## Abstract

The onset of necking in dynamically expanding ductile rings is delayed due to the stabilizing effect of inertia, and with increasing expansion velocity, both the number of necks incepted and the number of fragments increase. In general, neck retardation is expected to delay fragmentation as necking is often the precursor to fracture. However, in porous ductile materials, it is possible that fracture can occur without significant necking. Thus, the objective of this work is to unravel the complex interaction of initial porosity and inertia on the onset of necking and fracture. To this end, we have carried out a series of finite element calculations of unit cells with sinusoidal geometric perturbations and varying levels of initial porosity under a wide range of dynamic loading conditions. In the calculations, the material is modeled using a constitutive framework that includes many of the hardening and softening mechanisms that are characteristics of ductile metallic materials, such as strain hardening, strain rate hardening, thermal softening, and damage-induced softening. The contribution of the inertia effect on the loading process is evaluated through a dimensionless parameter that combines the effects of loading rate, material properties, and unit cell size. Our results show that low initial porosity levels favor necking before fracture, and high initial porosity levels favor fracture before necking, especially at high loading rates where inertia effects delay the onset of necking. The finite element results are also compared with the predictions of linear stability analysis of necking instabilities in porous ductile materials.

## 1 Introduction

Another approach to analyzing multiple necking and fragmentation in dynamically expanding circular rings is through unit cell finite element calculations introduced in Xue et al. [24]. The unit cell model is based on the idea that a radially expanding circular ring with periodic perturbations is simply a collection of unit cells with sinusoidal perturbations. By utilizing these unit cells of different sizes (defined in terms of length to diameter ratios) and initial conditions consistent with the expanding ring, Rodríguez-Martínez et al. [17] modeled the dependence of the average neck spacing on material properties and imposed loading rates. In these calculations, the material was assumed to be plastically incompressible and follow power law strain rate hardening. Material failure or fracture was not considered in this work and the necking strain, i.e., the strain at the ends of the unit cells at which the plastic flow localizes in the center of the cell was measured. These calculations show that the necking strain first decreases with increasing cell size and then starts to increase for cell sizes greater than a critical value. So that the minimum necking strain corresponds to an intermediate unit cell size. In Ref. [17], the unit cell size with the minimum necking strain, the critical cell size, was obtained for a range of imposed loading rates and for materials with different densities and strain rate sensitivities. These results were compared with the average neck spacing obtained from the finite element calculations of radially expanding circular rings and a very good quantitative agreement between the two approaches was noted. This suggests that, for a fixed loading rate and material properties, the critical cell size for which the deformation energy required to trigger a neck is minimum, represents the average neck spacing in the multiple necking pattern.

The unit cell finite element calculations introduced in Ref. [24] and utilized in Ref. [17] provide a simple computational tool to model the effect of initial porosity on multiple necking and fracture under dynamic loading conditions. To this end, we carry out three-dimensional unit cell finite element calculations for a wide range of the aspect ratio of the cells and amplitudes of the initial sinusoidal geometric perturbation. The material is modeled using an elastic-viscoplastic constitutive relation for a progressively cavitating ductile solid. The constitutive relation also includes strain hardening, strain rate hardening, and thermal softening. Fracture is assumed to occur when a critical level of porosity is reached as in Ref. [20]. We note that previous works on multiple necking and fragmentation assumed the material to be plastically incompressible with few notable exceptions such as Refs. [7,20,21] which considered porous plasticity, but in these works, the interplay between porosity and inertia was not explored. Here, the calculations are carried out for different levels of the initial porosity in the material. The contribution of the inertia effects to the loading process is evaluated through a dimensionless parameter that combines the effects of loading rate, material properties, and unit cell size. The unit cell finite element calculations are carried out for a range of imposed uniaxial velocities giving a wide variation in the value of the dimensionless inertia parameter. The finite element results are also compared with the predictions of linear stability analysis of necking instabilities in porous ductile materials developed in Ref. [21]. The linear stability analysis, despite several simplifying assumptions, provides a quick understanding of the fundamental problem and enables us to (i) obtain the fundamental dimensionless parameters that govern the necking instabilities; (ii) rationalize the stabilizing effects of stress multiaxiality and inertia; and (ii) provide additional verification of the effect of porosity on necking instabilities in ductile porous materials subjected to dynamic loading conditions.

