Open-cell cellular solids: A constitutive equation for hyperelasticity with deformation induced anisotropy

A constitutive theory is developed for an open-cell flexible cellular solid consisting of a network of struts each connecting two vertex points. A hypothesis is proposed that vertex points move affinely in the large-deformation regime, when the struts buckle, and that the force carried by a strut is a function of the longitudinal and rotational change of its vertex-to-vertex vector. The forces consist of one longitudinal force, parallel with the vertex-to-vertex vector of the strut and one transverse force. The overall stress response is initially dominated by the longitudinal force whilst the addition of the transverse force becomes significant at large deformations. The model contains three parameters: longitudinal stiffness, bending stiffness and critical stretch of a strut. These three parameters are calibrated against a simple compression test. The model is then validated against independent experiments in a simple tension, simple shear and a combined shear-compression test on an isotropic flexible polyether urethane foam. Excellent agreement is obtained between the experiments and the model.     

Simulation of the densification of real open-celled foam microstructures

Ubiquitous in nature and finding applications in engineering systems, cellular solids are an increasingly important class of materials. Foams are an important subclass of cellular solids with applications as packing materials and energy absorbers due to their unique properties. A better understanding of foam mechanical properties and their dependence on microstructural details would facilitate manufacture of tailored materials and development of constitutive models for their bulk response. Numerical simulation of these materials, while offering great promise toward furthering understanding, has also served to convincingly demonstrate the inherent complexity and associated modeling challenges.The large range of deformations which foams are subjected to in routine engineering applications is a fundamental source of complication in modeling the details of foam deformation on the scale of foam struts. It requires accurate handling of large material deformations and complex contact mechanics, both well established numerical challenges. A further complication is the replication of complex foam microstructure geometry in numerical simulations. Here various advantages of certain particle methods, in particular their compatibility with the determination of three-dimensional geometry via X-ray microtomography, are exploited to simulate the compression of “real” foam microstructures into densification. With attention paid to representative volume element size, predictions are made regarding bulk response, dynamic effects, and deformed microstructural character, for real polymeric, open-cell foams. These predictions include a negative Poisson's ratio in the stress plateau, and increased difficulty in removing residual porosity during densification.     

Impact of material processing and deformation on cell morphology and mechanical behavior of polyurethane and nickel foams

The cell morphology and mechanical behavior of open-cell polyurethane and nickel foams are investigated by means of combined 3D X-ray micro-tomography and large scale finite element simulations. Our quantitative 3D image analysis and finite element simulations demonstrate that the strongly anisotropic tensile behavior of nickel foams is due to the cell anisotropy induced by the deformation of PU precursor during the electroplating and heat treatment stages of nickel foam processing. In situ tensile tests on PU foams reveal that the initial main elongation axis of the cells evolves from the foam sheet normal direction to the rolling direction of the coils. Finite element simulations of the hyperelastic behavior of PU foams based on real cell morphology confirm the observation that cell struts do not experience significant elongation after 0.15 tensile straining, thus pointing out alternative deformation mechanisms like complex strut junctions deformation. The plastic behavior and the anisotropy of nickel foams are then satisfactorily retrieved from finite element simulations on a volume element containing eight cells with a detailed mesh of all the hollow struts and junctions. The experimental and computational strategy is considered as a first step toward optimization of process parameters to tailor anisotropy of cell shape and mechanical behavior for applications in batteries or Diesel particulate filtering.     

Modeling the dynamic response of visco-elastic open-cell foams

This article introduces a mesoscopic formulation for modeling the dynamic response of visco-elastic, open-cell solid foams. The effective material response is obtained by enforcing on a representative 3D unit cell the principle of minimum action for dissipative systems. The resulting model accounts explicitly for the foam topology, the elastic and viscous properties of the cell wall, and the inertial effects arising from non-affine motion within the cells. The microinertial effects become significant in retarding the foam collapse during exceedingly high strain-rate loading. As an application example, a heterogenous case of compressive deformation at high strain rate is simulated utilizing the present model as a constitutive update in a non-linear finite element analysis code. This FEM simulation shows the ability of the model to capture the progressive foam collapse during the dynamic compression as observed in experimental studies. Using the microscopic model, the inertial and viscous strain-rate effects are investigated through the foam density, viscosity, and relative density. Based on the physics incorporated into the local cell model, we provide insight into the physical mechanisms responsible for the experimentally observed strain-rate effects on the behavior of dynamically loaded foam materials.     

