Microelastic wave field signatures and their implications for microstructure identification

This work combines closed-form and computational analyses to elucidate the dynamic properties, termed signatures, of waves propagating through solids defined by the theory of elasticity with microstructure and the potential of such properties to identify microstructure evolution over a material’s lifetime. First, the study presents analytical dispersion relations and frequency-dependent velocities of waves propagating in microelastic solids. A detailed parametric analysis of the results show that elastic solids with microstructure recover traditional gradient elasticity under certain conditions but demonstrate a higher degree of flexibility in adapting to observed wave forms across a wide frequency spectrum. In addition, a set of simulations demonstrates the ability of the model to quantify the presence of damage, just another type of microstructure, through fitting of the model parameters, especially the one associated with the characteristic length scale of the underlying microstructure, to an explicit geometric representation of voids in different damage states.     

Velocity dispersion in an elastic plate with microstructure: effects of characteristic length in a couple stress model

Microstructural effects become important, when dimensions of the heterogeneous material are comparable to the length scale of microstructure and the state of stress needs to be defined in a non-local manner. Linear theory of elasticity, which is associated with the concept of homogeneity of material and local stresses, cannot describe the behavior of the materials with microstructures. In this study, Couple stress theory of elasticity has been employed to capture the size effects on the propagation of Lamb waves in an elastic plate with microstructure. Effects on the dispersion curves of Lamb waves are studied, when the characteristic length of the material is comparable to cell size. The governing equations of couple stress theory, involving stresses and couple stresses are solved to study the impact of different characteristic lengths, comparable with cell size. Since bone is a material with microstructure, so for numerical calculations and graphical representation of the results, the plate is considered to have mechanical properties typically used for bones.     

基于微形态模型的颗粒材料中波的频散现象研究

基于颗粒材料冲击与波动响应特性的调控波传播行为的超材料设计受到广泛关注,设计这类材料需要对颗粒材料的波传播机制及调控机理有深入认识. 波在颗粒材料中传播的频散现象及频率带隙等行为与材料的非均匀性密切相关,通常讨论频散现象是基于弹性理论框架建立微结构连续体或高阶梯度连续体等广义连续体模型来进行. 本研究基于细观力学给出了一个颗粒材料的微形态连续体模型. 在该模型中,考虑了颗粒的平动和转动,且颗粒间的相对运动分解为两部分:即宏观平均运动和细观真实运动. 基于此分解,提出了一个完备的变形模式,得到了对应于不同应变及颗粒间运动的宏细观本构关系. 结合宏观变形能的细观变形能求和表达式,获得了基于细观量表示的宏观本构模量. 应用所建议模型考察了波在弹性颗粒介质的传播行为,给出了不同形式的波的频散曲线,结果显示此模型具有预测频率带隙的能力.      

Analytical and experimental determination of the material intrinsic length scale of strain gradient plasticity theory from micro- and nano-indentation experiments

The enhanced gradient plasticity theories formulate a constitutive framework on the continuum level that is used to bridge the gap between the micromechanical plasticity and the classical continuum plasticity. They are successful in explaining the size effects encountered in many micro- and nano-advanced technologies due to the incorporation of an intrinsic material length parameter into the constitutive modeling. However, the full utility of the gradient-type theories hinges on one's ability to determine the intrinsic material length that scales with strain gradients, and this study aims at addressing and remedying this situation. Based on the Taylor's hardening law, a micromechanical model that assesses a nonlinear coupling between the statistically stored dislocations (SSDs) and geometrically necessary dislocations (GNDs) is used here in order to derive an analytical form for the deformation-gradient-related intrinsic length-scale parameter in terms of measurable microstructural physical parameters. This work also presents a method for identifying the length-scale parameter from micro- and nano-indentation experiments using both spherical and pyramidal indenters. The deviation of the Nix and Gao [Mech. Phys. Solids 46 (1998) 411] and Swadener et al. [J. Mech. Phys. Solids 50 (2002) 681; Scr. Mater. 47 (2002) 343] indentation size effect (ISE) models’ predictions from hardness results at small depths for the case of conical indenters and at small diameters for the case of spherical indenters, respectively, is largely corrected by incorporating an interaction coefficient that compensates for the proper coupling between the SSDs and GNDs during indentation. Experimental results are also presented which show that the ISE for pyramidal and spherical indenters can be correlated successfully by using the proposed model.     