## 2 Constitutive Framework

The constitutive framework used is the modified Gurson elastic-viscoplastic constitutive relation for a progressively cavitating solid [25,26] with the flow potential having the form
$Φ=σe2σ¯2+2q1f*cosh(3q2σh2σ¯)−1−(q1f*)2≤0$
(1)
where q1 = 1.25 and q2 = 1 are the material parameters introduced in Refs. [27,28], $σ¯$ is the matrix flow strength, and
$σe=32σ′:σ′,σh=13σ:1,σ′=σ−σh1$
(2)
with σ being the Cauchy stress tensor and 1 being the unit second-order tensor.
Following, Ref. [26], the function f* in Eq. (1) is given by
$f*={fiff
(3)
where f is the void volume fraction, fc is the critical void volume fraction to void coalescence, and ff is the void volume fraction at failure. The values of fc = 0.12 and ff = 0.25 are used in the calculations.
The rate of deformation tensor is written as the sum of an elastic part $de=L−1:σ^$, a plastic part dp, and a part due to thermal straining $dΘ=αΘ˙1$, so that
$d=L−1:σ^+αΘ˙1+dp$
(4)
Here, $σ^$ is the Jaumann rate of Cauchy stress, Θ is the temperature, α = 1 × 10−5 K−1 is the thermal expansion coefficient, and L = λ11 + 2μI is the tensor of isotropic elastic moduli, where λ = 40.38 GPa and μ = 26.92 GPa are the Lamé constants, and I is the fourth-order unit tensor. The plastic part of the rate of deformation tensor dp, as formulated in Ref. [29], is written as
$dp=[(1−f)σ¯ε¯˙pσ:∂Φ∂σ]∂Φ∂σ$
(5)
The effective plastic strain rate in the matrix material $ε¯˙p$ is given by
$ε¯˙p=ε˙0[σ¯g(ε¯p,Θ)]1/m$
(6)
with
$g(ε¯p,Θ)=σ0G(Θ)[1+ε¯p/ε0]N$
(7)
and
$G(Θ)=1+bGexp(−c[Θ0−273])[exp(−c[Θ−Θ0])−1]$
(8)
where $ε¯p=∫ε¯˙pdt$ is the effective plastic strain in the matrix material. In the calculations, the values of initial flow strength of the matrix material σ0 = 300 MPa, strain hardening exponent N = 0.1, strain rate sensitivity exponent m = 0.01, reference strain ɛ0 = 0.0043, reference strain rate $ε˙0=103s−1$, bG = 0.1406, and c = 0.00793 K−1 are used, and Θ0 = 293 K is the initial temperature of the material.
The evolution of the void volume fraction is governed by
$f˙=(1−f)dp:1$
(9)
where the value of f in the undeformed material, i.e., the value of f at time t = 0, represents the initial void volume fraction (or porosity), f0. For, f0 = 0, the material is fully dense and follows von Mises plasticity.
Adiabatic conditions are assumed so that
$ρcp∂Θ∂t=χσ:dp$
(10)
with ρ being the current density of the material (the initial density is ρ0 = 7600 kg/m3), cp = 465 J/kgK is the specific heat, and χ = 0.9 is the Taylor–Quinney coefficient.

The values of all the constitutive and material parameters are taken from Ref. [21].

## 3 Unit Cell Finite Element Model

We carry out three-dimensional unit cell finite element calculations as in Refs. [17,24] using the constitutive framework described in Sec. 2 to model necking and fracture in porous metallic bars subjected to dynamic loading conditions consistent with expanding rings. The finite element calculations are based on the dynamic principle of virtual work using a finite deformation Lagrangian convected coordinate formulation. Material points are referred to using a Cartesian coordinate system with positions in the reference configuration denoted as (X, Y, Z). The origin of the coordinate system is located at the center of mass of the unit cell. The unit cell modeled is a cylindrical bar with initial length L0 and diameter D with initial sinusoidal geometric imperfection:
$D(Y)=D0[1−A(1+cos(2πYL0))]$
(11)
where D0 is the initial unperturbed diameter of the unit cell and A is the amplitude of the imperfection. The coordinate along the axis of the unit cell Y in the reference configuration varies from −L0/2 to L0/2. Owing to the symmetry of the cross-section of the cylindrical unit cell, only 1/4 of the specimen is analyzed numerically (Fig. 1). Although three-dimensional unit cell finite element calculations are carried out in this work, the unit cell in Fig. 1 is axisymmetric about the Y-axis and can also be analyzed numerically using an axisymmetric finite element model.
Fig. 1
Fig. 1
Close modal
Two representative configurations of the unit cell with normalized cell lengths $L¯=L0/D0=0.5$ and 1 and with amplitude A = 0.02 of the sinusoidal imperfection are shown in Fig. 1. The finite element mesh of the unit cell with $L¯=0.5$ consists of 55,200 twenty node brick elements while that of the unit cell with $L¯=1$ consists of 53,800 twenty node brick elements. The initial and boundary conditions in velocity imposed on the unit cell are
$VY(X,Y,Z,0)=V0×YL0/2VY(X,±L0/2,Z,t)=±V0$
(12)
where VY is the velocity along the axis of the unit cell, t is the analysis time, and V0 is the magnitude of the imposed velocity. The unit cell finite element model considered in this work is based on the idea that a radially expanding circular ring with periodic perturbations can be represented as a collection of unit cells (shown in Fig. 1). Note that in a radially expanding ring, the geometric and loading symmetries nearly eliminate wave propagation along the circumferential direction. Following this, the initial condition in Eq. (12) is utilized to minimize wave propagation along the axial direction of the unit cell (which corresponds to the circumferential direction of the ring). In a radially expanding ring, wave propagation can occur in other directions, assuming this, no initial conditions are utilized along any other direction of the unit cell in this work. Also, in Ref. [17], the critical neck size as a function of the imposed strain rate in a plastically incompressible material obtained from full three-dimensional finite element calculations of radially expanding rings was found to be in quantitative agreement with the predictions of the unit cell finite element calculations with initial and boundary conditions same as in Eq. (12). Nonetheless, we carried out limited calculations to check the effect of initializing velocity along the radial direction on the predictions of the unit cell finite element calculations similar to Ref. [20]. Our results, which are not presented in this paper for the sake of brevity, show that the effect of the initial velocity along the radial direction has a negligible influence on the evolution of plastic strain and porosity. However, the initial velocity along the radial direction does slightly reduce the oscillations in the axial force for unit cells with small wavelengths.

