Void Growth and Coalescence in Sigmoidal Hardening Porous Plastic Solids under Tensile and Shear Loading

This work examines the void growth and coalescence in isotropic porous elastoplastic solids with sigmoidal material hardening via ﬁnite element three-dimensional unit cell calculations. The investigations are carried out for various combinations of stress triaxiality ratio ( T ) and Lode parameter ( L ) and considers a wide range of sigmoidal hardening behaviors with eﬀective hardening rates spanning two decades. The eﬀect of L is considered in the presence and in the absence of imposed shear stress. Our ﬁndings reveal that depending on the rate of sigmoidal hardening the cell stress-strain responses may exhibit two distinct transitions with respect to stress triaxiality T . Further, the sigmoidal hardening rate also inﬂuences porosity evolution which may show stagnation before a runaway growth up to ﬁnal failure. For a given T - L combination, an imposed shear stress exacerbates the onset of coalescence relative to its counterpart with no imposed shear stress. We ﬁnd that the residual cell ductility beyond the onset of coalescence is strongly inﬂuenced by the eﬀective material hardening rate at high triaxiality levels.


Introduction
Strain hardening in elastoplastic materials has important consequences in ductile failure.In power-law hardening materials, the stress triaxiality ahead of a crack tip increases rapidly with the strain hardening exponent (n), which can have implications on material failure by cleavage or void growth [1].On the other hand, higher values of n delay void growth under a constant stress state [2,3], thus creating a competitive scenario for ductile failure.While much work has been carried out to study void growth and coalescence in power-law hardening materials, the role of other material hardening characteristics has not been as well studied.Among those are materials exhibiting sigmoidal hardening, which is observed in some hexagonal close-packed metals (e.g., magnesium [4]), polymers [5], and even shape memory alloys [6]. Figure 1 shows illustrative sigmoidal stress-strain curves from polycrystal simulations of magnesium (Mg) alloys [7], which occurs because of profuse deformation twinning.While the underlying deformation mechanisms causing sigmoidal hardening may depend on the material, the broader stress-strain features include a two-stage hardening response characterized by an initial yield stress followed by an S-shaped curve culminating into a saturation-type behavior at large strains.
(a) (b) Fig. 1 Sigmoidal stress-strain responses of polycrystalline magnesium alloys with different initial textures (cases A-I).Panel (a) illustrates scenarios with similar hardening rates but different saturation stresses and (b) shows scenarios with similar saturation stress but different hardening rates.Data re-plotted from [7] A recent work [8] investigated void growth and coalescence in isotropic porous elastoplastic solids showing sigmoidal material hardening.Those calculations were performed for axisymmetric tensile stress states characterized by a constant stress triaxiality ratio, T = Σ m /Σ eq where Σ m and Σ eq are respectively the mean normal stress and the equivalent stress.Thus, the salient observations did not account for the intermediate principal stress (Σ 2 ), which samples stress states between purely axisymmetric and sheardominant.To consider the effect of Σ 2 , the Lode parameter is introduced: where Σ 1 ≥ Σ 2 ≥ Σ 3 are the principal stresses.The effect of T , at fixed L, on ductility (characterized by strain-to-failure, E c ) is well-known -an exponentially decreasing E c with increasing T .By way of contrast, the E c − L relationship at fixed T is not particularly well characterized even for conventional power-law hardening materials [9], let alone sigmoidally hardening materials.In power-law hardening materials, it appears that E c is non-monotonic with L with the minimum occurring in the regime −1.0 ≲ L ≲ 0 [10,11,12,13], although there is equally compelling evidence of monotonically increasing E c with increasing L [14,15].Recent micromechanical analysis [16] indicates that these trends are influenced by the initial porosity f 0 and T .A more recent computational study reveals that imposed boundary conditions also play a crucial role in the way E c − L relations manifest at fixed T [9].
With this background, we focus on three aspects pertaining to sigmoidal hardening materials.First, we investigate a wider and more realistic range of sigmoidal hardening parameters than those considered in Ref. [8].Second, we assess the void growth and coalescence trends for these materials over a range of T − L combinations.Third, we consider the T − L combinations without (ρ xy = 0) and with (ρ xy ̸ = 0) an imposed shear stress.

