1 Introduction

Ductile fracture of metals comprises two micromechanisms that operate simultaneously, namely void nucleation and void growth (Besson 2010; Benzerga et al. 2016), that do not operate to the same extent. High stress triaxiality levels, above the uniaxial tension threshold, lead to reduced ductility (failure strains) (Bao and Wierzbicki 2004; Papasidero et al. 2015), and are generally characterized by (roughly) homothetic growth of microvoids, seen as equiaxed dimples separated by thin intervoid material ligaments on the fracture surface. Failure under low stress triaxiality levels, below the uniaxial tension threshold, is mainly driven by void nucleation (Needleman 1987; Barsoum and Faleskog 2007), void flattening (rotating) and collapse with large strains (Tvergaard 2009; Nielsen et al. 2012; Longère et al. 2015; Torki and Benzerga 2018). A typical shear fracture surface contains fine elongated dimples, forming what is often referred to as intervoid shearing or void-sheet in which nucleated microvoids grow slightly and flatten in the direction of maximum stress/strain, see e.g. (Fleck and Hutchinson 1986; Bandstra et al. 1998; Nielsen et al. 2012; Tvergaard 2015; Noell et al. 2018; Chen and Osovski 2019a). Besides the mechanisms related to void closure (or collapse), void growth under shear is also observed and discussed in the early experiments of (Goods and Brown 1979; Fisher and Gurland 1981; Barsoum and Faleskog 2007), and more recently in (Fadida et al. 2020), who embedded an artificial pore in a shear-compression specimen gauge (low-negative stress triaxiality). Figure 1 shows the fracture surface of such a printed Ti6Al4V specimen. The embedded pore is flattened following the massive shear deformation in the gauge (“PORE” in Fig. 1), whereas the void-sheet in “A” (Fig. 1) reveals dimples that are elongated according to local distortions in the deformation field exerted by the curvature of the pore. In addition, patches of void-sheets are evidenced among the extensive wear markings away from the embedded void (see “B” in Fig. 1). Those microvoids are assumed to form during the loading process, thereby suggesting a positive local stress triaxiality in the deformation history of the gauge, despite the essentially compressive nature of the loading (Chen and Osovski 2020).

Fig. 1
figure 1

Fractograph of additive-manufactured Ti6Al4V shear compression specimen (SCS) with an embedded spherical pore (Fadida et al. 2020). The fracture surface is typical to such shear deformations, comprising small dimples that are deformed in the shear direction. The sample is tilted to demonstrate the deviations around the embedded pore. The fracture surface is marked with yellow dashed lines (GAUGE). A Marks the location of the extensive void nucleation around the pore. B Marks a patch of void-sheet among the extensive wear (W)

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Since positive stress-triaxiality is required for void nucleation and determines subsequent void growth, it can be speculated that a nucleated void modulates the local stress triaxiality under dominant shear deformation levels. In that respect, what one needs to consider are precisely those local stress triaxiality excursions, clearly related to local stress concentrations, that activate rapid void nucleation in an otherwise zero average triaxiality material ligament.

While the respective extent of each mechanism dictates the ultimate ductility to failure of a metal, one point has not been extensively considered in the various modelling attempts despite being experimentally supported, namely the two types of void nucleation mechanisms. The first type, operating in high purity single-phase metals, is that of homogeneous nucleation. By contrast, the overwhelming majority of engineering alloys contains second phase particles or fine precipitates that lead to heterogeneous void nucleation, the latter being necessarily of a stochastic nature.

Heterogeneous void nucleation is usually studied through explicit geometrical features in the ductile fracture process, such as void/inclusion/notch/crack, see e.g. (Berdin 2004; François et al. 2012; Bourih et al. 2018; Andersen et al. 2020), which makes the simulation more resource-demanding. Here, a new approach aiming at introducing void nucleation heterogeneity in thin intervoid ligaments is utilized and examined. Void nucleation heterogeneity is introduced via the material model parameters by assigning different nucleation strain values to each material point in the model. The heterogeneity’s effect on the material response is studied under shear deformation, with and without a structural void as a primary structural stress concentration. This work systematically investigates and compares cases of void nucleation and growth (damage) under low stress triaxiality (shear) in homogeneous and heterogeneous voiding micromechanisms, for two configurations, namely a uniform and a stress-concentrator containing model.