Deformation rate effects on failure modes of open-cell Al foams and textile cellular materials

The compressive behavior of open-cell aluminum alloy foam and stainless steel woven textile core materials have been investigated at three different deformation rate regimes. Quasi-static compressive tests were performed using a miniature loading frame, intermediate rates were achieved using a stored energy Kolsky bar, and high strain rate tests were performed using a light gas gun.In agreement with previous studies on foam materials, the strain rate was not found to have a significant effect on the plateau stress of metallic foams. For all the tests, real time imaging of the specimen combined with digital image correlation analysis allowed the determination of local deformation fields and failure modes. For the Kolsky bar tests, the deformations in the foam specimen were found to be more distributed than for the quasi-static test, which is attributed to moderate inertia effects. The differences in failure mode are more dramatic for the gas gun experiments, where a full compaction shock wave is generated at the impact surface. The stresses in front and behind the shock wave front were determined by means of direct and reverse gas gun impact tests, i.e., stationary and launched specimen, respectively. A one-dimensional shock wave model based on an idealized foam behavior is employed to gain insight into the stress history measurements. We show that the predictions of the model are consistent with the experimental observations. Woven textile materials exhibited moderate dependence of strength on the deformation rate in comparison with open-cell foam materials.     

Geometrical modelling of foam structures using implicit functions

Mechanical, thermo-mechanical, and fluid dynamic simulations of open-cell foams require an accurate geometry model. Usually, models are derived from computer- tomography (CT) data which do not allow analysing systematically variation and optimisation of the geometry. On the other hand, entirely computer generated models are mostly assembled of primitive objects like cylinders. This disregards strut thickness variations and node rounding which are observed in real open-cell foams. This paper presents an approach to generate models of ceramic open-cell foams using simple objects with variable thickness generated by implicit functions. This approach can also account for cavities within struts and nodes, which are observed in many real foam structures. The specific rounding at the foam nodes can be modelled by applying the transformation of Blinn. The quality of the generated foam models is verified using CT data of real foams.     

A robust hyper-elastic beam model under bi-axial normal-shear loadings

In this paper, a simple and robust constitutive model is proposed to simulate mechanical behaviors of hyper-elastic materials under bi-axial normal-shear loadings in the finite strain regime. The Mooney–Rivlin strain energy function is adopted to develop a two-dimensional (2D) normal-shear constitutive model within the framework of continuum mechanics. A motion field is first proposed for combined normal and shear deformations. The deformation gradient of the proposed field is calculated and then substituted into right Cauchy–Green deformation tensor. Constitutive equations are then derived for normal and shear deformations. They are two explicit coupled equations with high-level polynomial non-linearity. In order to examine capabilities of the developed hyper-elastic model, uniaxial tensile responses and non-linear stability behaviors of moderately thick straight and curved beams undergoing normal axial and transverse shear deformations are simulated and compared with experiments. Fused deposition modeling technique as a 3D printing technology is implemented to fabricate hyper-elastic beam structures from soft poly-lactic acid filaments. The printed specimens are tested under tensile/compressive in-plane and compressive out-of-plane forces. A finite element formulation along with the Newton–Raphson and Riks techniques is also developed to trace non-linear equilibrium path of beam structures in large defamation regimes. It is shown that the model is capable of predicting non-linear equilibrium characteristics of hyper-elastic straight and curved beams. It is found that the modeling of shear deformation and finite strain is essential toward an accurate prediction of the non-linear equilibrium responses of moderately thick hyper-elastic beams. Due to simplicity and accuracy, the model can serve in the future studies dealing with the analysis of hyper-elastic structures in which two normal and shear stress components are dominant.     

Effects of microstructure on uniaxial strength asymmetry of open-cell foams

Based on the elongated Kelvin model, the effect of microstructure on the uniaxial strength asymmetry of open-cell foams is investigated. The results indicate that this asymmetry depends on the relative density, the solid material, the cell morphology, and the strut geometry of open-cell foams. Even though the solid material has the same tensile and compressive strength, the tensile and compressive strength of open-cell foams with asymmetrical sectional struts are still different. In addition, with the increasing degree of anisotropy, the uniaxial strength as well as the strength asymmetry increases in the rise direction but reduces in the transverse direction. Moreover, the plastic collapse ratio between two directions is verified to depend mainly on the cell morphology. The predicted results are compared with Gibson and Ashby''s theoretical results as well as the experimental data reported in the literature, which validates that the elongated Kelvin model is accurate in explaining the strength asymmetry presented in realistic open-cell foams.     

