Micromechanical consideration of tensile crack behavior based on virtual internal bond in contrast to cohesive stress

A modified version of the virtual internal bond model (VIB) is presented. This involves the introduction of a R-bond restricting the relative rotation freedom of pairwise mass particle. Such a modification allows the VIB model to consider arbitrary values of the Poisson ratio. A linear elastic cohesive law considering both the R-bond and L-bond are assumed. The constitutive relationship is derived using the Cauchy–Born rules. The derived constitutive associates the bond stiffness with the Young’s modulus and Poisson ratio of materials. This gives the bond stiffness in terms of the Young’s modulus and Poisson ratio of materials.The modified VIB model is then used to analyze the tensile crack behavior. In contrast to the cohesive stress method, the deformation-governed concept will be used. The local materials failure is assumed to coincide with the reduction of the bond density due to the local deformation rather than by the local cohesive stress. A phenomenological relationship between the bond density and the deformation is established. The criterion which is applied to determined crack initiation and propagation is built into the constitutive model. As an example, the method is used to study the crack initiation and propagation behavior under tensile loading.     

Effects of cell shape and strut cross-sectional area variations on the elastic properties of three-dimensional open-cell foams

The Voronoi tessellation technique and the finite element (FE) method are utilized to investigate the microstructure-property relations of three-dimensional (3-D) cellular solids (foams) that have irregular cell shapes and non-uniform strut cross-sectional areas (SCSAs). Perturbations are introduced to a regular packing of seeds to generate a spatially periodic Voronoi diagram with different degrees of cell shape irregularity (amplitude a), and to the constant SCSA to generate a uniform distribution of SCSAs with different degrees of SCSA non-uniformity (amplitude b). Twenty FE models are constructed, based on the Voronoi diagrams for twenty foam samples (specimens) having the same pair of a and b, to obtain the mean values and standard deviations of the elastic properties. Spatially periodic boundary conditions are applied to each specimen. The simulation results indicate that for low-density imperfect foams, the elastic moduli increase as cell shapes become more irregular, but decrease as SCSAs get less uniform. When the relative density (R) increases, the elastic moduli of imperfect foams increase substantially, while the Poisson's ratios decrease moderately. The effect of the interaction between the two types of imperfections on foam elastic properties appears to be weak. In addition, it is found that the strut cross-sectional shape has a significant effect on the foam properties. Also, the elastic response of foams with the cell shape and SCSA imperfections appears to be isotropic regardless of changes in a, b and R and the strut cross-sectional shape.     

基于三维细观力学模型的开孔泡沫弹性性能预测

泡沫材料的宏观力学性能主要取决于基体材料的力学特性及其微细观结构特征,基于细观力学模型的分析方法是泡沫材料力学性能研究的重要途径。文中基于Matlab语言和Abaqus软件构建了描述中等孔隙率开孔弹性泡沫材料微结构特征的三维随机分布球形泡孔模型,并采用有限元方法对弹性泡沫压缩变形进行了模拟,并计算给出了不同孔隙率弹性泡沫材料弹性模量、剪切模量、体积模量以及泊松比的分布,建立了相应的唯象表达式。与理论模型及测试结果的比较表明,本文基于三维随机泡孔模型模拟结果构建的唯象表达式能够对弹性泡沫材料的弹性力学性能给出很好的预测。     

On the determination of the Young’s modulus of thin films using indentation tests

The main difficulty with the characterization of thin coatings using depth-sensing indentation tests is related to the determination of the contributions of the substrate and the film to the measured properties. In this study, three-dimensional numerical simulations of the Vickers hardness test are used in order to examine the influence of the elastic and plastic properties of the substrate and the film on the composite’s Young’s modulus results. The hardness of the film is equal to or higher than the substrate hardness. A study of the stress distributions and the indentation geometry of composites, film/substrate, was performed, taking into account the relative mechanical properties of the film and substrate. In addition, stress evolution during indentation was studied, in order to quantify the critical indentation depth under which the substrate is not elastically deformed. The accurate evaluation of the Young’s modulus of the films using weight functions is also examined: some of these have previously been proposed and one was introduced for this study. Two different fitting procedures were used to compare the results obtained from eight fictive film/substrate combinations using six weight functions. The first procedure, commonly used, considers the substrate’s modulus as a known parameter in the fitting process. In the second, the film and the substrate’s modulus are considered as unknown variables that are calculated simultaneously during the fitting process. The validity of the conclusions obtained using the fictive materials was checked by applying the weight functions to four real composites.     

