Korn Inequalities for a Reinforced Plate

Asymptotically optimal Korn inequalities are derived for a composite material that consists of two families of stiff rods surrounded by a homogeneous soft material. The composite plate is fixed through the protruding stiff rods only. The asymptotic behaviour is shown to be crucially different for families of connected rods and for those where the rods are isolated.     

Correction of Gauge Factor for Strain Gauges Used in Polymer Composite Testing

Strain gauges are used together with the corresponding gauge factor to relate the relative electrical resistance change of the strain gauge with the strain of the underlying material. The gauge factor is found from a calibration on a stiff material - steel. Nevertheless, the gauge factor depends on the stiffness of the calibration material and ideally the calibration should be done on a similar material as tested. In practice, the gauge factor found by the strain gauge manufacturer is often used. The paper documents that even for moderately stiff materials such as glass-fibre composites a significant error is found on the strain measurements obtained by the strain gauges. This is documented both experimentally and numerically. A stiffness, also test sample and strain gauge geometry dependent correction coefficient of the gauge factor is proposed. A correction coefficient covers material stiffnesses ranging from 1 GPa to 200 GPa.     

A Finite Element Analysis of a Viscoplastic Plate Subjected to Rapid Heating

Abstract

This paper models the response of a thin metallic plate that is subjected to a rapid heat input. In order to accurately model plate response, both the dynamic mechanical and transient heat transfer problems must be solved. The solution is complicated by nonlinearities due to radiation boundary conditions and material inelasticity. Furthermore, the viscoplastic constitutive equations that model the mechanical material behavior are numerically stiff. Nonlinear finite element algorithms are developed for both heat transfer and mechanical analyses. The algorithms are both stable and efficient for solving the problems considered herein. Example problems presented in the paper demonstrate the importance of including material nonlinearity in the model     

Long wavelength inner-resonance cut-off frequencies in elastic composite materials

We revisit an ancient paper (Auriault and Bonnet, 1985) which points out the existence of cut-off frequencies for long acoustic wavelength in high-contrast elastic composite materials, i.e. when the wavelength is large with respect to the characteristic heterogeneity length. The separation of scales enables the use of the method of multiple scale expansions for periodic structures, a powerful upscaling technique from the heterogeneity scale to the wavelength scale. However, the results remain valid for non-periodic composite materials which show a Representative Elementary Volume (REV). The paper extends the previous investigations to three-component composite materials made of hard inclusions, coated with a soft material, both of arbitrary geometry, and embedded in a connected stiff material. The equivalent macroscopic models are rigorously established as well as their domains of validity. Provided that the stiffness contrast within the soft and the connected stiff materials is of the order of the squared separation of scales parameter, it is demonstrated (i) that the propagation of long wave may coincide with the resonance frequencies of the hard inclusions/soft material system and (ii) that the macroscopic model presents a series of cut-off frequencies given by an eigenvalue problem for the resonating domain in the cell. These results are illustrated in the case of stratified composites and the possible microstructures of heterogeneous media in which the inner dynamics phenomena may occur are discussed.     

Experimental Study of Bending Behaviour of Reinforcements

In composite reinforcement shaping, textile preform undergo biaxial tensile deformation, in plane shear deformation, transverse compaction and out-of-plane bending deformations. Bending deformations have been neglected in some simulation codes up to now, but taking into account them would give more accurate simulations of forming especially for stiff and thick textiles. Bending behaviour is specific because the reinforcements are structural parts and out of plane properties cannot be directly deduced from in-plane properties, like for continuous material. Because the standard tests are not adapted for stiff reinforcements with non linear behaviour a new flexometer using optical measurements has been developed to test such reinforcements. This new device enables to carry out a set of cantilever tests with different histories of load. A series of tests has been performed to validate the test method and to show the capacities of the new flexometer to identify non linear non elastic behaviour.     

