Geometrically nonlinear isogeometric analysis of laminated composite plates based on higher-order shear deformation theory

In this paper, we present an effectively numerical approach based on isogeometric analysis (IGA) and higher-order shear deformation theory (HSDT) for geometrically nonlinear analysis of laminated composite plates. The HSDT allows us to approximate displacement field that ensures by itself the realistic shear strain energy part without shear correction factors (SCFs). IGA utilizing basis functions namely B-splines or non-uniform rational B-splines (NURBS) enables to satisfy easily the stringent continuity requirement of the HSDT model without any additional variables. The nonlinearity of the plates is formed in the total Lagrange approach based on the small strain assumptions. Numerous numerical validations for the isotropic, orthotropic, cross-ply and angle-ply laminated plates are provided to demonstrate the effectiveness of the proposed method.     

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 transient dynamic response of damped plates using a higher order shear deformation theory

Damped transient dynamic elasto-plastic analysis of plate is investigated. A finite element model based on a C ⁰ higher order shear deformation theory has been developed. Nine noded Lagrangian elements with five degrees of freedom per node are used. Selective Gauss integration is used to evaluate energy terms so as to avoid shear locking and spurious mechanisms. Von Mises and Tresca yield criteria are incorporated along with associated flow rules. Explicit central difference time stepping scheme is employed to integrate temporal equations. The mass matrix is diagonalized by using the efficient proportional mass lumping scheme. A program is developed for damped transient dynamic finite element analysis of elasto-plastic plate. Several numerical examples are studied to unfold different facets of damping of elasto-plastic plates.     

A new higher-order shear deformation theory and refined beam element of composite laminates

A new higher-order shear deformation theory based on global-local superposition technique is developed. The theory satisfies the free surface conditions and the geometric and stress continuity conditions at interfaces. The global displacement components are of the Reddy theory and local components are of the internal first to third-order terms in each layer. A two-node beam element based on this theory is proposed. The solutions are compared with 3D-elasticity solutions. Numerical results show that present beam element has higher computational efficiency and higher accuracy.The project supported by the National Natural Science Foundation of China (10172023)     

复合材料层合板精化高阶理论及其精化三角形板单元

提出一种新的精化高阶理论，该理论满足层间位移、应力连续条件，由此建立了三角形精化板单元。该单元满足单元间Ｃ１类弱连续条件，其收敛性得到保证，且具有简单、高效率的优点。     

An efficient coupled layerwise theory for static analysis of piezoelectric sandwich beams

Summary An efficient one-dimensional model is developed for the statics of piezoelectric sandwich beams. Third-order zigzag approximation is used for axial displacement, and the potential is approximated as piecewise linear. The displacement field is expressed in terms of three primary displacement variables and the electric potential variables by satisfying the conditions of zero transverse shear stress at the top and bottom and its continuity at layer interfaces. The deflection field accounts for the piezoelectric transverse normal strain. The governing equations are derived using a variational principle. The present results agree very well with the exact solution for thin and thick highly inhomogeneous simply supported hybrid sandwich beams. The developed theory can accurately model open and closed circuit boundary conditions. The first author is grateful to DST, Government of India, for financial support for this work.     

Thermal post-buckling analysis of imperfect laminated plates using a higher-order shear deformation theory

Thermal post-buckling analysis is presented for a simply supported, composite laminated plate subjected to uniform or non-uniform tent-like temperature loading. The initial geometrical imperfection of the plate is taken into account. The formulations are based on the Reddy's higher-order shear deformation plate theory, and include thermal effects. The analysis uses a mixed Galerkin-perturbation technique to determine thermal buckling loads and post-buckling equilibrium paths. Numerical examples cover the performances of perfect and imperfect, antisymmetrically angle-ply and symmetrically cross-ply laminated plates. The effects played by transverse shear deformation, thermal load ratio, plate aspect ratio, total number of plies, fiber orientation and initial geometrical imperfections are studied. Typical results are presented in dimensionless graphical form.     