As in Refs. [21,3033], eight-point Gaussian integration is used in each 20 node element for integrating the internal force contributions, and 27-point Gaussian integration is used for the element mass matrix. Lumped masses are used so that the mass matrix is diagonal. The discretized equations are integrated using the explicit Newmark β-method (β = 0) [34]. The constitutive updating is based on the rate tangent modulus method proposed in Ref. [35], while material failure is implemented via the element vanishing technique proposed in Ref. [36]. When the value of the void volume fraction f at an integration point reaches 0.9ff, the value of f is kept fixed so that the material deforms with a very low flow strength. The entire element is taken to vanish when three of the eight integration points in the element have reached this stage.

The unit cell finite element calculations are carried out for nine unit cells with normalized cell length $L¯$ (Fig. 1), varying from 0.25 to 3. For all unit cells, the initial length L0 = 1 mm is kept fixed, and the initial diameter is varied as $D0=L0/L¯$. For each unit cell, the calculations are carried out for three amplitudes of the sinusoidal geometric imperfection, A = 0.002, 0.005, and 0.02, four initial porosity levels, f0 = 0, 0.01, 0.05, and 0.08, and a range of imposed velocities V0. The initial porosity levels assumed in our calculations are rather high compared to conventionally processed engineering metals and alloys. However, with the emergence of manufacturing technologies such as additive manufacturing, it is becoming possible to control the initial porosity levels in a material. Our objective here is to unravel the interaction of initial porosity and inertia on the onset of multiple necking and fracture under dynamic loading conditions. This will provide an understanding of the extent to which the initial porosity level in a material can be controlled to engineer multiple necking and fracture patterns under dynamic loading conditions. From the unit cell finite element calculations, strain to failure dictated by necking and/or fracture is measured. Our finite element results presented in Sec. 5 show that the failure strain dictated by necking first decreases with increasing value of $L¯$ and then starts to increase for values of $L¯$ greater than a critical value. The value of $L¯$ for which the necking strain is minimum is referred to as the critical cell size $L¯c$. The value of $L¯c$, for which the deformation energy required to trigger failure is minimum, likely represents the average neck spacing in the multiple necking pattern as noted in Sec. 1.

## 4 Theoretical Model

In this section, we briefly summarize the main features of the state-of-the-art one-dimensional linear stability analysis developed in Ref. [21], to model necking instabilities in porous metallic bars subjected to dynamic loading conditions consistent with expanding ring. The theoretical model developed in Ref. [21] for the constitutive framework detailed in Sec. 2 includes both the effects of inertia and multiaxial stress state that develops inside the necked region.

