Transient response of FG higher-order nanobeams integrated with magnetostrictive layers using modified couple stress theory

The present paper studies the transient response of a functionally graded nanobeam integrated with magnetostrictive layers. The material properties of sandwich nanobeam are temperature dependent and assumed to vary in the thickness direction. In order to consider small-scale effects, the modified couple stress theory is also taken into consideration. Using a unified beam theory that contains various beam models and energy method as well as Hamilton's principle, the governing motion equations and related boundary conditions are obtained. The obtained results in this paper can be used as sensors and actuators in sensitive applications.     

复合材料层合梁接触问题的线性化与能量解法

本文将非线性的接触问题线性化,即假设已知带可调参数的接触区载荷分布模态以及接触区宽度,再反求圆柱压头的半径。首先根据叠加原理将受力状态分解成对称和反对称状态,然后用正交完备的三角级数和勒让德级数构造这两种受力状态的位移场,并应用最小势能原理确定位移场中的待定系数,从而确定层合梁的位移场和应力场。载荷分布模态中的可调参数可根据接触区表面的位移协调条件确定,从而求得圆柱压头的半径。最后,由于在给定接触区的条件下压头曲率与压头合力成正比,故可以得到压头曲率在不同的接触区下随载荷变化的直线族。根据这些直线族,可以由已知的压头曲率和外载荷确定接触区尺寸。计算结果显示,这种解法的收敛性非常好。根据物理方程与根据平衡方程得到的层合梁剪应力分布的一致性非常好。而且,在远离接触区处的应力分布同经典层合梁理论结果的一致性也非常好。      

Buckling characteristics of nanocrystalline nano-beams

The buckling and the postbuckling characteristics of nanocrystalline nano-beams with/without surface stress residuals are investigated. A hybrid model is proposed where a non-classical beam model is incorporated with a size-dependent micromechanical model. The micromechanical model has the merit of accounting for the beam material structure effects, i.e. the grain size and the grain boundary effects. To account for the beam size effects, the couple stress theory is implemented where some measures are added to capture the grain rigid rotation effects. The proposed hybrid model is harnessed to derive the governing equations of a nano-beam subjected to an axial compressive load accounting for the mid-plane stretching according to von-Karman kinematics and the surface stress residuals. Analytical solutions for the prebuckling and postbuckling configurations and natural frequencies as functions of the applied compressive axial load are derived. The effects of the beam material structure and the beam size on the beam’s prebuckling characteristics and the postbuckling configurations and natural frequencies are studied. The obtained results reveal that both the size and the material structure of nanobeams have great impacts on their buckling characteristics.     

考虑应变梯度效应的三点弯梁模型解析研究--第二部分:局部化带的位移及转角

基于梯度塑性理论,分析了应变软化及真实裂纹扩展阶段的局部化带的张拉位移和转角。在弹性阶段,可以由弹性理论来确定二者的关系。真实裂纹出现后,利用平衡条件、几何条件及梯度以来的应变软化本构关系,得到了真实裂纹长度与局部化带长度的关系。当真实裂纹刚出现时,局部化带长度达到最大值。在任何阶段,局部化带到中性轴的距离单调降低,局部化带的张拉位移和转角受梁深、带宽、弹模及下降模量等的影响。弹模及下降模量越大,带宽越小,则局部化带的张拉位移和转角都增加。而且,在前两个阶段,张拉位移都线性增加,但在后两个阶段,转角都非线性增加。     

The modified couple stress functionally graded Timoshenko beam formulation

In this paper, a size-dependent formulation is presented for Timoshenko beams made of a functionally graded material (FGM). The formulation is developed on the basis of the modified couple stress theory. The modified couple stress theory is a non-classic continuum theory capable to capture the small-scale size effects in the mechanical behavior of structures. The beam properties are assumed to vary through the thickness of the beam. The governing differential equations of motion are derived for the proposed modified couple-stress FG Timoshenko beam. The generally valid closed-form analytic expressions are obtained for the static response parameters. As case studies, the static and free vibration of the new model are respectively investigated for FG cantilever and FG simply supported beams in which properties are varying according to a power law. The results indicate that modeling beams on the basis of the couple stress theory causes more stiffness than modeling based on the classical continuum theory, such that for beams with small thickness, a significant difference between the results of these two theories is observed.     

