Features of plane wave propagation along the layers of a prestrained nanocomposite

Features of the propagation of longitudinal and transverse plane waves along the layers of nanocomposites with process-induced initial stresses are studied. The composite has a periodic structure: it is made by repeating two highly dissimilar layers. The layers exhibit nonlinear elastic behavior in the range of loads under consideration. A Murnaghan-type elastic potential dependent on the three invariants of the strain tensor is used to describe the mechanical behavior of the composite constituents. To simulate the propagation of waves, finite-strain theory is used for developing a problem statement within the framework of the three-dimensional linearized theory of elasticity assuming finite initial strains. The dependence of the relative velocities of longitudinal and transverse waves on two components of small initial stresses in each layer and on the volume fraction of the constituents is studied. It is established that there are thickness ratios of layers in some nanocomposites such that the wave velocities are independent of the initial stresses and equal to the respective wave velocities in composites without initial stresses __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 4, pp. 3–26, April 2007.     

On constitutive modelling of porous neo-Hookean composites

In this paper a hyperelastic constitutive model is developed for neo-Hookean composites with aligned continuous cylindrical pores in the finite elasticity regime. Although the matrix is incompressible, the composite itself is compressible because of the existence of voids. For this compressible transversely isotropic material, the deformation gradient can be decomposed multiplicatively into three parts: an isochoric uniaxial deformation along the preferred direction of the material (which is identical to the direction of the cylindrical pores here); an equi-biaxial deformation on the transverse plane (the plane perpendicular to the preferred direction); and subsequent shear deformation (which includes “along-fibre” shear and transverse shear). Compared to the multiplicative decomposition used in our previous model for incompressible fibre reinforced composites [Guo, Z., Peng, X.Q., Moran, B., 2006, A composites-based hyperelastic constitutive model for soft tissue with application to the human annulus fibrosus. J. Mech. Phys. Solids 54(9), 1952–1971], the equi-biaxial deformation is introduced to achieve the desired volume change. To estimate the strain energy function for this composite, a cylindrical composite element model is developed. Analytically exact strain distributions in the composite element model are derived for the isochoric uniaxial deformation along the preferred direction, the equi-biaxial deformation on the transverse plane, as well as the “along-fibre” shear deformation. The effective shear modulus from conventional composites theory based on the infinitesimal strain linear elasticity is extended to the present finite deformation regime to estimate the strain energy related to the transverse shear deformation, which leads to an explicit formula for the strain energy function of the composite under a general finite deformation state.     

反对称斜交叠层板双向弯曲问题的横向基理论解

研究反对称斜交叠层板的双向弯曲问题。根据横向基理论，在厚度方向，给出了三个方向位移沿厚度的分布规律；在面内方向，采用升阶谱位移函数；然后，应用最小势能原理导出叠层板的静力控制方程。通过对多种层数和铺层角的层板力学响应计算分析表明：在典型位置的应力分布，本文解与三维弹性解符合很好，并有较好的收敛性。     

瞬态波在层合板中传播的粘弹比拟法

本文发展了粘弹比拟理论,并将之用于求解半无限空间三层复合材料在垂直层合方向传播的瞬态波问题。对于层合板中应力波的传播问题,寻找到了一等效粘弹体,并用一种较好的Laplace变换的数值反演法求得了等效松弛函数和其它一些必要的辅助函数。用特征线法求得了等效粘弹体的应力和速度,进而得到了三层复合材料中心的应力、速度,进一步就得到了层中任意点的应力和速度。对于一个可由精确理论(射线理论)给出计算结果的问题,将粘弹比拟理论的结果和射线理论的结果进行了比较,结果表明,粘弹比拟理论对三层复合材料的瞬态波传播问题是相当成功的。     

Analytical solution to continuous contact problem for a functionally graded layer loaded through two dissimilar rigid punches

A continuous contact problem of functionally graded layer resting on an elastic semi-infinite plane, which is loaded with through two different blocks is addressed in this study. The elasticity theory and integral transformation techniques are used in solution of the problem. The problem is reduced to a system of singular integral equations, and solved numerically by the aid of appropriate Gauss–Chebyshev integration formula. It is assumed that the elastic semi-infinite homogeneous plane is isotropic and all surfaces are frictionless and continuous. The shear modulus and the mass density of the FG layer vary exponentially along the thickness direction.     