Problem Formulation
In contrast to the axisymmetric finite element simulations in [8], we adopt a three-dimensional unit cell to model combined tensile and shear loading, Fig. 2. The computational setup comprises a cubic unit cell (initial dimension, L 0 ) with a spherical void (initial radius, R 0 ) at its center.Thus, the initial porosity is f 0 = (4πR 3 0 /3L 3 0 ).We define ligament parameters χ x = r x /L x and χ z = r z /L z where r i (i = x, z) is the current void dimension in the i th direction and L i is the corresponding cell dimension.The unit cell is constrained by periodic kinematic boundary conditions.The unit cell faces are under traction via elastic springs connected to prescribed velocity boundary conditions to maintain a constant macroscopic triaxial stress state described by Σ = Σ xx e x ⊗ e x + Σ yy e y ⊗ e y + Σ zz e z ⊗ e z + Σ xy e x ⊗ e y in the lab frame (x,y,z).The macroscopic Cauchy stress components (Σ ij ) are volume-averaged quantities computed from the local (at each Gauss point) stress components (σ ij ) as Σ ij = 1 V V σ ij dV where V is the current unit cell volume.Likewise, the volume-averaged macroscopic strains (E ij ) are dV where e ij is a logarithmic strain component at each Gauss point.The macroscopic mean stress is Σ m = (1/3)tr Σ and the macrosopic equivalent stress is Σ eq = (3/2)Σ ′ : Σ ′ .Likewise, the macroscopic effective strain is E eq = (2/3)E ′ : E ′ .

Bottom-Front Edge
Bottom-Right Edge Fig. 2 The finite element unit cell with an initial void volume fraction of 0.01.The model contains ∼ 28, 850 linear brick elements (C3D8 in ABAQUS).
Here, given a second-order tensor A, we write its deviatoric part as A ′ = A : J with J = I − (1/3)I ⊗ I with I and I being the fourth-order and second-order identity tensors, respectively.

Results
In what follows, we discuss illustrative results that highlight key trends for 0.75 ≤ T ≤ 3.0 and −1.0 ≤ L ≤ 1.0.The bounds for T characterize stress states ranging in specimens with blunt notches ahead to those ahead of crack tips and those for L range from generalized tension (L = −1.0) to generalized shear (L = 0), and generalized compression (L = 1.0).The results for porous unit cells under uniaxial tension (T = 1/3) are not included for brevity as they are indistinguishable from their pristine counterparts.
To identify the onset of coalescence, we adopt the strain-based criterion [18], which identifies coalescence onset as the critical equivalent strain, E c , at which the cell straining transitions from a triaxial to uniaxial mode.Further, we define failure strain, E f , as the equivalent strain at which one or both the ligament parameters χ i reach a critical value χ f .In theory, failure occurs when χ = 1, however from a numerical standpoint we set χ f = 0.95.In situations where E f occurs before E c , we take E c = E f .

