Cell-based Finite Element Model Research Articles

Bandgap analysis of periodic composite plates considering fluid–structure coupling

This article presents a novel approach for predicting vibration bandgaps in periodic composite plates with fluid–structure interaction (FSI) using a unit cell-based finite element model. The novelty of our approach lies in the formulation of a fluid-induced added mass matrix, which integrates the Bloch periodic boundary condition, allowing for the incorporation of the fluid’s inertial effect in the context of unit cell-based bandgap analysis. We therefore construct a unit cell model comprising a composite Mindlin plate which integrates periodic FSI effects with the simultaneous application of Bloch conditions on both the structure and the fluid domains. Subsequently, we studied a set of periodic composite plates with FSI effects on one or both sides, thereby assessing the influence of the fluid properties such as density and the fluid domain dimension on the structure vibration. The bandgap prediction is compared with the frequency response simulations which involve diversified microstructure designs. The obtained results provide indications regarding the effectiveness and applicability of the proposed numerical methodology.

Estimation of interface properties in epoxy-based barium titanate nanocomposites

The study of the interface around a nanoparticle in a nanocomposite is crucial to understanding the performance of nanocomposites. In this work, an estimate of the interface thickness and interface permittivity is obtained based on the changes in dielectric properties in epoxy due to the introduction of nanofillers. Barium titanate (BaTiO3) nanofillers with an average diameter of 90 nm are used. Specimens with various filler loadings (1%–10% by volume) are used. Nanoparticles are used either as-received or surface-functionalized with 3-glycidoxypropyltrimethoxy-silane (GPS) before use. Fourier Transform Infra-red (FTIR) spectroscopy is used to study the surface chemistry of BaTiO3 nanofillers after GPS functionalization. Reasonable uniformity of dispersion is observed from Field Emission Scanning Electron Microscopy (FESEM) images. Complex permittivity is measured using Broadband Dielectric Spectroscopy over a wide frequency range (10−3 Hz to 10+6 Hz). A unit cell-based Finite Element model is used to compute the effective permittivity of a nanocomposite, for a given interface thickness and interface permittivity. This method combines the experimental data obtained from dielectric spectroscopy with a unit cell-based Finite Element Method (FEM) for electric field computation. A match between experimental and computational data indicates that the interfacial thickness around the nanoparticle is approximately 200 nm and interface permittivity is slightly higher than that of neat epoxy resin over the given frequency range.

Blast Alleviation of Sacrificial Cladding with Graded and Uniform Cellular Materials

Graded cellular material is a superb sandwich candidate for blast alleviation, but it has a disadvantage for the anti-blast design of sacrificial cladding, i.e., the supporting stress for the graded cellular material cannot maintain a constant level. Thus, a density graded-uniform cellular sacrificial cladding was developed, and its anti-blast response was investigated theoretically and numerically. One-dimensional nonlinear plastic shock models were proposed to analyze wave propagation in density graded-uniform cellular claddings under blast loading. There are two shock fronts in a positively graded-uniform cladding; while there are three shock fronts in a negatively graded-uniform cladding. Response features of density graded-uniform claddings were analyzed, and then a comparison with the cladding based on the uniform cellular material was carried out. Results showed that the cladding with uniform cellular materials is a good choice for the optimal mass design, while the density graded-uniform cladding is more advantageous from the perspective of the critical length design indicator. A partition diagram for the optimal length of sacrificial claddings under a defined blast loading was proposed for engineering design. Finally, cell-based finite element models were applied to verify the anti-blast response results of density graded-uniform claddings.

Revisiting the Deshpande-Fleck Foam Model

The self-similar isotropic hardening model developed by Deshpande and Fleck has been widely used. An important issue in this model is to determine the value of ellipticity. The ellipticity was treated as a constant in the subsequent yield, but different values were suggested in the literature. In this paper a cell-based finite element model based on the 3D Voronoi technique is used to verify the Deshpande-Fleck foam model. It is found that the ellipticity determined from uniaxial and hydrostatic compressions varies with the equivalent plastic strain.

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

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 band 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.