低密度开孔泡沫蠕变松弛特性分析

利用三维Voronoi模型和有限元方法分析了胞壁材料具有粘弹特性的低密度开孔泡沫的蠕变和应力松弛行为.采用了三参量标准线性固体模型来描述胞壁材料的粘弹特性.所得结果表明.低密度开孔泡沫具有与其胞壁材料相同的松弛时间,当相对密度较低时(低于1%)开孔泡沫的松弛模量与胞壁材料的松弛模量和泡沫相对密度平方成正比.此外,计算结果还表明,低密度开孔泡沫在较小的初始应力条件下具有与其胞壁材料相同的延迟时间.其蠕变柔度与胞壁材料的蠕变柔度和泡沫相对密度平方倒数基本成正比.但随着初始应力值的增大,泡沫的延迟时间将会显著增加.     

Very Large Poisson’s Ratio with a Bounded Transverse Strain in Anisotropic Elastic Materials

In a recent paper by Ting and Chen [18] it was shown by examples that Poisson’s ratio can have no bounds for all anisotropic elastic materials. With the exception of cubic materials, the examples presented involve a very large transverse strain. We show here that a very large Poisson’s ratio with a bounded transverse strain exists for all anisotropic elastic materials. The large Poisson’s ratio with a bounded transverse strain occurs when the axial strain is in the direction very near or at the direction along which Young’s modulus is very large. In fact the transverse strain has to be very small for the material to be stable. If the non-dimensionalized Young’s modulus is of the order δ⁻¹, where δ is very small, the axial strain, the transverse strain and Poisson’s ratio are of the order δ, δ^1/2 and δ^−1/2, respectively. Mathematics Subject Classifications (2000) 74B05, 74E10.T.C.T. Ting: Professor Emeritus of University of Illinois at Chicago and Consulting Professor of Stanford University.     

Finite deformation continuum model for single-walled carbon nanotubes

A continuum-based model for computing strain energies and estimating Young’s modulus of single-walled carbon nanotubes (SWCNTs) is developed by using an energy equivalence-based multi-scale approach. A SWCNT is viewed as a continuum hollow cylinder formed by rolling up a flat graphite sheet that is treated as an isotropic continuum plate. Kinematic analysis is performed on the continuum level, with the Hencky (true) strain and the Cauchy (true) stress being employed to account for finite deformations. Based on the equivalence of the strain energy and the molecular potential energy, a formula for calculating Young’s modulus of SWCNTs is derived. This formula, containing both the molecular and continuum scale parameters, directly links macroscopic responses of nanotubes to their molecular structures. Sample numerical results show that the predictions by the new model compare favorably with those by several existing continuum and molecular dynamics models.     

Effect of microstructure on fracture of brittle materials: Unified approach

A theoretical approach to the fracture of brittle solids based on crack opening displacement and energy rate criterion is presented. The approach allows for the prediction of elastic (Young’s modulus) and fracture (fracture strength and thermal shock) response of a brittle material containing spherical pores and polycrystalline solids containing anisotropic residual stresses.     

Modelling open cell-foams based on the Weaire–Phelan unit cell with a minimal surface energy approach

The complex architecture of open cell foams has most often been described by Kelvin cell models. It has been shown that the accuracy to predict the elastic properties of open cell foams increases with an increasing level of detail and resemblance to real foam microstructures. However, the Kelvin cell does not possess pentagonal faces which are the most abundant within real open cell foams. Therefore this study focuses on the use of the Weaire–Phelan unit cell to model the elastic properties of an open cell polyurethane foam. Optical and scanning electron microscopy were used to characterise the architecture of the open cell foam. Surface Evolver software was used to minimize the surface energy and introduce the typical architectural characteristics of the open cell foam to the FE-model. The E-modulus and Poisson coefficient of the Kelvin and Weaire–Phelan cell show a similar behaviour as a function of density. The Weaire–Phelan cell predicts however a higher dependency of the shear modulus on the density. When the influence of the elongation of the cells in the rise direction of the foam and the uncertainty of the solid material properties of the polyurethane is taken into account, a good accuracy of the Kelvin cell and Weaire–Phelan structure based FE-models versus experimental compression tests is found.     

The stiffness and strength of the gyroid lattice

Recently, a nanoscale lattice material, based upon the gyroid topology has been self-assembled by phase separation techniques (Scherer et al., 2012) and prototyped in thin film applications. The mechanical properties of the gyroid are reported here. It is a cubic lattice, with a connectivity of three struts per joint, and is bending-dominated in its elasto-plastic response to all loading states except for hydrostatic: under a hydrostatic stress it exhibits stretching-dominated behaviour. The three independent elastic constants of the lattice are determined through a unit cell analysis using the finite element method; it is found that the elastic and shear modulus scale quadratically with the relative density of the lattice, whereas the bulk modulus scales linearly. The plastic collapse response of a rigid, ideally plastic gyroid lattice is explored using the upper bound method, and is validated by finite element calculations for an elastic-ideally plastic lattice. The effect of geometrical imperfections, in the form of random perturbations to the joint positions, is investigated for both stiffness and strength. It is demonstrated that the hydrostatic modulus and strength are imperfection sensitive, in contrast to the deviatoric response. The macroscopic yield surface of the imperfect lattice is adequately described by a modified version of Hill’s anisotropic yield criterion. The article ends with a case study on the stress induced within a gyroid thin film, when the film and its substrate are subjected to a thermal expansion mismatch.     

