Dynamic crushing of cellular materials: a particle velocity?based analytical method and its application

Shilong Wang · Zhijun Zheng · Yuanyuan Ding · Changfeng Zhu · Jilin Yu

Abstract Cellular material under high-velocity impact exhibits a typical feature of layerwise collapse. A cell-based finite element model is employed herein to simulate the direct impact of a closed-cell foam, and one-dimensional velocity field distributions are obtained to characterize the crushing b and propagating through a cellular material. An explicit expression for the continuous velocity distribution is derived based on the features of the velocity gradient distribution. The velocity distribution function is adopted to determine the dynamic stress–strain states of cellular materials under dynamic loading. The local stress–strain history distribution reveals that sectional cells experience a process from the precursor elastic behavior to the shock stress state, passing through the dynamic initial crushing state. A power-law relation between the dynamic initial crushing stress and the strain rate is established, which confirms the strain rate effect of cellular materials. By extracting the critical points immediately before the unloading stage in the local dynamic stress–strain history curves, the dynamic stress–strain states of cellular materials are determined. They exhibit loading rate dependence but are independent of the initial impact velocity.Furthermore, with increase of the relative density, the dynamic hardening behavior of the cellular specimen is enhanced and the crushing process event is advanced. The particle velocity-based analytical method is applied to analyze the dynamic responses of cellular materials. This method is better than continuum-based shock models, since it does not require a preassumed constitutive relation. Therefore, the particle velocity-based analytical method proposed herein may provide new ideas to carry out dynamic experimental measurements, which is especially applicable to inhomogeneous materials.

Keywords Cellular material· Velocity distribution· Dynamic behavior· Strain rate effect· Dynamic stress–strain state

1 Introduction

Cellular materials, such as aluminum foams and honeycombs, have high porosity and exhibit excellent performance in terms of their energy absorption capability. In practical applications of impact energy absorbers and antiblast structures, adverse energies applied externally are attenuated by cellular materials with nearly unchanged support stress[1–4] to ensure that the contained payload is protected.Highly localized deformation is one typical feature of cellular materials under dynamic impact. Usually, a structural stress wave is adopted to characterize the propagation of the localized crushing b and in such materials [5]. Correspondingly, a series of shock models have been proposed to analyze the dynamic responses of cellular materials [6–11].However, since cellular materials are inhomogeneous and have intrinsic characteristic scales, the applicability of continuum-based shock models to analyze the dynamic response of this kind of material inevitably results in differences from the real behavior of wave propagation through cellular materials [8]. Moreover, these models describe the quasistatic mechanical behavior on the homogeneous scale, while the application of a shock wave within a homogenized medium to effectively characterize the dynamic crushing behavior of cellular materials requires further discussion.

Considering the difficulty in measuring the local stress and strain states of cellular materials under dynamic impact experimentally, Liao et al. [12] developed a local strain calculation method based on the optimal local deformation gradient. As a practical example, the strain field distributions of irregular honeycombs under in-plane loading were calculated, which can well capture the propagation of the stress wave [13]. To study the dynamic behavior of cellular materials experimentally, the dynamic evolution of the deformation in open-cell aluminum foams was monitored and the local physical quantities were measured with the aid of high-speed photography coupled with digital image correlation (DIC) [14]. Hugoniot relations between the dynamic response (i.e., shock wave speed, stress, and strain) and impact velocities were the n established. It is worth noting that a linear Hugoniot relation between the shock wave speed and corresponding impact velocity was obtained, as also reported in Refs. [15–17]. In combination with the conservation of mass and momentum across the shock front, the dynamic stress–strain states can the n be determined conveniently. However, such an analytic model is based on the assumption that the stress ahead of the shock front has the same level as the first local stress maximum of the quasistatic stress–strain curve. However, the stress immediately ahead of the shock front, i.e., the dynamic initial crushing stress, is actually not a constant value but has a distribution away from the distal end [18]. Besides, the dynamic initial crushing stress also exhibits strain rate sensitivity, which is essentially different from the quasistatic state [15, 19]. Therefore, the dynamic stress–strain states of cellular materials were obtained numerically using a different approach based on the strain field distribution by Zheng et al. [11]. The obvious distinction between the dynamic and quasistatic stress–strain relations was attributed to the different deformation mechanisms.Layerwise crushing b and s form in cellular materials depending on the speed of the shock wave during dynamic crushing,while shear bands occur randomly under quasistatic loading,which indicates that the dynamic behavior of cellular materials is velocity dependent. Substantially, the dynamic constitutive behavior of cellular materials is deformation mode dependent.