开孔泡沫铝填充圆管的准静态压缩行为

采用开孔结构泡沫铝填充到薄壁圆形铝管中,制备出开孔泡沫铝夹芯铝管,并进行压缩实验,研究了这种结构材料的压缩力学行为和变形特征以及材料的结构特征参数对压缩力学性能和能量吸收特性的影响。在压缩过程中,泡沫铝夹芯铝管的载荷-位移曲线呈现出弹性段、波动的屈服平台段和压实段3个阶段特征;铝管的径厚比及泡沫铝本身的参数和强度对填充管的屈服强度、平均压溃力和吸能特性均有着非常显著的影响。填充泡沫铝后铝管的压缩变形方式发生改变,管壁只发生向外翻折变形,产生的环状褶皱减少。     

Fracture toughness of open-cell Kelvin foam

Brittle fracture behavior of a perfect open-cell Kelvin foam is considered. The foam is modeled as a spatial lattice consisting of brittle elastic struts rigidly connected to each other at the nodal points. The fracture toughness is determined from the analysis of a quasi-plane problem for a slice of the foam with an embedded finite length crack generated by broken struts. The crack plane is chosen on the basis of a previous study of crack nucleation phenomenon, and the crack length, which assures the self-similar K-field in the tip vicinity, is established by numerical experiments. For the considered densities range the crack includes several hundreds of broken struts and, consequently, the portion of the foam to be considered in the analysis has a very large number of nodal degrees of freedom. The computational cost is reduced significantly by using for the analysis the representative cell method based on the discrete Fourier transform. As a result, the initial problem for the foam slice is reduced to the problem for the repetitive cell which includes 12 struts.     

Computational aspects of a pseudo-elastic constitutive model for muscle properties in a soft-bodied arthropod

In this paper we discuss the computational implementation of a new constitutive model that describes the muscle properties in a soft-bodied arthropod. Qualitatively, the muscle tissues behave similar to particle-reinforced rubber and are capable of large non-linear elastic deformations, show a hysteretic behavior, and display stress softening during the first few cycles of repeated loading. Such behavior can be described by the framework of pseudo-elastic transversely isotropic hyperelasticity. The computational model assumes compressible overall response, and is based upon a multiplicative split of the deformation gradient tensor into volumetric and isochoric parts. Details regarding the implementation of the computational model in the context of an implicit finite element solution procedure are presented. In particular, an explicit expression is provided for the material tangent stiffness tensor. Results obtained utilizing the new implementation are also presented.     

Homogenization estimates for fiber-reinforced elastomers with periodic microstructures

This work presents a homogenization-based constitutive model for the mechanical behavior of elastomers reinforced with aligned cylindrical fibers subjected to finite deformations. The proposed model is derived by making use of the second-order homogenization method [Lopez-Pamies, O., Ponte Castañeda, P., 2006a. On the overall behavior, microstructure evolution, and macroscopic stability in reinforced rubbers at large deformations: I—theory. J. Mech. Phys. Solids 54, 807–830], which is based on suitably designed variational principles utilizing the idea of a “linear comparison composite.” Specific results are generated for the case when the matrix and fiber materials are characterized by generalized Neo-Hookean solids, and the distribution of fibers is periodic. In particular, model predictions are provided and analyzed for fiber-reinforced elastomers with Gent phases and square and hexagonal fiber distributions, subjected to a wide variety of three-dimensional loading conditions. It is found that for compressive loadings in the fiber direction, the derived constitutive model may lose strong ellipticity, indicating the possible development of macroscopic instabilities that may lead to kink band formation. The onset of shear band-type instabilities is also detected for certain in-plane modes of deformation. Furthermore, the subtle influence of the distribution, volume fraction, and stiffness of the fibers on the effective behavior and onset of macroscopic instabilities in these materials is investigated thoroughly.     

Multi-scale modeling of the mechanical response of plain weave composites and cellular solids

Multi-component materials with customized mechanical properties, such as textile composites and sandwich materials (cellular core with metallic or composite skin), show a great prospective for use in aerostructures. Understanding of the mechanical response of these materials is still in progress. In the present paper, the tensile response of plain weave composites as well as the compressive response of cellular solids are investigated using a multi-scale damage model. The model, implemented by means of the FE method, is based on homogenized progressive damage modeling of a representative unit-cell. Four failure modes have been considered in the failure analysis of the tows, while material property degradation was performed using a damage mechanics approach which takes into account strain softening. For the cellular solids, two different types of FE models were considered namely, a beam model and a shell model. Failure analysis and material property degradation of the struts were integrated into a bilinear material model. Simulations show a non-linear tensile response of the plain weave mainly attributed to matrix cracking and shear failures occurring at warp tows and resin-rich areas. For the cellular solid, preliminary elastic analyses show a customizability of the normal stiffness with regard to strut’s dimensions.     

