Ductile fracture of materials with randomly distributed voids

Abstract

A reliable determination of the onset of void coalescence is critical to the modelling of ductile fracture. Numerical models have been developed but rely mostly on analyses on single defect cells, thus underestimating the interaction between voids. This study aims to provide the first extensive analysis of the response of microstructures with random distributions of voids to various loading conditions and to characterize the dispersion of the results as a consequence of the randomness of the void distribution. Cells embedding a random distribution of identical spherical voids are generated within an elastoplastic matrix and subjected to a macroscopic loading with constant stress triaxiality and Lode parameter under periodic boundary conditions in finite element simulations. The failure of the cell is determined by a new indicator based on the loss of full rankedness of the average deformation gradient rate. It is shown that the strain field developing in random microstructures and the one in unit cells feature different dependencies on the Lode parameter L owing to different failure modes. Depending on L, the cell may fail in extension (coalescence) or in shear. Moreover the random void populations lead to a significant dispersion of failure strain, which is present even in simulations with high numbers of voids.

Appendices

About the $\delta$ indicator

This appendix provides several complements about the $\delta$ indicator. Its expression is first derived by computing the homogeneous deformation of Green matrix. An example of application is then presented. Finally a sensitivity analysis regarding the threshold coefficients is carried out.

1.1 Derivation of the expression for $\delta$

Consider a perfectly plastic volume element (neglecting here the elasticity) which deforms homogeneously when subjected to the loading conditions (13). In order to simply represent the porous nature of the cell, the material behavior will obey Green’s (1972) isotropic yield criterion (also used by Fritzen et al. (2013)):

(23)

with C a constant ($C=0$ corresponds to a von Mises material, and for $C=1/2$, there is no lateral contraction of the cube in tension). The other equations in Eq. (4) are unchanged, but they are applied here to macroscopic quantities.

As the material behavior is isotropic, stays diagonal in the diagonalizing basis of . Then can be written as:

(24)

where $b_2$ and $b_3$ are functions to be determined. As , and is diagonal, can be written as:

(25)

For a perfectly plastic Green material, the behavior law in (4) reads:

(26)

is diagonal so there are three constants $\alpha _1$, $\alpha _2$ and $\alpha _3$ such that:

(27)

(28)

Combining (25) and (28) yields the system:

$$\begin{aligned}&\frac{{\dot{\epsilon }}}{1+{\dot{\epsilon }}t}={\dot{p}}\alpha _1 \end{aligned}$$

(29)

$$\begin{aligned}&{\dot{b}}_2={\dot{p}}\alpha _2 b_2 \end{aligned}$$

(30)

$$\begin{aligned}&{\dot{b}}_3={\dot{p}}\alpha _3 b_3 \end{aligned}$$

(31)

The plastic multiplier is then ${\dot{p}}=\frac{{\dot{\epsilon }}/\alpha _1}{1+{\dot{\epsilon }}t}$ and the differential equations can be solved with the initial conditions $b_2(0)=1$, $b_3(0)=1$:

$$\begin{aligned} b_2=(1+{\dot{\epsilon }}t)^{\alpha _2/\alpha _1}\quad b_3=(1+{\dot{\epsilon }}t)^{\alpha _3/\alpha _1} \end{aligned}$$

(32)

Finally,

(33)

The function comparing the behavior of and the homogeneous plastic deformation case is then:

(34)

The $\delta$ criterion used throughout the article is recovered by setting $C=0$, which corresponds to the simplified case of a von Mises material. In this case, the criterion depends no more on the applied .

The evolution of $\delta$ for a simulation with $T=1$ and $L=-1$ (coalescence in uniaxial strain state) is shown in Fig. 15, for two values of C: 0 and 1/2. For both values of C, the vanishing of $\delta _C$ is simultaneous with the stabilization of transverse displacement. However the sharp drop of $\delta _C$ allows a more precise numerical determination of the onset of coalescence than the more progressive stabilization of the transverse strain. For $C=1/2$, $\delta _{1/2}$ is approximately constant at the beginning of the simulation, so that the hypothesis of homogeneous flow in a Green volume element (taking into account the porosity) well represents the overall behavior of the cell with a von Mises matrix. However, with $C=0$, $\delta _0$ does not depend anymore on the stress state, while still keeping the sudden drop of $\delta _C$ necessary for the determination of the coalescence onset.

Fig. 15

Detection of failure through simple extension criterion (stabilization of the transverse strain) or vanishing of $\delta$ function. Microstructure R1 under the loading condition $T=1$, $L=-1$

Fig. 16

Effect of varying threshold conditions for the failure indicator on the $E_c-L$ curve. All simulations on the R1 microstructure, at $T=1$

1.2 Sensibility analysis regarding the threshold coefficients

Finally we verify that the $\delta$ indicator is a reliable indicator of failure by assessing its sensitivity to the choice of the empirically chosen threshold values. As the Eq. (16) shows, the determination of the onset of coalescence relies on two thresholds: a relative one A, which compares the current value of $\delta$ to its maximum, and an absolute one B mostly active in shear-like conditions. The values for those were chosen as $A=0.05$ and $B=0.005$ but a robust indicator should not be too sensitive to these values.

Figure 16 compares the effect of different A and B values on the $E_c-L$ curve (common triaxiality $T=1$, microstructure R1). At constant B, the effect of A is only visible in the HLEMZ and the LLEMZ, and generally negligible. At constant A, B only affects the coalescence strain values in the SMZ. Although a change in B can modify the strain by 0.05, the global aspect of the curve is preserved. The determination of failure by the indicator therefore appears robust with respect to changes in the coefficients.

