A generalized multilength scale nonlinear composite plate theory with delamination

A new type of plate theory for the nonlinear analysis of laminated plates in the presence of delaminations and other history-dependent effects is presented. The formulation is based on a generalized two length scale displacement field obtained from a superposition of global and local displacement effects. The functional forms of global and local displacement fields are arbitrary. The theoretical framework introduces a unique coupling between the length scales and represents a novel two length scale or local-global approach to plate analysis. Appropriate specialization of the displacement field can be used to reduce the theory to any currently available, variationally derived, displacement based (discrete layer, smeared, or zig-zag) plate theory.The theory incorporates delamination and/or nonlinear elastic or inelastic interfacial behavior in a unified fashion through the use of interfacial constitutive (cohesive) relations. Arbitrary interfacial constitutive relations can be incorporated into the theory without the need for reformulation of the governing equations. The theory is sufficiently general that any material constitutive model can be implemented within the theoretical framework. The theory accounts for geometric nonlinearities to allow for the analysis of buckling behavior.The theory represents a comprehensive framework for developing any order and type of displacement based plate theory in the presence of delamination, buckling, and/or nonlinear material behavior as well as the interactions between these effects.The linear form of the theory is validated by comparison with exact solutions for the behavior of perfectly bonded and delaminated laminates in cylindrical bending. The theory shows excellent correlation with the exact solutions for both the inplane and out-of-plane effects and the displacement jumps due to delamination. The theory can accurately predict the through-the-thickness distributions of the transverse stresses without the need to integrate the pointwise equilibrium equations. The use of a low order of the general theory, i.e. use of both global and local displacement fields, reduces the computational expense compared to a purely discrete layer approach to the analysis of laminated plates without loss of accuracy. The increased efficiency, compared to a solely discrete layer theory, is due to the coupling introduced in the theory between the global and local displacement fields.     

A new modified couple stress theory for anisotropic elasticity and microscale laminated Kirchhoff plate model

A new modified couple stress theory for anisotropic elasticity is proposed. This theory contains three material length scale parameters. Differing from the modified couple stress theory, the couple stress constitutive relationships are introduced for anisotropic elasticity, in which the curvature (rotation gradient) tensor is asymmetric and the couple stress moment tensor is symmetric. However, under isotropic case, this theory can be identical to modified couple stress theory proposed by Yang et al. (Int J Solids Struct 39:2731–2743, 2002). The differences and relations of standard, modified and new modified couple stress theories are given herein. A detailed variational formulation is provided for this theory by using the principle of minimum total potential energy. Based on the new modified couple stress theory, composite laminated Kirchhoff plate models are developed in which new anisotropic constitutive relationships are defined. The First model contains two material length scale parameters, one related to fiber and the other related to matrix. The curvature tensor in this model is asymmetric; however, the couple stress moment tensor is symmetric. Under isotropic case, this theory can be identical to the modified couple stress theory proposed by Yang et al. (Int J Solids Struct 39:2731–2743, 2002). The present model can be viewed as a simplified couple stress theory in engineering mechanics. Moreover, a more simplified model of couple stress theory including only one material length scale parameter for modeling the cross-ply laminated Kirchhoff plate is suggested. Numerical results show that the proposed laminated Kirchhoff plate model can capture the scale effects of microstructures.     

