Jinho Oh

Higher order zig-zag plate theory under thermo-electric-mechanical loads combined

A higher order zig-zag plate theory is developed to refine the predictions of the mechanical, thermal, and electric behaviors partially coupled. The in-plane displacement fields are constructed by superimposing linear zig-zag field to the smooth globally cubic varying field through the thickness. Smooth parabolic distribution through the thickness is assumed in the out-of-plane displacement in order to consider transverse normal deformation and stress. The layer-dependent degrees of freedom of displacement fields are expressed in terms of reference primary degrees of freedom by applying interface continuity conditions as well as bounding surface conditions of transverse shear stresses. Artificial shear correction factors are not needed in the present formulation. Thus the proposed theory has only seven primary unknowns and they do not depend upon the number of layers. Through the numerical examples of partially coupled analysis, the accuracy and efficiency of the present theory are demonstrated. The present theory is suitable in the predictions of deformation and stresses of thick smart composite plate under mechanical, thermal, and electric loads combined.

Enhanced lower-order shear deformation theory for fully coupled electro-thermo-mechanical smart laminated plates

Lower-order shear deformation theories are adequate to predict the global behavior of a smart plate. They cannot, however, predict accurate deformation and stress distributions through the thickness of laminated smart plates. Thus higher-order zigzag theories have been proposed to accurately calculate them. In most cases, a simplified higher-order zigzag theory requires C1 shape functions in finite-element implementation that are not so common for plate and shell analysis in commercial FE software. This presents the practical limitation of simplified zigzag theories to the commercial FE package. In fact, an iso-parametric C0 plate model is standard for the analysis and design of composite laminated plates and shells. In this paper, an enhanced lower-order shear deformation theory (ELSDT) is developed to provide a simple yet accurate tool for the analysis of smart structures under combined loads (including thermal and electrical loads as well as mechanical loads). It is systematically derived by minimizing the least-square errors between the first-order theory and the higher-order theory. This makes it possible to transform the strain energy of a higher-order zigzag theory to that of a lower-order zigzag theory. The resulting lower-order theory, which is referred to as the ELSDT, requires the C0 shape function only, and it is applicable to fully coupled mechanical, electric, and thermal problems. First a higher-order zigzag theory is established, which includes both a linear zigzag function and a cubic polynomial for in-plane displacements, a quadratic polynomial in the out-of-plane displacement, and a layerwise function for the electric potential. The ELSDT is then constructed via the aforementioned procedure. The accuracy and robustness of the present theory are demonstrated through numerical examples.

A FINITE ELEMENT ANALYSIS BASED ON HIGHER-ORDER ZIG-ZAG SHELL THEORY FOR LAMINATED COMPOSITES WITH MULTIPLE DELAMINATIONS

A new three-node triangular shell element based on higher order zig-zag theory is developed for laminated composite shells with multiple delaminations. The present higher order zig-zag shell theory is described in a general curvilinear coordinate system and in general tensor notation. All the complicated curvatures of surface including twisting curvatures can be described in an exact manner in the present shell element because this element is based on geometrically exact surface representation. The displacement field of the proposed finite element includes slope of deflection, which requires C1 continuity between element interfaces. Thus the nonconforming shape function of Spechts three-node triangular plate bending element is employed to interpolate out-of-plane displacement. The present element passes the bending and twisting patch tests in flat surface configurations. The developed element is evaluated through the buckling problems of composite cylindrical shells with multiple delaminations. Through the numerical examples it is demonstrated that the proposed shell element is efficient because it has minimal degrees of freedom per node. The accuracy of the present element is demonstrated in the prediction of buckling loads and buckling modes of shells with multiple delaminations. The present shell element should serve as a powerful tool in the prediction of buckling loads and modes of multi-layered thick laminated shell structures with arbitrary-shaped multiple delaminations.

A Coupled Analysis of Smart Plate Under Electro-Mechanical Loading Using Enhanced Lower-Order Shear Deformation Theory

Enhanced lower order shear deformation theory is developed in this study. Generally, lower order theories are not adequate to predict accurate deformation and stress distribution through the thickness of laminated plate. For the accurate prediction of detailed stress and deformation distributions through the thickness, higher order zigzag theories have been proposed. However, in most cases, simplified zigzag higher order theory requires , shape functions in finite element implementation. In commercial FE softwares, , shape functions are not so common in plate and shell analysis. Thus zigzag theories are useful for the highly accurate prediction of thick composite behaviors but they are not practical in the sense that they cannot be used conveniently in the commercial package. In practice, iso-parametric plate model is the standard model for the analysis and design of composite laminated plates and shells. Thus in the present study, an enhanced lower order shear deformation theory is developed. The proposed theory requires only shape function in FE implementation. The least-squared energy error between the lower order theory and higher order theory is minimized. An enhanced lower order shear deformation theory(ELSDT) in this paper is proposed for smart structure under complex loadings. The ELSDT is constructed by the strain energy transformation and fully coupled mechanical, electric loading cases are studied. In order to obtain accurate prediction, zigzag in-plane displacement and transverse normal deformation are considered in the deformation Held. In the electric behavior, open-circuit condition as well as closed-circuit condition is considered. Through the numerous examples, the accuracy and robustness of present theory are demonstrated.

Dynamic Analysis for Delaminated Composites with Arbitrary Shaped Multiple Delaminations Based on Higher-Order Zig-Zag Theory

A finite element based on the efficient higher order zig-zag theory with multiple delaminations is developed to refine the predictions of frequency and mode shapes. Displacement fields through the thickness are constructed by superimposing linear zig-zag field to the smooth globally cubic varying field. The layer-dependent degrees of freedom of displacement fields are expressed in terms of reference primary degrees of freedom by applying interface continuity conditions as well as bounding surface conditions of transverse shear stresses including delaminated interfaces. Thus the proposed theory is not only accurate but also efficient. This displacement field can systematically handle the number, shape, size, and locations of delaminations. Through the dynamic version of variational approach, the dynamic equilibrium equations and variationally consistent boundary conditions are obtained. Through the natural frequency analysis and time response analysis of composite plate with delaminations, the accuracy and efficiency of the present finite element are demonstrated. The present finite element is suitable in the predictions of the dynamic response of the thick composite plate with multiple delaminations.

Dynamic Analysis for Delaminated Composites Using DKQ Concept Based on Higher-Order Zig-Zag Theory

* Associate Professor, AIAA member † Graduate Research Assistant ‡ Graduate Research Assistant ABSTRCT A finite element based on the efficient higher order zig-zag theory with multiple delaminations is developed to refine the predictions of frequency and mode shapes. Displacement fields through the thickness are constructed by superimposing linear zig-zag field to the smooth globally cubic varying field. The layerdependent degrees of freedom of displacement fields are expressed in terms of reference primary degrees of freedom by applying interface continuity conditions as well as bounding surface conditions of transverse shear stresses including delaminated interfaces. Thus the proposed theory is not only accurate but also efficient. This displacement field can systematically handle the number, shape, size, and locations of delaminations. Throught the dynamic version of variational approach, the dynamic equilibrium equations and variationally consistent boundary conditions are obtained. Through the natural frequency analysis and time response analysis of composite plate with delaminations, the accuracy and efficiency of the present finite element are demonstrated. The present finite element is suitable in the predictions of the dynamic response of the thick composite plate with multiple delaminations.