A spectral finite element for axial-flexural-shear coupled wave propagation analysis in lengthwise graded beam

A new spectral element (SE) is formulated to analyse wave propagation in anisotropic inhomogeneous beam. The inhomogeneity is considered in the longitudinal direction. Due to this particular pattern of inhomogeneity, the governing partial differential equations (PDEs) have variable coefficients and an exact solution for arbitrary variation of material properties, even in frequency domain, is not possible to obtain. However, it is shown in this work that for exponential variation of material properties, the equations can be solved exactly in frequency domain, when the same parameter governs the variation of elastic moduli and density. The SE is formed using this exact solution as interpolating polynomial. As a result a single element can replace hundreds of finite elements (FEs), which are essential for all wave propagation analysis and also for accurate representation of the inhomogeneity. The developed element is used for eliciting several advantages of the gradation, including mode selection, mode blockage and smoothening of stress waves.     

Modeling microarchitecture and mechanical behavior of nacre using 3D finite element techniques Part I Elastic properties

Three dimensional finite element models of nacre were constructed based on reported microstructural studies on the 'brick and mortar' micro-architecture of nacre. 3D eight noded isoparametric brick elements were used to design the microarchitecture of nacre. Tensile tests were simulated using this model. The tests were conducted at low stresses of 2 MPa which occur well within the elastic regime of nacre and thus effects related to locus and extent of damage were ignored. Our simulations show that using the reported values of elastic moduli of organic (0.005 GPa) and aragonitic platelets (205 GPa), the displacements observed in nacre are extremely large and correspond to a very low modulus of 0.011 GPa. The reported elastic modulus of nacre is of the order of 50 GPa. The reason for this inconsistency may arise from two possibilities. Firstly, the organic layer due to its multilayered structure is possibly composed of distinct layers of different elastic moduli. The continuously changing elastic modulus within the organic layer may approach modulus of aragonite near the organic-inorganic interface. Simulations using variable elastic moduli for the organic phase suggest that an elastic modulus of 15 GPa is consistent with the observed elastic behavior of nacre. Another explanation for the observed higher elastic modulus may arise from localized platelet-platelet contact. Since the observed modulus of nacre lies within the above two extremes (i.e. 15 GPa and 205 GPa) it is suggested that a combination of the two possibilities, i.e. a higher modulus of the organic phase near the organic-inorganic interface and localized platelet-platelet contact can result in the observed elastic properties of nacre.     

Elastic and viscoelastic mechanical properties of brain tissues on the implanting trajectory of sub-thalamic nucleus stimulation

Corresponding to pre-puncture and post-puncture insertion, elastic and viscoelastic mechanical properties of brain tissues on the implanting trajectory of sub-thalamic nucleus stimulation are investigated, respectively. Elastic mechanical properties in pre-puncture are investigated through pre-puncture needle insertion experiments using whole porcine brains. A linear polynomial and a second order polynomial are fitted to the average insertion force in pre-puncture. The Young’s modulus in pre-puncture is calculated from the slope of the two fittings. Viscoelastic mechanical properties of brain tissues in post-puncture insertion are investigated through indentation stress relaxation tests for six interested regions along a planned trajectory. A linear viscoelastic model with a Prony series approximation is fitted to the average load trace of each region using Boltzmann hereditary integral. Shear relaxation moduli of each region are calculated using the parameters of the Prony series approximation. The results show that, in pre-puncture insertion, needle force almost increases linearly with needle displacement. Both fitting lines can perfectly fit the average insertion force. The Young’s moduli calculated from the slope of the two fittings are worthy of trust to model linearly or nonlinearly instantaneous elastic responses of brain tissues, respectively. In post-puncture insertion, both region and time significantly affect the viscoelastic behaviors. Six tested regions can be classified into three categories in stiffness. Shear relaxation moduli decay dramatically in short time scales but equilibrium is never truly achieved. The regional and temporal viscoelastic mechanical properties in post-puncture insertion are valuable for guiding probe insertion into each region on the implanting trajectory.     

On completeness of shape functions for finite element analysis

Some elements commonly used for analysis are examined for examined for completeness of polynomial interpolation and computational efficiency. Extensions to n-dimensional space are shown to be natural consequences of the interpolation, thus all elements considered here allow for finite element approximation in higher than three-dimensional spaces (e.g. space–time interpolations). From the study it is concluded that ‘serendipity’ class elements from the most efficient elements up to third-degree polynomial approximations. The method used here to develop the serendipity shape functions allows for different orders of interpolation along each edge. Thus, in zones where high accuracy is required meshes can now be easily changed from linear to quadratic or higher-order elements. Computations on some simple problems have demonstrated this to be a superior method than using large numbers of low ordered elements.     