A new simple third-order shear deformation theory of plates

An improved simple third-order shear deformation theory for the analysis of shear flexible plates is presented in this paper. This new plate theory is composed of three parts: the simple third-order kinematics of displacements reduced from the higher-order displacement field derived previously by the author; a system of 10th-order differential equilibrium equations in terms of the three generalized displacements of bending plates; five boundary conditions at each edge of plate boundaries. Although the resulting displacement field is the same as that proposed by Murthy, the variational consistent governing equations and the associated proper boundary conditions are derived and identified in this work for the first time in the literature. The applications and accuracy of the present shear deformation theory of plates are demonstrated by analytically solving the differential governing equations of a twisting plate, a bending beam and two bending plates to which the 3-D elasticity solutions are available, and excellent agreements are achieved even for the torsion of a plate with square cross-section as well the local effects of stresses at plate boundaries can be characterized accurately. These analytical solutions clearly show that the simple third-order shear deformation theory developed in this work indeed gives better results than the first-order shear deformation theories and other simple higher-order shear deformation theories, since the present third-order shear flexible theory is based on a more rigorous kinematics of displacements and consists of not only a system of variational consistent differential equations, but also a group of consistent boundary conditions associated with the differential equations. The present simple third-order shear deformation theory can easily be applied to the static and dynamic finite element analysis of laminated plates just like the applications of other popular shear flexible plate theories, and improved results could be obtained from the present simple third-order shear deformable theories of plates.     

Dynamic analysis of composite plate with multiple delaminations based on higher-order zigzag theory

In a recent paper, Cho and Kim [Journal of Applied Mechanics] proposed a higher-order cubic zigzag theory of laminated composites with multiple delaminations. The proposed theory is not only accurate but also efficient because it work with a minimal number of degrees of freedom with the application of interface continuity conditions as well as bounding surface conditions of transverse shear stresses including delaminated interfaces. In this work, we investigate the dynamic behavior of laminated composite plates with multiple delaminations. A four-node finite element based on the efficient higher-order zigzag plate theory of laminated composite plates with multiple delaminations is developed to refine the prediction of frequencies, mode shape, and time response. Through the dynamic version of the variational approach, the dynamic equilibrium equations and variationally consistent boundary conditions are obtained. Natural frequency prediction and time response analysis of a composite plate with multiple delaminations demonstrate the accuracy and efficiency of the present finite element method. To prevent penetration violation at the delamination interfaces, unilateral contact constraints by Lagrange multiplier method are applied in the time response analysis. The present finite element is suitable for the prediction of dynamic response of thick composite plates with multiple and arbitrary shaped delaminations.     

Nonlinear dynamic response of piezoelastic laminated plates considering interfacial damage effects

This paper presents a nonlinear model for cross-ply piezoelastic laminated plates containing the damage effect of the intralayer materials and interlaminar interfaces. The model is based on the general six-degrees-of-freedom plate theory, the discontinuity of displacement, and electric potential on the interfaces are depicted by three shape functions, which are formulated according to solutions about three equilibrium equations and conservation of charge. By using the Hamilton variation principle, the three-dimensional nonlinear dynamic equations of piezoelastic laminated plates with damage are presented. Then using the finite difference method and the Newmark scheme, an analytical solution is presented. In numerical examples, the effects of different damage models, damage evolution, amplitude and frequency of electric loads on the nonlinear dynamic response of piezoelectric laminated plate with interfacial imperfections are investigated.     

Buckling and post-buckling of laminated composite plates

This paper addresses the buckling and post-buckling of laminated composite plates using higher order shear deformation theory associated with Green–Lagrange non-linear strain–displacement relationships. All higher order terms arising from nonlinear strain–displacement relations are included in the formulation. The present plate theory satisfies zero transverse shear strain conditions at the top and bottom surfaces of the plate in von Karman sense. A C⁰ isoparametric finite element is developed for the present nonlinear model.     

脱层梁屈曲的高阶剪切理论

脱层的存在将会大大降低层合结构的屈曲载荷。该文将含任意位置脱层的两端固支梁分成多段子层,用厚度的三次多项式模拟脱层梁屈曲时子层的轴向位移,利用变分原理和欧拉方程导出了脱层梁的屈曲方程和定解条件,并用状态空间方法进行求解。通过与一阶剪切理论和经典理论的比较,指出了它们各自的适用范围;考虑了脱层梁三种不同的屈曲模态。分析了脱层长度、深度、位置和材料的铺层方向对脱层梁屈曲载荷的影响;最后给出了多处简单脱层的屈曲分析。     