Macroscopic responses in the absence of applied shear stress
For brevity, we present the results for ŝ = {0.5, 0.8} and ê = {0.046,0.125, 0.475, 1.0} for a range of T -L combinations.Fig. 1 summarizes the stress ratios for the T -L combinations keeping ρ xy = 0. Fig. 4 shows the cell responses of materials with ŝ = 0.5 but varying ê values.The results for L = 1 are not shown here to avoid overcrowding as the strains accumulated in those cases are much larger and rapid failure may not occur.At a fixed T , L = −1 presents the most severe scenario insofar as the porosity evolution is concerned, which in turn governs stress softening and hence, E c .Beyond this expected trend at this level of f 0 [16], sigmoidal hardening produces a rich suite of responses that intimately depend on T and whose qualitative features are persistent across the range of L.  Consider RH materials first (Fig. 4a and 4b).Under moderate T levels (≲ 1.0) the strain (E peak ) corresponding to the peak stress is approximately equal to the material saturation strain (ε f ), i.e., E peak ≈ ε f .By way of comparison, E c , (marked by x) is much larger, i.e., E c ≫ E peak .With increasing L the gap between E peak and E c increases at these T levels.At higher T levels the effect of L is diminished such that at T = 3.0, E c ≈ E peak .
In IH materials (Fig. 4c) the gap between E peak and E c is smaller compared to RH materials at T ≲ 1.0.Interestingly, at T = 2.0 the cell responses exhibit strain hardening leading up to the coalescence onset; i.e., E c ≈ E peak .In contrast, at T = 3.0 gradual stress softening occurs immediately following the initial yield with no evidence of sigmoidal hardening and E c ≫ E peak .
In SH materials (Fig. 4d), E c ≈ E peak for T ≲ 1.0 with a strain hardening response leading up to the coalescence onset.For T ≳ 2.0 the responses are qualitatively similar to those of IH materials at T = 3.0 together with E c ≫ E peak .
The porosity evolution in RH, IH, and SH materials (Fig. 4e-4h) is largely insensitive to L up to E peak .Beyond that, it shows sensitivity to L for T ≲ 2.0.Note that, at a given L the critical porosity (f c ) at E c is relatively insensitive to ê at moderate triaxiality levels (T ≲ 1.0) but depends strongly on T at higher T levels.Fig. 5 collates the results for ŝ = 0.8.While the broad trends are similar to those seen in ŝ = 0.5 at T ≤ 1.0, important differences can be spotted at higher T levels.Unlike ŝ = 0.5, only one transition is seen as a function of T .For a given ê, the cell response transitions from E c ≫ E peak at moderate T levels (T ≲ 1.0) to a response characterized by E c ≈ E peak at high T levels ≳ 2.0.Moreover, at T ≳ 2.0 the initial yield continues to occur at σ i (i.e., Σeq = 1) unlike in the case of ŝ = 0.5 where Σeq < 1.
In Fig. 5e-5h, the porosity appears to follow an exponential evolution.However, a closer look (Fig. 6) reveals that porosity initially begins to increase, then stagnates (or slows down) over an extended strain range, and then increases rapidly again, see RH responses Fig. 6a and 6b.[8] observed similar responses in RH materials for L = −1.0 and found that they occurred due to large differences in the magnitude of the flow stresses between the polar and the equatorial regions of the void as a result of the sigmoidal material hardening.What is interesting to note here is that it prevails over a much wider range of ŝ and ê values (Fig. 6c) and over a broader range of T across the entire range of L. In comparison, the stagnation tendency is much less in SH materials (Fig. 5h).Fig. 7 illustrates the evolution of the ligament parameters (χ x and χ z ) for ŝ = 0.8.Note that χ is indicative of ligament thinning between adjacent voids.As expected, at L = −1.0 the lateral void growth is symmetric (χ x = χ z ) irrespective of T .For L = 0, initially χ z > χ x but with progressive deformation χ x increases rapidly and overtakes χ z .This occurs because, while r x < r z at all times, with deformation L x ≪ L z owing to the Poisson's effect combined with ρ xx ≪ ρ zz .As seen in Fig. 7b, all combinations of T and ê show this behavior but the effect is most discernible for T = 1 and ê = 1.0.In such scenarios, E f occurs when χ x = χ f .Another notable characteristic evident in RH materials is that, for both values of L the χ evolution is markedly slowed down in the regime between the initial yield and the second stage hardening.It coincides with the slowing down of the porosity evolution, cf.Fig. 6.This behavior is similar to the evolution of χ isotropic cases at L = −1.0[8].We note in passing that ŝ = 0.9 shows similar characteristics as ŝ = 0.8.Fig. 8a-8c shows the effect of the Lode parameter on E c for the RH, IH, and SH material types with varying ŝ values at different T levels.Results corresponding to L = 1.0 are not included as in all the cases, E c is attained when χ = χ crit , which occurs at very large values of E eq .As seen, L plays a role in E c at T = 0.75 and T = 1.0 in RH (Fig. 8a) and IH (Fig. 8b) materials.In fact, the range of E c for −1.0 ≤ L ≤ +0.5 is the same for RH and IH materials.In contrast, E c of SH materials is much less sensitive to L, particularly at T ≥ 1.0 but increases with increasing ŝ, Fig. 8c.For a fixed T − L combination, a higher ŝ (at a fixed ê) results in higher E c .Fig. 8d-8i capture the ligament parameter values corresponding to E c .For any T , L = −1.0gives χ x c = χ z c , as expected.Interestingly, while χ x c increases with increasing L, the changes in χ z c are relatively modest (Fig. 8g-8i).As a result, for L > −1.0, χ x c > χ z c despite the fact that ρ zz > ρ xx .As can be seen, this trend prevails over the range of material parameters (ŝ, ê) and triaxiality levels considered here.