A crack in a piezoelectric layer bonded to a dissimilar elastic layer under transient load

Summary  A piezoelectric layer bonded to the surface of an elastic structure is considered. The piezoelectric and the elastic layers are infinite along the x-axis and have finite thickness in the y-direction. The polarization direction of the piezoelectric material is along the y-axis. By means of the method of singular integral equations, the solution in a Laplace transform plane is demonstrated. Laplace inversion yields the results in the time domain. Numerical values of the crack tip fields under in-plane transient electromechanical loading are obtained. The influence of layers thickness on stress and electric displacement intensity factors is investigated. Received 16 March 2000; accepted for publication 16 August 2000     

Buckling of an orthotropic graded coating with an embedded crack bonded to a homogeneous substrate

To simulate buckling of nonuniform coatings, we consider the problem of an embedded crack in a graded orthotropic coating bonded to a homogeneous substrate subjected to a compressive loading. The coating is graded in the thickness direction and the material gradient is orthogonal to the crack direction which is parallel with the free surface. The elastic properties of the material are assumed to vary continuously along the thickness direction. The principal directions of orthotropy are parallel and perpendicular to the crack orientation. The loading consists of a uniform compressive strain applied away from the crack region. The graded coating is modeled as a nonhomogeneous medium with an orthotropic stress–strain law. Using a nonlinear continuum theory and a suitable perturbation technique, the plane strain problem is reduced to an eigenvalue problem describing the onset of buckling. Using integral transforms, the resulting plane elasticity equations are converted analytically into singular integral equations which are solved numerically to yield the critical buckling strain. The Finite Element Method was additionally used to model the crack problem. The main objective of the paper is to study the influence of material nonhomogeneity on the buckling resistance of the graded layer for various crack positions, coating thicknesses and different orthotropic FGMs.     

Opening‐Mode Crack in a Unidirectional Composite Material

The plane static elastic problem of stress concentration in a unidirectional discrete infinite composite weakened by fiber breaks on a line normal to the reinforcement direction (an analog of the Griffith problem of elasticity theory) is considered. The composite is subjected to uniform stresses at infinity, and the crack edges are loaded symmetrically by the normal pressure. The problem reduces to constructing a polynomial with known values at the points of fiber breaks. The stress distribution along the line of breaks is obtained in the form of a fractional rational function of fiber number.     

Stress in elastic plates reinforced by fibres lying in concentric circles

We Consider fibre-reinforced elastic plates with the reinforcement continuously distributed in concentric circles ; such a material is locally transversely isotropic, with the circumferential direction as the preferred direction. For an annulus bounded by concentric circles, the exact solution of the traction boundary value problem is obtained. When the extension modulus in the fibre direction is large compared to other extension and shear moduli, the material is strongly anisotropic. For this case a simpler approximate solution is obtained by treating the material as inextensible in the fibre direction. It is shown that the exact solution reduces to the inextensible solution in the appropriate limit. The inextensible theory predicts the occurrence of stress concentration layers in which the direct stress is infinite ; for materials with small but finite extensibility these layers correspond to thin regions of high stress and high stress gradient. A boundary layer theory is developed for these regions. For a typical carbon fibre-resin composite, the combined boundary layer and inextensible solutions give an excellent approximation to the exact solution. The theory is applied to the problem of an isotropic plate, under uniform stress at infinity, containing a circular hole which is strengthened by the addition of an annulus of fibre-reinforced material.     

On the behavior of stress waves in composite materials—II. Theoretical and experimental studies on the effects of constituent debonding

The theory derived in Part I is compared to data obtained from flyer-plate experiments on laminated composites. The composites were constructed of alternating layers of aluminum and polymethyl methacrylate. Impact occurred on a flat plane oriented perpendicular to the interface planes of the composite constituents. As opposed to the behavior of a fully bonded composite which allows only one longitudinal stress wave to propagate through its interior, two longitudinal stress waves were observed in the debonded composite.Reasonable agreement was achieved in the comparisons of theory to experiment after adjustment was made for the effect of residual bond strength at the constituent interfaces.     