高阶剪切变形理论下三边夹紧一边铰支复合材料层板的几何非线性分析

首先用虚位移原理推导出以位移形式表达的Reddy型高阶剪变形理论复合材料层板的非线性控制方程及相应的边界条件。选定的五个位移函数均满足三边夹紧一边铰支边界条件，用Galerkin方法把无量纲化之后的控制方程转化为一组非线性代数方程组，用线性化的方法和可调节参数的修正迭代法求解这组方程。最后求出了不同复合材料的挠度和弯矩值。     

ON THE REFINED FIRST-ORDER SHEAR DEFORMATION PLATE THEORY OF KARMAN TYPE

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非线性压电效应下压电层合板的弯曲

考虑非线性压电效应,即电致弹性和电致伸缩效应情况下压电层合板的弯曲。从非线性压电方程和几何方程导出了压电层合板合应力、合力矩与应变之间的广义本构关系,这些关系关于电场是非线性的。利用Ritz法和双傅立叶级数得到四边简支对称压电层合板在高电场作用下的非线性解并进行计算。结果表明,只考虑线性压电效应只能适应于作用电场较低或基础层的刚度比压电层的刚度要大得多的情况。     

Shear deformable composite plates with nonlinear curvatures: modeling and nonlinear vibrations of symmetric laminates

Moving from a general plate theory, a modified general shear deformable laminated plate theory (MGFSDT) exhibiting nonlinear curvatures but still allowing for some worth features of linear curvature models (von Karman) is formulated. Starting from MGFSDT partial differential equations, a minimal discretized model (Duffing equation) for symmetric cross-ply laminates nonlinear vibrations, whose coefficients account for shear deformability and nonlinear curvatures, is obtained via the Galerkin procedure. The variable features of such a Duffing model as obtainable via alternative kinematic assumptions at the continuum level are highlighted. Through the comparison of a number of underlying models in different technical situations, information on the influence of shear deformability on system nonlinear response and on the influence of nonlinear curvatures are obtained. Frequency–response curves through a multiple scale analysis are presented for different continuous models, kinds of material, mode numbers and boundary conditions.     

A new trigonometric shear deformation theory for isotropic,laminated composite and sandwich plates

A new trigonometric shear deformation theory for isotropic and composite laminated and sandwich plates, is developed. The new displacement field depends on a parameter “m”, whose value is determined so as to give results closest to the 3D elasticity bending solutions. The theory accounts for adequate distribution of the transverse shear strains through the plate thickness and tangential stress-free boundary conditions on the plate boundary surface, thus a shear correction factor is not required. Plate governing equations and boundary conditions are derived by employing the principle of virtual work. The Navier-type exact solutions for static bending analysis are presented for sinusoidally and uniformly distributed loads. The accuracy of the present theory is ascertained by comparing it with various available results in the literature. The results show that the present model performs as good as the Reddy’s and Touratier’s shear deformation theories for analyzing the static behavior of isotropic and composite laminated and sandwich plates.     

Nonlinear bending of size-dependent circular microplates based on the modified couple stress theory

The present study proposes a nonclassical Kirchhoff plate model for the axisymmetrically nonlinear bending analysis of circular microplates under uniformly distributed transverse loads. The governing differential equations are derived from the principle of minimum total potential energy based on the modified couple stress theory and von Kármán geometrically nonlinear theory in terms of the deflection and radial membrane force, with only one material length scale parameter to capture the size-dependent behavior. The governing equations are firstly discretized to a set of nonlinear algebraic equations by the orthogonal collocation point method, and then solved numerically by the Newton–Raphson iteration method to obtain the size-dependent solutions for deflections and radial membrane forces. The influences of length scale parameter on the bending behaviors of microplates are discussed in detail for immovable clamped and simply supported edge conditions. The numerical results indicate that the microplates modeled by the modified couple stress theory causes more stiffness than modeled by the classical continuum plate theory, such that for plates with small thickness to material length scale ratio, the difference between the results of these two theories is significantly large, but it becomes decreasing or even diminishing with increasing thickness to length scale ratio.     

