Comparison of energy release rate and minimum strain energy density for potential failure of composite with bi-material interface

An analytical method is developed to describe the fields of stress and displacement in a bi-material strip specimen with an edge interfacial crack. All of the basic governing equations, boundary conditions on crack surfaces and conditions of continuity along the interface are satisfied by the eigenfunction expansion method. The other boundary conditions are satisfied by the generalized variational principle. The stress intensity factors are calculated for determining the energy release rate and minimum strain energy density factor S_min that is used the strain energy density criterion. Problems with oscillatory singularity and contact zone are discussed. Not only the effects of bi-material modulus ratio, thickness ratio, Poisson's ratio and crack length to S_min, but also the influences of bi-material modulus ratio, thickness ratio to phase angle are presented. Among these parameters, particular situations where S_min become jeopardously high and lead to failure are discussed.     

具损伤压电智能层合板的动力分析

基于应变能等效原理、高阶剪切变形理论和Hamilton变分原理,考虑复合材料铺设层内的损伤效应,建立了具损伤压电智能层合板的运动控制方程,并运用Galerkin方法进行求解.数值算例中,讨论了损伤效应、厚跨比及压电层厚度与层合板总厚度之比对四边简支压电智能层合板自由振动频率的影响和外部控制电压对其动力响应的影响.     

An intermediate model method for obtaining a discrete relaxation spectrum from creep data

A convenient method is described for obtaining a discrete stress relaxation spectrum from linear viscoelastic creep data by means of a three-stage process. In stage one, a discrete retardation spectrum is fitted to the creep data using a least squares procedure, subject to the constraint that the discrete spectrum must be a specified order of polynomial function of the retardation time. In stage two, the resulting generalised Voigt model is solved numerically for an imposed step in strain, to determine the stress relaxation modulus function of time. In stage three, a discrete relaxation spectrum is fitted to the calculated stress relaxation modulus function. Although three stages are involved instead of the usual two, the procedure has been found to have certain practical advantages. These advantages make it suitable for the generation of relaxation spectra needed in viscoelastic stress analyses of solids, for example by the finite element method. In order to illustrate the proposed procedure it is applied to both artificial data and experimental creep data for poly(methyl methacrylate) at 70°C and at the glass transition.     

Experimental determination of the relaxation modulus of a linear-isotropic-viscoelastic material

It is well known that the relaxation modulus of a linear-isotropic-viscoelastic material under uniaxial load is the stress history of a unit-step strain history. A unit-stepped strain function cannot be obtained with the common instron testing machine. Instead, an easily obtainable strain function is described from which the relaxation modulus is derived. Experiments were conducted to illustrate this method. Experimental-data-reduction techniques are described. Experimentally measured output stress vs. time is fitted by a ‘smooth’ polynomial using a least-square criterion. Then by differentiation of this polynomial at proper times, the relaxation function is obtained.     

Effective elastic modulus of film-on-substrate systems under normal and tangential contact

Load and depth sensing indentation methods have been widely used to characterize the mechanical properties of the thin film-substrate systems. The measurement accuracy critically depends on our knowledge of the effective elastic modulus of this heterogeneous system. In this work, based on the exact solution of the Green's function in Fourier space, we have derived an analytical relationship between the surface tractions and displacements, which depends on the ratio of the film thickness to contact size and the generalized Dundurs parameters that describe the modulus mismatch between the film and substrate materials. The use of the cumulative superposition method shows that the contact stiffness of any axisymmetric contact is the same as that of a flat-ended punch contact. Therefore, assuming a surface traction of the form of [1−(r/a)²]^−1/2 with radial coordinate r and contact size a, we can obtain an approximate representation of the effective elastic moduli, which agree extremely well with the finite element simulations for both normal and tangential contacts. Motivated by a recently developed multidimensional nanocontact system, we also explore the dependence of the ratio of tangential to normal contact stiffness on the ratio of film thickness to contact radius and the Dundurs parameters. The analytical representations of the correction factors in the relationship between the contact stiffness and effective modulus are derived at infinite friction conditions.     

