A modified couple stress model for bending analysis of composite laminated beams with first order shear deformation

Based on a modified couple stress theory, a model for composite laminated beam with first order shear deformation is developed. The characteristics of the theory are the use of rotation–displacement as dependent variable and the use of only one constant to describe the material’s micro-structural characteristics. The present model of beam can be viewed as a simplified couple stress theory in engineering mechanics. An example as a cross-ply simply supported beam subjected to cylindrical bending loads of f_w = q₀ sin (πx/L) is adopted and explicit expression of analysis solution is obtained. Numerical results show that the present beam model can capture the scale effects of microstructure, and the deflections and stresses of the present model of couple stress beam are smaller than that by the classical beam mode. Additionally, the present model can be reduced to the classical composite laminated Timoshenko beam model, Isotropic Timoshenko beam model of couple stress theory, classical isotropic Timoshenko beam, composite laminated Bernoulli–Euler beam model of couple stress theory and isotropic Bernoulli–Euler beam of couple stress theory.     

A model of composite laminated Reddy beam based on a modified couple-stress theory

In this paper, a new anisotropic constitutive relation based on a modified couple-stress theory is defined for composite laminated Reddy beam. The theory contains only one material length-scale parameter in each ply of the composite laminated beam. The example of a cross-ply simply supported beam subjected to transverse load q₀sin(πx/L) is presented. Numerical results show that the proposed beam model can capture the scale effect of the microstructure. The proposed model can be reduced to several models of the modified couple-stress theory by adopting the assumptions in Timoshenko beam, Bernoulli–Euler beam and material isotropy. It can be seen that the deflections and stresses obtained by the proposed beam model are smaller than those based on Timoshenko and Bernoulli–Euler beam assumptions.     

基于新修正偶应力的增强型Reddy层合板热尺度效应分析

基于新修正偶应力理论,提出复合材料增强型Reddy层合板热尺度效应模型.该模型只含有一个材料长度参数l,同时将首次引入厚度方向的旋转变量.通过虚功原理推导出平衡方程,并且利用纳维方法,分析热载作用下细观复合材料层合/夹层方板的位移和应力.数值计算表明:该模型能够很好地捕捉板的热尺度效应,随着材料长度参数增大,板的热尺度...     

A model of composite laminated Reddy plate of the global-local theory based on new modified couple-stress theory

In this article, based on the global-local theory, a model for a composite laminated Reddy plate of new modified couple-stress theory is developed for the first time. This model satisfies free surface conditions and the geometric and stresses continuity conditions at interfaces. Differing from existing modified couple-stress theory, an anisotropic constitutive relation is suggested. There is only one micro material characteristic length constant in each layer of the composite laminated plate. Principle of virtual work is employed to derive the equilibrium equations and the corresponding boundary conditions. With the example of a cross-ply simple-supported Reddy plate subjected to the bending load q₀ = sin?(πx/L)sin?(πy/L), the transverse shear stresses at interfaces are addressed. Additionally, the numerical results show that the present plate model can capture the scale effect, especially the scale effect of the transverse shear stresses at interfaces of the composite laminated plate.     

基于新修正偶应力理论的斜交铺设层合Kirchhoff板模型与尺度效应

基于新修正偶应力理论, 建立了只含有1个尺度参数的复合材料斜交铺设层合Kirchhoff板模型, 并分析了铺设角对尺度效应的影响。采用虚功原理推导了任意铺设角的层合Kirchhoff板的弯曲方程和边界条件。通过新模型给出了四边简支反对称角铺设微尺度层合Kirchhoff板的解析解。结果表明:细观尺度各向异性层合板的尺度效应并不仅仅受其几何尺寸的影响, 还受铺设角的影响;铺设角对尺度效应可能产生非常显著的影响。      

A nonlinear modified couple stress-based third-order theory of functionally graded plates

