macroscopic onset-of-failure curves as predicted by theSOM and V AR calculations, for an initially transversely, isotropic porous medium with ideally-plastic matrix phase and initial porosityfo= 10%.

The plot shows the critical equivalent strainε^creq as a function of the applied strain-rate triaxialityXE.

cylindrical voids and initial porosity fo = 10%. The main observation in the context of this figure is that for negative strain-rate triaxialities, theSOM is more unstable than the correspondingV AR method. In contrast, for positive XE, the V AR method becomes unstable earlier than the SOM. Furthermore, it is shown that instabilities occur at low strain-rate triaxialities, where the shear strains are dominant over the dilatational strains.

In summary, it is important to mention that based on the analysis made in the context of finite nonlinearities in the previous section, theSOMis expected to predict more accurately the evolution of porosity and aspect ratio when compared with theV ARmethod and consequently be more accurate in determining instabilities in porous media. Nonetheless, both methods exhibit a similar qualitative behavior, while they improve substantially on the GU R model by being capable of capturing the effects of the void shape, which has been shown to be very significant in the prediction of instabilities in ideally-plastic solids.