Effect of different meshing parameters and boundary conditions

In this section, we review the simulation hypotheses and assess their influence on the results presented up to now, showing therefore how representative the results are and how far they can be generalized. First we verify that finite element discretization effects can be neglected, and investigate the effect of different boundary conditions.

1.1 Effect of the meshing parameters

All the simulations described up to now were carried out on meshes of cells with the same meshing parameter. To determine the influence of mesh size on coalescence results, the same microstructure R1 was meshed with different meshing parameters $h_{cell}/r_0\in \{1.25, 1, 0.875, 0.625\}$ (with the notation of Sect. 2.1). The maximum element size near the voids is also adapted to keep the ratio $h_{cell}/h_{void}=5$ constant. The same loading condition $T=1,\,L=-1$ is applied to the four meshes. Figure 17 shows that stress values during the simulations differ between the meshes, but the relative difference between the finest and coarsest meshes is about $5\%$, which remains acceptable. The onset of coalescence $E_c$ which is our main quantity of interest, is almost identical between the meshes, at $E_c=0.33\pm 1\%$. Therefore the influence of mesh refinement for random microstructure cells appear limited (although there was only a ratio of 2 between the element sizes of the coarsest and the finest mesh), which justifies the value $h_{cell}=0.08$ adopted throughout this study.

Fig. 17

Cauchy stress during the simulation for several meshes of the R1 random microstructure with different meshing parameters. Loading condition: $T=1, L=-1$

Fig. 18

Conditions on average deformation gradient obtained by fixing some degrees of freedom on vertices of the cubic cell

Fig. 19

Influence of boundary conditions on the response of the cell. All computations at triaxiality $T=1$

1.2 Effect of the boundary conditions

We here investigate the influence of boundary conditions. The results from Sect. 4 are first compared to those obtained with different boundary conditions. Namely we investigate the influence of conditions on the average gradient, and of planar faces conditions. The consistency of results at $L=-1$ is also checked by a comparison with simulations on axisymmetric cells.

The conditions imposed on the average gradient to prevent rigid body motion are first investigated. In Sect. 2.3 we imposed symmetric, as for Ling et al. (2016). However another reasonable choice would be to fix some degrees of freedom at the vertices of the cubic cell, as depicted in Fig. 18, which is the standard method for boundary value problems. A vertex is already fixed in order to prevent translations, but by fixing two degree of freedom on a second one, and a last one on a third vertex, all rotations are fixed. This can be reformulated as:

$$\begin{aligned} {\bar{F}}_{12}={\bar{F}}_{13}={\bar{F}}_{23}=0 \end{aligned}$$

(35)

i.e. is an upper triangular matrix. Due to the mixed conditions imposed by the macroscopic spring element, the results from the symmetric case cannot be easily transposed to the triangular case. These two choices lead to distinct proportional loading path classes and should therefore be compared.

On the microstructure R1, at fixed triaxiality $T=1$, simulations were performed for several Lode parameters to compare the two sets of conditions on (Fig. 19a). The evolution of $E_c$ is close between the two types of conditions, and the same ductility zones can be identified for the triangular gradient condition. However, in that case, cusps seem to be less pronounced than for a symmetric gradient; this may be due to the different treatment of shear components by the two conditions. Therefore the influence of the conditions on remains limited.

Fig. 20

Evolution of the strain at coalescence with respect to T for the cubic and 2D axisymmetric unit cells for porosity values $f_0=6\%$ and $1\%$ (constant Lode parameter $L=-1$)

We then compare the effects of periodic and parallel faces boundary conditions. Parallel faces conditions mean that the cubic cell retains parallel flat faces throughout the computation (for instance all the points on the $x_0=0$ face have the same x-displacement). This condition is more constraining than periodic boundary conditions. As the comparison in Fig. 19b shows, the two types of conditions lead to qualitatively different responses. For the parallel faces, no separation between three ductility zones can be seen (except near $L=0$) and the response of the random microstructure is closer to that typical of the unit cell. Moreover no decrease of ductility near $L=0$ is observed for the unit cell. Results for the unit cell differ between the parallel faces and periodic boundary conditions, because in the periodic case, faces are allowed not to remain strictly parallel and planar. On the contrary, boundary conditions made of parallel sides strongly hinder the shear mode failure and only the extension mode remains possible. The competition between these two modes tends to postpone failure (see the cusps on Fig. 4). Therefore, the reduced competition between modes may explain an earlier coalescence for parallel unit cells. The preceding results show that boundary conditions exert a strong influence on the response of the cell.

Finally, the consistency of results obtained at $L=-1$ is checked. As this type of loading is axisymmetric, a computation with a 2D axisymmetric unit cell was also performed for comparison. Such unit cells are frequent in ductile fracture studies (Morin et al. (2015) for instance). The diameter and the height of the cylinder were chosen equal to $L_{cube}$. The porosity is still $6\%$, so the radius of the void was modified to $0.22 L_{cube}$. The boundary conditions for this cell differ slightly from those described in Sect. 2.3: they are no more periodic and are replaced by straight edges conditions. Besides the virtual constant triaxiality element is not linked to the average deformation gradient but to the displacement of the top left node.

Figure 20 compare results for the unit cell and the 2D axisymmetric cell at varying stress triaxiality for $L=-1$. In this type of loading, the unit cell was shown in Sect. 4 to exhibit the same behavior as random microstructures, compatible with a Rice-Tracey evolution. At initial porosity $f_0=6\%$, the axisymmetric cell presents however a significantly higher exponent (in absolute value) for the evolution of $E_c$ with respect to T. This effect seems due to the relatively high porosity in the axisymmetric cell: as depicted in Fig. 20, the evolution of $E_c$ with respect to T for the low porosity $f_0=1\%$ axisymmetric cell is much closer to the one predicted by Rice and Tracey.