In this paper a general nonlinear third-order plate theory that accounts for (a) geometric nonlinearity, (b) microstructure-dependent size effects, and (c) two-constituent material variation through the plate thickness (i.e., functionally graded material plates) is presented using the principle of virtual displacements. A detailed derivation of the equations of motion, using Hamilton’s principle, is presented, and it is based on a modified couple stress theory, power-law variation of the material through the thickness, and the von Kármán nonlinear strains. The modified couple stress theory includes a material length scale parameter that can capture the size effect in a functionally graded material. The governing equations of motion derived herein for a general third-order theory with geometric nonlinearity, microstructure dependent size effect, and material gradation through the thickness are specialized to classical and shear deformation plate theories available in the literature. The theory presented herein also can be used to develop finite element models and determine the effect of the geometric nonlinearity, microstructure-dependent size effects, and material grading through the thickness on bending and postbuckling response of elastic plates.     

基于一种新修正偶应力理论的平面正交各向异性功能梯度梁静弯曲模型及尺度效应

基于一种新修正偶应力理论建立了微尺度平面正交各向异性功能梯度梁模型。模型中包含两个材料尺度参数,因此能够分别描述在两个正交方向上由尺度效应带来的不同大小弯曲刚度增强。基于最小势能原理推导了平衡方程和边界条件,并以自由端受集中载荷作用的悬臂梁为例给出了弯曲问题的解析解。该梁模型的控制方程以及解的形式和经典梁模型是一致的,只是在刚度项中增加了一项和尺度效应有关的项。算例结果表明:采用本文模型所预测的梁挠度总是小于经典理论的结果,即捕捉到了尺度效应。尺度效应会随着梁几何尺寸的减小而增大,并在梁的几何尺寸远大于尺度参数时逐渐消失。     

A non-classical Mindlin plate model based on a modified couple stress theory

A non-classical Mindlin plate model is developed using a modified couple stress theory. The equations of motion and boundary conditions are obtained simultaneously through a variational formulation based on Hamilton??s principle. The new model contains a material length scale parameter and can capture the size effect, unlike the classical Mindlin plate theory. In addition, the current model considers both stretching and bending of the plate, which differs from the classical Mindlin plate model. It is shown that the newly developed Mindlin plate model recovers the non-classical Timoshenko beam model based on the modified couple stress theory as a special case. Also, the current non-classical plate model reduces to the Mindlin plate model based on classical elasticity when the material length scale parameter is set to be zero. To illustrate the new Mindlin plate model, analytical solutions for the static bending and free vibration problems of a simply supported plate are obtained by directly applying the general forms of the governing equations and boundary conditions of the model. The numerical results show that the deflection and rotations predicted by the new model are smaller than those predicted by the classical Mindlin plate model, while the natural frequency of the plate predicted by the former is higher than that by the latter. It is further seen that the differences between the two sets of predicted values are significantly large when the plate thickness is small, but they are diminishing with increasing plate thickness.     

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 modified couple stress model for bending of normal deformable viscoelastic nanobeams resting on visco-Pasternak foundations

ABSTRACT

The modified couple stress theory (MCST) is utilized to investigate the bending of viscoelastic nanobeams laying on visco-Pasternak elastic foundations based on a new shear and normal deformations beam theory. This model consists of the material length scale coefficient that captures the size impact on small-scale beams. The simply supported beam is made of viscoelastic material, subjected to time harmonic transverse load. The nanobeam is presumed to be laying on double layers of foundations. The first layer is modeled as Kelvin–Voigt viscoelastic model and the second is taken as a shear layer. Based on the proposed beam theory and MCST, the differential motion equations are deduced using Hamilton’s principle. To check the validity of the obtained formulations, the predicted results are compared with those available in the open literature. In addition, the influences of various parameters such as the material length scale parameter, length-to-depth ratio, viscoelastic damping structure, the stiffness and damping coefficients of the viscoelastic substrate, and shear and normal strains on the deflection and stresses are illustrated.     