The linear stability analysis technique to study the formation of dynamic necking instabilities [37,38] involves testing the stability of the fundamental solution of the problem $S1$ at any time t1 by introducing into the governing equations a small perturbation of the form
$δS(Y,t)t1=δS1eiξY+η(t−t1)$
(13)
where $δS1$ is the perturbation amplitude, ξ is the wavenumber, and η is the growth rate of the perturbation at time t1. The wavenumber is related to the normalized perturbation wavelength as $L¯=(1/D0)(2π/ξ)$. The normalized perturbation wavelength is analogous to the normalized unit cell length in the finite element calculations.
The perturbed solution of the problem is given by
$S=S1+δS$
(14)
with $|δS|≪|S1|$. By inserting Eq. (14) into the governing equations of the problem and keeping only the first-order terms in the increments $δS1$, linearized equations are obtained. A non-trivial solution for $δS1$ can only be obtained if the determinant of the system of linearized equations is equal to zero. Application of this condition leads to a fourth-degree polynomial in η
$B4(S1,ξ)η4+B3(S1,ξ)η3+B2(S1,ξ)η2+B1(S1,ξ)η+B0(S1,ξ)=0$
(15)
with coefficients $Bi(S1,ξ)$ that depend on the fundamental solution and the wavenumber. Equation (15) has four roots in η, two real and two complex conjugates. The requisite for unstable growth of $δS$ is given by Re(η) > 0, and hence, the root that has the greater positive real part η+ is considered for the analysis. The stabilizing effect of inertia and multiaxial stress state on small and large wavenumbers, respectively, promotes the growth of intermediate modes [8,3841]. To track the history of the growth rate of all the growing modes during the post-homogeneous deformation process (i.e., for strains greater than the Considère strain), we use the cumulative instability index $I=∫0tη+dt$ [21,38,42,43]. The index I accumulates the growth rate of all the growing modes during the loading process, i.e., we introduce the perturbation at different times and sum the growth rate obtained for each loading time. At a given loading time t, i.e. at a given strain ɛ, the mode that grows the fastest, i.e., the mode with the greatest value of I, is referred to as critical wavenumber or critical perturbation mode ξc. Likewise, the greatest value of I is referred to as critical cumulative instability index Ic. The critical wavenumber enables us to calculate the critical normalized perturbation wavelength $L¯c=(1/D0)(2π/ξc)$, which is analogous to the critical unit cell size $L¯c$ obtained from the finite element calculations.

## 5 Results and Discussions

This section is divided into two sub-sections. Section 5.1 presents the key results and discussion of the unit cell finite element calculations and Sec. 5.2 compares the unit cell finite element results with the predictions of the linear stability analysis. Both, the unit cell finite element calculations and the linear stability analyses are carried out for a range of imposed velocities V0, giving a variation in the value of the dimensionless inertia parameter Π defined as
$Π=(V0D0L0)2ρ0σ0$
(16)
in the range of 0.05 to 0.3. The parameter Π is the inertial resistance to motion and allows for the quantification of the dynamic effects [11,17,21,44].

### 5.1 Key Finite Element Results.

The evolution of the effective plastic strain $ε¯midp$ and the porosity f in the mid-section of two unit cells, $L¯=1.0$ and 0.5, with the effective plastic strain at the end of the unit cells $ε¯endp$ subjected to Π = 0.1 are shown in Fig. 2. For both the unit cells, the values of A = 0.002 and f0 = 0.01. As shown in the figure, at the early stages of deformation, the values of $ε¯midp$ evolve linearly with $ε¯endp$ (Fig. 2(a)), with limited increase in the value of f (Fig. 2(b)). However, with continued deformation, the plastic flow localizes in the mid-section of the unit cells, and the value of $ε¯midp$ increases asymptotically while the value of $ε¯endp$ remains nearly constant. The value of $ε¯endp$ at which $dε¯midp/dε¯endp→∞$ is defined here as the effective plastic strain at the onset of necking. Following the onset of necking, the value of f also increases asymptotically. The onset of necking and the asymptotic increase in the value of f occur at a smaller value of $ε¯endp$ in the unit cell with $L¯=1.0$ than in the unit cell with $L¯=0.5$. Furthermore, in the unit cell with $L¯=1.0$, the value of $ε¯endp$ at the onset of necking is smaller in the center of the mid-section of the unit cell compared to the surface of the mid-section of the unit cell. This is because in the unit cell with $L¯=1.0,$ plastic flow first localizes in the center and subsequently propagates toward the surface of the mid-section of the unit cell. On the contrary, in the unit cell with $L¯=0.5$, plastic flow first localizes at the surface and subsequently propagates toward the center of the mid-section of the unit cell. Similarly, the onset of the asymptotic growth in the value of f first occurs in the center of the mid-section of the unit cell with $L¯=1.0$, while the onset of the asymptotic growth in the value of f first occurs at the surface of the mid-section of the unit cell with $L¯=0.5$. Here, we identify the onset of fracture as the smallest value of $ε¯endp$ at which f > fc.

Fig. 2
Fig. 2
Close modal

In Figs. 2(a) and 2(b), the value of $ε¯endp$ at which necking initiates in the center of the mid-section of the unit cell is marked with open symbols while the closed symbols correspond to the value of $ε¯endp$ at which necking initiates at the surface of the mid-section of the unit cell. Also, in Figs. 2(a) and 2(b), the value of $ε¯endp$ at which fracture initiates in the center of the mid-section of the unit cell is marked with encircled open symbols while the encircled closed symbols correspond to the value of $ε¯endp$ at which fracture initiates at the surface of the mid-section of the unit cell. The variation in the location of the initiation of plastic flow localization for the two unit cells in Fig. 2 is further illustrated via contour plots of effective plastic strain $ε¯p$ in Fig. 3. As seen in the figure, in the unit cell with $L¯=1.0$ (Fig. 3(a)), the value of $ε¯p$ is maximum in the center of the mid-section of the unit cell whereas in the unit cell with $L¯=0.5$ (Fig. 3(b)), the value of $ε¯p$ is maximum at the surface of the mid-section of the unit cell.