Effective elastic properties of auxetic microstructures: anisotropy and structural applications

Materials presenting a negative Poisson’s ratio (auxetics) have drawn attention for the past two decades, especially in the field of lightweight composite structures and cellular media. Studies have shown that auxeticity may result in higher shear modulus, indentation toughness and acoustic damping. In this work, three auxetic periodic microstructures based on 2D geometries are considered for being used as sandwich-core materials. Elastic moduli are computed for each microstructure by using finite elements combined with periodic homogenization technique. Anisotropy of elastic properties is investigated in and out-of-plane. Comparison is made between auxetics and the classical honeycomb cell. A new 3D auxetic lattice is proposed for volumic applications. Cylindrical and spherical elastic indentation tests are simulated in order to conclude on the applicability of such materials to structures. Proof is made that under certain conditions, auxetics can be competitive with honeycomb cells in terms of indentation strength. Their relatively soft response in tension can be compensated, in some situations, by high shear moduli.     

Implementation of a lamina-based constitutive relationship for analysing thick composites

The finite element implementation of a lamina-based elastic constitutive relation appropriate for describing the three-dimensional (3-D) behaviour of fibre composite laminated media is presented. Unconstrained by the two-dimensional restrictions accompanying plate and shell theories, this approach resolves the macro 3-D deformation fields found in 'thick' laminated or filament wound composites without requiring each lamina to be individually discretized to assign material properties. In rectangular isoparametric solid elements where laminae locally parallel an element 'face', the implementation represents 'exactly' the elastic laminate behaviour by replacing the discrete lamina stiffnesses with continuous polynomial moduli functions. Attributable directly to the lamina constitutive relationship used, appropriate for fibre-dominated material systems, the implementation automatically preserves interlaminar continuity of tractions and displacements within each element for laminates assembled from a single lamina type. The order of the effective moduli functions is determined by the element kinematics and by requiring identical net mid-surface elemental forces and moments. In general, this representation substantially reduces the number of stored variables and through-thickness integrations required, and allows 'exact' integration by higher-order Gaussian quadrature. Additionally a single element may span any portion or number of laminae, thus allowing nearly arbitrary meshing and solution resolution. Several numerical examples using 3-D 8-node isoparametric solid elements demonstrate the approach's convergence and overall behaviour.     

计算复合材料有效弹性模量的重心有限元方法

采用几何法构造出任意边数多边形单元的重心插值形函数, 应用Galerkin法提出了求解弹性力学问题的重心有限元方法。用重心有限元方法对SiC/Ti和B/Al 2种纤维复合材料横向截面的有效弹性模量进行了预报。计算模型取纤维呈六边形排列且为各向同性的代表性单胞, 对其杨氏模量、 剪切模量和体积模量在较大的体积分数范围内进行了数值模拟。通过与解析公式和传统有限元的计算结果对比, 重心有限元方法的计算结果符合解析公式解的趋势, 与传统有限元的计算结果吻合较好。与传统有限元方法相比, 重心有限元方法的单元划分不受三角形或四边形的形状限制, 能够再现材料的真实结构。由于单元较大且数目较少, 本文方法具有很高的计算效率。      

Characterization of the anisotropic materials capable of exhibiting an isotropic Young or shear or area modulus

By means of the decomposition of an anisotropic elastic tensor into symmetric traceless tensors, the general intrinsic expressions involving no redundant elastic coefficients are obtained for the orientation distribution functions of the Young, shear and area moduli of an anisotropic material. Necessary and sufficient conditions are established for each of these moduli to be isotropic. It is found that an anisotropic material exhibiting an isotropic Young or shear or area modulus can be only either transversely isotropic or orthotropic.     