A non-classical third-order shear deformation plate model based on a modified couple stress theory

A non-classical third-order shear deformation plate model is developed using a modified couple stress theory and Hamilton’s principle. The equations of motion and boundary conditions are simultaneously obtained through a variational formulation. This newly developed plate model contains one material length scale parameter and can capture both the size effect and the quadratic variation of shear strains and shear stresses along the plate thickness direction. It is shown that the new third-order shear deformation plate model recovers the non-classical Reddy-Levinson beam model and Mindlin plate model based on the modified couple stress theory as special cases. Also, the current non-classical plate model reduces to the classical elasticity-based third-order shear deformation plate model when the material length scale parameter is taken to be zero. To illustrate the new model, analytical solutions for the static bending and free vibration problems of a simply supported plate are obtained by directly applying the general forms of the governing equations and boundary conditions of the model. The numerical results show that the deflection and rotations predicted by the new plate model are smaller than those predicted by its classical elasticity-based counterpart, while the natural frequency of the plate predicted by the former is higher than that by the latter. It is further seen that the differences between the two sets of predicted values are significant when the plate thickness is small, but they are diminishing with increasing plate thickness.     

冰雹冲击复合材料层合板失效模式分析与数值模拟

针对复合材料层合板结构,建立了冰雹冲击复合材料层合板的有限元模型,在充分考虑冲击过程中冰雹的流体特性下,给出了冰雹和复合材料层合板的材料模型和损伤准则,利用显式有限元分析工具LS-DYNA研究了不同冰雹冲击速度下复合材料层合板的临界破坏速度和破坏形式。结果表明,文中给出的冰雹、复合材料层合板的材料模型和损伤准则能够合理地再现冰雹冲击复合材料板的过程;复合材料层合板(AS4/8552)在冰雹高速冲击下首先发生的是基体开裂,当冰雹速度到达125m/s时,层合板上表面纤维发生断裂,但在整个冰雹冲击过程中层合板没有发生压缩失效     

A simple unsymmetric 4-node 12-DOF membrane element for the modified couple stress theory

In this work, the recently proposed unsymmetric 4-node 12-DOF (degree-of-freedom) membrane element (Shang and Ouyang, Int J Numer Methods Eng 113(10): 1589-1606, 2018), which has demonstrated excellent performance for the classical elastic problems, is further extended for the modified couple stress theory, to account for the size effect of materials. This is achieved via two formulation developments. Firstly, by using the penalty function method, the kinematic relations between the element's nodal drilling DOFs and the true physical rotations are enforced. Consequently, the continuity requirement for the modified couple stress theory is satisfied in weak sense, and the symmetric curvature test function can be easily derived from the gradients of the drilling DOFs. Secondly, the couple stress field that satisfies a priori the related equilibrium equations is adopted as the energy conjugate trial function to formulate the element for the modified couple stress theory. As demonstrated by a series of benchmark tests, the new element can efficiently capture the size-dependent responses of materials and is robust to mesh distortions. Moreover, as the new element uses only three conventional DOFs per node, it can be readily incorporated into the standard finite element program framework and commonly available finite element programs.     

A 3‐node C0 triangular element for the modified couple stress theory based on the smoothed finite element method

In this paper, a 3‐node C⁰ triangular element for the modified couple stress theory is proposed. Unlike the classical continuum theory, the second‐order derivative of displacement is included in the weak form of the equilibrium equations. Thus, the first‐order derivative of displacement, such as the rotation, should be approximated by a continuous function. In the proposed element, the derivative of the displacement is defined at a node using the node‐based smoothed finite element method. The derivative fields, continuous between elements and linear in an element, are approximated with the shape functions in element. Both the displacement field and the derivative field of displacement are expressed in terms of the displacement degree of freedom only. The element stiffness matrix is calculated using the newly defined derivative field. The performance of the proposed element is evaluated through various numerical examples.     

开孔碳纤维层合板层间应力分析

针对具有典型铺设角 的开孔碳纤维层合板, 采用三维有限元数值模拟方法, 分析了在单向拉伸载荷作用下孔边附近的层间应力, 讨论了界面层参数对层间应力的影响, 详细给出了典型铺设角之间层间应力的分布规律和最大层间应力产生的位置。结果表明: 对于相同铺设角的界面层, 沿厚度方向的位置影响层间应力的大小, 但不影响分布趋势; 而铺层顺序(如 或 )对层间应力的大小和分布趋势影响则较小。最大层间正应力产生于 的界面层, 位于与拉伸方向成90°的位置, 是外加拉伸应力的51%; 最大层间剪应力产生于 的界面层, 最大层间环向剪应力位于与拉伸方向成74°的位置, 是外加拉伸应力的64%; 而最大层间径向剪应力位于与拉伸方向成66°的位置, 是外加拉伸应力的25%。      