Fig. 3
Fig. 3
Close modal

For the two cases shown in Fig. 2, the onset of necking precedes the onset of fracture. We now present two examples where the onset of fracture precedes necking for the same imposed inertia parameter Π = 0.1. Figure 4 shows the evolution of $ε¯midp$ and f with $ε¯endp$ in two unit cells. For one unit cell, $L¯=3.0$ and f0 = 0.08, and for the second unit cell, $L¯=0.25$ and f0 = 0.05, while the value of A = 0.002 is same for both the unit cells. As shown in Fig. 4(a), the value of $ε¯midp$ evolves gradually with the value of $ε¯endp$ in both the unit cells and the fracture criteria is met, i.e., f > fc, see Fig. 4(b), before the necking criteria, i.e., $dε¯midp/dε¯endp→∞$. In the unit cell with $L¯=3.0$ and f0 = 0.08, the value of $ε¯endp$ at the onset of fracture is slightly smaller in the center of the mid-section of the unit cell compared to the surface. So that, in this unit cell fracture initiates in the center and subsequently propagates toward the surface. On the contrary, in the unit cell with $L¯=0.25$ and f0 = 0.05, fracture initiates at the surface and subsequently propagates toward the center. The variation in the location of fracture initiation for the two unit cells in Fig. 4 is further illustrated via contour plots of f in Fig. 5. As seen in the figure, in the unit cell with $L¯=3.0$ and f0 = 0.08 (Fig. 5(a)), the value of f is maximum in the center of the mid-section, whereas in the unit cell with $L¯=0.25$ and f0 = 0.05 (Fig. 5(b)), the value of f is maximum at the surface of the mid-section.

Fig. 4
Fig. 4
Close modal
Fig. 5
Fig. 5
Close modal

We now analyze the variation of the effective plastic strain at the end of the unit cell at failure $(ε¯endp)f$ dictated either by the onset of localization or fracture in the center or at the surface of the mid-section of the unit cell (whichever occurs first), as a function of the normalized unit cell length $L¯$. The variation of $(ε¯endp)f$ with $L¯$ for four initial porosity levels f0 and for each f0, for three values of imposed Π are shown in Fig. 6. In Fig. 6, the values of $(ε¯endp)f$ set by the onset of localization in the center of the mid-section of the unit cell are marked by the open symbols while the closed symbols mark the onset of localization at the surface of the mid-section of the unit cell. Similarly, the values of $(ε¯endp)f$ set by the onset of fracture in the center of the mid-section of the unit cell are marked by the encircled open symbols while the encircled closed symbols mark the onset of fracture at the surface of the mid-section of the unit cell.

Fig. 6
Fig. 6
Close modal

The variation of $(ε¯endp)f$ with $L¯$ for fully dense unit cells, i.e., f0 = 0 for three values of imposed Π are shown in Fig. 6(a). Since void nucleation is ignored in our calculations, there is no porosity-induced fracture in the absence of initial porosity, and the values of $(ε¯endp)f$ are solely dictated by the onset of localization. As shown in Fig. 6(a), for a fixed value of Π, the value of $(ε¯endp)f$ first decreases with increasing $L¯$, and then starts to increase for values of $L¯$ greater than a critical value. The value of $L¯$ for which $(ε¯endp)f$ is minimum (i.e., the value of $L¯$ for which $d(ε¯endp)f/dL¯=0$) is the critical cell size $L¯c$. The greater values of $(ε¯endp)f$, for small and large normalized cell lengths, are due to the stabilizing effects of stress multiaxiality and inertia, respectively [8,39]. For large normalized cell lengths, an increase in the value of imposed Π results in a significant increase in the value of $(ε¯endp)f$. This is because the stabilizing effect of inertia increases with increasing value of $L¯$. In contrast, for small normalized cell lengths, there is a very little or no effect of the value of Π on $(ε¯endp)f$, because necking and fracture for small normalized cell lengths are controlled by stress multiaxiality. For smaller values of $L¯$, for example $L¯=0.25$ and 0.5, localization initiates at the surface while for greater values of $L¯$ localization initiates in the center of the mid-section of the unit cells. Due to the stabilizing effects of inertia, the value of $L¯c$ decreases with increasing value of Π, while the value of $(ε¯endp)f$ corresponding to $L¯c$ increases. For example, the values of $L¯c$ for Π = 0.05 and 0.3 are ≈2.5 and 1.25, respectively, while the values of $(ε¯endp)f$ corresponding to $L¯c$ for Π = 0.05 and 0.3 are ≈0.14 and 0.5, respectively. The values of $L¯c$ for Π = 0.05 and 0.3 in Fig. 6(a) are comparable to the values reported in Ref. [17] for unit cell calculations where the material was assumed to follow von Mises plasticity with strain rate hardening.

