Coupled mechanics and finite element for non‐linear laminated piezoelectric shallow shells undergoing large displacements and rotations

A theoretical framework is presented for analysing the coupled non‐linear response of shallow doubly curved adaptive laminated piezoelectric shells undergoing large displacements and rotations. The formulated mechanics incorporate coupling between in‐plane and flexural stiffness terms due to geometric curvature, coupling between mechanical and electric fields, and encompass geometric non‐linearity effects due to large displacements and rotations. The governing equations are formulated explicitly in orthogonal curvilinear co‐ordinates and are combined with the kinematic assumptions of a mixed‐field shear‐layerwise shell laminate theory. Based on the above formulation, a finite element methodology together with an incremental‐iterative technique, based on Newton–Raphson method is formulated. An eight‐node coupled non‐linear shell element is also developed. Various evaluation cases on laminated curved beams and cylindrical panels illustrate the capability of the shell finite element to predict the complex non‐linear behaviour of active shell structures including buckling, which is not captured by linear shell models. The numerical results also show the inherent capability of piezoelectric shell structures to actively induce large displacements through piezoelectric actuators, by jumping between multiple equilibrium states. Copyright © 2005 John Wiley & Sons, Ltd.     

Buckling of thin‐walled cylindrical shells under axial compression

Lightweight thin‐walled cylindrical shells subjected to external loads are prone to buckling rather than strength failure. The buckling of an axially compressed shell is studied using analytical, numerical and semi‐empirical models. An analytical model is developed using the classical shell small deflection theory. A semi‐empirical model is obtained by employing experimental correction factors based on the available test data in the theoretical model. Numerical model is built using ANSYS finite element analysis code for the same shell. The comparison reveals that the analytical and numerical linear model results match closely with each other but are higher than the empirical values. To investigate this discrepancy, non‐linear buckling analyses with large deflection effect and geometric imperfections are carried out. These analyses show that the effects of non‐linearity and geometric imperfections are responsible for the mismatch between theoretical and experimental results. The effect of shell thickness, radius and length variation on buckling load and buckling mode has also been studied. Copyright © 2009 John Wiley & Sons, Ltd.     

An eight‐node hybrid‐stress solid‐shell element for geometric non‐linear analysis of elastic shells

This paper presents eight‐node solid‐shell elements for geometric non‐linear analysis of elastic shells. To subdue shear, trapezoidal and thickness locking, the assumed natural strain method and an ad hoc modified generalized laminate stiffness matrix are employed. A selectively reduced integrated element is formulated with its membrane and bending shear strain components taken to be constant and equal to the ones evaluated at the element centroid. With the generalized stresses arising from the modified generalized laminate stiffness matrix assumed to be independent from the ones obtained from the displacement, an extended Hellinger–Reissner functional can be derived. By choosing the assumed generalized stresses similar to the assumed stresses of a previous solid element, a hybrid‐stress solid‐shell element is formulated. Commonly employed geometric non‐linear homogeneous and laminated shell problems are attempted and our results are close to those of other state‐of‐the‐art elements. Moreover, the hybrid‐stress element converges more readily than the selectively reduced integrated element in all benchmark problems. Copyright © 2002 John Wiley & Sons, Ltd.     

A refined non‐linear non‐conforming triangular plate/shell element

A refined non‐conforming triangular plate/shell element for geometric non‐linear analysis of plates/shells using the total Lagrangian/updated Lagrangian approach is constructed in this paper based on the refined non‐conforming element method for geometric non‐linear analysis. The Allman's triangular plane element with vertex degrees of freedom and the refined triangular plate‐bending element RT9 are used to construct the present element. Numerical examples demonstrate that the accuracy of the new element is quite high in the geometric non‐linear analysis of plates/shells. Copyright © 2003 John Wiley & Sons, Ltd.     