Recently, the stream of literature on the evolution of the dynamic behavior of cellular materials was systematically reviewed to highlight the commonality and contrasts in terms of the phenomena and mechanisms, and their modeling [20].This historical perspective shows that much know ledge has been acquired on the dynamic crushing behavior of cellular materials. However, the re is no effective and convenient way to determine the dynamic stress–strain state of cellular materials yet, especially for experimental tests, since experimental data are difficult to measure due to issues associated with the highly localized deformation and the limitations of experimental techniques. The dynamic rigid-plastic hardening (D-R-PH)idealization [11] can be expressed as where D is the dynamic strain hardening parameter andis the dynamic initial crush stress. Consequently, the two parameter dynamic material model exhibits powerful ability to characterize the dynamic stress–strain states and may lead to ideas to determine the dynamic mechanical behavior of cellular materials using a feasible method. Following this route, the effects of the relative density, entrapped gas, and hardening behavior of the base material on the shock properties of cellular materials were investigated practically by carrying out virtual tests [21], providing further understanding of the mechanisms underlying such effects of mesostructural features. Since the shock model is established on the condition of shock initiation, the intermediate state of cellular materials cannot be well elucidated by such theoretical idealization.

The treatment of the macroscopic discontinuity at the crushing front constitutes the basis of the description of the dynamic behavior of cellular materials using shock theory[22]. However, the shock front where crushed cells start to form always has a finite thickness [18, 23] in reality. To portray the dynamic behavior of cellular materials more realistically, the Lagrangian analysis method can be adopted to investigate the local dynamic response of three-dimensional(3D) Voronoi structures under a Taylor impact [19]. By considering the critical points on local dynamic stress–strain curves, a unique dynamic stress–strain state curve that is related to different loading rates can be extracted without any preassumed constitutive relations. When this Lagrangian analysis method is directly applied to experimental tests on foams, a tremendous amount of data on the historical particle velocity distribution is required to ensure the accuracy of results, since differential equations must essentially be solved, which is hard to realize due to the limitations of experimental techniques. Therefore, a convenient method that could be used to draw ideas effectively from dynamic tests on cellular materials is pressingly desired.

In this paper, the dynamic behavior of cellular materials is further studied using the particle velocity distribution to reveal the mesodeformation mechanism. The aim is to propose an analytical model for the velocity distribution for application in calculations of the history distribution of the dynamic local stress and local strain. Finally, the dependence of the dynamic behavior of cellular materials on the relative density is investigated using the proposed analytical model.

2 Numerical simulations

2.1 Cell?based finite element model

A closed-cell foam model (Fig.1a) is generated using the 3D Voronoi technique [24]. Firstly, a set of distinct and isolated points are randomly generated in a 3D region based on the principle that the distance between any two points is not less than a given distance. Secondly, a Delaunay tessellation[25] is constructed by generating a triangulation that maximizes the minimum angle of all the angles of the triangles.Thirdly, Voronoi polyhedra are obtained by generating the intersections of the planes orthogonal to the edges of all the Delaunay tetrahedra passing through the midpoints of these edges. Finally, a Voronoi diagram is obtained and trimmed to obtain a Voronoi structure. The Voronoi configuration can characterize the mesostructure of some kinds of cellular foam well and thus is widely employed in numerical simulations to underst and the mechanical behavior of cellular materials [17, 26–29].

Fig. 1 a Finite element model of closed-cell foam, b direct impact scenario

The cellular specimen model used in this study contains 600 nuclei in a volume of 20mm × 20mm × 30mm with cell irregularity of 0.4. The average cell size dc, defined as the diameter of a sphere whose volume is equal to the average volume of the cells within the Voronoi structure, is about 3.2mm. The cell-wall material of the cellular specimen is assumed to be elastic–perfectly plastic with density ρs= 2700kg/m3, Young’s modulus E = 69GPa, Poisson’s ratio ν = 0.3, and yield stress σys= 170MPa. The cellwall thickness of the cellular specimen is uniform and is dependent on the relative density of the specimen ρ, which is defined as the ratio of the cellular specimen density ρfto the cell-wall material density ρs. In this study, the relative density of the cellular specimen is set as 0.1, which gives a cellwall thickness as 0.095mm. The ABAQUS/Explicit finite element code [30] is used to perform the numerical simulations. The cell walls of the Voronoi structure are meshed with shell elements of type S3R and S4R. After mesh convergence analysis, the element size is taken as 0.3mm [21].Thus, the cellular specimen used in study has about 30,872 S3R elements and 109,119 S4R elements.