FeCrAl合金材料织构对其宏观力学本构关系的影响研究

FeCrAl合金具有优良的高温抗氧化性和耐辐照性能,是事故容错核燃料包壳的重要候选材料. 其在加工过程和热处理过程中易形成α纤维织构（<110>//RD）和γ纤维织构（<111>//ND）,会影响材料的宏观力学性能与深加工成形能力. 本研究针对具有不同织构的多晶FeCrAl合金,建立了代表性体元模型, 使用晶体塑性有限元方法,在ABAQUS/Explicit中模拟材料单轴加载下的宏观应力应变曲线,分析不同织构对FeCrAl合金宏观力学本构关系的影响. 计算结果表明,对于具有α织构、γ织构和晶粒无择优取向的材料,在轧向上的应力应变曲线差异较小. γ织构会引起材料强烈的各向异性,在轧面法向上的屈服强度远高于轧向和横向上的屈服强度,这是因为晶粒的<111>方向平行于加载方向,滑移系难以启动. 提高γ纤维织构的比例,将增大轧面法向上的屈服强度. 本研究可以为优化FeCrAl合金材料织构、加工条件和材料力学性能提供参考.     

FeCrAl合金材料织构对其宏观力学本构关系的影响研究

FeCrAl合金具有优良的高温抗氧化性和耐辐照性能,是事故容错核燃料包壳的重要候选材料. 其在加工过程和热处理过程中易形成α纤维织构（<110>//RD）和γ纤维织构（<111>//ND）,会影响材料的宏观力学性能与深加工成形能力. 本研究针对具有不同织构的多晶FeCrAl合金,建立了代表性体元模型, 使用晶体塑性有限元方法,在ABAQUS/Explicit中模拟材料单轴加载下的宏观应力应变曲线,分析不同织构对FeCrAl合金宏观力学本构关系的影响. 计算结果表明,对于具有α织构、γ织构和晶粒无择优取向的材料,在轧向上的应力应变曲线差异较小. γ织构会引起材料强烈的各向异性,在轧面法向上的屈服强度远高于轧向和横向上的屈服强度,这是因为晶粒的<111>方向平行于加载方向,滑移系难以启动. 提高γ纤维织构的比例,将增大轧面法向上的屈服强度. 本研究可以为优化FeCrAl合金材料织构、加工条件和材料力学性能提供参考.     

基于不同板理论的嵌于柔性层中薄膜的皱曲分析

论文对于柔性层-薄膜-柔性层三层结构系统,基于经典板理论、一阶剪切变形理论和高阶剪切变形理论,分别推导给出薄膜皱曲的控制方程.对于两个柔性层,则把它们处理成具有有限厚度的平面应变弹性体.针对上下柔性层固支边界或自由边界条件,利用线性扰动方法得到柔性层对薄膜的横向压力差,最终获得确定薄膜具有周期性正弦型皱曲的临界载荷方程...     

Onset of failure in finitely strained layered composites subjected to combined normal and shear loading

A limiting factor in the design of fiber-reinforced composites is their failure under axial compression along the fiber direction. These critical axial stresses are significantly reduced in the presence of shear stresses. This investigation is motivated by the desire to study the onset of failure in fiber-reinforced composites under arbitrary multi-axial loading and in the absence of the experimentally inevitable imperfections and finite boundaries.By using a finite strain continuum mechanics formulation for the bifurcation (buckling) problem of a rate-independent, perfectly periodic (layered) solid of infinite extent, we are able to study the influence of load orientation, material properties and fiber volume fraction on the onset of instability in fiber-reinforced composites. Two applications of the general theory are presented in detail, one for a finitely strained elastic rubber composite and another for a graphite-epoxy composite, whose constitutive properties have been determined experimentally. For the latter case, extensive comparisons are made between the predictions of our general theory and the available experimental results as well as to the existing approximate structural theories. It is found that the load orientation, material properties and fiber volume fraction have substantial effects on the onset of failure stresses as well as on the type of the corresponding mode (local or global).     