二维复合材料结构模态分析的边界元法

将边界元方法用于分析二维复合材料结构的自由振动模态，利用特解处理体积力（惯性力）仅需静态基本解就可求解问题，对一各向同性悬臂梁，用该法得到的结果与用有限元或各向性边界元法得到的结果符合得很好，但该法可解各向异性问题，对层状复合材料简支梁，用该法得到了数值结果与用一维层状复合材料梁的理论解的比较表明，当结构的长厚比大于２０时，二者符合得很好，当结构的长厚比小于２０时，一维层状材料梁的理论将产生很大的     

A gradient model of light-induced bending in photochromic liquid crystal elastomer and its nonlinear behaviors

Photochromic liquid crystal elastomer was recently reported to be able to deform largely and bend under illumination. In this paper, considering the opto-chemical process and the nematic-isotropic phase transition, we introduce the light and temperature into the constitutive relation of the liquid crystal elastomers, and propose a model for the light-induced bending. The dynamic deflection curve equation of the light-induced bending is derived based on the Hamilton principle. In the equation, the effect of light is introduced as an effective optical bending moment, which is caused by the inhomogeneous light-induced strain and Young's modulus. Several simulation examples are given to show the light-induced bending under different boundary conditions and various illumination or temperature controlling. Under the condition of deep nematic phase and weak enough illumination, the approximate analytical expression of the effective moment and the stress distribution can be obtained. Rich nonlinear behaviors are found in this model. The effective moment is a non-monotonic function of time, thickness ratio, and light intensity when the thickness ratio is not very large. The stress distribution through the thickness is nonlinear with two or three zero-stress planes.     

热应力作用下层合路面结构层间剥离内力分析

温度变化会造成环氧树脂-混凝土特种路面结构边缘处产生脱层现象.基于层合结构热应力的两变量解析模型,推导了层间剥离力矩和剥离剪力.该方法避免了采用弹性理论求解局部复杂应力状态的困难,为解决层合结构边界脱层问题提供了新途径;由此,研究了环氧树脂层的厚度和弹性模量对层间剥离内力和混凝土层最大拉应力的影响.计算结果表明,通过控制环氧树脂层厚度和弹性模量,可有效降低剥离内力,避免剥离破坏.     

Second strain gradient elasticity of nano-objects

Mindlin's second strain gradient continuum theory for isotropic linear elastic materials is used to model two different kinds of size-dependent surface effects observed in the mechanical behaviour of nano-objects. First, the existence of an initial higher order stress represented by Mindlin's cohesion parameter, b₀, makes it possible to account for the relaxation behaviour of traction-free surfaces. Second, the higher order elastic moduli, c_i, coupling the strain tensor and its second gradient are shown to significantly affect the apparent elastic properties of nano-beams and nano-films under uni-axial loading. These two effects are independent from each other and allow for separated identification of the corresponding material parameters. Analytical results are provided for the size-dependent apparent shear modulus of a nano-thin strip under shear. Finite element simulations are then performed to derive the dependence of the apparent Young modulus and Poisson ratio of nano-films with respect to their thickness, and to illustrate hole free surface relaxation in a periodic nano-porous material.     

A constant-contour-length reptation model without independent alignment or consistent averaging approximations for chain retraction

A reptation model for the primitive chain that does not assume independent alignment or consistent-averaging for the retraction process, or equilibrium relaxation for the reptation process is proposed and compared to the analytical expressions of Doi and Edwards in single-step, double-step strains and steady-state shear flow. The Doi and Edwards model with independent alignment approximation underpredicts the magnitude of the relaxation modulus by 25%, and consistently overpredicts the magnitude of the damping function; for steady shear flow, it predicts the correct shape for the steady-state viscosity and the first normal stress difference coefficient, although the magnitude is incorrect. The analytical expressions of Doi and Edwards without independent alignment approximation are excellent approximations to the damping function. In double-step strains, the expressions of Doi assuming consistent averaging, but no independent alignment, predict well the stress decay following the second strain. Linear response theory is found to be invalid for describing the stress relaxation following single-step strain for the models considered. Similar to the Doi and Edwards model, no overshoot for the first normal stress difference is observed for the simulation model. Unlike the Doi equation derived without the independent alignment approximation but restricted to double-step strains, the simulation model proposed here can be easily generalized to complex flow fields. No contour length fluctuation or constraint release is considered in this model, and chain retraction is assumed to be instantaneous.     

