Hybrid‐stabilized solid‐shell model of laminated composite piezoelectric structures under non‐linear distribution of electric potential through thickness

Eighteen‐node solid‐shell finite element models have been developed for the analysis of laminated composite plate/shell structures embedded with piezoelectric actuators and sensors. The explicit hybrid stabilization method is employed to formulate stabilization vectors for the uniformly reduced integrated 18‐node three‐dimensional composite solid element. Unlike conventional piezoelectric elements, the concept of the electric nodes introduced in this paper can effectively eliminate the burden of constraining the equality of the electric potential for the nodes lying on the same electrode. Furthermore, the non‐linear distribution of electric potential in the piezoelectric layer is expressed by introducing internal electric potential, which not only can simplify modelling but also obtains the same as the exact solution. Copyright © 2003 John Wiley & Sons, Ltd.     

Integration of active vibration control methods with finite element models of smart laminated composite structures

Vibration control problems can be directly and systematically solved in a single analysis stage using commercial finite element programs. Integration of control methods into the finite element solutions (ICFES) can be achieved in ANSYS. In this work, first, the direct velocity feedback (DVF) control is tested on a 3-DOF mechanical system under a step input. The simulation results obtained by the ICFES are compared with the analytical results obtained by the Laplace transform method. Then, active control of free and forced vibrations in a smart laminated composite structure (SLCS) with two different lay-ups is studied numerically and experimentally. The SLCS consists of a symmetric laminated glass–epoxy composite beam with [0/90]_s and [45/−45]_s lay-ups and a piezoelectric actuator. For the vibration suppression, the DVF control tested on a mechanical system is applied to the SLCS. In addition, displacement feedback (DF) control is studied. Experiments are conducted to verify the natural frequencies and the closed loop time responses. Analytical results for the mechanical system and experimental results for the SLCS match well to the corresponding results obtained using the ICFES technique.     

Efficient and accurate multilayer solid‐shell element: non‐linear materials at finite strain

We present in this paper an efficient and accurate low‐order solid‐shell element formulation for analyses of large deformable multilayer shell structures with non‐linear materials. The element has only displacement degrees of freedom (dofs), and an optimal number of enhancing assumed strain (EAS) parameters to pass the patch tests (both membrane and out‐of‐plane bending) and to remedy volumetric locking. Based on the mixed Fraeijs de Veubeke‐Hu‐Washizu (FHW) variational principle, the in‐plane and out‐of‐plane bending behaviours are improved and the locking associated with (nearly) incompressible materials is avoided via a new efficient enhancement of strain tensor. Shear locking and curvature thickness locking are resolved effectively by using the assumed natural strain (ANS) method. Two non‐linear 3‐D constitutive models (Mooney–Rivlin material and hyperelastoplastic material at finite strain) are applied directly without requiring the enforcement of the plane‐stress assumption. In particular, we give a simple derivation for the hyperelastoplastic model using spectral representations. In addition, the present element has a well‐defined lumped mass matrix, and provides double‐side contact surfaces for shell contact problems. With the dynamics referred to a fixed inertial frame, the present element can be used to analyse multilayer shell structures undergoing large overall motion. Numerical examples involving static analyses and implicit/explicit dynamic analyses of multilayer shell structures with both material and geometric non‐linearities are presented, and compared with existing results obtained from other shell elements and from a meshless method. It is shown that elements that did not pass the out‐of‐plane bending patch test could not provide accurate results, as compared to the present element formulation, which passed the out‐of‐plane bending patch test. The present element proves to be versatile and efficient in the modelling and analyses of general non‐linear composite multilayer shell structures. Copyright © 2005 John Wiley & Sons, Ltd.     