In Fig. 6(b), we show the variation of $(ε¯endp)f$ with $L¯$ for unit cells with initial porosity level f0 = 0.01 for three values of imposed Π. The value of f0 = 0.01 is not sufficient to induce fracture without the onset of localization so that similar to f0 = 0, for f0 = 0.01 as well, the values of $(ε¯endp)f$ are dictated by the onset of localization. The dependence of the value of $(ε¯endp)f$ on the value of $L¯$ for all three values of Π for unit cells with f0 = 0.01 is roughly the same as for the unit cells with f0 = 0. The values of the $L¯c$ identified from the Figs. 6(a) and 6(b) as a function of the imposed Π are also consistent with the average neck spacing obtained from the analyses of multiple necking in long cylindrical bars modeled with Gurson-type plasticity and the same initial porosity f0 = 0.01 and subjected to initial and loading conditions consistent with the expanding ring [20,21]. In Ref. [21], where the same constitutive model and material parameters reported in Sec. 2 were used, for Π = 0.05, 0.1, and 0.3, the average neck spacing was predicted to be 1.7, 1.3, and 0.9, respectively. While in Ref. [20], where specific material parameters for aluminum and copper were used, for Π = 0.05, the average neck spacing was predicted to be 2.85 for copper and 2.31 for aluminum, and for Π = 0.1, the average neck spacing was predicted to be 2.21 for copper.

The variation of $(ε¯endp)f$ with $L¯$ for unit cells with initial porosity levels f0 = 0.05 and 0.08 for three values of imposed Π is shown in Figs. 6(c) and 6(d), respectively. For high initial porosity levels, depending on the values of $L¯$ and imposed Π, the onset of fracture can precede the onset of necking. The value of the $(ε¯endp)f$ for a fixed $L¯$ and imposed Π, when dictated by the onset of fracture (for high initial porosity levels), is always less than the value of $(ε¯endp)f$ when dictated by the onset of necking (for low initial porosity levels). As shown in Figs. 6(c) and 6(d), for unit cells with f0 = 0.05 and 0.08, fracture precedes necking for smaller values of $L¯$ for all three imposed values of Π, while for greater values of $L¯$, the propensity for fracture preceding necking increases with increasing value of f0 and Π. For a smaller value of $L¯$, when fracture precedes necking, the value of $(ε¯endp)f$ is either insensitive to the value of $L¯$ or it decreases with decreasing value of $L¯$. This is in contrast to the circumstances where necking precedes fracture as shown in Figs. 6(a) and 6(b). Also, in contrast to the circumstances where necking precedes fracture, for greater value of $L¯$ when fracture precedes necking, the value of $(ε¯endp)f$ becomes less sensitive to both the values of $L¯$ and imposed Π. So that for sufficiently high initial porosity levels and greater values of imposed Π a critical value of $L¯$, i.e., $L¯c$ does not exist.

The onset of fracture as modeled here, only depends on the evolution of porosity, which in turn predominantly depends on the accumulated plastic strain for a fixed state of stress. This explains the decrease in the dependence of $(ε¯endp)f$ on the values of $L¯$ and Π, for greater values of $L¯$ and Π, when fracture precedes necking (Fig. 6(d)). On the other hand, the decrease in the value of $(ε¯endp)f$ for a smaller value of $L¯$ can be explained with the results presented in Fig. 7. In this plot, we show the evolution of effective plastic strain $ε¯midp$ and hydrostatic stress σh in the mid-section of the unit cells with $ε¯endp$, in two unit cells, $L¯=0.25$ and 0.5, with f0 = 0.08 and subjected to Π = 0.1. As shown in the figure, both the values of $ε¯midp$ (Fig. 7(a)), and σh (Fig. 7(b)), in the mid-section of the unit cell increase as the value of $L¯$ decreases, favoring the growth of porosity and promoting early fracture.