Evaluation of thermo-viscoelastic property of CFRP laminate based on a homogenization theory

The thermo-viscoelastic constitutive equation of unidirectional carbon fiber reinforced plastic (CFRP) is evaluated using a numerical approach based on the finite element method (FEM) and homogenization theory. The constitutive equation of the CFRP is considered in the Laplace-transformed domain, and it is discussed on the basis of the correspondence principle, which is satisfied by each of the Laplace-transformed elastic moduli. Homogenization theory is employed to estimate the ‘homogenized elastic moduli’ of the composite composed of matrix resin and carbon fibers. Using the approximation of a generalized Maxwell model, the relaxation moduli of CFRP are obtained by numerical computation using the FEM. From the relaxation modulus of epoxy resin and elastic moduli of carbon fiber, thermo-viscoelastic properties of CFRP laminates at several temperatures can be estimated using the FEM with homogenization theory. The effectiveness of the present study is verified by comparing the experimental results and numerical calculations for the relaxation moduli of the CFRP laminates.     

复合材料圆管构件等效模量的计算方法

针对任意壁厚的复合材料圆管构件的等效弹性模量和剪切模量提出了高阶理论计算方法, 它考虑了构件的横向剪切效应以及层合材料的三维本构关系, 并在相同壁厚条件下对三种缠绕方式( [0°/(±θ)_n ]S, [ (±θ)_n ]S 和[ 90°/(±θ)_n ]S) 的等效模量进行了预测, 并且与经典层合板理论的预测结果进行了比较, 该方法可用于复合材料杆件结构的设计中。      

Ancestrally high elastic modulus of gecko setal beta-keratin.

Typical bulk adhesives are characterized by soft, tacky materials with elastic moduli well below 1MPa. Geckos possess subdigital adhesives composed mostly of beta-keratin, a relatively stiff material. Biological adhesives like those of geckos have inspired empirical and modelling research which predicts that even stiff materials can be effective adhesives if they take on a fibrillar form. The molecular structure of beta-keratin is highly conserved across birds and reptiles, suggesting that material properties of gecko setae should be similar to that of beta-keratin previously measured in birds, but this has yet to be established. We used a resonance technique to measure elastic bending modulus in two species of gecko from disparate habitats. We found no significant difference in elastic modulus between Gekko gecko (1.6 GPa +/- 0.15s.e.; n=24 setae) and Ptyodactylus hasselquistii (1.4 GPa +/- 0.15s.e.; n=24 setae). If the elastic modulus of setal keratin is conserved across species, it would suggest a design constraint that must be compensated for structurally, and possibly explain the remarkable variation in gecko adhesive morphology.     

2.5维C/SiC复合材料弹性参数不确定性识别方法研究

提出了一种2.5维C/SiC编织复合材料弹性参数不确定性识别方法。采用刚度平均法获得复合材料等效弹性参数理论预测值。选取对结构动态特性影响较大的3个弹性参数E11,E22和G12作为待识别参数;在确定性识别结果基础上,采用拉丁超立方体采样构造随机试验样本,开展不确定性参数识别方法仿真研究。仿真结果表明,针对考虑弹性参数不确定性的2.5维C/SiC复合材料,采用所提出的方法能够准确识别材料弹性参数的均值与标准差,建立反映实际结构动态特性统计意义的精确动力学模型。     

Topology optimization of continuum structures with bi-modulus materials

Since the elasticity of bi-modulus materials is stress dependent, it is difficult to apply most conventional topology optimization methods to such bi-modulus structures owing to great computational expense. Therefore, this study employs the material-replacement method to improve the computational efficiency for topology optimization of bi-modulus structures. In this method, first, the bi-modulus material is replaced by two isotropic materials which have the same tensile or compressive modulus. Secondly, the isotropic materials for finite elements are determined by the local stress/strain states. The local elemental stiffness can be modified according to the current modulus and stress state of the element. Thirdly, the relative densities of elements, acting as the design variables, are updated using the optimality criterion method. Finally, the distributions of elemental densities and moduli are obtained for further applications. Several typical numerical examples are used to demonstrate the effectiveness of the proposed method.     

A comparative study of hyperelastic and hypoelastic material models with constant elastic moduli for large deformation problems

Many finite element programs including commercial codes for large deformation analysis employ incremental formulations of rate-type constitutive equations which are based on hyperelastic or hypoelastic material models with constant elastic moduli. In this paper, a comparative study is carried out for hyperelastic and hypoelastic material models with constant elastic moduli of a face-centered cubic single crystal of copper. A strain energy function from the inter-atomic potential for single-crystal copper is also considered for the hyperelastic material model to obtain physically based elastic deformations. Numerical results show that constant elastic moduli of hypoelastic material models can cause considerable errors in stress and strain increments when the changes in volume and cross-sectional area of a material are not negligible.     