A direct impact scenario is applied in the numerical simulations. During numerical testing, the specimen impinges onto a fixed rigid plate with initial impact velocity V0, as shown in Fig.1b. A linear multipoint constraint is applied to the surfaces in the impact direction to avoid rollover of the specimen under high-velocity impact. The general contact is applied to satisfy the complex contact behaviors among any possible contact surfaces with a friction coefficient of 0.2.

2.2 Calculation method for velocity field distribution

The cellular specimen under dynamic impact exhibits highly localized deformation; i.e., the collapse of cells initiates within a narrow b and while the cells ahead of the collapse b and remain almost undeformed. As the loading event continues, the collapse b and spreads from the impact end to the free end layer by layer, as shown in Fig.2. The propagation of the collapse b and happens in nearly flat planes with negligible lateral expansion. Herein, the investigation of the impact behavior of the cellular specimen can be simplified to the issue of one-dimensional wave propagation. As a basic physical quantity, the velocity field can be applied to characterize the status of the cellular structure. Besides, the velocity components along the impact direction can be extracted directly from the numerical results, enabling convenient and reliable calculation of the local velocity field. A middle section parallel to the impact direction is selected to investigate the deformation evolution during impact. A sequence of local velocity field distributions in the framework of the Lagrangian coordinate is determined, and the corresponding mesodeformation patterns of the cut section of the cellular specimen with initial impact velocity V0= 250m/s extracted,as shown in Fig.2. It transpires that the shock-like deformation is captured well by the velocity field distribution. During crushing, the kinetic energy of the cellular specimen is progressively dissipated in the form of plastic collapse of cell walls, which results in a gradual reduction of the impact velocity of the undeformed portion.

Fig. 2 Velocity distribution fields in impact direction of cellular specimen together with corresponding deformation configurations

Fig. 3 1D velocity distribution and corresponding velocity gradient

The one-dimensional (1D) velocity distribution can be determined by averaging the velocity components along the longitudinal direction of the specimen at each Lagrangian position within a range of one cell size at a specific moment,as shown in Fig.3. A three-point centered difference is adopted to determine the local velocity gradient distribution.It is demonstrated that the macroscopic deformation status of the cellular specimen under impact can be basically separated into three regions: the crushed section, the crushing initiation section, and the undeformed section. The crushing initiation section spans a range of about one cell diameter,where the cells transform dramatically from an undeformed to compacted state, as depicted in Fig.3. Macroscopically,the propagation of the crushing front behaves as a shock wave, so it is usually treated as a singular plane to propose shock models in the framework of continuum mechanics.To date, a series of shock models have been proposed to characterize the crushing behavior of cellular materials [10,11, 31].

3 Particle velocity?based analysis

The 1D velocity gradient distribution at a specific moment exhibits a bell-shaped feature, as illustrated in Fig.3.According to this feature, the Gaussian unimodal distribution is applied to characterize the velocity gradient distribution, expressed as

where α, b, and ware substantially understood as the change of the instantaneous impact velocity, the Lagrangian location of the extreme value on the velocity gradient distribution, and the width of the crushing initiation section, respectively.It is worth noting that this Gaussian distribution description is not the only possible form of fitting function. However,the chosen function should offer a simple expression, easy solution, and physical clarity. Moreover, the sensitivity of the dynamic response to the fitting accuracy of the velocity gradient is presented in “Appendix A.” Integrating Eq.(2)with respect to the Lagrangian location and considering the boundary condition v = 0 at X = 0 yields a spatial velocity distribution of

Fig. 4 1D velocity distribution obtained from numerical simulations and corresponding fitting

where erf(?) is the error function andThe analytic model in Eq.(3) exhibits adaptable ability to capture the features of the velocity distribution of the cellular specimen at different impact moments extracted directly from the numerical results, as shown in Fig.4.