Dispersive Rayleigh-Wave Propagation in Microstructured Solids Characterized by Dipolar Gradient Elasticity

It is well known that the classical theory of elasticity predicts Rayleigh-wave motions, which are not dispersive at any frequency. Of course, at high frequencies, this is a result that contradicts experimental data and also does not agree with results of the discrete particle theory (atomic-lattice approach). To remedy this shortcoming, the Mindlin–Green–Rivlin theory of dipolar gradient elasticity is employed here to analyze waves of the Rayleigh type propagating along the surface of a half-space. The analysis shows that these waves are indeed dispersive at high frequencies, a result that can be useful in applications of high-frequency surface waves, where the wavelength is often on the micron order. Provided that certain relations hold between the various microstructure parameters entering the theory employed here, the dispersion curves of these waves have the same form as that given by previous analyses based on the atomic-lattice theory. In this way, the present analysis gives also means to obtain estimates for microstructure parameters of the gradient theory.     

A classification of higher-order strain-gradient models in damage mechanics

Summary In a previous contribution, higher-order strain-gradient models for linear elasticity have been studied in statics and dynamics [9]. In this paper, the extension towards damage mechanics is made. A damage model is derived from a discrete microstructure. In the homogenisation process, higher-order strain gradients appear both in the linear and in the nonlinear parts of the constitutive equation. Similar to the elastic models, stabilising and destabilising gradients can be distinguished. The stabilising or destabilising effect of each gradient term is determined. Opposite (competing) effects on the stability are found for the gradients of the elastic and the gradients in the damage response. Various truncations of the two strain-gradient series are studied, with the aim to arrive at a continuum model that fulfills the following requirements (i) it is derivable from a discrete microstructure, (ii) it is able to describe wave dispersion in elastic and damaging media properly, and (iii) it can be used to model strain-softening phenomena, i.e. it is a regularised model. The response of the various models is studied analytically and numerically. For the analytical investigation, dispersive waves are studied and critical wave lengths are derived. Numerical simulations are carried out with the element-free Galerkin method. This combined analytical/numerical approach allows to establish the role of the critical wave length both for mechanically stable and mechanically unstable models. For stabilised models, the critical wave length sets the width of the damaging zone. On the other hand, for destabilised models, the critical wave length sets a periodicity in the response that leads to divergence of the numerical scheme. The influence of the individual gradient terms on the stability and the structural ductility is verified in static and dynamic analyses. We thank Akke Suiker and Andrei Metrikine of Delft University of Technology for stimulating discussions throughout this study.     

基于广义应变梯度理论的纳米梁挠曲电效应研究

挠曲电效应是应变梯度与电极化的耦合，它存在于所有的电介质材料中。在纳米电介质结构的挠曲电效应研究中，应变梯度弹性对挠曲电响应的影响一直以来被低估甚至被忽略了。根据广义应变梯度理论，应变梯度弹性中独立的尺度参数只有三个，而文献中所采用的一个或两个尺度参数的应变梯度理论只是它的简化形式。基于该理论，论文建立了考虑广义应变梯度弹性的三维电介质结构的理论模型，并以一维纳米梁为例研究了其弯曲问题的挠曲电响应及其能量俘获特性。结果表明，纳米梁的挠曲电响应存在尺寸效应，并且弹性应变梯度会影响结构挠曲电的尺寸效应，特别是当结构的特征尺寸低于尺度参数时。论文的工作为更进一步理解纳米尺度下的挠曲电机理和能量俘获特性提供理论基础和设计依据。     

Gradient elasticity in statics and dynamics: An overview of formulations,length scale identification procedures,finite element implementations and new results

In this paper, we discuss various formats of gradient elasticity and their performance in static and dynamic applications. Gradient elasticity theories provide extensions of the classical equations of elasticity with additional higher-order spatial derivatives of strains, stresses and/or accelerations. We focus on the versatile class of gradient elasticity theories whereby the higher-order terms are the Laplacian of the corresponding lower-order terms. One of the challenges of formulating gradient elasticity theories is to keep the number of additional constitutive parameters to a minimum. We start with discussing the general Mindlin theory, that in its most general form has 903 constitutive elastic parameters but which were reduced by Mindlin to three independent material length scales. Further simplifications are often possible. In particular, the Aifantis theory has only one additional parameter in statics and opens up a whole new field of analytical and numerical solution procedures. We also address how this can be extended to dynamics. An overview of length scale identification and quantification procedures is given. Finite element implementations of the most commonly used versions of gradient elasticity are discussed together with the variationally consistent boundary conditions. Details are provided for particular formats of gradient elasticity that can be implemented with simple, linear finite element shape functions. New numerical results show the removal of singularities in statics and dynamics, as well as the size-dependent mechanical response predicted by gradient elasticity.     

Buckling analysis of cylindrical thin-shells using strain gradient elasticity theory

The stability problem of cylindrical shells is addressed using higher-order continuum theories in a generalized framework. The length-scale effect which becomes prominent at microscale can be included in the continuum theory using gradient-based nonlocal theories such as the strain gradient elasticity theories. In this work, expressions for critical buckling stress under uniaxial compression are derived using an energy approach. The results are compared with the classical continuum theory, which can be obtained by setting the length-scale parameters to zero. A special case is obtained by setting two length scale parameters to zero. Thus, it is shown that both the couple stress theory and classical continuum theory forms a special case of the strain gradient theory. The effect of various parameters such as the shell-radius, shell-length, and length-scale parameters on the buckling stress are investigated. The dimensions and constants corresponding to that of a carbon nanotube, where the length-scale effect becomes prominent, is considered for this investigation.     