Elasto‐plastic finite element analysis of shells with damage due to microvoids

This paper presents a non‐linear finite element analysis for the elasto‐plastic behaviour of thick/thin shells and plates with large rotations and damage effects. The refined shell theory given by Voyiadjis and Woelke (Int. J. Solids Struct. 2004; 41 :3747–3769) provides a set of shell constitutive equations. Numerical implementation of the shell theory leading to the development of the C⁰ quadrilateral shell element (Woelke and Voyiadjis, Shell element based on the refined theory for thick spherical shells. 2006, submitted) is used here as an effective tool for a linear elastic analysis of shells. The large rotation elasto‐plastic model for shells presented by Voyiadjis and Woelke (General non‐linear finite element analysis of thick plates and shells. 2006, submitted) is enhanced here to account for the damage effects due to microvoids, formulated within the framework of a micromechanical damage model. The evolution equation of the scalar porosity parameter as given by Duszek‐Perzyna and Perzyna (Material Instabilities: Theory and Applications, ASME Congress, Chicago, AMD‐Vol. 183/MD‐50, 9–11 November 1994; 59–85) is reduced here to describe the most relevant damage effects for isotropic plates and shells, i.e. the growth of voids as a function of the plastic flow. The anisotropic damage effects, the influence of the microcracks and elastic damage are not considered in this paper. The damage modelled through the evolution of porosity is incorporated directly into the yield function, giving a generalized and convenient loading surface expressed in terms of stress resultants and stress couples. A plastic node method (Comput. Methods Appl. Mech. Eng. 1982; 34 :1089–1104) is used to derive the large rotation, elasto‐plastic‐damage tangent stiffness matrix. Some of the important features of this paper are that the elastic stiffness matrix is derived explicitly, with all the integrals calculated analytically (Woelke and Voyiadjis, Shell element based on the refined theory for thick spherical shells. 2006, submitted). In addition, a non‐layered model is adopted in which integration through the thickness is not necessary. Consequently, the elasto‐plastic‐damage stiffness matrix is also given explicitly and numerical integration is not performed. This makes this model consistent mathematically, accurate for a variety of applications and very inexpensive from the point of view of computer power and time. Copyright © 2006 John Wiley & Sons, Ltd.     

A non‐linear interface element for 3D mesoscale analysis of brick‐masonry structures

This paper presents a novel interface element for the geometric and material non‐linear analysis of unreinforced brick‐masonry structures. In the proposed modelling approach, the blocks are modelled using 3D continuum solid elements, whereas the mortar and brick–mortar interfaces are modelled by means of the 2D non‐linear interface element. This enables the representation of any 3D arrangement for brick‐masonry, accounting for the in‐plane stacking mode and the through‐thickness geometry, and importantly it allows the investigation of both the in‐plane and the out‐of‐plane responses of unreinforced masonry panels. A co‐rotational approach is employed for the interface element, which shifts the treatment of geometric non‐linearity to the level of discrete entities, and enables the consideration of material non‐linearity within a simplified local framework employing first‐order kinematics. In this respect, the internal interface forces are modelled by means of elasto‐plastic material laws based on work‐softening plasticity and employing multi‐surface plasticity concepts. Following the presentation of the interface element formulation details, several experimental–numerical comparisons are provided for the in‐plane and out‐of‐plane static behaviours of brick‐masonry panels. The favourable results achieved demonstrate the accuracy and the significant potential of using the developed interface element for the non‐linear analysis of brick‐masonry structures under extreme loading conditions. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

A finite element formulation for piezoelectric shell structures considering geometrical and material non‐linearities

In this paper, we present a non‐linear finite element formulation for piezoelectric shell structures. Based on a mixed multi‐field variational formulation, an electro‐mechanical coupled shell element is developed considering geometrically and materially non‐linear behavior of ferroelectric ceramics. The mixed formulation includes the independent fields of displacements, electric potential, strains, electric field, stresses, and dielectric displacements. Besides the mechanical degrees of freedom, the shell counts only one electrical degree of freedom. This is the difference in the electric potential in the thickness direction of the shell. Incorporating non‐linear kinematic assumptions, structures with large deformations and stability problems can be analyzed. According to a Reissner–Mindlin theory, the shell element accounts for constant transversal shear strains. The formulation incorporates a three‐dimensional transversal isotropic material law, thus the kinematic in the thickness direction of the shell is considered. The normal zero stress condition and the normal zero dielectric displacement condition of shells are enforced by the independent resultant stress and the resultant dielectric displacement fields. Accounting for material non‐linearities, the ferroelectric hysteresis phenomena are considered using the Preisach model. As a special aspect, the formulation includes temperature‐dependent effects and thus the change of the piezoelectric material parameters due to the temperature. This enables the element to describe temperature‐dependent hysteresis curves. Copyright © 2011 John Wiley & Sons, Ltd.     

变厚度扁球壳的非线性分析

本文利用边界元方法分析了变厚度扁球壳的非线性问题。引入了与变厚因素有关的等效载荷(?),利用板的初等函数基本解,而不是直接寻求变厚扁球壳的基本解,使原问题的难度大大减少,采用了连续性强、精度高的样条函数插值,即使在划分节点取得较少的情况下,结果的精度是令人满意的。     

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.     