Fig. 7
Fig. 7
Close modal

Next, we analyze the influence of the imperfection amplitude A on the predictions of the unit cell finite element calculations. The variation of $(ε¯endp)f$ with $L¯$ for unit cells with initial porosity level f0 = 0.05 for three values of A and an imposed Π = 0.1 are shown in Fig. 8(a). As shown in the figure, an increase in the value of A results in a decrease in the value of $(ε¯endp)f$ for all the values of $L¯$. The decrease in the value of $(ε¯endp)f$ with increasing value of A is particularly significant for smaller values of $L¯$. For smaller values of $L¯$, the range of the values of $L¯$ for which fracture precedes necking increases with increasing value of A. For example, for A = 0.002 fracture precedes necking for $L¯≤0.5$ while for A = 0.02 fracture precedes necking for $L¯≤0.75$. Figure 8(b) shows the variation of $(ε¯endp)f$ with $L¯$ for unit cells with f0 = 0.05 for three values of A and an imposed Π = 0.3. An increase in the value of imposed Π delays the onset of localization, especially, for greater values of $L¯$, thus, favoring the onset fracture before the onset of necking. For greater values of $L¯$, the range of the values of $L¯$ for which fracture precedes necking decreases with increasing value of A. This is contrary to the effect of A on the range of $L¯$ undergoing fracture for smaller values of $L¯$. We note that an increase in the value of A results in a slight increase in the value of the critical cell size $L¯c$ for both the values of imposed Π. However, the effect of A on the values of $L¯c$ and $(ε¯endp)f$ corresponding to $L¯c$ decreases with the decreasing value of A.

Fig. 8
Fig. 8
Close modal

### 5.2 Comparison Between Finite Element Calculations and Linear Stability Analysis.

In this section, we compare the predictions of the unit cell finite element calculations and the linear stability analysis. For this, we first need to define a criterion for the perturbation mode to turn into a necking mode (e.g., see Refs. [20,40]). To this end, we follow the concept of effective instability introduced in Ref. [45], which assumes that necking is triggered when the cumulative instability index I reaches a value Ineck. The value of Ineck can be determined from the results of the finite element calculations as shown in Ref. [46]. In Ref. [46], it was proposed that Ineck corresponds to the critical cumulative instability index Ic, see Sec. 4, obtained from the linear stability analysis performed for the necking strain corresponding to $L¯c$ obtained from the finite element calculations. Here, we rely on the finite element results for fully dense (f0 = 0) unit cells subjected to Π = 0.1 (Fig. 6(a)), to determine the value of Ineck. For fully dense material subjected to Π = 0.1, the unit cell finite element calculations show that the value of critical normalized cell length $L¯c=2$, and the corresponding value of necking strain $(ε¯endp)f=0.214$. Next, using this necking strain as the strain for which the critical cumulative index is calculated in the linear stability analysis, the value of Ineck is estimated to be 1. We assume that the value of Ineck is the same for all perturbation wavelengths ($L¯$), imposed inertia parameters (Π), and initial porosity levels (f0). This is a rather strong assumption, as discussed later, but it enables us to calibrate the linear stability analysis in a simple manner and check its predictive capabilities.

The variation of $(ε¯endp)f$ with $L¯$ predicted using the unit cell finite element calculations, and the variation of the strain for which the condition Ineck = 1 is met with the perturbation wavelength $L¯$ predicted using the linear stability analysis are compared in Fig. 9. The comparison in Fig. 9 is carried out for initial porosity levels f0 = 0, 0.05, and 0.08 and imposed inertia parameters Π = 0.05 and 0.3. As shown in Fig. 9(a), for Π = 0.05 and f0 = 0, there is a good qualitative and quantitative agreement between the predictions of the linear stability analysis and the unit cell finite element calculations for all the values of $L¯$. However, for greater values of f0 and smaller values of $L¯$, the linear stability analysis over predicts the value of $(ε¯endp)f$. This is because for high initial porosity levels the unit cells with smaller values of $L¯$ undergo fracture before necking which is more sensitive to the complex state of stress that develops in the three-dimensional unit cells (recall from Sec. 4 that the linear stability analysis models small perturbations of the fundamental one-dimensional solution). There is also a good qualitative agreement between the predictions of the linear stability analysis and unit cell finite element calculations for Π = 0.3 and f0 = 0 (Fig. 9(b)). The linear stability analysis, however, under predicts the values of $(ε¯endp)f$ for greater values of $L¯$. This suggests that the value of Ineck is not constant and it depends on both the wavelength $L¯$ and the value of imposed Π (a detailed analysis of the dependence of Ineck on the wavelength and inertia is left for a future work). Also, similar to Π = 0.05 (Fig. 9(a)), for Π = 0.3 (Fig. 9(b)) as well, the linear stability analysis over predicts the value of $(ε¯endp)f$ for greater values of f0 where the unit cells undergo fracture before necking.