Impact Resistance and Design of Graded Cellular Cladding

One-dimensional impact response of graded cellular rods with monotonously increasing density distribution, which benefits in buffering the impact object in proximal end, is investigated by using shock models and cell-based finite element models. Based on the rigid–plastic hardening (R-PH) idealization, a theoretical model of the shock front propagation behavior in graded cellular rods subjected by impact is developed and solved numerically with a fourth-order Runge–Kutta scheme. Finite element analysis is performed using 2D random Voronoi technique, and the comparisons illustrate that the theoretical predictions and finite element results are in good agreement. The anti-impact responses of graded cellular materials with different gradient parameters are carried out, and it is demonstrated that the design indicators, including support stress, critical length and maximum deceleration of the object, are highly related to the gradient parameters of graded cellular materials. A design strategy for the impact alleviation of graded cellular materials is proposed. Thus, the gradient parameters can be finally determined to meet the requirement of the actual design indicators and it can also offer an optimal critical length for final graded cellular materials.

Modification and verification of the Deshpande–Fleck foam model: A variable ellipticity

The ellipticity involved in the Deshpande–Fleck foam model describing the constitutive behaviour of cellular materials was usually considered to be constant, but some very different values were suggested in the literature. A cell-based finite element model of closed-cell foam under uni-/multi-axial compression is employed to verify the Deshpande–Fleck foam model. The ellipticity is determined by applying uniaxial and hydrostatic compression tests and it is found to vary with the equivalent plastic strain, i.e., the ellipticity decreases with the equivalent plastic strain and then increases sustainably before full densification. According to the understandings from the numerical results, a fitting relation between the ellipticity and the equivalent plastic strain is suggested. The ellipticities of an open-cell foam and a closed-cell foam studied experimentally in the literature are fitted well with this relation. A modification to the Deshpande–Fleck foam model with a variable ellipticity is thus proposed. The modified Deshpande–Fleck foam model brings much accurate predictions with using a rigid–plastic hardening (R-PH) idealisation model, which describes the stress–strain relation of cellular material under uniaxial compression. Good agreement is also observed between the experimentally measured stress–strain responses and the predictions of the modified Deshpande–Fleck foam model, especially when considering the effect of the plastic Poisson's ratio with a non-associated flow rule. The findings herein are helpful to improve the prediction accuracy of the Deshpande–Fleck foam model.

A simplified model and its asymptotic solution for the crashworthiness design of graded cellular material

Cellular materials have strong energy absorption capacity and have been extensively used in crashworthiness design of structures. The mechanical responses of cellular materials are related to their relative density. Introducing a density gradient to cellular materials may further improve their crashworthiness. A backward design method based on a nonlinear plastic shock model was developed to meet the crashworthiness requirement of protecting a mass when impinging a graded cellular rod. For the case of constant impact force, an asymptotic solution to the density distribution of graded cellular rod is obtained in this paper. The density distributions between the explicit asymptotic solution and the exact solution numerically obtained with a fourth-order Runge-Kutta scheme have been compared. The first-order approximate solution is a linear approximation, and the second-order approximate prediction is basically consistent with the numerical solution. For the impact force history, the first-order approximate prediction has a large deviation, and the second-order approximate prediction has a slight deviation. For practical application the second-order approximate solution seems to be accurate enough. Cell-based finite element models are employed to verify the asymptotic solution. The finite element models have been constructed and computed via ABAQUS/Explicit code. Samples of graded cellular rods designed from the asymptotic solution are tested numerically. The predictions using the second-order approximate solution of density distribution is very close to the finite element results. The design using the first-order approximate solution is not accurate enough, but in fact it is not so bad. Compared the uniform density distribution, the second-order approximate prediction is good for the crashworthiness optimization. The design of density distribution for a specific impact scenario is found to be inappropriate for much high velocity impact. In summary, a case study shows that the second-order approximate solution is very close to the exact solution and it is good enough for the design. The backward design method and the asymptotic solution of density distribution are helpful to guide the crashworthiness design of graded cellular materials.