ㄇ形梁剪力滞效应的解析解

精确分析ㄇ形梁在纵横向荷载共同作用下,其横断面上正应力分布规律对于计算其有效宽度有重要意义,应用力法原理,先将ㄇ形梁和翼板截开成矩形截面梁和平面应力板,在截面上代之以赘余的分布剪力,对于平面应力板,通过利用板变形的对称性来简化其边界条件,然后假设一个满足板的控制方程的Airy应力函数求得板的应力和位移,再利用Timoshenko梁理论求得梁的挠度和转角,根据截面上梁与板的纵向位移相等的变形协调条件便可最终确定截面上的分布剪力,给出的数值算例验证了方法的有效性,并与铜陵长江公路大桥主梁的模型有限元结果和试验结果作了对比,解析解法还可用来检验其他各种数值计算方法的精度,并可推广到其他多跨薄壁结构梁桥的膜应力分析中。     

复合材料叠层梁的冲击响应特性

用平面应力模型来模拟复合材料叠层梁，得到了正交各向异性叠层梁的频率和模态函数，并用间接模态叠加法（IMSM）求得了叠层梁的瞬态响应。针对正交各向异性单层梁，把研究结果与Timoshenko梁结果进行了的比较，说明了厚度效应对冲击响应的重要影响。对叠层梁受冲击过程中其内部的应力特征及波传播现象进行了分析。     

Free vibration of magneto-electro-elastic nanobeams based on modified couple stress theory in thermal environment

In this paper, the size-dependent free vibration of magneto-electro-elastic (MEE) nanobeams in thermal environment is investigated. Size effects are taken into account using the modified couple stress theory, which is capable of accounting for higher-order electromechanical coupling, and the equations are developed on the basis of Euler–Bernoulli beam model and using von Karman nonlinear strain. The vibration of hinged–hinged nanobeams is investigated by way of example. Effects of various parameters such as temperature, thickness, and length on natural frequencies are demonstrated, and it is indicated that increased length and decreased thickness lead to decreased nanobeam natural frequencies.     

The calculation of adhesive fracture energies in mode I: revisiting the tapered double cantilever beam (TDCB) test

Analytical corrections have been derived for a beam theory analysis for the adhesively bonded tapered double cantilever beam test specimen to account for the effects of beam root rotation and for the real, as opposed to idealised, profile of the beam as required experimentally. A number of adhesive-substrate combinations were tested according to a new test protocol and the new analysis method for data reduction is compared critically with the existing simple beam theory and experimental compliance approaches. Correcting the beam theory for root rotation effects is shown to be more important than correcting only for the effects of shear deformation of the substrates. Results from a finite element analysis, using a cohesive zone model, also showed close agreement with the proposed new corrected beam theory analysis method.     

Stress intensity factor solutions for estimation of fatigue lives of laser welds in lap-shear specimens

Fatigue behavior of laser welds in lap-shear specimens of high strength low alloy (HSLA) steel is investigated based on experimental observations and two fatigue life estimation models. Fatigue experiments of laser welded lap-shear specimens are first reviewed. Analytical stress intensity factor solutions for laser welded lap-shear specimens based on the beam bending theory are derived and compared with the analytical solutions for two semi-infinite solids with connection. Finite element analyses of laser welded lap-shear specimens with different weld widths were also conducted to obtain the stress intensity factor solutions. Approximate closed-form stress intensity factor solutions based on the results of the finite element analyses in combination with the analytical solutions based on the beam bending theory and Westergaard stress function for a full range of the normalized weld widths are developed for future engineering applications. Next, finite element analyses for laser welded lap-shear specimens with three weld widths were conducted to obtain the local stress intensity factor solutions for kinked cracks as functions of the kink length. The computational results indicate that the kinked cracks are under dominant mode I loading conditions and the normalized local stress intensity factor solutions can be used in combination with the global stress intensity factor solutions to estimate fatigue lives of laser welds with the weld width as small as the sheet thickness. The global stress intensity factor solutions and the local stress intensity factor solutions for vanishing and finite kinked cracks are then adopted in a fatigue crack growth model to estimate the fatigue lives of the laser welds. Also, a structural stress model based on the beam bending theory is adopted to estimate the fatigue lives of the welds. The fatigue life estimations based on the kinked fatigue crack growth model agree well with the experimental results whereas the fatigue life estimations based on the structural stress model agree with the experimental results under larger load ranges but are higher than the experimental results under smaller load ranges.     