Closed-form solutions for axially functionally graded Timoshenko beams having uniform cross-section and fixed–fixed boundary condition

In this paper, we study the free vibration of axially functionally graded (AFG) Timoshenko beams, with uniform cross-section and having fixed–fixed boundary condition. For certain polynomial variations of the material mass density, elastic modulus and shear modulus, along the length of the beam, there exists a fundamental closed form solution to the coupled second order governing differential equations with variable coefficients. It is found that there are an infinite number of non-homogeneous Timoshenko beams, with various material mass density, elastic modulus and shear modulus distributions having simple polynomial variations, which share the same fundamental frequency. The derived results can be used as benchmark solutions for testing approximate or numerical methods used for the vibration analysis of non-homogeneous Timoshenko beams. They can also be useful for designing fixed–fixed non-homogeneous Timoshenko beams which may be required to vibrate with a particular frequency.     

Prediction of Young’s modulus of graphene sheets and carbon nanotubes using nanoscale continuum mechanics approach

Analytical formulations are presented to predict the elastic moduli of graphene sheets and carbon nanotubes using a linkage between lattice molecular structure and equivalent discrete frame structure. The obtained results for a graphene sheet show an isotropic behavior, in contrast to limited molecular dynamic simulations. Young’s modulus of CNT represents a high dependency of stiffness on tube thickness, while dependency on tube diameter is more tangible for smaller tube diameters. The presented closed-form solution provides an insight to evaluate finite element models constructed by beam elements. The results are in a good agreement with published data and experimental results.     

Analysis on the variances of material and structural properties based on random field theory

Based on the random field theory (RFT) and the stochastic finite element method (SFEM), the variances of the mechanical properties of materials and structures are studied. Manufacturing processes can easily lead to the spatial variations of the load and the material properties such as moduli and density. Characterizing the elastic moduli, load and density with one-dimensional random fields, the analytical solutions for the coefficient of variations (COVs) of effective material moduli, displacement and natural frequencies of beams are obtained. Then, with the fiber and matrix properties, volume fraction modeled by two-dimensional random fields and the fiber angle as a single random variable, a Monte Carlo simulation (MCS) is performed to generate the variances of effective modulus of fiber-reinforced composite laminar plate. Compared with the previous numerical conclusions, the present results reveal that the variances of effective material properties and structural displacement are greatly dependent on both the random fields and the sizes of structures in theory.     

Functional grading in hierarchical honeycombs: Density specific elastic performance

The introduction of hierarchy into structures has been credited with improving their elastic and other properties. Similarly, functional grading has been demonstrated to increase the damage tolerance of honeycomb structures, although with the penalty of reduced Young’s modulus or increased density. The combination of both hierarchy and functional grading has not been reported for honeycomb structures, although it is known in natural materials. A parametric numerical modelling study has been made of the in-plane elastic properties of honeycombs and how they are affected by functional grading and hierarchy, and importantly to establish whether it is possible to avoid reductions in Young’s modulus. A set of analytical models has been developed to describe functional grading and hierarchy in honeycombs, based upon beam mechanics and the transform section method. The conditions for transition of a hierarchical honeycomb in behaviour from that of a discrete structure to that of a continuum are established. Furthermore, conditions are established for which hierarchical honeycombs, uniform or functionally graded, can surpass in-plane Young’s moduli of conventional honeycombs a by factor of up to 2, on an equal density basis.     

An n-layered spherical inclusion model for predicting the elastic moduli of concrete with inhomogeneous ITZ

An n-layered spherical inclusion model is presented in this paper for predicting the elastic moduli of concrete with inhomogeneous interfacial transition zone (ITZ). In this model, concrete is represented as a three-phase composite material, composed of the aggregate, bulk paste, and an inhomogeneous ITZ. An analytical solution for the ITZ volume fraction is derived for the general aggregate gradation. By constituting a semi-empirical initial cement gradient model, the local water/cement ratio, degree of hydration, and porosity at the ITZ are estimated. The inhomogeneous ITZ is then divided into a series of homogenous concentric shell elements of equal thickness. The elastic moduli of concrete are determined by solving the n-layered spherical inclusion problem. Finally, the validity of the model is verified with three independent sets of experimental data and the effects of the maximum aggregate diameter, aggregate gradation, and ITZ thickness on the Young’s modulus of concrete are evaluated in a quantitative manner. The paper concludes that the proposed n-layered spherical inclusion model can be used to predict the elastic moduli of concrete.