Implementation of a high order lattice spring model for elasticity

In a lattice spring model (LSM), the material is discretised into particles linked by springs. However, LSMs always adopt linear springs, which results in a stiff approximation of the corresponding elastic solution. In this work, a high order LSM is proposed to overcome this limitation by introducing additional degrees of freedoms (DOFs) to the particles. Based on the energy minimisation principle and the local strain technique, equations for the stiffness matrices of high order LSM are derived. Relationships between micro spring parameters and macro material constants are derived from the Cauchy-born rules and the hyperelastic theory. Numerical examples show that the high order LSM can provide a better solution than that of the linear LSM and that the LSM is more suitable for modelling singularity and fracture problems.     

强动载荷下结构的柔性防护和刚性防护

通过研究柔性防护和刚性防护方面的实例,结合作者的研究结果,对应力波效应和材料动态特性效应如何影响强动载荷下的结构安全防护进行了分析和讨论。研究表明,在研究强动载荷下的结构安全防护时,不论结构承受的是爆炸冲击波还是弹体的直接撞击,都应该考虑结构与周围介质中的波传播效应与材料的应变率效应,以及这两者之间的互相联系和互相耦合。     

ELASTO-PLASTIC BRANCHED ANALYSIS FOR AN INTERFACE CRACK BASED ON CRACK ENERGY DENSITY

The elasto-plastic finite element analyses for an interface crack indissimilar material,based on the crack energy density(CED)concept,are investigatedin mode Ⅰ loading condition.It is confirmed that the values of CED almost remainstable when the notch radius ρ is sufficiently small,both in elastic and elasto-plasticcase.Numerical results for both elastic and elasto-plastic cases show that under themode Ⅰ loading condition,when the crack propagates to the more stiff material with asmall angle,the total CED will become larger than that along the interface.If thecrack heads into the more compliant material,the CED will become less than that alongthe interface.     

The delamination of a growing elastic sheet with adhesion

We study the onset of delamination blisters in a growing elastic sheet adhered to a flat stiff substrate. When the ends of the sheet are kept fixed, its growth causes residual stresses that lead to delamination. This instability can be viewed as a discontinuous buckling between the complete adhered solution and the buckled solution. We provide an analytic expression for the critical deformation at which the instability occurs. We show that the critical threshold scales with a single dimensionless parameter that comprises information from the geometry of the sheet, the mechanical parameters of material and the adhesive features of the substrate.     

Multiple bifurcations in wrinkling analysis of thin films on compliant substrates

Wrinkling phenomena of stiff thin films on compliant substrates are investigated based on a non-linear finite element model. The resulting non-linear equations are then solved by the Asymptotic Numerical Method (ANM) that gives interactive access to semi-analytical equilibrium branches, which offers considerable advantage of reliability compared with classical iterative algorithms. Bifurcation points are detected through computing bifurcation indicators well adapted to the ANM. The effect of boundary conditions and material properties of the substrate on the bifurcation portrait is carefully studied. The evolution of wrinkling patterns and post-bifurcation modes including period-doubling has been observed beyond the onset of the primary sinusoidal wrinkling mode in the post-buckling range.     

Interplay between phononic bandgaps and piezoelectric microstructures for energy harvesting

The paper introduces a multifunctional structural design combining superior mechanical wave filtering properties and energy harvesting capabilities. The proposed concept is based on the ability of most periodic structures to forbid elastic waves from propagating within specific frequency ranges known as phononic bandgaps. The bandgap density and the resulting filtering effect are dramatically enhanced through the introduction of a microstructure consisting of stiff inclusions which resonate at specific frequencies and produce significant strain and energy localization. Energy harvesting is achieved as a result of the conversion of the localized kinetic energy into electrical energy through the piezoelectric effect featured by the material in the microstructure. The idea is illustrated through the application to hexagonal truss-core honeycombs featuring periodically distributed stiff cantilever beams provided with piezoelectric electrodes. The multifunctional capability results from the localized oscillatory phenomena exhibited by the cantilevers for excitations falling in the neighborhood of the bending fundamental frequencies of the beams. This application is of particular interest for advanced aerospace and mechanical engineering applications where distinct capabilities are simultaneously pursued and weight containment represents a critical design constraint. The scalability of the analysis suggests the possibility to miniaturize the design to the microscale for microelectromechanical systems (MEMS) applications such as self-powered microsystems and wireless sensors.     