Numerical simulation of debonding of adhesively bonded joint

This paper describes a numerical method to simulate the debonding of adhesively bonded joints. Assuming that the adhesive thickness and the adhesive Young’s modulus are small with respect to the characteristic length of the joint and to the Young’s modulus of the adherents, a simplified model is derived in the case of large displacements using the asymptotic expansion technique. Then, the problem of the crack growth is stated, in the case of a stable growth, as the search of the local minima of the total energy of the joint, sum of the mechanical energy and the Griffith’s fracture energy. This is made using the Newton’s method. To this end, the expressions of the first and second derivatives of the mechanical energy with respect to a crack front displacement are derived analytically. Finally, numerical examples are presented, highlighting the unstable character of the crack growth at initiation.     

Stress analysis of a functional graded material plate with a circular hole

This paper is to study the two-dimensional stress distribution of a functional graded material plate (FGMP) with a circular hole under arbitrary constant loads. With using the method of piece-wise homogeneous layers, the stress distribution of the functional graded material plate having radial arbitrary elastic properties is derived based on the theory of the complex variable functions. As examples, numerical results are presented for the FGMPs having given radial Young’s modulus or Poisson’s ratio. It is shown that the stress is greatly reduced as the radial Young’s modulus increased, but it is less influenced by the variation of the Poisson’s ratio. Moreover, it is also found that the stress level varies when the radial Young’s modulus increased in different ways. Thus, it can be concluded that the stress around the circular hole in the FGMP can be effectively reduced by choosing the proper change ways of the radial elastic properties.     

Effect of CNT length and CNT-matrix interphase in carbon nanotube (CNT) reinforced composites

The load transfer mechanisms and effective moduli of single-walled nanotube (SWNT) reinforced composites are studied using a continuum model. A “critical” fiber length is defined for full load transfer by numerically evaluating the strain-energy-changes for different fiber lengths. The effective longitudinal Young’s modulus and bulk modulus of the composite are derived. The effect of the interphase is also discussed. The results indicate the fiber length is critical both to the load transfer efficiency and effective moduli of the composite. The SWNT-matrix interphase plays an important role in load transfer efficiency but affects the effective moduli only slightly.     

泡沫铝合金动态力学性能实验研究

利用分离式霍布金森压杆(SHPB)实验技术和MTS材料实验机对两组不同孔径、不同密度的开孔泡沫铝合金进行了准静态和动态压缩实验研究。实验结果表明:泡沫铝合金的静态和动态变形过程均具有泡沫材料变形的三个阶段特征。开孔泡沫铝合金的变形是均匀变化过程，并不出现局部的变形带。与相对密度对力学性能的影响相比，孔径大小的影响可以忽略不计。在考察的应变率范围内，屈服应力对应变率并不很敏感。     

Non-linear finite element analysis of cone penetration in layered sandy loam soil – Considering precompression stress state

Axisymmetric finite element (FE) method was developed to simulate cone penetration process in layered granular soil. The FE was modeled using ABAQUS/Explicit, a commercially available package. Soil was considered as a non-linear elastic plastic material which was modeled using variable elastic parameters of Young’s Modulus and Poisson’s ratio and Drucker–Prager criterion with yield stress dependent material hardening property. The material hardening parameters of the model were estimated from the USDA-ARS National Soil Dynamics Laboratory – Auburn University (NSDL-AU) soil compaction model. The stress–strain relationship in the NSDLAU compaction model was modified to account for the different soil moisture conditions and the influence of precompression stress states of the soil layers. A surface contact pair (‘slave-master’) algorithm in ABAQUS/Explicit was used to simulate the insertion of a rigid cone (RAX2 ABAQUS element) into deformable and layered soil medium (CAX4R ABAQUS element). The FE formulation was verified using cone penetration data collected on a soil chamber of Norfolk sandy loam soil which was prepared in two compaction treatments that varied in bulk density in the hardpan layer of (1) 1.64 Mg m⁻³ and (2) 1.71 Mg m⁻³. The FE model successfully simulated the trend of cone penetration in layered soils indicating the location of the sub-soil compacted (hardpan) layer and peak cone penetration resistance. Modification of the NSDL-AU model to account for the actual soil moisture content and inclusion of the influence of precompression stress into the strain behavior of the NSDL-AU model improved the performance of FE in predicting the peak cone penetration resistance. Modification of the NSDL-AU model resulted in an improvement of about 42% in the finite element-predicted soil cone penetration forces compared with the FE results that used the NSDL-AU ‘virgin’ model.     