Analysis on nonlinear dynamics of a deploying composite laminated cantilever plate

This paper presents the analysis on the nonlinear dynamics of a deploying orthotropic composite laminated cantilever rectangular plate subjected to the aerodynamic pressures and the in-plane harmonic excitation. The third-order nonlinear piston theory is employed to model the transverse air pressures. Based on Reddy’s third-order shear deformation plate theory and Hamilton’s principle, the nonlinear governing equations of motion are derived for the deploying composite laminated cantilever rectangular plate. The Galerkin method is utilized to discretize the partial differential governing equations to a two-degree-of-freedom nonlinear system. The two-degree-of-freedom nonlinear system is numerically studied to analyze the stability and nonlinear vibrations of the deploying composite laminated cantilever rectangular plate with the change of the realistic parameters. The influences of different parameters on the stability of the deploying composite laminated cantilever rectangular plate are analyzed. The numerical results show that the deploying velocity and damping coefficient have great effects on the amplitudes of the nonlinear vibrations, which may lead to the jumping phenomenon of the amplitudes for first-order and second-order modes. The increase of the damping coefficient can suppress the increase of the amplitudes of the nonlinear vibration.     

基于各向异性修正偶应力理论的Reddy型层合板的自由振动

本文基于各向异性修正偶应力理论建立了只含一个尺度参数的Reddy型复合材料层合板的自由振动模型。同见诸于文献的细观尺度Kirchhoff薄板偶应力模型相比,本文提出的新模型能够更精确的预测细观尺度下的中、厚层合板的自振频率。基于Hamilton原理推导了细观尺度下Reddy型复合材料层合板的运动微分方程以及边界条件,并以正交铺设的四边简支复合材料层合方板为例进行了解析求解,分析了尺度参数对自振频率的影响并对比了Kirchhoff、Mindlin和Reddy等三种板模型计算结果的异同。算例结果表明本文所给出的模型能够捕捉到复合材料层合板自由振动问题的尺度效应。另外,在细观尺度下Kirchhoff板模型所预测的自振频率相对于Mindlin板模型和Reddy板模型总是过高,且越接近厚板三者的差别就越大,这与经典理论中三种板模型的对比情况是一致的。     

On the Generalization of Reissner Plate Theory to Laminated Plates,Part I: Theory

This is the first part of a two-part paper presenting the generalization of Reissner thick plate theory (Reissner in J. Math. Phys. 23:184–191, 1944) to laminated plates and its relation with the Bending-Gradient theory (Lebée and Sab in Int. J. Solids Struct. 48(20):2878–2888, 2011 and in Int. J. Solids Struct. 48(20):2889–2901, 2011). The original thick and homogeneous plate theory derived by Reissner (J. Math. Phys. 23:184–191, 1944) is based on the derivation of a statically compatible stress field and the application of the principle of minimum of complementary energy. The static variables of this model are the bending moment and the shear force. In the present paper, the rigorous extension of this theory to laminated plates is presented and leads to a new plate theory called Generalized-Reissner theory which involves the bending moment, its first and second gradients as static variables. When the plate is homogeneous or functionally graded, the original theory from Reissner is retrieved. In the second paper (Lebée and Sab, 2015), the Bending-Gradient theory is obtained from the Generalized-Reissner theory and comparison with an exact solution for the cylindrical bending of laminated plates is presented.     

A unified nonlinear formulation for plate and shell theories

A general geometrically exact nonlinear theory for the dynamics of laminated plates and shells under-going large-rotation and small-strain vibrations in three-dimensional space is presented. The theory fully accounts for geometric nonlinearities by using the new concepts of local displacements and local engineering stress and strain measures, a new interpretation and manipulation of the virtual local rotations, an exact coordinate transformation, and the extended Hamilton principle. Moreover, the model accounts for shear coupling effects, continuity of interlaminar shear stresses, free shear-stress conditions on the bonding surfaces, and extensionality. Because the only differences among different plates and shells are the initial curvatures of the coordinates used in the modeling and all possible initial curvatures are included in the formulation, the theory is valid for any plate or shell geometry and contains most of the existing nonlinear and shear-deformable plate and shell theories as special cases. Five fully nonlinear partial-differential equations and corresponding boundary and corner conditions are obtained, which describe the extension-extension-bending-shear-shear vibrations of general laminated two-dimensional structures and display linear elastic and nonlinear geometric coupling among all motions. Moreover, the energy and Newtonian formulations are completely correlated in the theory.     