Boundary element method for model analysis of 2-D composite structure

The boundary element method is used for the modal analysis of free vibration of 2-D composite structures in this paper. Since the particular solution method is used to treat the terms of body forces (inertial forces) in the equation of motion, only static fundamental solutions are needed in solving the problem. For an isotropic cantilever beam, the numerical results obtained by using the BEM presented in this paper are in good agreement, with, those of using FEM or other BEM, but this BEM can also be used to analyze problems for anisotropic materials. For simply supported composite laminated beams, the comparisons of the numerical reslts obtained by this method with the analytical results obtained by 1-D laminated beam theory indicate that if the ratio of length/thickness is greater than 20, the results of the two methods are in good agreement, but if the ratio of length/thickness is less than 20, big errors will occur for 1-D laminated beam theory.     

考虑膜厚影响薄膜磁致伸缩系数、泊松比和杨氏模量的同时测量

基于经典层合板理论,建立了一个能同时测量薄膜-基底系统中薄膜的磁致伸缩系数、杨氏模量和泊松比的板模型.以前的研究计算薄膜磁致伸缩系数时,大多假设薄膜的弹性属性与相应的块材一致,由此导致的磁致伸缩系数计算是不准确的.在目前大多数方法中仅仅在使用一个单一的弹性各向同性基底中能够避免这个问题.该文模型在各向异性基底下同样适用,并且不要求薄膜的厚度远远小于基底厚度,因此也能够用来计算磁致伸缩应变和设计微电机械系统和生物微电机械系统.对已有的铁基非晶薄膜的实验数据,在不同磁场强度下,磁致伸缩系数的计算结果与已有模型进行了比较,它们之间的差异得到了解释.同时,还可以得到薄膜的弹性常数.     

Stress analysis in rolling contact problem of a finite thickness FGM layer

The rolling contact problem of a non-homogeneous layer is considered here. The graded layer possesses a variable elastic modulus with an exponential distribution. The Poisons ratio is assumed to be constant. A rigid cylindrical indenter is rolling over the surface of the graded layer with a constant velocity. First, the Navier equations of equilibrium are solved in the Fourier domain. Later, the boundary and the continuity conditions are satisfied in order to extract the governing singular integral equations. The numerical solution of the integral equations is provided by means of the Gauss–Chebyshev integration method. Finally, the sensitivity of the solution is analyzed for the effective parameters namely: the stiffness ratio, the layer thickness and the coefficient of friction. The results indicate that a minimum value of the coating thickness is required to alleviate the severe stress gradients in the critical locations. If the coating thickness decreases by a 50% then the Von Mises stress will increases about 20%. Also, a softening graded layer can result in a lower stress level over the interface which may enhance the bonding strength.

     

Stability of parametric vibrations of hybrid composite plates

In this paper, the dynamic stability of laminated hybrid composite plates subjected to periodic uniaxial stress and bending stress is studied. The governing equations of motion of Mathieu-type are established by using the Galerkin method with reduced eigenfunctions transforms. Based on Bolotin's method the regions of dynamic instability of laminated hybrid composite plates are determined by solving the eigenvalue problems. The effects of layer thickness ratio, layer number, core material and load parameter on the dynamic instability of laminated hybrid composite plates are investigated and discussed.     

Normal stress differences in large-amplitude oscillatory shear flow for the corotational ��ANSR�� model

Using the integral form of a nonlinear corotational model, we derive explicit analytical expressions for the zeroth, second, and fourth harmonics of the second normal stress difference in large-amplitude oscillatory shear (LAOS). This model yields an arbitrary normal stress ratio (ANSR) in any simple shearing deformation, including LAOS. This corotational ANSR model adds one parameter to the corotational Maxwell model, a time constant ?? ₀ controlling the ratio ??₂/??₁ for both the real and imaginary parts of each harmonic of the normal stress difference. The explicit analytical expressions for all harmonics of the alternating shear stress and first normal stress difference responses in LAOS match those obtained previously for the corotational Maxwell model. We evaluate the corotational ANSR model for the case of a single Maxwell relaxation time fluid.     