Stress concentrations in fractured compact bone simulated with a special class of anisotropic gradient elasticity

A new format of anisotropic gradient elasticity is formulated and implemented to simulate stress concentrations in cortical bone. The higher-order effect of the underlying microstructure in cortical bone is accounted for through the introduction of two length scale parameters and associated strain gradient terms which modify the response of the standard elastic macroscopic continuum: one internal length related to the longitudinal fibres and the other related to the transversal Haversian systems. Thus, anisotropic material behaviour is not only included in the anisotropy of the elastic effective stiffness properties, but also in the anisotropic sources of heterogeneity. The model is validated numerically in tests with bone fractures in the longitudinal and the transversal directions. It was found that the dominant length scale effects are those that coincide with the direction of fracture, as defined by the orientation of a pre-existing crack.     

Numerical and analytical study to estimate the effect of two length scales upon the permeability of a fibrous porous medium

A fibrous porous medium with two length scales is modeled as a bed of porous cylinders aligned perpendicular to the flow of viscous fluid. The flow behavior is described using Stokes and Darcy flow equations in the regions around (higher length scale) and within the cylinders (lower length scale) respectively. The typical ratio of higher and lower length-scale regions enable us to invoke lubrication approximation and simplify the equations to develop a closed form solution for the overall permeability of this dual-scale porous medium. A parametric analysis is performed to explore the dependence of permeability on factors such as the volumetric ratio of higher and lower length-scale regions, permeability and size of inclusions in the smaller length-scale region. The analytical model is compared with the numerical results and the trend is compared with the experiments.     

SHEAR-HORIZONTAL WAVES IN A ROTATED Y-CUT QUARTZ PLATE WITH AN ISOTROPIC ELASTIC LAYER OF FINITE THICKNESS

We study shear-horizontal (SH) waves in a rotated Y-cut quartz plate carrying an isotropic elastic layer of finite thickness.The three-dimensional theories of anisotropic elasticity and isotropic elast...     

Surface waves at the interface between an inviscid fluid and a dipolar gradient solid

This paper is about the dispersion analysis of surface waves propagating at the interface between an inviscid fluid and a higher gradient homogeneous elastic solid modelled as a dipolar gradient continuum. In order to compare the results, a second gradient model is also evaluated. The analysis is carried out by finding the roots of the secular equation, and by carefully studying their physical meaning. As it is well known, higher gradient continua are dispersive, i.e. phase and group velocities are frequency dependent. As a consequence, the existence of surface waves will indeed depend on frequency. In order to investigate the behaviour of surface waves in this specific fluid–solid configuration, a complete dispersion analysis is performed, with a particular focus on the frequency range in which the phase velocity of shear waves is lower than the speed of waves of the fluid. Surface waves of the type Leaky Rayleigh and Scholte–Stoneley are observed in this frequency range. This work extends the knowledge on surface waves in the case of higher gradient solids and applications of these results can be found in the field of non-destructive damage evaluation in micro structured materials, composites, metamaterials and biological tissues.     

Love–Bishop rod solution based on strain gradient elasticity theory

In the present work, the propagation of longitudinal stress waves is investigated with a strain gradient elasticity theory given by Lam et al. In principle, the analysis of wave motion is based on the Love rod model including the lateral deformation effects, but in the same time is also taken into account the shear strain effects with Bishop?s correction. By applying Hamilton?s principle, a general explicit strain gradient elasticity solution is developed for the longitudinal stress waves, and it is compared with the special solutions based on the modified couple stress and classical theories. This work gives useful information with regard to the meaning of the three scale parameters in the strain gradient elasticity theory used here.     

The Peierls stress in non-local elasticity

The effect of non-locality on the Peierls stress of a dislocation, predicted within the framework of the Peierls-Nabarro model, is investigated. Both the integral formulation of non-local elasticity and the gradient elasticity model are considered. A modification of the non-local kernel of the integral formulation is proposed and its effect on the dislocation core shape and size, and on the Peierls stress are discussed. The new kernel is longer ranged and physically meaningful, improving therefore upon the existing Gaussian-like non-locality kernels. As in the original Peierls-Nabarro model, lattice trapping cannot be captured in the purely continuum non-local formulation and therefore, a semi-discrete framework is used. The constitutive law of the elastic continuum and that of the glide plane are considered both local and non-local in separate models. The major effect is obtained upon rendering non-local the constitutive law of the continuum, while non-locality in the rebound force law of the glide plane has a marginal effect. The Peierls stress is seen to increase with increasing the intrinsic length scale of the non-local formulation, while the core size decreases accordingly. The solution becomes unstable at intrinsic length scales larger than a critical value. Modifications of the rebound force law entail significant changes in the core configuration and critical stress. The discussion provides insight into the issue of internal length scale selection in non-local elasticity models.     