A brittle material with tunable elasticity: Crêpe paper

We study the mechanical response, and tearing features of crêpe paper, a two-dimensional, very anisotropic material, with one direction much less stiff than the other one. Depending on how the soft direction has been pre-stretched or not, the apparent Young modulus of the material can be varied over a broad range, while its fracture energy remains unaltered. The classical tearing concertina problem shows that a macroscopic measurement (the shape of the teared region) provides a direct access to the fracture properties of the material (effective Young's modulus, and fracture energy). The overall discussion is conducted in the frame of Griffith's theory of fracture.     

Exact analytical solutions for the local and global buckling of sandwich beam-columns under various loadings

Sandwich structures are widely used in many industrial applications, due to the attractive combination of a lightweight and strong mechanical properties. This compromise is realized thanks to the presence of different parts in the composite material, namely the skins and possibly core reinforcements or thin-walled core structure which are both thin/slender and stiff relative to the other parts, namely the homogeneous core material, if any. The buckling phenomenon thus becomes mainly responsible for the final collapse of such sandwiches. In this paper, classical sandwich beam-columns (with homogeneous core materials) are considered and elastic buckling analyses are performed in order to derive the critical values and the associated bifurcation modes under various loadings (compression and pure bending). The two faces are represented by Euler–Bernoulli beams, whereas the core material is considered as a 2D continuous solid. A set of partial differential equations is first obtained from a general bifurcation analysis, using the above assumptions. Original closed-form analytical solutions of the critical loading and mode of a sandwich beam-column are then derived for various loading conditions. Finally, the proposed analytical formulae are validated using 2D linearized buckling finite element computations, and parametric analyses are performed.     

Mechanical model for staggered bio-structure

A generic mechanical model for bio-composites, including stiff platelets arranged in a staggered order inside a homogeneous soft matrix, is proposed. Equations are formulated in terms of displacements and are characterized by a set of non-dimensional parameters. The displacements, stress fields and effective modulus of the composite are formulated. Two analytical models are proposed, one which includes the shear deformations along the entire medium and another simplified model, which is applicable to a slender geometry and yields a compact expression for the effective modulus. The results from the models are validated by numerical finite element simulations and found to be compatible with each other for a wide range of geometrical and material properties. Finally, the models are solved for two bio-structures, nacre and a collagen fibril, and their solutions are discussed.     

Effect of yield surface vertex on crack-tip fields in mode III

An asymptotic crack-tip analysis of stress and strain fields is carried out for an antiplane shear crack (Mode III) based on a corner theory of plasticity. Because of the nonproportional loading history experienced by a material element near the crack tip in stable crack growth, classical flow theory may predict an overly stiff response of the elastic plastic solid, as is the case in plastic buckling problems. The corner theory used here accounts for this anomalous behavior. The results are compared with those of a similar analysis based on the J₂ flow theory of plasticity.     

Numerical solution of stiff systems from nonlinear dynamics using single-term Haar wavelet series

The paper presents single-term Haar wavelet series (STHWS) approach to the solution of nonlinear stiff differential equations arising in nonlinear dynamics. The properties of STHWS are given. The method of implementation is discussed. Numerical solutions of some model equations are investigated for their stiffness and stability and solutions are obtained to demonstrate the suitability and applicability of the method. The results in the form of block-pulse and discrete solutions are given for typical nonlinear stiff systems. As compared with the TR BDF2 method of Shampine and Gill’s method, the STHWS turns out to be more effective in its ability to solve systems ranging from mildly to highly stiff equations and is free from stability constraints.     