The paper is organized as follows: the numerical model is first described, including the material model parameters, boundary conditions, statistical assumptions, and model geometry. The results section examines the effect of void nucleation heterogeneity on the porosity evolution in the sheet model. Then, this phenomenon is analyzed around a structural stress concentration (circular pore). Next, a parametric study of the sensitivity of the results to the void nuicleation model parameters is presented. The main results of this study are discussed next, followed by a concluding section.

2 Micromechanical model description

A periodic two-dimensional simulation cell (Fig. 2), consisting of a voided matrix that is subjected to quasi-static simple-shear deformation, is modeled under plane stress using finite-element ABAQUS/EXPLICIT software (Simulia 2020). The aim is to explain the mechanical interaction between stress concentrations (microvoids) during a massive strain localization scenario. The mesh used for the calculations is constructed of approximately 70,000 6-nodes triangle elements (see CPS6M in (Simulia 2020)) while ensuring the smallest elements around the central pore are approximately \(0.025d\) in size, where \(d = 0.2L\), see Fig. 2 for notation. Triangle elements ensure the stability of the simulation over some accuracy compared to the hex elements, especially when dealing with a localization scenario that can be enforced artificially by shear-locking, for example. Periodic boundary conditions are applied using the displacement constraints in Eq. (1) to the exterior edges nodes of the model.

$$ \begin{gathered} {\varvec{u}}_{{\left( {DC} \right)}} - {\varvec{u}}_{{\left( {AB} \right)}} = {\varvec{u}}_{D} - {\varvec{u}}_{A} \hfill \\ {\varvec{u}}_{{\left( {AD} \right)}} - {\varvec{u}}_{{\left( {BC} \right)}} = {\varvec{u}}_{D} - {\varvec{u}}_{C} \hfill \\ {\varvec{u}}_{D} + {\varvec{u}}_{B} = {\varvec{u}}_{A} + {\varvec{u}}_{C} \,\,\,\,\,\,\,\,\,;\,\,\,\,\,\,\,\,\,{\varvec{u}}_{D} = 0 \hfill \\ \end{gathered} $$
(1)
Fig. 2
figure 2

Two configurations of the unit cell model. In a, the “sheet” model represent a uniform material. In b, the second configuration comprises a single circular pore in the center of the cell. Periodic-boundaries condition is applied to the exterior edges of the cell. Corner D is fixed, and by moving corner “A” in the horizontal direction, a simple shear deformation mode is enforced. Under these conditions, the sheet model in (a) results in zero stress triaxiality and zero Lode parameter even for large deformations

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where \({\varvec{u}}\) is a displacement vector of a node on the exterior boundaries of the simulation cell. \({\varvec{u}}_{{\left( {DC} \right)}}\) and \({\varvec{u}}_{{\left( {AB} \right)}}\) represent a pair of counter nodes along the edge (DC) and (AB), respectively. Simple shear deformation is applied to the cell by prescribing horizontal displacement of corner A in Fig. 2, during which strain localization is expected to be established between voids over the periodic boundaries.

The following model material description was implemented as a VUMAT user-subroutine. The plastic and elastic strains and the porosity are iteratively calculated by fulfilling the material constitutive relations, such as the yield potential and the hardening law, using the backward-Euler scheme. The total strain increment in each time step is decomposed into elastic and plastic parts, based on the following expressions:

$$ \sigma_{ij} = \sigma_{ij}^{trial} - C_{ijkl} \Delta \varepsilon_{kl}^{pl} \,\,; \quad \sigma_{ij}^{trial} = \sigma_{ij}^{old} + C_{ijkl} \Delta \varepsilon_{kl}^{{}} \,\,;\quad \Delta \varepsilon_{ij}^{pl} = \lambda \frac{\partial \Phi }{{\partial \sigma_{ij} }} $$
(2)

where \(\sigma_{ij}^{old}\) is the Cauchy stress tensor of the previous step, and \(\sigma_{ij}^{trial}\) is the initial guess of the iterative elastic/plastic strain decomposition procedure, assuming the entire strain increment \((\Delta \varepsilon_{ij} )\) to be elastic \((\Delta \varepsilon_{ij}^{pl} = 0)\). \(C_{ijkl}\) is the elastic moduli tensor and \(\lambda\) is the plastic multiplier. The hardening law is assumed to be isotropic, of the form:

$$ \sigma_{eq} = \sigma_{Y} \left( {1 + \frac{E}{{\sigma_{Y} }}\varepsilon_{eq} } \right)^{N} $$
(3)

where \(\sigma_{eq}\) and \(\varepsilon_{eq}\) are the matrix flow stress and strain. The change in the equivalent flow stress is achieved by equating the microscopic plastic work rate to the equivalent macroscopic work through \(\sigma_{eq} \Delta \varepsilon_{eq} \left( {1 - f} \right) = \sigma_{ij} \Delta \varepsilon_{ij}\), where \(f\) is the porosity. The selected material properties, given in Table 1, are representative of mild steels (Nahshon and Hutchinson 2008; Chen and Osovski 2019a, 2019b).

Table 1 Plastic hardening material model parameters
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The GTN (Gurson 1977; Tvergaard and Needleman 1984; Needleman 1987) flow potential modifies the von Mises flow to account for porosity as a damage state variable:

$$ \Phi = \left( {\frac{{\sigma_{VM} }}{{\sigma_{eq} }}} \right)^{2} + 2f^{*} q_{1} \cosh \left( {\frac{{3\sigma_{h} }}{{2\sigma_{eq} }}} \right) - \left( {q_{1}^{2} f^{{*^{2} }} + 1} \right) $$
(4)

where \(\sigma_{VM}\) is the von Mises equivalent stress, and \(\sigma_{h}\) is the hydrostatic component of the Cauchy stress tensor, defined as \(\sigma_{h} = \frac{1}{3}\sigma_{kk}\) and \(\sigma_{VM} = \sqrt {\frac{3}{2}S_{ij} S_{ij} }\) where \(S_{ij} = \sigma_{ij} - \sigma_{h} \delta_{ij}\). \(f^{*}\) represents the porosity (damage state-variable) in the model. The material is initially damage-free, \(f_{0} = 0\), and \(q_{1} = 1.5\).

Void nucleation \((\dot{f}_{n} )\) is assumed in every plastically deforming element, which subsequently activates porosity growth \((\dot{f}_{g} )\), see the evolution functions (with the first and second invariants of the strain rate tensor) in Eqs. (5) and (6). Together, those functions determine the total porosity rate such that \(\dot{f} = \dot{f}_{n} + \dot{f}_{g}\).

$$ \dot{f}_{n} = \,\,\,\,\,\,\,\left\{ \begin{gathered} \frac{{\dot{\varepsilon }_{eq} f_{N} }}{{s_{N} \sqrt {2\pi } }}e^{{ - \frac{1}{2}\left( {\frac{{\varepsilon_{eq} - \varepsilon_{N} }}{{s_{N} }}} \right)^{2} }} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\sigma_{h} > 0 \hfill \\ 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\sigma_{h} \le 0 \hfill \\ \end{gathered} \right. $$
(5)
$$ \dot{f}_{g} = \,\,\,\,\,\,\left( {1 - f} \right)tr\left( {\Delta \varepsilon_{ij}^{pl} } \right) $$
(6)

where \(\Delta \varepsilon_{ij}^{pl}\) is the plastic-strain-increment tensor, \(\varepsilon_{N}\) and \(s_{N}\) are the mean value and the standard deviation of the nucleation-strain normal distribution, and \(f_{N}\) is the volume fraction of nucleating voids. Equation (5) evaluates the rate of void nucleation \((\dot{f}_{n} )\) with equivalent plastic strain \((\varepsilon_{eq} )\) following a gaussian distribution scheme, calculated in each material point in the model, see Fig. 3a. The statistical basis of the void nucleation term in equation (5) refers to each material integration point. Thus, if all integration points have an equal \(\varepsilon_{N}\) value, the void nucleation is inherently homogeneous in the simulation cell. Heterogeneous void nucleation is achieved by statistically (randomly) distributing \(\varepsilon_{N}\) so that each material point is assigned with a different value, see a representative case in Fig. 3b–c. Material points with low \(\varepsilon_{N}\) values experience accelerated damage accumulation at small equivalent plastic strains, illustrating an initial statistical defect distribution in the matrix (Fig. 3a). Note that the results in this paper are verified for mesh convergence, meaning that finer meshing yields a similar behavior.