Dynamic impact of an elastically supported beam—large area contact

The dynamic contact problem of a rigid, smooth striker impacting an elastically supported beam is solved. Use is made of the superposition of an elastic layer solution together with an elementary beam theory solution that incorporates the dynamic effects. The problem is formulated in such a manner as to require the solution of a Volterra integral equation for each time increment. Numerical results are presented for various parameters.     

Anticlastic effects and the transition from narrow to wide behavior in orthotropic beams

The classical assumptions of standard beam theory include that of uniaxial stress for isotropic beams, or vanishing transverse moment for beams of layered orthotropic materials. Wide beams, on the other hand, display the familiar effect in which the transverse “anticlastic” curvature is assumed to vanish. The transition between these behaviors is not well characterized, and is addressed in the present work by means of a solution for orthotropic plates with two free lateral edges, so as to represent a beam of any aspect ratio. It is found that the transition is independent of the thickness, but depends on the length/width aspect ratio and the material properties. The calculations show that the deviation from narrow beam behavior occurs at a length/width aspect ratio of about 10 for isotropic and many orthotropic materials, and depends on the orthotropic material properties. Comparisons with experimental data in the literature support these conclusions.     

Postbuckling characteristics of nanobeams based on the surface elasticity theory

In the present paper, an attempt is made to numerically investigate the postbuckling response of nanobeams with the consideration of the surface stress effect. To accomplish this, the Gurtin–Murdoch elasticity theory is exploited to incorporate surface stress effect into the classical Euler–Bernoulli beam theory. The size-dependent governing differential equations are derived and discretized along with various end supports by employing the principle of virtual work and the generalized differential quadrature (GDQ) method. Newton’s method is applied to solve the discretized nonlinear equations with the aid of an auxiliary normalizing equation. After solving the governing equations linearly, to obtain each eigenpair in the nonlinear model, the linear response is used as the initial value in Newton’s method. Selected numerical results are given to show the surface stress effect on the postbuckling characteristics of nanobeams. It is found that by increasing the thickness of nanobeams, the postbuckling equilibrium path obtained by the developed non-classical beam model tends to the one predicted by the classical beam theory and this anticipation is the same for all selected boundary conditions.     

基于有限元法的周期压电智能梁滤波特性分析

周期结构具有通频和禁频特性,使其在动态载荷的滤波器、具有主动控制功能的结构研究中得到了重要应用。基于Timoshenko梁理论,考虑基梁和压电片的转动惯量和剪切效应,采用有限元法和传递矩阵法推导了波在周期性地粘贴压电片的Timoshenko梁中的传播模型,分析了几何尺寸和材料特性对其频带性质的影响,并与Bernoulli-Euler梁理论得到的结果进行了对比。研究表明,当基梁与压电层厚度比达到40时,禁带带宽减小了54%,因此对于周期结构中的深梁,应舍弃Bernoulli-Euler梁理论而采用Timoshenko梁理论建立的模型;对于不同尺寸和材料特性的压电周期结构,频带性质会有很大不同,可以通过调整结构的参数来改变其频带性质,从而改变波动在结构中的传播特性。     

An improved theory and finite-element model for laminated composite and sandwich beams using first-order zig-zag sublaminate approximations

A new beam finite element based on a new discrete-layer laminated beam theory with sublaminate first-order zig-zag kinematic assumptions is presented and assessed for thick and thin laminated beams. The model allows a laminate to be represented as an assemblage of sublaminates in order to increase the model refinement through the thickness, when needed. Within each sublaminate, discrete-layer effects are accounted for via a modified form of DiSciuva's linear zig-zag laminate kinematics, in which continuity of interfacial transverse shear stresses is satisfied identically. In the computational model, each finite element represents one sublaminate. The finite element is developed with the topology of a fournoded rectangle, allowing the thickness of the beam to be discretized into several elements, or sublaminates, if necessary, to improve accuracy. Each node has three engineering degrees of freedom, two translations and one rotation. Thus, this element can be conveniently implemented into general purpose finite-element codes. The element stiffness coefficients are integrated exactly, yet the element exhibits no shear locking due to the use of a consistent interdependent interpolation scheme. Numerical performance of the current element is investigated for an arbitrarily layered beam, a symmetrically layered beam and a sandwich beam with low and high aspect ratios. The comparisons of numerical results with elasticity solutions show that the element is very accurate and robust.     