Shock wave speed and stress-strain relation of aluminium honeycombs under dynamic compression

The propagation of layer-wise crushing bands in cellular materials under dynamic impact can be described by the plastic shock wave model. A cell-based finite element model of irregular aluminum honeycomb is constructed to carry out several constant-velocity compression tests. The shock wave speed is obtained by the one-dimensional stress distribution in the specimen along the loading direction. The relation between the shock wave speed and impact velocity is obtained and analyzed. It is found that the relation tends to be linear with the increase of the impact velocity. But the shock wave speed tends to be a constant value with the decrease of the impact velocity. A piecewise model is proposed to describe the dynamic stress-strain relation of aluminum honeycombs based on a piecewise hypothesis of the relation between the shock wave speed and the impact velocity together with the one-dimensional shock wave theory. Different stress-strain relations corresponding to different impact velocity regions and different deformation modes are obtained.

Micromechanical modeling of 3D printable interpenetrating phase composites with tailorable effective elastic properties including negative Poisson's ratio

3D printable two-phase interpenetrating phase composites (IPCs) are designed by embedding a 3D periodic re-entrant lattice structure (as the reinforcing phase) in a matrix phase. These IPCs display the cubic or tetragonal symmetry. A micromechanical model is developed to evaluate effective elastic properties of the IPCs. Effective Young's moduli, shear moduli and Poisson's ratios (PRs) of each IPC are determined from the effective stiffness and compliance matrices of the composite, which are obtained through a homogenization analysis using a unit cell-based finite element (FE) model incorporating periodic boundary conditions. The FE simulation results are also compared with those based on various analytical bounding techniques in micromechanics, including the Voigt–Reuss, Hashin–Shtrikman, and Tuchinskii bounds. The effective properties of the IPC can be tailored by adjusting five geometrical parameters, including two strut lengths, two re-entrant angles and one strut diameter, and elastic properties of the two constituent materials. The numerical results reveal that IPCs with a negative PR can be generated by using a compliant matrix material and large re-entrant angles. In addition, it is found that the two re-entrant angles can greatly affect other effective elastic properties of the IPC: the effective shear modulus can be enhanced, while the effective Young's modulus can be enhanced or compromised with the increase of the re-entrant angles. Furthermore, it is seen that by adjusting one of the two re-entrant angles or one of the two strut lengths, the material symmetry exhibited by the IPC can be changed from cubic to tetragonal.

Strain-rate effect on initial crush stress of irregular honeycomb under dynamic loading and its deformation mechanism

The seemingly contradictory understandings of the initial crush stress of cellular materials under dynamic loadings exist in the literature, and a comprehensive analysis of this issue is carried out with using direct information of local stress and strain. Local stress/strain calculation methods are applied to determine the initial crush stresses and the strain rates at initial crush from a cell-based finite element model of irregular honeycomb under dynamic loadings. The initial crush stress under constant-velocity compression is identical to the quasi-static one, but less than the one under direct impact, i.e. the initial crush stresses under different dynamic loadings could be very different even though there is no strain-rate effect of matrix material. A power-law relation between the initial crush stress and the strain rate is explored to describe the strain-rate effect on the initial crush stress of irregular honeycomb when the local strain rate exceeds a critical value, below which there is no strain-rate effect of irregular honeycomb. Deformation mechanisms of the initial crush behavior under dynamic loadings are also explored. The deformation modes of the initial crush region in the front of plastic compaction wave are different under different dynamic loadings.

Stress Distribution in Graded Cellular Materials Under Dynamic Compression

Dynamic compression behaviors of density-homogeneous and density-graded irregular honeycombs are investigated using cell-based finite element models under a constant-velocity impact scenario. A method based on the cross-sectional engineering stress is developed to obtain the one-dimensional stress distribution along the loading direction in a cellular specimen. The cross-sectional engineering stress is contributed by two parts: the node-transitive stress and the contact-induced stress, which are caused by the nodal force and the contact of cell walls, respectively. It is found that the contact-induced stress is dominant for the significantly enhanced stress behind the shock front. The stress enhancement and the compaction wave propagation can be observed through the stress distributions in honeycombs under high-velocity compression. The single and double compaction wave modes are observed directly from the stress distributions. Theoretical analysis of the compaction wave propagation in the density-graded honeycombs based on the R-PH (rigid-plastic hardening) idealization is carried out and verified by the numerical simulations. It is found that stress distribution in cellular materials and the compaction wave propagation characteristics under dynamic compression can be approximately predicted by the R-PH shock model.