变厚度扁球壳大挠度问题的样条配点法

本文首先导出了变厚度扁球壳在均布压力作用下的轴对称大挠度方程,然后应用逐步加载法将此线性化,最后利用样条配点法解线性微分方程组。文中给出了特征曲线。     

Simulation and experimental studies of the repaired composite laminate

Firstly the compress experiment is undertaken to investigate the efficiency of repaired panels in this paper, and then modeling of the mechanical behavior of the repaired composite panel under compressive static load is conducted by using of the finite element method. The effect of geometric non‐linearity on the stress‐strain response is considered in the numeric analysis. Fatherly, the user material subroutine (UMAT) is integrated with the ABAQUS package with the geometric non‐linearity effect for studying the damage initiation and its progression in the composite structure, and quadrilateral, linear, thick shell elements (S8R) are adopted. Finally, the predicted strain distribution, damage evolution and strength of the laminate are compared with the test results.     

Curved quadratic triangular degenerated‐ and solid‐shell elements for geometric non‐linear analysis

Compared to the large number of curved quadrilateral degenerated‐ and solid‐shell elements, there are only a very few curved triangular degenerated‐ and solid‐shell elements. Based on the assumed natural strain sampling scheme previously developed for a quadratic degenerated‐shell element for linear analysis, this paper devises geometric non‐linear six‐node degenerated‐shell and twelve‐node solid‐shell elements. Both elements can be curved and are only equipped with the standard nodal d.o.f.s. Careful consideration has been exercised to circumvent various locking phenomena that plague degenerated‐ and solid‐shell elements. Numerical examples are presented to illustrate their efficacy. Copyright © 2003 John Wiley & Sons, Ltd.     

Stress integration procedures for a biaxial isotropic material model of biological membranes and for hysteretic models of muscle fibres and surfactant

Stress calculation for a biaxial isotropic material model of a biological membrane and for hysteretic models of muscle fibres and surfactant is presented in the paper. The non‐linear elastic membrane model is defined by uniaxial and biaxial stress–stretch relations, while the hysteretic models of tissue fibres and surfactant are described by the stress–stretch and surface tension–surfactant area ratio constitutive relationships, respectively. The conditions when tissue is or is not covered by surfactant are considered. It is assumed that the material is subjected to cyclic loading. Quasi‐static and steady conditions are considered. The models are implemented in large strain finite element incremented‐iterative analysis of shell deformations. Numerical examples demonstrate characteristics of the computational procedures and structural response of biological membranes when subjected to cyclic loading. Hysteretic response of biological membranes subjected to cyclic loading is caused by hysteresis of fibres and hysteresis of surfactant. The hysteretic effects may play an important role in the physiology of human body. Copyright © 2006 John Wiley & Sons, Ltd.     

Nonlinear buckling analysis of hygrothermoelastic composite shell panels using finite element method

Hygrothermal stresses due to the change in environmental condition may induce buckling and dynamic instability in the composite shell structures. In the present investigation, the hygrothermoelastic buckling behavior of laminated composite shells are numerically simulated using geometrically nonlinear finite element method. The orthogonal curvilinear coordinate is used for modeling a general doubly curved deep or shallow shell surface. The geometrically nonlinear finite element formulation is based on general nonlinear strain–displacement relations in the orthogonal curvilinear coordinate system. The present theory can be applicable to thin and moderately thick shells. The mechanical linear and nonlinear stiffnesses, and the nonmechanical nonlinear geometric stiffness matrices and the hygrothermal load vector are presented. It is also observed that during the present numerical solution of nonlinear equilibrium equation, in order to construct the nonlinear stiffness matrices for the first load step, the initial deformation can be assumed as zero or any computer generated small random number or the properly scaled fundamental buckling mode shape. To verify the present formulations and finite element code, the present results are compared well with those available in the open literature. Parametric studies such as thickness ratio and shallowness ratio on buckling are performed for spherical, truncated conical and cylindrical composite shell panels. The buckling behavior and deflection shapes are characterized by multiple wrinkles along unreinforced direction at higher moisture concentrations or temperature rise.     