渐开线齿轮的接触分析

以Hertz应力表达式为基础,将单齿啮合的渐开线齿轮等效为相互挤压的两个圆柱体,推导了齿轮接触应力的理论表达式．建立一套单齿对啮合的有限元分析方法,包括引入渐开线和齿根过渡曲线方程及对应的自变量区间,建立参数化齿廓;模型轮缘厚度取3倍的模数,周向宽度取3倍的齿厚;在接触面上进行网格细化处理;在对应的主、从动齿轮内缘分别施加均匀切向力和固定约束等．计算结果显示,有限元解和理论解吻合较好,最大偏差不超过5%,该方法适用于不同的啮合轮齿参数．由有限元方法得到的渐开线轮齿的接触应力符合Hertz理论中的半椭圆分布规律,Mises应力和剪应力分布也符合接触力学理论,但应力分布的对称中心存在偏离或者偏斜现象,分析认为,是由于轮齿弯曲变形造成的．     

Buckling of functionally graded circular columns including shear deformation

An analysis of the stability of circular cylindrical columns/beams composed of functionally graded materials is made where shear deformation is taken into account. In this study, a new approach is carried out. Different from the assumption of uniform shear stress at the cross-section adopted in the Timoshenko beam theory, proposed model provides a new approach for treating the problem. Based on the traction-free surface condition, two coupled governing equations for the deflection and rotation are derived, and a single governing equation is further obtained. A comparison of buckling loads derived from the proposed circular column model and the Timoshenko and Euler–Bernoulli theories of beams is made. Moreover, the effects of radial gradient on buckling loads of elastic columns with circular cross-section made of functionally graded materials are elucidated. Finally, the stability of double-walled carbon nanotubes is considered and critical stress is determined and compared with existing results. The results obtained by the proposed model show very good agreement with the results of the Timoshenko beam theory or Reddy–Bickford beam theory.     

双层微梁固有特性的尺寸效应

微尺寸梁存在明显尺寸效应,应变梯度理论可以描述这种尺寸效应。该文基于修正偶应力理论,应用双层梁与单层梁的等效关系,给出了双层微梁的动力学模型,具体求解了简支双层微梁的固有频率,并分析了微梁特征尺寸及双材料参数对双层微梁固有特性的影响规律。结果表明,当双层微梁的厚度接近材料内秉特征尺寸参数时,其固有频率值明显大于传统理论下的值;当双层微梁的厚度远大于材料内秉特征尺寸时,其固有频率值与传统理论下的值基本一致。双层微梁无量纲固有频率表现出明显尺寸效应,并随双材料参数的改变表现出一定的差异。当一层梁厚度远大于另一层厚度时,双层微梁可简化为单层微梁。     

Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory

Investigated herein is the free vibration characteristics of microbeams made of functionally graded materials (FGMs) based on the strain gradient Timoshenko beam theory. The material properties of the functionally graded beams are assumed to be graded in the thickness direction according to the Mori–Tanaka scheme. Using Hamilton’s principle, the equations of motion together with corresponding boundary conditions are obtained for the free vibration analysis of FGM microbeams including size effect. A detailed parametric study is performed to indicate the influences of beam thickness, dimensionless length scale parameter, and slenderness ratio on the natural frequencies of FGM microbeams. Moreover, a comparison between the various beam models on the basis of the classical theory (CT), modified couple stress theory (MCST), and strain gradient theory (SGT) is presented for different values of material property gradient index. It is observed that the value of gradient index play an important role in the vibrational response of the microbeams of lower slenderness ratios. It is further observed that by increasing the length-to-thickness ratio of the microbeam, the value of dimensionless natural frequency tends to decrease for all amounts of the gradient index.