Macroscopic responses in the presence of applied shear stress
In this section, we present illustrative results with ρ xy ̸ = 0 for ŝ = 0.8 for varying ê and compare them against the corresponding cases with ρ xy = 0. Fig. 2 summarizes the applied stress ratios for particular T -L combinations.In the present work, we adopt the stress ratios such that Σ yy ≥ {Σ xx , Σ zz }.Note that the angle φ between the primary loading direction and the direction of the maximum principal stress varies for each case, which is different from the problem of constant φ considered by [11] and [13] but similar to the work of [17].
Table 2 Stress ratios for T -L combination for combined tension and shear loading.φ denotes the angle between the loading axis and the maximum principal stress direction.Fig. 9a, 9c, and 9e collate the cell responses at T = 0.75 for RH (ê = 0.125), IH (ê = 0.475), and SH (ê = 1.0) materials.Note that the peak stress in RH and SH materials is nearly the same as the peak stress for the pristine matrix material.In comparison, the SH material exhibits lower peak stress, an effect of the rate of sigmoidal hardening dictated solely by ê (as ŝ is constant) and is agnostic to ρ xy .While the broad trends for ρ xy ̸ = 0 are similar to ρ xy = 0, some important differences are seen.For all three material types, ρ xy has no effect on the cell responses at L = 1.0.With decreasing L, however, imposing shear stress expedites the rapid porosity evolution and hence, lowers E c .At T = 2.0 (Fig. 9b, 9d, and 9f) the same materials exhibit different qualitative features compared to T = 0.75, particularly for L ∈ {−1.0, 0.0}.For the RH (Fig. 9b) and IH (Fig. 9d) materials, the difference in E c without and with shear stress is practically indistinguishable.Notwithstanding this, beyond the onset of coalescence, the stress softening and porosity evolution show a more gradual evolution in the presence of shear stress.The slower porosity evolution could result in a delayed final failure thereby improving the overall material resistance.In comparison, the SH material (Fig. 9f) shows a somewhat higher E c for ρ xy ̸ = 0 than for ρ xy = 0 in contrast to the lower triaxiality case (cf.Fig. 9e).These differences in E c can be important at such high T levels.The overall trends of E c and χ i c , are similar to those in Fig. 8 and are discussed in the context of sigmoidal material parameters in the next section.

Discussion
In the preceding section, we consider the Lode parameter effect on the porous response of sigmoidally hardening materials in the absence or presence of imposed shear stress.In the latter case, the angle φ made by the loading axis with the major principal stress direction comes into play (cf.Fig. 2).Generally (but not always), a fixed value of φ is chosen such that E c is minimized for a particular T -L combination [11] and is obtained by trial and error [13].Here, we have not performed such a study.
Another aspect pertains to material ductility.In theory, ductility (failure strain) is the strain at which stress drops to zero, which coincides with vanishing inter-void ligament, i.e., χ = 1.As alluded to earlier, E c indicates the strain at which the process of coalescence initiates and is often adopted as a ductility criterion.A recent computational study [9] shows that the trends of ductility with L can significantly vary depending on the choice of ductility criterion and unit cell boundary conditions, and therefore, material ductility is open to interpretation.This is in addition to the dependence of ductility on the initial porosity [16,19].Recently, Benzerga and colleagues [20,21,22,19] have introduced the concept of unhomogeneous yielding (UY).For axisymmetric stress states, the strain corresponding to UY is identical to E c .On the other hand, for stress states that deviate from axisymmetry, the strain associated with the first occurrence of UY may be different from E c .While UY onset appears to be a useful indicator, an unambiguous representation of ductility under general proportional loading states seems to be an open question.
With these caveats, we discuss material failure trends from the viewpoint of material hardening (ŝ and ê).Fig. 10 collates the combined role of loading material parameters on E c for ρ xy = 0. Details aside, these results indicate that E c depends on the nature of hardening rather the effective sigmoidal hardening rate ĥ = ŝ/ê.In other words, ĥ does not serve as a unique descriptor of the sigmoidal material hardening.That is, two materials with different {ŝ, ê} combinations giving the same ĥ will not give the same E c .The calculations suggest that in such a scenario, a material with higher ê and ŝ values tends to show better ductility than its counterpart with a lower ê and ŝ values.
Fig. 11 illustrates how ê affect E c for ŝ = 0.8.When ρ xy = 0, E c increases with ê for T = 0.75 (and T = 1.0, not shown).While not shown here, at T = 3.0 the trend of E c depends on ŝ.For ŝ = 0.5, E c is insensitive to ê while for ŝ = 0.9 E c increases with ê.The case of ŝ = 0.8 demonstrates an intermediate trend in that E c increases with ê but saturates beyond ê ∼ 0.475.The increase in E c gets progressively larger for higher ŝ values.
When a shear stress is imposed (ρ xy ̸ = 0), the E c − ê trends follow those for ρ xy = 0.Over the range of ê, the effect of L is as follows.At T = 0.75, E c values at L = −1.0 (Fig. 11a) and L = 0 (Fig. 11b) are lower than their corresponding values for ρ xy = 0 but ρ xy has no effect on E c for L = 0.5 (Fig. 11c).On the other hand, at T = 2.0 is unaffected by L.Over the range of L, the relative increase in E c with ê at T = 0.75 is ∼ 25%.On the other hand, at T = 2.0 the relative increase is ∼ 100% over the same range of ê.That is, for a fixed ŝ an SH material exhibits an improvement in E c compared to an RH material and this improvement increases with increasing T .Notably, these relative improvements E c with ê are independent of L and ρ xy .
Fig. 11d-11f shows the corresponding plots of E f defined earlier.The trends are identical to E c .At first glance, it appears that there is no perceptible quantitative difference between E c and E f .However, the difference becomes clearer in Fig. 12 where we plot the relative cell ductility defined as Êcell = (E f −E c )/E c versus ĥ = ŝ/ê.For T = 0.75, and 1.0, Êcell is insensitive to ĥ.At higher T levels though, Êcell shows a non-monotonic trend.As seen, for a given L, Êcell is the largest for ĥ ≪ 1 and tends to be the lowest for ĥ ∼ 1 (IH materials).In other words, for IH materials, ultimate failure occurs soon after the coalescence process begins.By way of contrast, for ĥ ≪ 1 (SH materials) or ĥ ≫ 1 (RH materials) ultimate failure can be delayed, particularly at high T levels.This behavior is asymmetric with respect to ĥ and the largest gains (for a fixed L) are seen for ĥ ≪ 1.Given that the trends of E c and E f are agnostic to ρ xy (at least the shear ratios considered here), the Êcell trends are also unaffected by ρ xy .