Fig. 3
figure 3

a Void nucleation evolution with plastic strain \(\varepsilon\), calculated at each material point, equation (4). In b, a representative case in which the mean nucleation strain parameter \((\varepsilon_{N} )\) is statistically distributed, see the histogram in c, thus void nucleation evolution is heterogeneous. The red line in c marks the average value of \(\varepsilon_{N}\) in the matrix

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The description of the final stage to failure in the porosity evolution, i.e. coalescence to fracture, is described next. Void coalescence occurs in advanced stages of the loading process when the porosity has reached the critical porosity value (\(f_{C}\)), beyond which the porosity is accelerated towards the fracture threshold (\(f_{F}\)) while reducing the yield stress of the material (Table 2). This coalescence scenario is described by the following evolution function (Tvergaard and Needleman 1984; Koplik and Needleman 1988):

$$ f^{*} = \left\{ {\begin{array}{*{20}c} f & {f < f_{C} } \\ {f_{C} + \frac{{{1 \mathord{\left/ {\vphantom {1 {q_{1} }}} \right. \kern-\nulldelimiterspace} {q_{1} }} - f_{C} }}{{f_{F} - f_{C} }}\left( {f - f_{C} } \right)} & {f_{C} \le f \le f_{F} } \\ \end{array} } \right. $$
(7)
Table 2 GTN model material model parameters
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The values assumed for the GTN model, shown in Table 2, are inspired from those used in the literature for steel and other engineering materials, see e.g. (Tvergaard and Needleman 1984; Benseddiq and Imad 2008; Nahshon and Hutchinson 2008; Kiran and Khandelwal 2014; Verleysen and Peirs 2017).

The stress state characteristics, i.e., the Lode parameter (L) and stress triaxiality (T), are calculated using:

$$ L = - \frac{{27J_{3} }}{{2\sigma_{eq}^{3} }};\,T = \frac{{\sigma_{h} }}{{\sigma_{eq} }} $$
(8)

where \(J_{3} = \det \left( {S_{ij} } \right)\) is the third invariant of the deviatoric stress tensor.

The average mechanical measures are calculated via:

$$ \overline{X} = \frac{{\sum\limits_{i} {X_{i} \,V_{i} } }}{{\sum\limits_{i} {V_{i} } }} $$
(9)

where \(X_{i}\) can be any state variable calculated in element \(i\) (stress triaxiality, Lode parameter, equivalent strain or porosity), and \(V_{i}\) is the element area. The term porosity refers to the secondary void porosity (excluding the primary void). Failure is identified as the first element to reach \(f_{F}\) in the model.

3 Results

3.1 Sheet model

The sheet model (Fig. 2a) represents a uniform material with no geometrical stress concentrations. This model emphasizes the void nucleation heterogeneity effect on the mechanical response and porosity evolution with statistical void nucleation. The following results compare a statistically uniform distribution of \(\varepsilon_{N}\) bounded between 0 and 0.3 (Fig. 3c) and a homogeneous matrix with a fixed nucleation strain \(\left( {\varepsilon_{N} = 0.15} \right)\), thus preserving the average nucleation strain value.

Void-nucleation heterogeneity slightly affects the average Lode parameter and the average stress triaxiality (Fig. 4a) compared to the case of homogeneous nucleation. However, a sharp rise in the Lode parameter with a simultaneous reduction in triaxiality is observed around 0.13 average plastic strain. In both cases (homogeneous and heterogeneous), the average porosity level mainly comprises void nucleation, whereas void growth is negligible (Fig. 4b). The average porosity becomes almost linear with the average strain (Fig. 4b), a case that might be obtained from a statistically uniform distribution of the void nucleation mean strain.

Fig. 4
figure 4

The strain nucleation parameter \(\varepsilon_{N}\) is randomly distributed in the periodic cell. The average stress triaxiality and Lode parameter in a, and the average secondary void porosity in b, are plotted versus the average equivalent plastic strain. A linear fit of the heterogeneous case average porosity evolution with average strain, \(\overline{f} = 0.06\overline{\varepsilon }\)(R2 = 0.994) is given in a red dashed line in b. The histograms in c and d compare the porosity distributions at specific average strains

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The porosity histograms (Fig. 4c, d) highlight the induced heterogeneity effect on the porosity distribution development. In the homogeneous void nucleation matrix, the porosity develops uniformly at the onset of the deformation when the average equivalent plastic strain is \(\overline{\varepsilon } = 0.03\) (see the narrow peak in the homogeneous case in Fig. 4c). With further deformation at \(\overline{\varepsilon } = 0.1\), an additional porosity peak develops (Fig. 4d). Such an evolution does not occur in the heterogeneous case for which higher porosities are attained without emerging peaks (compare Fig. 4c and d).