Crashworthiness of graded cellular materials: A design strategy based on a nonlinear plastic shock model

Dynamic behaviors and crashworthiness design of density graded cellular materials are investigated by using theoretical method and finite element simulation. A nonlinear plastic shock model is employed to guide the gradient design of cellular rod under mass impact and cell-based finite element models are used to verify the design strategy. Effects of impact-force parameter on the design of relative density distribution of graded cellular material and on the residual velocity of the striker for different lengths of graded cellular rods are explored. Three cases of design strategy with different impact-force parameters are analyzed and verified. The results reveal that the design strategy is reliable when the impact-force parameter is not less than zero. If the relative density distribution of graded cellular rod increases monotonically, the actual deformation mode of graded cellular rod is identical with the assumed one in the theoretical derivations. It is noticed that a portion of the graded cellular rods close to the distal end is not fully compacted. A scheme with restricting the shock strain not less than a specific value is proposed to shorten the graded cellular rods. Therefore, the desirable crashworthiness of graded cellular materials can be obtained by designing the density distribution of graded cellular materials.

Dynamic material parameters of closed-cell foams under high-velocity impact

Cellular materials under high-velocity impact have highly localized deformation with the cells layer-wise collapse, which is usually characterized by the propagation of shock wave. Researches have shown that the shock wave speed is strongly dependent on the impact velocity, but the effect of the meso-structural and base-material parameters is unclear. In this study, the dynamic material parameters of closed-cell foams are investigated with cell-based finite element models. The one-dimensional velocity distribution along the loading direction is used to capture the propagation of shock front. The shock wave speed is thus determined and it exhibits a linear relationship with impact velocity when the impact velocity is high enough. The difference between the shock wave speed and the impact velocity is a dynamic material parameter of cellular material and the effect of meso-structural and base-material parameters on this dynamic material parameter is investigated with dimensional analysis. An expression of the dynamic material parameter with respect to the relative density and the base-material parameters is obtained. It shows that the dynamic material parameter intensively relies on the relative density and increases linearly with the relative density. The investigation of dynamic stresses with the aid of a shock model shows that the initial crushing stress increases in a power-law tendency with the increase of relative density. As a result, a stress–strain relation involving the relative density of material cellular and the yield stress and density of base material is obtained for the closed-cell foams considered. The effects of hardening behaviors of cell-wall material and gas trapped within cells are also considered. It is found that the dynamic material parameter exhibits nearly linear increase with the increase of hardening parameters of base material and initial gas pressure, while the dynamic initial crushing stress is independent of the strain-hardening parameter and the entrapped gas pressure but increases with the strain-rate hardening parameter linearly. These findings may be helpful for guiding the crashworthiness design of cellular materials and structures.

Impact Resistance of Graded Cellular Metals Using Cell-Based Finite Element Models

A varying cell-size method based on Voronoi technique is extended to construct 3D graded cellular models. The dynamic behaviors of graded cellular structures with different density gradients are then investigated with finite element code ABAQUS/Explicit. Results show that graded cellular materials have better performance as energy absorbers. Graded cellular structures with large density near the distal end can protect strikers, and those with low density near the distal end can protect structures at the distal end. It is concluded that graded cellular materials with suitable design may have excellent performance in crashworthiness.

Blast Alleviation of Cellular Sacrificial Cladding: A Nonlinear Plastic Shock Model

The anti-blast behavior of cellular sacrificial cladding is investigated based on a continuum-based nonlinear plastic shock model. A rate-independent, rigid–plastic hardening (R-PH) model with two material parameters, namely the initial crushing stress and the strain hardening parameter, is employed to idealize the cellular material. The governing equation of the motion of cover plate is obtained and solved numerically with a fourth-order Runge–Kutta scheme. A comparison of the crushing percentage contours of sacrificial cladding based on the R-PH model and the rigid–perfectly plastic–locking (R-PP-L) model is carried out. Results transpire that the R-PP-L model is not accurate enough to evaluate the energy absorption. Dimensional analysis is employed to study the critical length of cellular sacrificial cladding and an empirical expression is determined by the controlling valuable method. An asymptotic solution is also obtained by applying the regular perturbation theory. Finally, the design criteria of cellular sacrificial cladding based on the R-PH shock model is verified by a cell-based finite element model.