Once the form of the velocity distribution function has been determined, the variations of the associated parameters with time can be investigated by applying Eq.(3) to the velocity distributions extracted from the numerical results,as shown in Fig.5. The results indicate that both parameters a and b exhibit a nonlinear trend with impact time,while the parameter w increases linearly with the development of the crushing process. During impact, kinetic energy is gradually dissipated by collapse of cell walls, and the decreasing impact velocity attenuates the propagation of the crushing front while the width of the crushing initiation region exp and s. When the impact velocity is decreased to a critical velocity, the deformation exhibits a combination of layerwise crushing b and s and r and om shear b and s,which is known as the transition mode. In such case, the 1D simplification may overaverage the behaviors of the cellular specimen at moderate impact velocity. Herein, the method based on the spatial velocity distribution function to study the mechanical behavior of cellular materials is much more reliable for the loading case with sufficiently high impact velocity.

Considering the boundary condition b(0) = 0, loworder polynomial functions, i.e., a(t) = a1t2+ a2t + a3,b(t) = b1t2+ b2t, and w(t) = w1t + w2, are adopted to fit the variations of the parameters with impact time, as demonstrated in Fig.5, with fitting parameters of a1= 7447mm·ms?3,a2= 201.8mm·m s?2, a3= ?120.9mm·m s?1,b1= ?848.1mm·m s?2, b2= 291.4mm·m s?1,w1= 0.7331mm·ms?1, and w2= 1.659mm. After determining the quantitative relations of the three parameters in Eq.(3), the general form of the spatial velocity distribution function can be rewritten as

where the seven undetermined coefficients correctly characterize the response of cellular materials with different properties (the relative density of the cellular material, and the properties of the base material and mesostructure). Note that Eq.(4) cannot satisfy the initial condition v(0, X) = ?V0.This incompatibility is attributed to the dispersion in the data due to the boundary effect as well as the neglect of the propagation and reflection of elastic wave in the early stage of the crushing event (see “Appendix B”). Therefore, the velocity-based model is highly applicable for investigating the dynamic behavior of cellular materials with a relatively stable crushing front. Together with the jump conditions(i.e., conservation of mass and momentum) across the shock front in Lagrangian form, the analytical velocity distribution can be applied to study the local dynamic evolution characteristics of the stress and strain, dynamic initial crushing stress, and dynamic stress–strain states of cellular materials during impact by extracting the velocity profiles at nodes from numerical or experimental results.

Fig. 5 Variations of parameters a a, b b, and c w in Eq.(3) with time

4 Results and discussion

4.1 Dynamic local stress and strain

In the theory of 1D stress wave propagation, the conservation equations for mass and momentum can be expressed in Lagrangian form as [32]

where ing, σ, v and ρfare strain, stress, particle velocity, and initial density, respectively. Here, it is stipulated that the stress and strain are positive for the compressive case. The n,the dynamic behavior of the local strain in cellular materials can be obtained by combining Eqs.(4) and (5) with the initial condition ing(0, X) = 0. Similarly, the local stress can be determined from Eqs.(4) and (6) together with the boundary condition σ(t, L0) = 0 (where L0is the initial length of the cellular specimen). Here, it is noteworthy that, for a cellular material examined using a cell-based model, the definition of local stress or local strain is the average response at a considered location over the range of an intrinsic characteristic length (i.e., the average cell size). In actual dynamic tests of cellular materials, velocity data recorded to analyze the response are not sufficient to guarantee the accuracy of such results due to the limitations of existing experimental techniques. Compared with the investigation of the dynamic behavior of cellular materials using the Lagrangian analysis method [19], the analytical method developed herein only requires particle velocity data at a few Lagrangian positions for different impact moments to determine the spatial velocity distribution function expressed in Eq.(4). The strain history and stress history can the n be conveniently acquired using Eqs.(5) and (6). Accordingly, the solving method based on the spatial velocity distribution proposed herein provides a new approach to carry out experimental studies on the dynamic behavior of cellular materials.