Delamination effects on cracked steel members reinforced by prestressed composite patch

Prestressed composite patch bonded on cracked steel section is a promising technique to reinforce cracked details or to prevent fatigue cracking on steel structural elements. It introduces compressive stresses that produce crack closure effect. Moreover, it modifies the crack geometry by bridging the crack lips and reduces the stress range at crack tip. Fatigue tests were performed on notched steel plate reinforced by CFRP strips as a step toward the validation of crack patching for fatigue life extension of riveted steel bridges. A debond crack in the adhesive–plate interface was observed by optical technique. Debond crack total strain energy release rate is computed by the modified virtual crack closure technique. A parametric analysis is performed in order to investigate the influence of some design parameters such as the composite patch Young’s modulus, the adhesive thickness and the pretension level on the adhesive–plate interface debond.     

空心颗粒填充复合材料的建模与弹塑性仿真计算

空心玻璃微珠（hollow glass microballon,简称HGM）填充复合材料称为复合泡沫材料,近来已被广泛应用在工业中。利用RSA(Random Sequential Adsorption)方法生成了含不同体积比的HGMs填充代表体元模型,然后用有限元方法计算得出了材料的应力-应变关系,将材料属性简化为双线性随动强化模型,分析了HGM填充比、壁厚对材料的有效弹性常数、屈服极限的影响,并分析了材料内部细观应力场和塑性应变分布情况。结果发现,HGM壁厚比对材料有效弹性模量和屈服极限的增减起着决定性作用,而对于任意填充模式来说,其比模量和比强度总是大于纯树脂,这一点体现了该材料轻质的优良特性。材料基体的应力集中部位分布以及塑性应变区域的分布也取决于HGM的壁厚比。     

Effective elastic behavior of some models for perfect cellular solids

The effective elastic behavior of some models for low density cellular solids, or solid foams, are calculated using analytical and numerical techniques. The models are perfect in the sense that imperfections or irregularities as often encountered in real foams have been removed. We believe that the present models can serve as references to which more advanced models which include imperfections and irregularities can be compared. The work in this paper does not address buckling or yielding in cell walls, which play an increasingly important role as foam stresses increase.     

The effect of cell irregularity on the high strain compression of 2D Voronoi honeycombs

The in-plane compression of low-density irregular Voronoi honeycombs with periodic boundary conditions has been simulated to engineering strains of 0.6 using finite element analysis. Different degrees of geometric irregularity in the honeycomb cells, as quantified using a regularity parameter, have been employed. The stress–strain predictions reveal that, for a fixed relative density, a more irregular honeycomb has a higher tangential modulus at low strain but supports a lower compressive stress at higher strain (above approximately 0.04) when compared with a more regular honeycomb. A combined ‘springs in parallel’ and ‘springs in series’ model has also been compared quantitatively with the simulation stress–strain results, the relative importance of the ‘springs in series’ mechanism having been found to increase with the irregularity of the honeycomb and, in many cases, with the applied compressive strain. In addition, the dependency of the Poisson’s ratio, the maximum bending strain in the cell walls, and the mean junction rotation upon the applied compressive strain have also been determined for a range of honeycomb irregularities.     

Compressive response of open-cell foams. Part I: Morphology and elastic properties

This study is concerned with the understanding and modeling of the compressive response of open cell foams. The response starts with a nearly linear elastic regime which terminates into a limit load followed by an extensive load plateau. The plateau, which is responsible for the excellent energy absorption capacity of foams, is followed by a second stiff branch. Results from polyester urethane open cell foams with relative densities of about 0.025 are used to illustrate this behavior using experiments coupled with several levels of modeling. The experiments include characterization of the microstructure and the properties of the base material and measurement of the compressive response of the foams of various cell sizes.A sequence of models for predicting the complete response of such foam is developed. The foam is idealized to be periodic using the space-filling Kelvin cell assigned the major geometric characteristics found in the foams tested. The cells are elongated in the rise direction, the ligaments are assumed to be straight, to have Plateau border cross-sections and nonuniform cross-sectional area distribution. The ligaments are modeled as shear-deformable extensional beams and the base material is assumed to be linearly elastic. Prediction of the initial elastic moduli are addressed in Part I. Closed form expressions for the material constants are presented as well as results using a FE model of the characteristic cell. Comparison between measurements and predictions is very favorable. The paper finishes with results from a limited parametric study of the elastic moduli. The results demonstrate that inclusion of the geometric complexities mentioned above is essential for successful prediction of the moduli of such foams. The nonlinear parts of the response including the foam crushing behavior are addressed in Part II.