Static analysis of 2-2 cement-based piezoelectric composites

Exact static analysis of 2-2 cement-based piezoelectric composites is presented based on the theory of piezo-elasticity in this paper. Displacement method was used, and the solutions of four different kinds of piezoelectric composites under load were obtained. The effects of polarization direction of piezoelectric layers and material parameters on the solutions are discussed. The relationship between the blocking force and the applied voltage for the actuators was obtained. The solutions were compared with both numerical and experimental results, and good agreement was found.     

A new approach for developing dynamic theories for structural elements : Part 2: Application to thermoelastic layered composites

In this study a continuum theory is proposed which predicts the dynamic behavior of thermoelastic layered composites consisting of two alternating layers. In constructing the theory, it is noted that the governing equations for a single layer, derived in Part I, hold in each phase of the layered composite. The theory is completed by supplementing these equations with continuity conditions and using a smoothing operation. The derivation of the continuity conditions is based on the assumption that the layers are perfectly bonded at interfaces. To assess the theory, spectra from the exact and the derived theory are compared for waves propagating in various directions of the composite. The match between the two is excellent. For waves propagating normal to layering the theory predicts both the banded and periodic structure of the spectra. The region of validity of the theory on the wave number-frequency plane can be enlargened by increasing the orders of the theory and the continuity conditions.     

Mechanics of an elastic solid reinforced with bidirectional fiber in finite plane elastostatics: complete analysis

A continuum-based model is presented for the mechanics of bidirectional composites subjected to finite plane deformations. This is framed in the development of a constitutive relation within which the constraint of material incompressibility is augmented. The elastic resistance of the fibers is accounted for directly via the computation of variational derivatives along the lengths of bidirectional fibers. The equilibrium equation and necessary boundary conditions are derived by virtue of the principles of virtual work statement. A rigorous derivation of the corresponding linear theory is developed and used to obtain a complete analytical solution for small deformations superposed on large. The proposed model can serve as an alternative 2D Cosserat theory of nonlinear elasticity.     

Setting up a theory of stability of fibrous and laminated composites

The results obtained in setting up a theory of stability of fibrous and laminated composites in the case where the plane Π is in an arbitrary position are analyzed. The plane Π is formed by the points of a buckling mode that have equal phases relative to the line of compression. This theory follows from the linearized three-dimensional theory of stability of deformable bodies and is used to determine the critical compressive load and the associated position of the plane Π. Numerical examples are presented. A brief historical sketch is given     

Structure of the constitutive relations for hereditarily elastic materials reinforced by hard fibers

The problem of simplifying the nonlinear hereditary elasticity relations is considered for strongly anisotropic materials such as fiber-reinforced composites. This is done using their property that the material stiffness is high along the reinforcement and is low in the cross-sectional direction. The material is considered transversally isotropic. The simplification is performed by analyzing asymptotic representations of creep relations. Relations of various degrees of accuracy for various types of composites and stress states are obtained. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 3, pp. 120–127, May–June, 2005.     

The stability of the interface between two bodies compressed along interface cracks. 2. Exact solutions for the case of equal roots

The problem of buckling of the interface between two bodies is considered for the case where several plane cracks are located in the interface and the bodies are compressed along the cracks (along the interface of two different materials). The studies were carried out for a plane problem using the three-dimensional linearized theory of stability of deformable bodies. The complex variables and potentials of the above-mentioned linearized theory are used. This problem is reduced to the problem of linear conjugation of two analytical functions of a complex variable. The exact solution of the above-mentioned buckling problem is obtained for the case where the roots of the basic equation are equal. Some mechanical effects are analyzed under general conditions (elastic, elastoplastic, compressible, incompresible, isotropic, and orthotropic bodies). S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 5, pp. 66–73, May, 2000.