Sandwich shell finite element for dynamic explicit analysis

The contribution of this paper consists of new development of transverse shear stresses through the thickness and finding an expression for the critical time step for explicit time integration of layered shells. This work presents the finite element (FE) formulation and implementation of a higher‐order shear deformable shell element for dynamic explicit analysis of composite and sandwich shells. The formulation is developed using a displacement‐based third‐order shear deformation shell theory. Using the differential equilibrium equations and the interlayer requirements, special treatment is developed for the transverse shear, resulting in a continuous, piecewise quartic distribution of the transverse shear stresses through the shell thickness. Expressions are developed for the critical time step of the explicit time integration for orthotropic homogeneous and layered shells based on the developed third‐order formulation. To assess the performance of the present shell element, it is implemented in the general non‐linear explicit dynamic FE code DYNA3D. Several problems are solved and results are presented and compared to other theoretical and numerical results. Copyright © 2002 John Wiley & Sons, Ltd.     

A large deformation solid‐shell concept based on reduced integration with hourglass stabilization

In this paper a new eight‐node (brick) solid‐shell finite element formulation based on the concept of reduced integration with hourglass stabilization is presented. The work focuses on static problems. The starting point of the derivation is the three‐field variational functional upon which meanwhile established 3D enhanced strain concepts are based. Important additional assumptions are made to transfer the approach into a powerful solid‐shell. First of all, a Taylor expansion of the first Piola–Kirchhoff stress tensor with respect to the normal through the centre of the element is carried out. In this way the stress becomes a linear function of the shell surface co‐ordinates whereas the dependence on the thickness co‐ordinate remains non‐linear. Secondly, the Jacobian matrix is replaced by its value in the centre of the element. These two assumptions lead to a computationally efficient shell element which requires only two Gauss points in the thickness direction (and one Gauss point in the plane of the shell element). Additionally three internal element degrees‐of‐freedom have to be determined to avoid thickness locking. One important advantage of the element is the fact that a fully three‐dimensional stress state can be modelled without any modification of the constitutive law. The formulation has only displacement degrees‐of‐freedom and the geometry in the thickness direction is correctly displayed. Copyright © 2006 John Wiley & Sons, Ltd.     

Influence functions of a thin shallow meniscus-shaped mirror

Thin shallow spherical shell theory is used to derive the general influence function, owing to uniform and/or discrete (actuators) loads, for a thin shallow meniscus-shaped mirror of uniform thickness with a central hole and supported at discrete points. Small elastic deformations are considered. No symmetry on the load distribution constrains the model. Explicit analytical expressions of the set of equations are given for calculating the influence functions. Results agree with the finite element analysis (FEA) to within 1%. When the FEA requires megabytes of RAM memory, the analytical method needs only kilobytes and typically runs 30 times faster. This is a crucial advantage for the iterative optimization of mirror supports such as large passive or active meniscus-shaped primary mirror supports or Cassegrain/Gregorian adaptive secondary actuator configurations. References are given on estimating the shear effects (thick mirror), the thickness variation effect, and the influence of the size of the support pads.     

A mixed shell formulation accounting for thickness strains and finite strain 3d material models

A non‐linear quadrilateral shell element for the analysis of thin structures is presented. The Reissner–Mindlin theory with inextensible director vector is used to develop a three‐field variational formulation with independent displacements, stress resultants and shell strains. The interpolation of the independent shell strains consists of two parts. The first part corresponds to the interpolation of the stress resultants. Within the second part independent thickness strains are considered. This allows incorporation of arbitrary non‐linear 3d constitutive equations without further modifications. The developed mixed hybrid shell element possesses the correct rank and fulfills the in‐plane and bending patch test. The essential feature of the new element is the robustness in the equilibrium iterations. It allows very large load steps in comparison with other element formulations. We present results for finite strain elasticity, inelasticity, bifurcation and post‐buckling problems. Copyright © 2007 John Wiley & Sons, Ltd.     

Generalized coupled thermoelasticity of functionally graded cylindrical shells

In this paper, the response of a circular cylindrical thin shell made of the functionally graded material based on the generalized theory of thermoelasticity is obtained. The governing equations of the generalized theory of thermoelasticity and the energy equations are simultaneously solved for a functionally graded axisymmetric cylindrical shell subjected to thermal shock load. Thermoelasticity with second sound effect in cylindrical shells based on the Lord–Shulman model is compared with the Green–Lindsay model. A second‐order shear deformation shell theory, that accounts for the transverse shear strains and rotations, is considered. Including the thermo‐mechanical coupling and rotary inertia, a Galerkin finite element formulation in space domain and the Laplace transform in time domain is used to formulate the problem. The inverse Laplace transform is obtained using a numerical algorithm. The shell is graded through the thickness assuming a volume fraction of metal and ceramic, using a power law distribution. The effects of temperature field for linear and non‐linear distributions across the shell thickness are examined. The results are validated with the known data in the literature. Copyright © 2006 John Wiley & Sons, Ltd.