Dynamic Behavior of Cellular Materials under Combined Shear-Compression

A 3D cell-based finite element model is employed to investigate the dynamic biaxial behavior of cellular materials under combined shear-compression. The biaxial behavior is characterized by the normal stress and shear stress, which could be determined directly from the finite element results. A crush plateau stress is introduced to illustrate the critical crush stress, and the result shows that the normal plateau stress declines with the increase of the shear plateau stress, which climbs with the increase of loading angle. An elliptical criterion of normal plateau stress vs. shear plateau stress is obtained by the nonlinear regression method.

Dynamic stress–strain states for metal foams using a 3D cellular model

Dynamic uniaxial impact behaviour of metal foams using a 3D cell-based finite element model is examined. At sufficiently high loading rates, these materials respond by forming ‘shock or consolidation waves’ (Tan et al., 2005a, 2005b). However, the existing dynamic experimental methods have limitations in fully informing this behaviour, particularly for solving boundary/initial value problems. Recently, the problem of the shock-like response of an open-cell foam has been examined by Barnes et al. (2014) using the Hugoniot-curve representations. The present study is somewhat complementary to that approach and additionally aims to provide insight into the ‘rate sensitivity’ mechanism applicable to cellular materials. To assist our understanding of the ‘loading rate sensitivity’ behaviour of cellular materials, a virtual ‘test’ method based on the direct impact technique is explored. Following a continuum representation of the response, the strain field calculation method is employed to determine the local strains ahead of and behind the resulting ‘shock front’. The dynamic stress–strain states in the densification stage are found to be different from the quasi-static ones. It is evident that the constitutive behaviour of the cellular material is deformation-mode dependent. The nature of the ‘rate sensitivity’ revealed for cellular materials in this paper is different from the strain-rate sensitivity of dense metals. It is shown that the dynamic stress–strain states behind a shock front of the cellular material lie on a unique curve and each point on the curve corresponds to a particular ‘impact velocity’, referred as the velocity upstream of the shock in this study. The dynamic stress–strain curve is related to a layer-wise collapse mode, whilst the equivalent quasi-static curve is related to a random shear band collapse mode. The findings herein are aimed at improving the experimental test techniques used to characterise the rate-sensitivity behaviour of real cellular materials and providing data appropriate to solving dynamic loading problems in which cellular metals are utilised.

Dynamic Crushing of Voronoi Honeycombs: Local Stress-Strain States

Dynamic stress-strain states in Voronoi honeycombs are investigated by using cell-based finite element models. Two different loading scenarios are considered: the high-constant-velocity compression and the direct impact. The 2D local engineering strain fields are calculated. According to the feature of shock front propagation, the 1D distribution of local engineering strain in the loading direction is deduced from the 2D strain fields, which provide evidences of the existence of discontinuities at shock front in cellular materials and thus enhance the basis of the continuum-based shock models. A method to quantitatively clarify the local stress-strain states ahead of and behind the shock front is developed. The results show that the dynamic stress-strain states in the densification stage obtained from both loading scenarios are different from the quasi-static stress-strain relation. The stress ahead of the shock front obtained from the high-constant-velocity compression scenario is slightly smaller than the quasi-static yield stress, but that obtained from the direct impact scenario is larger than the quasi-static yield stress. The possible mechanisms of deformation and wave propagation are explored.

Mass Impact of Density-Graded Cellular Metals in a Temperature Field

We consider the problem of a density-graded cellular rod in a temperature gradient field axially subjected to a mass impact. Two-dimensional cell-based finite element models and one-dimensional shock models are employed to explore the mechanisms of deformation and wave propagation. The yield stress distribution in a cellular specimen depends on both the density gradient field and the temperature gradient field. The stress distribution and the yield stress distribution are analyzed. For the increasing yield stress along the impact direction, one shock front propagates from the proximal end to the distal end of the specimen. For the decreasing yield stress along the impact direction, two shock fronts propagate in opposite directions and the one close to the proximal end ceases at a particular time. The predicted stresses of the extended shock models are compared well with the numerical results.