MULTI-PULSE ORBITS AND CHAOTIC DYNAMICS OF A COMPOSITE LAMINATED RECTANGULAR PLATE

This paper presents an analysis on the nonlinear dynamics and multi-pulse chaotic motions of a simply-supported symmetric cross-ply composite laminated rectangular thin plate with the parametric and forcing excitations. Firstly, based on the Reddy’s third-order shear deformation plate theory and the model of the von Karman type geometric nonlinearity, the nonlinear governing partial difirential equations of motion for the composite laminated rectangular thin plate are derived by using the Hamilton’s principle. Then, using the second-order Galerkin discretization, the partial differential governing equations of motion are transformed to nonlinear ordinary differential equations. The case of the primary parametric resonance and 1:1 internal resonance is considered. Four-dimensional averaged equation is obtained by using the method of multiple scales. From the averaged equation obtained here, the theory of normal form is used to give the explicit expressions of normal form. Based on normal form, the energy phase method is utilized to analyze the global bifurcations and multi-pulse chaotic dynamics of the composite laminated rectangular thin plate. The theoretic results obtained above illustrate the existence of the chaos for the Smale horseshoe sense in a parametrical and forcing excited composite laminated thin plate. The chaotic motions of the composite laminated rectangular thin plate are also found by using numerical simulation, which also indicate that there exist different shapes of the multi-pulse chaotic motions for the composite laminated rectangular thin plate.     

On the refined first-order shear deformation plate theory of Kármán type

A new refined first-order shear-deformation plate theory of the Kármán type is presented for engineering applications and a new version of the generalized Kármán large deflection equations with deflection and stress functions as two unknown variables is formulated for nonlinear analysis of shear-deformable plates of composite material and construction, based on the Mindlin/Reissner theory. In this refined plate theory two rotations that are constrained out in the formulation are imposed upon overall displacements of the plates in an implicit role. Linear and nonlinear investigations may be made by the engineering theory to a class of shear-deformation plates such as moderately thick composite plates, orthotropic sandwich plates, densely stiffened plates, and laminated shear-deformable plates. Reduced forms of the generalized Kármán equations are derived consequently, which are found identical to those existe in the literature. Foundation item: the National Natural Science Foundation of China (59675027) Biography: Zhang Jianwu (1954-)     

Nonlinear vibrations of a composite laminated cantilever rectangular plate with one-to-one internal resonance

The nonlinear vibrations of a composite laminated cantilever rectangular plate subjected to the in-plane and transversal excitations are investigated in this paper. Based on the Reddy??s third-order plate theory and the von Karman type equations for the geometric nonlinearity, the nonlinear partial differential governing equations of motion for the composite laminated cantilever rectangular plate are established by using the Hamilton??s principle. The Galerkin approach is used to transform the nonlinear partial differential governing equations of motion into a two degree-of-freedom nonlinear system under combined parametric and forcing excitations. The case of foundational parametric resonance and 1:1 internal resonance is taken into account. The method of multiple scales is utilized to obtain the four-dimensional averaged equation. The numerical method is used to find the periodic and chaotic motions of the composite laminated cantilever rectangular plate. It is found that the chaotic responses are sensitive to the changing of the forcing excitations and the damping coefficient. The influence of the forcing excitation and the damping coefficient on the bifurcations and chaotic behaviors of the composite laminated cantilever rectangular plate is investigated numerically. The frequency-response curves of the first-order and the second-order modes show that there exists the soft-spring type characteristic for the first-order and the second-order modes.