基于三维梁理论的复合材料层合管等效抗弯刚度

针对任意铺层形式和任意壁厚复合材料圆形层合管件,提出一种等效抗弯刚度的计算方法。此方法采用符合复合材料圆形管件梁真实变形的变形理论,考虑横向剪切变形,非均匀扭转效应,主、次挠曲效应和层合材料的三维弹性效应,按照壳壁中实际应力状态,建立了复合材料圆形管件等效抗弯刚度的计算模型。通过与4种铺层管件的三点弯曲实验结果以及经典层合板理论计算的等效抗弯刚度进行对比,验证了计算模型的正确性。通过退化与各向同性材料抗弯刚度的计算方法进行对比,分析了计算模型的适用性。      

复合材料层合板的混沌运动分析

建立了四边简支纤维增强复合材料层合薄板在受到x轴方向面内激励和横向激励联合作用下的动力学方程,考虑了复合材料层合板主参数共振和1:1内共振的情况,利用二阶Galerkin离散和多尺度方法得到四维直角坐标形式下复合材料层合板系统的平均方程。然后利用规范形理论化简平均方程得到简单规范形。应用Kovacic等人提出的高维Melnikov方法研究复合材料层合薄板的脉冲混沌运动,理论分析发现在参数激励和外激励作用下系统存在smale马蹄意义下的混沌运动并存在脉冲跳跃轨道。     

A nonlinear Timoshenko beam formulation based on the modified couple stress theory

This paper presents a nonlinear size-dependent Timoshenko beam model based on the modified couple stress theory, a non-classical continuum theory capable of capturing the size effects. The nonlinear behavior of the new model is due to the present of induced mid-plane stretching, a prevalent phenomenon in beams with two immovable supports. The Hamilton principle is employed to determine the governing partial differential equations as well as the boundary conditions. A hinged–hinged beam is chosen as an example to delineate the nonlinear size-dependent static and free-vibration behaviors of the derived formulation. The solution for the static bending is obtained numerically. The solution for the free-vibration is presented analytically utilizing the method of multiple scales, one of the perturbation techniques.     

Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams

A class of higher-order continuum theories, such as modified couple stress, nonlocal elasticity, micropolar elasticity (Cosserat theory) and strain gradient elasticity has been recently employed to the mechanical modeling of micro- and nano-sized structures. In this article, however, we address stability problem of micro-sized beam based on the strain gradient elasticity and couple stress theories, firstly. Analytical solution of stability problem for axially loaded nano-sized beams based on strain gradient elasticity and modified couple stress theories are presented. Bernoulli–Euler beam theory is used for modeling. By using the variational principle, the governing equations for buckling and related boundary conditions are obtained in conjunctions with the strain gradient elasticity. Both end simply supported and cantilever boundary conditions are considered. The size effect on the critical buckling load is investigated.     

Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory

Dynamic stability of microbeams made of functionally graded materials (FGMs) is investigated in this paper based on the modified couple stress theory and Timoshenko beam theory. This non-classical Timoshenko beam model contains a material length scale parameter and can interpret the size effect. The material properties of FGM microbeams are assumed to vary in the thickness direction and are estimated though Mori–Tanaka homogenization technique. The higher-order governing equations and boundary conditions are derived by using the Hamilton’s principle. The differential quadrature (DQ) method is employed to convert the governing differential equations into a linear system of Mathieu–Hill equations from which the boundary points on the unstable regions are determined by Bolotin’s method. Free vibration and static buckling are also discussed as subset problems. A parametric study is conducted to investigate the influences of the length scale parameter, gradient index and length-to-thickness ratio on the dynamic stability characteristics of FGM microbeams with hinged–hinged and clamped–clamped end supports. Results show that the size effect on the dynamic stability characteristics is significant only when the thickness of beam has a similar value to the material length scale parameter.