通孔泡沫铝的动态压缩行为

在SHPB装置上对渗流法制备的通孔泡沫铝进行了动态压缩实验,研究了相对密度为0.341~0.419的通孔泡沫铝在10-3~2000 s-1应变率范围内的压缩响应特征和应变率相关性,并用扫描电镜（scanning electron microscope,SEM)分析了泡沫铝的压缩变形特征。实验结果表明,通孔泡沫铝有明显应变率效应,随应变率上升,泡沫铝流动应力提高。SEM观察结果揭示,在动态压缩下,通孔泡沫铝宏观上均匀变形,微观变形机制以泡孔横向伸展坍塌为主。     

A large deformation framework for compressible viscoelastic materials: Constitutive equations and finite element implementation

For modeling the constitutive properties of viscoelastic solids in the context of small deformations, the so-called three-parameter solid is often used. The differential equation governing the model response may be derived in a thermodynamically consistent way considering linear spring-dashpot elements. The main problem in generalizing constitutive models from small to finite deformations is to extend the theory in a thermodynamically consistent way, so that the second law of thermodynamics remains satisfied in every admissible process. This paper concerns with the formulation and constitutive equations of finite strain viscoelastic material using multiplicative decomposition in a thermodynamically consistent manner. Based on the proposed constitutive equations, a finite element (FE) procedure is developed and implemented in an FE code. Subsequently, the code is used to predict the response of elastomer bushings. The finite element analysis predicts displacements and rotations at the relaxed state reasonably well. The response to coupled radial and torsional deformations is also simulated.     

A theory of amorphous solids undergoing large deformations,with application to polymeric glasses

This paper develops a continuum theory for the elastic–viscoplastic deformation of amorphous solids such as polymeric and metallic glasses. Introducing an internal-state variable that represents the local free-volume associated with certain metastable states, we are able to capture the highly non-linear stress–strain behavior that precedes the yield-peak and gives rise to post-yield strain softening. Our theory explicitly accounts for the dependence of the Helmholtz free energy on the plastic deformation in a thermodynamically consistent manner. This dependence leads directly to a backstress in the underlying flow rule, and allows us to model the rapid strain-hardening response after the initial yield-drop in monotonic deformations, as well as the Bauschinger-type reverse-yielding phenomena typically observed in amorphous polymeric solids upon unloading after large plastic deformations. We have implemented a special set of constitutive equations resulting from the general theory in a finite-element computer program. Using this finite-element program, we apply the specialized equations to model the large-deformation response of the amorphous polymeric solid polycarbonate, at ambient temperature and pressure. We show numerical results to some representative problems, and compare them against corresponding results from physical experiments.     

On the compressive strength of open-cell metal foams with Kelvin and random cell structures

Two families of finite element models of anisotropic, aluminum alloy, open-cell foams are developed and their predictions of elastic properties and compressive strength are evaluated by direct comparison to experimental results. In the first family of models, the foams are idealized as anisotropic Kelvin cells loaded in the <100> direction and in the second family more realistic models, based on Surface Evolver simulations of random soap froth with N³ cells are constructed. In both cases the ligaments are straight but have nonuniform cross sectional area distributions that resemble those of the foams tested. The ligaments are modeled as shear deformable beams with elasto-plastic material behavior. The calculated compressive response starts with a linearly elastic regime. At higher stress levels, inelastic action causes a gradual reduction of the stiffness that eventually leads to a stress maximum, which represents the strength of the material. The periodicity of the Kelvin cell enables calculation of the compressive response up to the limit stress with just a single fully periodic characteristic cell. Beyond the limit stress, deformation localizes along the principal diagonals of the microstructure. Consequently beyond the limit stress the response is evaluated using finite size 3-D domains that allow the localization to develop. The random models consist of 3-D domains of 216, 512 or 1000 cells with periodicity conditions on the compressed ends but free on the sides. The compressive response is also characterized by a limit load instability but now the localization is disorganized resembling that observed in experiments. The foam elastic moduli and strengths obtained from both families of models are generally in very good agreement with the corresponding measurements. The random foam models yield 5–10% stiffer elastic moduli and slightly higher strengths than the Kelvin cell models. Necessary requirements for this high performance of the models are accurate representation of the material distribution in the ligaments and correct modeling of the nonlinear stress–strain response of the aluminum base material.     

The elastic–plastic behaviour of foam under shock loading

An experimental investigation of the elastic–plastic nature of shock wave propagation in foams was undertaken. The study involved experimental blast wave and shock tube loading of three foams, two polyurethane open-cell foams and a low-density polyethylene closed-cell foam. Evidence of precursor waves was observed in all three foam samples under various compressive wave loadings. Experiments with an impermeable membrane are used to determine if the precursor wave in an open-cell foam is a result of gas filtration or an elastic response of the foam. The differences between quasi-static and shock compression of foams is discussed in terms of their compressive strain histories and the implications for the energy absorption capacity of foam in both loading scenarios. Through a comparison of shock tube and blast wave loading techniques, suggestions are made concerning the accurate measurements of the principal shock Hugoniot in foams.