The influence of the width of the statistical distribution is shown next, in which the \(\varepsilon_{N}\) values are gradually restricted to tighter ranges around 0.15 (Fig. 5b–c). Higher heterogeneity (wider span of \(\varepsilon_{N}\), see Fig. 5a–c), reduces both the absolute value of the stress triaxiality and Lode parameter, as well as the porosity level (Fig. 5e). Note that since the void growth component is almost negligible (Fig. 5e), the results up to this point pertain primarily to the void nucleation mechanism.

Fig. 5
figure 5

Nucleation strain mean value parameter \((\varepsilon_{N} )\) histograms. Frequency is the element count normalized by the total number of elements. The three cases a–c, differ in the range of \(\varepsilon_{N}\) that is assigned to the elements, while all cases share the same average value 0.15. The evolving average stress triaxiality and Lode parameter under simple shear deformation are shown in d, and the secondary void volume fraction is in e

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3.2 Pore-containing model

The pore-containing model (Fig. 2b) is analyzed next. Failure is defined as first element deletion in the matrix, e.g. the porosity in that element reaches the fracture threshold. Compared to the sheet model, the average stress triaxiality and the Lode parameter (Fig. 6a) are drastically affected by the embedded pore, to the extent that the homogeneous and the statistically heterogeneous nucleation are now virtually indistinguishable. Nevertheless, the average porosity (f in Fig. 6b) seems to be lowered by the induced heterogeneity, increasing the sustainable strain before failure when compared to the homogeneous case. In all cases, the failure occurs close to the free surfaces of the void where the deformation is the largest.

Fig. 6
figure 6

A comparison between homogeneous and heterogeneous nucleation in the pore containing model. The stress triaxiality T and the Lode parameter L are given in a as a function of the average cell plastic strain. The secondary void porosity evolution P with average strain is shown in b, together with its components of void nucleation N and void growth G

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Three stages are distinguished in the failure process. Up to an average equivalent strain of approximately 0.03, the porosity is mainly comprised of void nucleation (N in Fig. 6b) with negligible void growth (G in Fig. 6b). Then, the void nucleation rate is moderated with the uprising void growth. Eventually, above an average plastic strain of 0.1, the porosity accumulation rate (with average plastic strain) tends to level off for both homogeneous and heterogeneous cases.

The transition in the porosity evolution for an average strain of 0.1 is reflected by the central void shape (Fig. 7), signaling the initiation of strain localization through the periodic boundaries. It is known, from previous studies, that voids elongate and rotate under shear deformation, and with further shearing, the elongated void evolves into an S-shape, during which its tips are reoriented towards the maximum strain path (Chen and Osovski 2019a), the localization path. Whereas void nucleation (Fig. 7a–b) is higher in the region of the strain localization band, void growth (Fig. 7c–d) is more concentrated around the elongated sections of the void. With that, the statistical heterogeneous void nucleation slightly affects the size evolution of the central void (Fig. 7e).

Fig. 7
figure 7

Contour plots of void nucleation and void growth in the voided sheet for a similar average strain of 0.11 in the a, c homogeneous and the b, d heterogeneous cases, calculated using Eq. (5) and (6), respectively. The maximum strain around the void is 1.2, emphasizing the heterogeneity imposed by the central void. In e, the central void normalized volume is shown to decrease with deformation, although the average triaxiality in the cell rises

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The normalized distributions (by total element count) of the resulting stress triaxiality and void evolution parameters in the matrix are analyzed next at equivalent plastic strains of 0.03 (Fig. 8a–c) and 0.1 (Fig. 8d–f). The induced heterogeneity has a negligible effect on the stress triaxiality distribution (Fig. 8a, d) which is mainly affected by the embedded central pore. Nevertheless, the porosity nucleation and growth components show significant discrepancies between the homogeneous and heterogeneous cases at large shear strains. When the porosity starts to develop at an average plastic strain of 0.03, the resulting void nucleation and growth distributions (Fig. 8b, c) are not drastically influenced by the void nucleation heterogeneity. However, for higher average plastic strain (0.1 in Fig. 8e, f), the induced heterogeneity filters out what seems to be a side-effect of the homogeneous void nucleation in the matrix, seen by the higher peaks in the homogeneous case.