Non‐linear finite element modal approach for the large amplitude free vibration of symmetric and unsymmetric composite plates

A theoretical analysis is presented for the large amplitude vibration of symmetric and unsymmetric composite plates using the non‐linear finite element modal reduction method. The problem is first reduced to a set of Duffing‐type modal equations using the finite element modal reduction method. The main advantage of the proposed approach is that no updating of the non‐linear stiffness matrices is needed. Without loss of generality, accurate frequency ratios for the fundamental mode and the higher modes of a composite plate at various values of maximum deflection are then determined by using the Runge–Kutta numerical integration scheme. The procedure for obtaining proper initial conditions for the periodic plate motions is very time consuming. Thus, an alternative scheme (the harmonic balance method) is adopted and assessed, as it was employed to formulate the large amplitude free vibration of beams in a previous study, and the results agreed well with the elliptic solution. The numerical results that are obtained with the harmonic balance method agree reasonably well with those obtained with the Runge–Kutta method. The contribution of each linear mode to the maximum deflection of a plate can also be obtained. The frequency ratios for isotropic and composite plates at various maximum deflections are presented, and convergence of frequencies with the number of finite elements, number of linear modes, and number of harmonic terms is also studied. Copyright © 2005 John Wiley & Sons, Ltd.     

A FEM model for the active control of curved FGM shells using piezoelectric sensor/actuator layers

In this paper, a generic finite element formulation is developed for the static and dynamic control of FGM (functionally graded material) shells with piezoelectric sensor and actuator layers. The properties of the FGM shell are graded in the thickness direction according to a volume fraction power‐law distribution. The proposed finite element model is based on variational principle and linear piezoelectricity theory. A constant displacement and velocity feedback control algorithm coupling the direct and inverse piezoelectric effects is applied in a closed‐loop system to provide feedback control of the integrated FGM shell structure. Both static and dynamic control of FGM shells are simulated to demonstrate the effectiveness of the proposed active control scheme within a framework of finite element discretization and piezoelectric integration. Copyright © 2002 John Wiley & Sons, Ltd.     

Genetic algorithm based optimal design for vibration control of composite shell structures using piezoelectric sensors and actuators

The present article deals with the design of optimal vibration control of smart fiber reinforced polymer (FRP) composite shell structures using genetic algorithm (GA) based linear quadratic regulator (LQR) and layered shell coupled electro-mechanical finite element analysis. Open loop procedure has been used for optimal placement of actuators considering the control spillover of the higher modes to prevent closed loop instability. An improved real coded GA based LQR control scheme has been developed for designing an optimal controller in order to maximize the closed loop damping ratio while keeping actuators voltages within limit. Results show that increased closed loop-damping has been achieved with a large reduction of control effort considering control spillover.     

Hybrid‐stress solid elements for shell structures based upon a modified variational functional

In this paper, we start with a modified generalized laminate stiffness matrix that serves as a remedy to resolve the thickness locking and some abnormalities encountered by solid‐shell elements in laminate analyses. A modified Hellinger–Reissner functional having displacement and a set of generalized stresses as independent fields is devised. Based upon the functional, eight‐node and 18‐node hybrid‐stress solid‐shell elements are proposed. A number of benchmark tests on homogenous and laminated plates/shells are conducted. The accuracy of the elements is promising. Copyright © 2002 John Wiley & Sons, Ltd.     

一种基于Layerwise理论的压电复合材料层合圆柱壳机电耦合有限单元

基于Reddy的Layerwise理论,对含压电铺层的复合材料层合壳的静力响应特性进行了理论研究。基于Layerwise理论,推导了含压电层的复合材料层合壳的应变分量与电场强度表达式。利用Hamilton原理和变分法,推导了压电智能层合壳的欧拉-拉格朗日方程,并采用有限元解法,建立了相应的有限元控制方程及其机电耦合刚度矩阵。通过算例结果与文献中的精确解和试验值进行了对比,表明相较于传统的经典层合板壳理论,本文理论方法的有效性和优势性;并分析了径厚比等参量对两端简支压电智能层合壳静力响应值的影响规律。      

Free vibration of composite and sandwich laminates with a higher-order facet shell element

A simple C⁰ isoparametric finite element formulation based on a shear deformable model of higher-order theory using a higher-order facet shell element is presented for the free vibration analysis of isotropic, orthotropic and layered anisotropic composite and sandwich laminates. This theory incorporates a realistic non-linear variation of displacements through the shell thickness, and eliminates the use of shear correction coefficients. The validity and efficiency of the present formulation is established by obtaining solutions to a wide range of problems and comparing them with the available three-dimensional closed-form and finite element solutions. In addition, other plate and shell solutions of different kind and available in the literature are also compiled and tabulated for the sake of completeness. The parametric effects of degree of orthotropy, length-to-thickness ratio, plate aspect ratio, number of layers and fibre orientation upon the frequencies and mode shapes are discussed.     