Effect of ionic interaction on linear and nonlinear viscoelastic properties of ethylene based ionomer melts

The effect of ionic interaction on linear and nonlinear viscoelastic properties was investigated using poly(ethylene-co-methacrylic acid) (E/MAA) and its ionomers which were partially neutralized by zinc or sodium. Dynamic shear viscosity and step-shear stress relaxation studies were performed. Stress relaxation moduli G(t, y) of the E/MAA and its sodium or zinc ionomers were factorized into linear relaxation moduli G°(t) and damping functions h(y). The relaxation modulus at the smallest strain in each ionomer agreed with the linear relaxation modulus calculated from storage modulus G and loss modulus G. In the linear region, the ionic interaction shifted the relaxation time longer with keeping the same relaxation time distribution as E/MAA. In the nonlinear region, the ionic interaction had no influence on h(y) when the ion content was low. At higher ion content, however, the ion bonding enhanced the strain softening of h(y).     

Third-order polynomial model for analyzing stickup state laminated structure in flexible electronics

Laminated hard-soft integrated structures play a significant role in the fabrication and development of flexible electronics devices. Flexible electronics have advantageous characteristics such as soft and light-weight, can be folded, twisted, flipped inside-out, or be pasted onto other surfaces of arbitrary shapes. In this paper, an analytical model is presented to study the mechanics of laminated hard-soft structures in flexible electronics under a stickup state. Third-order polynomials are used to describe the displacement field, and the principle of virtual work is adopted to derive the governing equations and boundary conditions. The normal strain and the shear stress along the thickness direction in the bi-material region are obtained analytically, which agree well with the results from finite element analysis. The analytical model can be used to analyze stickup state laminated structures, and can serve as a valuable reference for the failure prediction and optimal design of flexible electronics in the future.     

Constitutive relationships for polymeric materials with power-law distributions of relaxation times

The linear relaxation modulus of polydisperse polymer melts and solutions can often be approximated by a power law,ct ^–m over some range of time,t. If, in addition, the nonlinear rheology is given by a separable integral equation, with a strain-dependent factor typical of those observed experimentally, then some commonly observed empirical rules and equations can be readily derived as approximations, namely the Cox-Merz relationship between complex viscosity and steady-state shear viscosity, Bersted's predictions of steady shear stress and first normal-stress difference from a truncated spectrum of linear relaxation times, and the observation of Koyama and coworkers that the ratio of the nonlinear to the linear time-dependent elongational viscosity is independent of strain rate, over a range of strain rates outside the linear regime.     

Interrelation of creep and relaxation for nonlinearly viscoelastic materials: application to ligament and metal

Creep and stress relaxation are known to be interrelated in linearly viscoelastic materials by an exact analytical expression. In this article, analytical interrelations are derived for nonlinearly viscoelastic materials which obey a single integral nonlinear superposition constitutive equation. The kernel is not assumed to be separable as a product of strain and time dependent parts. Superposition is fully taken into account within the single integral formulation used. Specific formulations based on power law time dependence and truncated expansions are developed. These are appropriate for weak stress and strain dependence. The interrelated constitutive formulation is applied to ligaments, in which stiffness increases with strain, stress relaxation proceeds faster than creep, and rate of creep is a function of stress and rate of relaxation is a function of strain. An interrelation was also constructed for a commercial die-cast aluminum alloy currently used in small engine applications.     

Simulation of the linear rheological behavior of silica-filled,star-shaped SBR compounds

The rheological behavior of star-shaped SSBR/silica 60 phr compounds with different filler surface areas was experimentally studied and simulated using constitutive modeling. Rheological behavior was characterized in small amplitude oscillatory shear (SAOS) and stress relaxation after a small step shear. Unfilled SBR and SBR filled with four different silica grades with BET surface areas of 55, 135, 160, and 195 m²/g were used. A clear trend in rheological behavior was observed with surface area. A frequency sweep in the SAOS regime indicated an increase in dynamic properties with surface area. Additionally, linear stress relaxation tests at a strain level of 0.05 showed an increase in relaxation modulus with surface area and the presence of a plateau in the relaxation modulus at large times in compounds containing silica with high surface areas. The Leonov and Simhambhatla-Leonov models, modified to incorporate multimode particle network relaxation, were successfully used to simulate the frequency dependence of the storage modulus and the time evolution of the linear relaxation modulus for all samples. However, simulations of the frequency dependence of the loss modulus showed poor results in comparison with experimental data for the filled compounds.