Fig. 8
figure 8

Probability density (frequency) graphs of a, d the stress triaxiality, b, e void nucleation, and c, f secondary void growth. Taken at the same average plastic strain of 0.03 and 0.1 in the pore-containing model. Note that the void growth component is lower by two orders of magnitude than the void nucleation values

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3.3 Sensitivity to the nucleation model parameters

In the following, the sheet and pore models are examined over a range of nucleation parameters. In specific, the triaxiality, Lode parameter, and nucleated porosity are calculated for cases in the range of \(f_{n} \in \left( {0.025,\,0.1} \right)\) and \(s_{n} \in \left( {0.05,01} \right)\), see Fig. 9. In these cases, the nucleation-strain is \(\varepsilon_{n} = 0.15\).

Fig. 9
figure 9

Triaxiality, Lode parameter, and secondary void porosity evolution with plastic strain of the a–c sheet model and the d–f pore model. The nucleation strain in all cases equals 0.15

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The distinction between homogeneous and heterogeneous nucleation, in terms of triaxiality and Lode parameter, is more emphasized in the sheet model (Compare Fig. 9a with Fig. 9b). Porosity accumulation with the average plastic strain is shown in Fig. 9c for the sheet model, and in Fig. 9d for the pore model. The linear-like relation between the porosity and the plastic strain in the heterogeneous sheet model that was shown earlier in Fig. 4b, is verified in Fig. 9c for other model parameters. In both the sheet and pore models, the statistically induced heterogeneity effect for each specific set of parameters is similar to its homogeneous counterpart. Note that the value of \(s_{n}\) influences the porosity evolution in the sheet model so that the homogeneous and heterogeneous void nucleation are apparently different (Fig. 9c). However, both end with similar porosity when localization has been reached.

4 Discussion

The effect of void nucleation heterogeneity is studied in two configurations. The first analyzes a uniform sheet as a representative of the initial state of the material. The second case repeats the previous analysis, while the cell contains a central pore, acting as a stress concentrator. This stage may be deemed to represent the later stages of the ductile fracture process.

Concerning the sheet model, the induced heterogeneity in void nucleation mainly affects the mechanical strength (or damage accumulation) with some adjustments to the stress state characteristics (i.e., stress triaxiality and Lode parameter). In addition, the void growth component is minimal, so that failure is essentially dominated by structural softening and void nucleation.

For the case of a pore-containing cell, the void nucleation spatial heterogeneity mainly affects the average plastic strain at failure, while the average mechanical fields (i.e., stress state and damage evolution) are similar to those of an equivalent homogeneous case. With that, an increase in void growth is observed, which in turn decreases the void nucleation rate (Fig. 6b). This result stems from the local positive triaxiality that develops at the elongated sides of the geometrical pore when sheared, as evident by the experiments of (Fadida et al. 2020, "A" in Fig. 1).

To summarize, the homogeneous vs. heterogeneous nucleation assumptions have a larger influence when considering a uniform material, without structural stress concentrations. As a general consideration, the two cases studied in this work correspond to the initial macroscopic heterogeneous nucleation stage in homogeneous materials, evolving towards the configuration of a pore-containing solid in which such larger voids act as local stress concentrations. The damage and fracture processes can be viewed as a gradual evolution from the former to the latter material state.

5 Concluding remarks

  • The influence of void nucleation heterogeneity on the mechanical properties of the bulk material is studied using periodic unit cells under simple shear deformation. Large pores determine the bulk stress characteristics (stress triaxiality and Lode parameter), while void nucleation heterogeneity influences the material strength.

  • The two configurations analyzed here can be conceptually considered as representing the damage evolution of an initially homogeneous material, with statistically distributed heterogeneous void nucleation, towards the state of a large void-containing material that is approaching its final failure.

  • For similar average strains, the homogeneous case accumulates more damage compared to the heterogeneous one. Thus, the homogeneous assumption yields a conservative estimate as void nucleation heterogeneity increases the material's ductility.