Conclusions
In this work, we numerically investigate the void growth and coalescence in plastically isotropic materials exhibiting sigmoidal stress-strain responses.The results extend well beyond those presented in a recent work [8] in two respects -(i) a wider and more realistic range of material hardening characteristics are considered, and (ii) the effect of intermediate principal stress is explored.Several important conclusions are drawn: 1.The cell responses of the three material types (RH, IH, and SH) may be described in terms of two distinct transitions as a function of T .For T ≤ T low , a sigmoidal hardening response is observed, with E c ≫ E peak .This is followed by a response characterized by strain hardening, with E c ≈ E peak in the regime T low < T ≤ T high .Finally at T > T high , the strain hardening response is replaced by a lack of strain hardening and E c ≫ E peak .The occurrence of both transitions (i.e., T low and T high ) is determined by the sigmoidal hardening rate.RH materials only exhibit the first transition, while SH materials lack the first transition at least in the regime of T considered here.IH materials demonstrate both transitions.2. In the absence of an imposed shear stress, porosity evolution exhibits stagnation, which coincides with the stagnation in the thinning of lateral ligaments.In RH materials, it is observed over the entire range of T investigated in this work whereas in IH materials the behavior tends to be tempered at high T values.SH materials do not show any perceptible stagnation.3. The broad trends of the stress-strain responses and porosity evolution in the presence of shear stress are similar to those in its absence.With decreasing L, the presence of a shear stress lowers E c at moderate levels of triaxiality.At high triaxiality levels, SH materials exhibit substantial improvement in E c relative to RH materials irrespective of whether a shear stress is imposed or not.A much lower improvement is observed at lower triaxiality levels.4. At high triaxiality levels, the relative cell ductility shows an improvement in the values of the effective material hardening rates that are either smaller or larger than unity.While this does not indicate a Fig. 12 Relative cell ductility as a function of the effective material hardening rate ( ĥ) for ρxy = 0.
high overall ductility, it suggests that such materials can sustain non-negligible strains beyond those that correspond to the onset of the coalescence process.The largest improvement in the relative cell ductility is seen for RH materials.

Fig. 9
Fig. 9 Effect of imposed shear stress (solid lines) on macroscopic cell responses under prescribed T − L combinations in (a-b) RH, (c-d) IH, and (e-f) SH materials.Dashed lines are the corresponding results with ρxy = 0.The results are shown for ŝ = 0.8.

Fig. 10
Fig. 10 Effect of ê on the strain to coalescence (Ec) for illustrative cases of L. ρxy = 0

Table 1
Stress ratios for T -L combination for tensile loading.