Fig. 9
Fig. 9
Close modal

In Fig. 10, we compare the variation in the values of $L¯c$ and $ε¯c$ with imposed Π predicted using the unit cell finite element calculations and the linear stability analysis. For the finite element results, the value of $L¯c$ is the critical cell size and the value of $ε¯c$ is the value of $(ε¯endp)f$ corresponding to $L¯c$. While, for the linear stability analysis, the value of $L¯c$ is the critical wavelength and the value of $ε¯c$ is the critical strain (i.e., the minimum strain in the $ε−L¯$ curves). The comparisons in Fig. 10 are shown for initial porosity levels f0 = 0 and 0.08. For all the cases, the predictions of the linear stability analysis are obtained using two values of the cumulative instability index at necking Ineck = 1 and 6.72. The value of Ineck = 6.72 was obtained in Ref. [21] by calibrating Ineck to the finite element results of multiple necking in a long cylindrical bar, free of geometric imperfections, and subjected to initial and boundary conditions consistent with dynamically expanding rings. The geometric imperfection included in the unit cell calculations carried out in this work results in a decrease in the value of Ineck from 6.72 to 1 (this suggests that Ineck also depends on the amplitude of the geometric imperfection, see Ref. [46]).

Fig. 10
Fig. 10
Close modal

As shown in Fig. 10, both the unit cell finite element calculations and the linear stability analysis predict that the value of $L¯c$ decreases and the value of $ε¯c$ increases with increasing value of imposed Π. However, the dependence of the value of $L¯c$ on the value of imposed Π decreases for greater values of Π, in agreement with the finite element results pertaining to multiple necking in a ring or a long cylindrical bar in Refs. [17,21]. Both, the unit cell finite element calculations and the linear stability analysis also predict slightly smaller values of $L¯c$ and $ε¯c$ for f0 = 0.08 than for f0 = 0. This suggests that increasing the initial porosity level in the material decreases (or increases) the average neck spacing (the number of necks) and the localization strain in multiple necking patterns. The linear stability analysis predictions of $L¯c$ and $ε¯c$ using Ineck = 1 are in general good quantitative agreement with the finite element results whereas linear stability analysis using Ineck = 6.72 under predicts the value of $L¯c$ and over predicts the value of $ε¯c$. This is consistent with the finite element results in Fig. 7, which show that an increase in the imperfection amplitude results in an increase in the value of $L¯c$ and a decrease in the value of $(ε¯endp)f$.

## 6 Conclusions

In this work, we have unraveled the complex interaction of initial porosity and inertia on the onset of necking and fracture in ductile materials subjected to dynamic loading conditions consistent with expanding rings. To this end, we have carried out a series of three-dimensional finite element calculations of unit cells with sinusoidal geometric perturbations. In the calculations, the material is modeled using a constitutive framework that includes many of the hardening and softening mechanisms that are characteristic of ductile metallic materials, such as strain hardening, strain rate hardening, thermal softening, and damage-induced softening. The contribution of the inertia effect to the loading process is evaluated through a dimensionless inertia parameter that combines the effects of loading rate, material properties, and cell size. The calculations are carried out for varying levels of initial porosity and a wide range of the value of imposed inertia parameter. From all the unit cell finite element calculations, strain to failure dictated by necking and/or fracture are measured. The finite element results are also compared with the predictions of linear stability analysis of necking instabilities in porous ductile materials. The key conclusions of this work are as follows:

• Low initial porosity levels favor the onset of necking before fracture, and high initial porosity levels favor the onset of fracture before necking, especially, at high loading rates where inertia effects delay the onset of necking.

• For the levels of initial porosity and imposed inertia parameter where the onset of necking precedes fracture, a critical unit cell size (cell size with minimum necking strain) as a function of initial porosity and imposed inertia parameter can be identified.

• For the levels of initial porosity and imposed inertia parameter where the onset of necking precedes fracture, the critical unit cell size obtained from finite element calculations and the critical wavelength for necking obtained from linear stability analysis decreases with increasing initial porosity levels and the value of the inertia parameter.

• For sufficiently high initial porosity levels and imposed inertia parameter where the onset of fracture precedes necking, the value of strain at failure obtained from finite element calculations is found to be less sensitive to the size of the unit cell. So that, it is not possible to identify the critical cell size.

• The linear stability analysis, despite several simplifying assumptions, shows a very good correlation with the predictions of unit cell finite element calculations under the circumstances where the onset of necking precedes fracture.

The most important message of this paper is that when necking precedes fracture, for fully dense materials or materials with low initial porosity, there exists a critical cell size or a critical wavelength for which the deformation energy required to trigger a neck is minimum. This critical cell size or critical wavelength likely represents the average neck spacing in the multiple necking pattern of dynamically expanding rings. However, for materials with high initial porosity, an increase in inertial loading may favor early fracture, precluding the development of a necking pattern. Under these circumstances, when fracture precedes necking, no critical cell size or critical wavelength dictates the fragmentation of dynamically expanding ring.

## Acknowledgment

The finite element calculations reported on were carried out using high-performance research computing resources provided by Texas A&M University. J.A.R.-M. acknowledges the financial support provided by the European Research Council under the European Union’s Horizon 2020 research and innovation programme (Project PURPOSE, Grant agreement 758056).

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