Adhesive element modelling and weighted static shape control of composite plates with piezoelectric actuator patches

A novel finite element model is presented for static and dynamic analysis of composite plates integrated with a laminated piezoelectric layer, a host laminated composite plate and an adhesive layer between them. A new adhesive element is developed which includes both peel and shear effects in the adhesive layer based on first‐order shear deformation plate theory. The thin adhesive layer between the piezoelectric layer and the host plate is modelled by assuming that it carries constant shear and peel strains throughout its thickness. In addition, a weighted static shape control scheme for finding the optimal voltage distribution for static shape control is given. By selecting different weighting matrices, a variety of items such as displacements, slopes, curvatures, strains and even generalized forces, can be included in finding the optimal actuating voltage for static shape control. The present model is validated by comparing with those results available in the literature. The numerical results show that the weighted linear least method can give a satisfactory voltage distribution to best match the desired shape. Copyright © 2004 John Wiley & Sons, Ltd.     

A mixed solid‐shell element for the analysis of laminated composites

This paper presents a versatile low order locking‐free mixed solid‐shell element that can be readily employed for a wide range of linear elastic structural analyses, that is, from thick isotropic structures to multilayer anisotropic composites. This solid‐shell element has eight nodes with only displacement degrees of freedom and few assumed stress parameters that provide very accurate interlaminar stress calculations through the element thickness. These elements can be stacked on top of each other to model multilayer structures, fulfilling the interlaminar stress continuity at the interlayer surfaces and zero traction conditions on the top and bottom surfaces of the laminate. The element formulation is based on the well‐known Fraeijs de Veubeke–Hu–Washizu mixed variational principle with enhanced assumed strains formulation and assumed natural strains formulation to alleviate the different types of locking phenomena in solid‐shell elements. The distinct feature of the present formulation is its ability to accurately calculate the interlaminar stress field in multilayer structures, which is achieved by the introduction of a constraint equation on the interlaminar stresses in the Fraeijs de Veubeke–Hu–Washizu principle‐based enhanced assumed strains formulation. The intelligent computer coding of the present formulation makes the present element appropriate for a wide range of structural analyses. To assess the present formulation's accuracy, a variety of popular numerical benchmark examples related to element convergence, mesh distortion, and shell and laminated composite analyses are investigated and the results are compared with those available in the literature. These benchmark examples reveal that the proposed formulation provides very good results for the structural analysis of shells and multilayer composites. Copyright © 2011 John Wiley & Sons, Ltd.     

An assumed strain triangular curved solid shell element formulation for analysis of plates and shells undergoing finite rotations

A formulation for 36‐DOF assumed strain triangular solid shell element is developed for efficient analysis of plates and shells undergoing finite rotations. Higher order deformation modes described by the bubble function displacements are added to the assumed displacement field. The assumed strain field is carefully selected to alleviate locking effect. The resulting element shows little effect of membrane locking as well as shear locking, hence, it allows modelling of curved shell structures with curved elements. The kinematics of the present formulation is purely vectorial with only three translational degrees of freedom per node. Accordingly, the present element is free of small angle assumptions, and thus it allows large load increments in the geometrically non‐linear analysis. Various numerical examples demonstrate the validity and effectiveness of the present formulation. Copyright © 2001 John Wiley & Sons, Ltd.     

基于传递矩阵法分析ACLD圆柱壳振动控制问题的一种完全力电耦合模型

目前在ACLD薄壳结构的动力学分析中,通常采用一种忽略压电约束层面内电场强度,仅考虑在厚度方向为常量分布的法向电场强度的简化力电耦合模型。文中首先从理论上分析了简化力电耦合模型的局限性,进而提出了一种新的完全力电耦合模型,以此为基础导出了该模型下ACLD圆柱壳的一阶常微分矩阵状态方程,并结合传递矩阵法和齐次扩容精细积分法求解该方程。相对于传统三维模型方法,所建立的新模型和求解方法不仅大幅度简化了计算,而且适用于分析部分覆盖和任意支承条件下ACLD圆柱壳的振动控制问题。最后通过数值算例对完全力电耦合模型和简化力电耦合模型进行了比较,结果表明：简化力电耦合模型仅适用于低频分析;完全力电耦合模型具有更宽的频率适应性