3D Heterogeneous Stiffness Reconstruction Using MRI and the Virtual Fields Method

The first extension of the virtual fields method to the reconstruction of heterogeneous stiffness properties from 3D bulk full-field displacement data is presented in this paper. Data are provided by Magnetic Resonance Imaging (MRI). Two main issues are addressed: 1. the identification of the stiffness ratio between two different media in a heterogeneous solid; 2. the reconstruction of stiffness heterogeneities buried in a heterogeneous solid. The approach is based on a finite element discretization of the equilibrium equations. It is tested on experimental full-field data obtained on a phantom with the stimulated echo MRI technique. The phantom is made of a stiff spherical inclusion buried within a lower modulus material. Preliminary independent tests showed that the material of the inclusion was four times stiffer than the surrounding material. This ratio value is correctly identified by our approach directly on the phantom with the MRI data. Moreover, the modulus distribution is promisingly reconstructed across the whole investigated volume. However, the resulting modulus distribution is highly variable. This is explained by the fact that the approach relies on a second order differentiation of the data, which tends to amplify noise. Noise is significantly reduced by using appropriate filtering algorithms.     

多柔体系统数值分析的模型降噪方法

多柔体系统的动力学方程通常是一组刚性微分方程, 目前普遍采用的刚性微分方程数值解法主要通过数值阻尼滤除系统响应中的高频分量, 其求解效率难以令人满意. 为了降低多柔体系统动力学方程的刚性, 从而可采用ODE45等常规微分方程求解器进行求解, 研究了在建模过程中滤除高频振荡分量的方法. 在以当前时刻为起点的短时间内对柔性体的应力进行均匀化, 用均匀化后的应力计算柔性体的变形虚功率, 由此得到的系统动力学方程的解中不含过高频率的弹性振动, 并且可以通过调节均匀化时间区间的长度参数控制滤波的范围. 数值算例表明: 这种模型降噪方法的计算效率和精度均不低于刚性微分方程求解器, 并且在刚性微分方程求解器失效的情况下模型降噪方法仍有良好的精度和效率. 本文所提的模型降噪方法可成为求解多柔体系统动力学方程的新途径.      

硬壳层非饱和软基路堤的填土高度研究

硬壳层在软土路堤工程中具有很好的承载传递特性,应充分发挥其有利作用以优化路堤填土高度设计与施工管理.考虑硬壳层的应力扩散作用和自重反压护道作用,基于非饱和土平面应变抗剪强度统一解与吸力角双曲线模型,推导了实际地应力状态下硬壳层非饱和软土地基的临塑/临界载荷计算公式,进而建立了路堤临塑/临界填土高度的解析解,并给出解析公式的适用条件与应用步骤,最后探讨了各因素对路堤临界填土高度的影响规律.所得解析公式能综合反映4种因素的影响,具有很好的可比性,且对比验证了解析公式的正确性及给定公式适用条件的必要性与合理性.研究结果表明:硬壳层的影响具有整体性,应同时考虑其厚度和弹性模量的变化,可忽略路堤等效宽度的影响;侧压力系数的影响明显,侧压力系数为1.0对应的路堤填土高度明显偏大,应原位测试确定地基土的真实侧压力系数;采用不同强度准则的计算结果差异显著,应充分考虑中间主应力对公式适用范围及路堤填土高度的影响;高、低基质吸力的影响特性不同,低吸力时路堤填土高度线性增加且与截距无关,高吸力时路堤填土高度受双重作用相对大小的控制,或先逐渐减小后缓慢增大、或逐渐增大,并与截距密切相关.      

On the rigidity of certain surfaces with folds and applications to shell theory

In the asymptotic theory of thin elastic shells the rigidity of the mid-surface with kinematic boundary conditions plays an important role. Rigidity is understood in the sense of infinitesimal (linearized) rigidity, i.e., the displacements vanish provided the variation of the first fundamental form vanishes. In this case the surface is also called “stiff”, as it cannot undergo pure bendings. A stiff surface is imperfectly stiff or perfectly stiff when the origin respectively does or does not belong to the essential spectrum of the boundary-value problem. These questions are investigated in the framework of Douglis-Nirenberg elliptic systems, with boundary conditions and transmission conditions at the folds. The index properties ensures quasi-stiffness, i.e. stiffness up to a finite number of degrees of freedom. The concept of perfect stiffness is linked with estimates for the rigidity system at an appropriate level of regularity for the data and the solution. It is proved that surfaces with folds are never perfectly stiff. It is also shown that the transmission conditions at the folds contain more conditions than those satisfying the Shapiro-Lopatinskii property. This leads to certain rigidity properties of the folds. Some examples are given.