蜂窝材料的弹性波传播特性

通过研究蜂窝材料的弹性波频散关系,分析了其弹性波传播特性. 采用基于小波理论的分析方法,将材料参数和位移均展开为双正交周期小波基函数的形式,利用Bloch定理将波动方程转化为特征值方程,求解该方程得到3种典型结构------正方、三角与六角排列的铝(Al)和聚丙烯(PP)蜂窝材料的频散关系,并进行了比较分析. 结果显示:结构形式的不同显著地影响其波动特性,而制作材料的不同则没有影响;3种结构形式都不存在完全带隙,但正方和三角形结构在一定的传播方向范围内存在方向带隙,而六角形结构则在任何方向都不存在方向带隙;与正方结构相比,三角结构在相同孔隙率下,在更广的传播方向内和更低的频率下,能产生较宽的方向带隙.      

Micromechanics based second gradient continuum theory for shear band modeling in cohesive granular materials following damage elasticity

Gradient theories, as a regularized continuum mechanics approach, have found wide applications for modeling strain localization failure process. This paper presents a second gradient stress–strain damage elasticity theory based upon the method of virtual power. The theory considers the strain gradient and its conjugated double stresses. Instead of introducing an intrinsic material length scale into the constitutive law in an ad hoc fashion, a microstructural granular mechanics approach is applied to derive the higher-order constitutive coefficients such that the internal length scale parameter reflects the natural granularity of the underlying material microstructure. The derivations of the required damage constitutive relationships, the strong form governing equations as well as its weak form for the second gradient model are described. The recently popularized Element-Free Galerkin (EFG) method is then employed to discretize the weak form equilibrium equation for accommodating the resultant higher-order continuity requirements and further handling the mesh sensitivity problem. Numerical examples for shear band simulations show that the proposed second gradient continuum model can produce stable, accurate as well as mesh-size independent solutions without a priori assumption of the shear band path.     

Nonlinear vibration analysis of micro-plates based on strain gradient elasticity theory

In this study, non-linear free vibration of micro-plates based on strain gradient elasticity theory is investigated. A general form of Mindlin’s first-strain gradient elasticity theory is employed to obtain a general Kirchhoff micro-plate formulation. The von Karman strain tensor is used to capture the geometric non-linearity. The governing equations of motion and boundary conditions are obtained in a variational framework. The Homotopy analysis method is employed to obtain an accurate analytical expression for the non-linear natural frequency of vibration. For some specific values of the gradient-based material parameters, the general plate formulation can be reduced to those based on some special forms of strain gradient elasticity theory. Accordingly, three different micro-plate formulations are introduced, which are based on three special strain gradient elasticity theories. It is found that both geometric non-linearity and size effect increase the natural frequency of vibration. In a micro-plate having a thickness comparable with the material length scale parameter, the strain gradient effect on increasing the non-linear natural frequency is higher than that of the geometric non-linearity. By increasing the plate thickness, the strain gradient effect decreases or even diminishes. In this case, geometric non-linearity plays the main role on increasing the natural frequency of vibration. In addition, it is shown that for micro-plates with some specific thickness to length scale parameter ratios, both geometric non-linearity and size effect have significant role on increasing the frequency of non-linear vibration.     

超材料混凝土的带隙特征及对冲击波的衰减效应

借鉴超材料的研究思路，在混凝土中引入谐振骨料，设计出具有消波特性的超材料混凝土。首先，通过结构动力学方法计算超材料混凝土的有效质量，从而建立了超材料混凝土带隙起始频率及截止频率的简化模型，并给出了带隙起始频率及截止频率的理论表达式。然后，分析了涂层弹性模量、芯柱密度、基体密度、骨料体积占比和芯柱边长与软涂层厚度比对超材料混凝土带隙特征的影响。最后，采用数值模拟的方法，对比了超材料混凝土和普通混凝土对冲击波的衰减效应。研究结果表明:(1)低弹性模量涂层能够形成低频带隙，但带隙宽度较窄，而高弹性模量涂层能够形成较宽的带隙，但带隙起始频率较高；(2)通过选择高密度芯柱材料和低密度基体材料，可以得到低频、宽带隙特征；(3)通过增大骨料体积占比和芯柱边长与软涂层厚度比可以实现扩宽带隙的目的；(4)与普通混凝土相比，超材料混凝土对冲击波具有更好的衰减作用。