The time history of local stress is calculated from Eqs.(4) and (6) with zero stress at the free end (σ(0,t) = 0), as shown in Fig.6. These results indicate that each cross-section corresponding to a certain Lagrangian position will experience three stages successively:the elastic stage, the collapse stage, and the compacted stage. Once an impact event begins, collapse of cell walls happens immediately after the precursor elastic behavior, and the stress increases to the level of the initial crushing stress. Due to the attenuation of the impact strength and the interaction of the elastic unloading wave, the stress in the compacted region gradually decreases, exhibiting a negative correlation with distance from the impacted end, until the crushing event terminates completely. Note also that the upper envelope curve of the local stress histories coincides with the boundary stress history extracted directly from the numerical results. For the direct impact case, the compaction region through which the crushing front passes becomes part of the supported rigid plate. The inertia effect can be ignored, and the stress has a uniform distribution in this stationary region. Therefore, the reliability of the stress–time distribution determined by the analytical analysis method is further validated. However,the re is nearly no oscillation phenomenon in the elastic stage compared with the trend of the distribution of the local stress history obtained by the Lagrangian analysis method [19]. Actually, the velocity distribution exhibits a transition from the instantaneous impact velocity in the compaction region to the final stationary state in the undeformed region, as shown in Fig.4, which can probably be attributed to elastic bending waves [18]. The fitting used in the analytical method described herein eliminates the oscillation in the velocity distribution curve, resulting in linear behavior in this stage, as depicted in Fig.6.

Fig. 6 Comparison of stress behind shock front obtained directly from finite element results with local stress calculated numerically

Similarly, using Eqs.(4) and (5), the time-history distribution of the strain at any Lagrangian position can be obtained by considering the initial condition ing(X, 0) = 0, as shown in Fig.7. The locations with near-zero strain indicate cells that remain undeformed before arrival of the crushing front. The local strain field calculation method [12] is further adopted to verify the reliability of the local strain field distribution calculated by the analytical method proposed herein, as shown in Fig.8. These results show that the strain fields determined by the two methods are in good agreement.This indicates the validity of this velocity-based analytical method for investigation of the dynamic response of cellular materials. When the crushing front arrives at a specific section, the sectional strain increases dramatically to the level of the densification strain, which is related to the intensity of the instantaneous impact velocity, as shown in Fig.9. This demonstrates that the densification strain behind the crushing front increases with increase of the impact velocity and finally asymptotically approaches a constant value that is dependent on the relative density of the cellular specimen.It is also interesting to note that, when the impact velocity is about 130m/s, the densification strain increases slowly and basically coincides with the results determined from the D-R-PH shock model [11]. When the impact velocity is below 130m/s, the densification strain is lower than the results predicted using the D-R-PH shock model, and the deviation becomes more significant as the impact velocity decreases to moderate values. Since the deformation mode of the cellular material is strongly dependent on the impact velocity, the validity of the shock model under moderatevelocity impact is questionable. A critical impact velocity to indicate the applicability of the D-R-PH model must be defined. For a Voronoi structure with relative density of 0.1,this critical impact velocity is taken as 130m/s, although this is not a general result, since the velocity depends on the methods that are adopted to determine the dynamic behavior of cellular materials quantitatively, as described in “Appendix C.”

Fig. 7 Variation of local strain at Lagrangian locations with time

Fig. 8 Comparison of local strain field distributions obtained by the velocity-based analytical method and local strain field calculation method

Fig. 9 Variation of densification strain with impact velocity

Fig. 10 Variation of local stress with impact time

The variation of the stress at certain Lagrangian positions is shown in Fig.10. As the crushing front approaches, the location immediately ahead enters an initial crushing state and serves as a precursor for the incoming collapse behavior of the cells. Once the crushing front passes by, the compacted region behind the crushing front remains stationary and the shock stress (the stress behind the crushing front)decreases with the drop of the impact velocity. One phenomenon to be noted is that a smooth trend in the local stresses occurs before the crushing front arrives compared with the oscillation in this stage obtained using the Lagrangian analysis method. As discussed above, the fitting treatment applied to the velocity distribution essentially neglects the effect of the elastic loading and unloading waves.

4.2 Strain rate effect of cellular material

The deformation mode of cellular materials exhibits strong impact velocity sensitivity, changing from random shear bands under quasistatic compression to layered crushing b and s under high-velocity impact. Nevertheless, the impact velocity, which is not a material parameter, only serves as an extrinsic factor to relate with some intrinsic parameters of the materials (such as micro-inertia, cell morphology, or strain rate sensitivity of cell wall). The strain rate sensitivity of the peak stress and crushing stress of honeycombs was quantified numerically and analytically [33]. However,for cellular materials, the re are conflicting conclusions regarding whether or not their dynamic behavior are sensitive to the strain rate. Dynamic tests were conducted on aluminum foams by Deshpande and Fleck [34] using a split Hopkinson pressure bar (SHPB) apparatus; the results indicated that the plateau stress of aluminum foams shows no strain rate sensitivity, even when the loading rate increases to