Coupled refined layerwise theory for dynamic free and forced response of piezoelectric laminated composite and sandwich beams

This work extends a previously presented coupled refined layerwise theory to dynamic analysis of piezoelectric laminated composite and sandwich beams. Contrary to most of the available theories, all the kinematic and stress boundary conditions are satisfied at the interfaces of the piezoelectric layers with the non-zero longitudinal electric field. Moreover, both electrical transverse normal strains and transverse flexibility are taken into account for the first time in the present theory. In the presented formulation a high-order polynomial, an exponential expression and a layerwise term containing the electric field are included in the describing expression of the in-plane displacement of the beam. For the transverse displacement, the coupled refined model uses a combination of continuous piecewise fourth-order polynomials with a layerwise representation of electrical unknowns. The electric field is also approximated as linear across the thickness direction of piezoelectric layers. One of advantages of the present theory is that the mechanical number of the unknown parameters is very small and is independent of the number of the layers. For validation of the proposed model, various free and forced vibration tests for thin and thick laminated/sandwich piezoelectric beams are carried out. For various electrical and mechanical boundary conditions, excellent correlation has been found between the results obtained from the proposed formulation with those resulted from the three-dimensional theory of piezoelasticity.     

Large amplitude free vibration of shear deformable laminated composite annular sector plates by a sector p-element

A sector p-element is presented for the large amplitude free vibration analysis of laminated composite annular sector plates. The effects of out-of-plane shear deformations, rotary inertia, and geometric non-linearity are taken into account. The shape functions are derived from the shifted Legendre orthogonal polynomials. The element stiffness and mass matrices are integrated analytically with the aid of symbolic computing. The method consists of modeling the annular sector plate as one element. The accuracy of the solution is improved simply by increasing the polynomial order. The time-dependent coefficients are described by a truncated Fourier series. The equations of free motion are obtained using the harmonic balance method and solved by the linearized updated mode method. Results for the linear and non-linear frequencies of clamped laminated composite annular sector plates are obtained. The case of a clamped isotropic annular sector plate is also shown. The linear frequencies are found to converge rapidly downwards as the polynomial order is increased. Comparisons of the linear frequencies with published results show excellent agreement. The effects of sector angle, inner-to-outer radius ratio, thickness-to-outer radius ratio, moduli ratio, number of plies, and layup sequence on the backbone curves are also investigated. It is shown that the hardening behavior increases or decreases depending on geometric and lamination parameters.     

Thermal buckling of functionally graded circular plates based on higher order shear deformation plate theory

In this research, thermal buckling of circular plates compose of functionally graded material (FGM) is considered. Equilibrium and stability equations of a FGM circular plate under thermal loads are derived, based on the higher order shear deformation plate theory (3rd order plate theory). Assuming that the material properties vary as a power form of the thickness coordinate variable z and using the variational method, the system of fundamental partial differential equations is established. A buckling analysis of a functionally graded circular plate (FGCP) under various types of thermal loads is carried out and the result are given in closed-form solutions. The results are compared with the critical buckling temperature obtained for FGCP based on first order (1st order plate theory) and classical plate theory (0 order plate theory) given in the literature. The study concludes that higher order shear deformation theory accurately predicts the behavior of FGCP, whereas the first order and classical plate theory overestimates buckling temperature.     

基于高阶位移模式的压电层合板动力有限元模型

建立了含压电片层合板的有限元动力学模型。以位于压电层上下表面处的电场强度和层间电压为未知量,给出了三次函数的电势分布模式,采用Reddy的高阶剪切理论描述板的位移场,假设板厚度方向的正应力为零给出了减缩的本构方程,采用有限元方法,基于Hamilton原理导出结构的动力学方程,然后用静态缩聚的方法压缩掉电场自由度和次要的位移自由度。最后用四边形矩形单元求解了一对称铺层和非对称铺层悬臂板的固有频率,并与ANSYS结果对比验证了本文模型的精确性。     

C~0型高阶锯齿理论及复合材料厚梁热应力分析

基于已有锯齿理论构造单元时,需使用满足单元间C1连续的插值函数,难于构造多节点高阶单元,而且精度较低。针对已有锯齿理论存在的问题,本文首先发展了C0型锯齿理论。通过虚位移原理推导出在热载荷作用下复合材料梁的平衡方程,并给出了简支复合材料层合梁解析解。基于发展的锯齿理论分析了复合材料夹层梁和层合梁热膨胀问题,并与其他理论结果对比。数值结果表明,发展的C0型锯齿理论能克服已有锯齿理论的难题。     

Analysis of interlaminar stress and nonlinear dynamic response for composite laminated plates with interfacial damage

By considering the effect of interfacial damage and using the variation principle, three-dimensional nonlinear dynamic governing equations of the laminated plates with interfacial damage are derived based on the general sixdegrees-of-freedom plate theory towards the accurate stress analysis. The solutions of interlaminar stress and nonlinear dynamic response for a simply supported laminated plate with interfacial damage are obtained by using the finite difference method, and the results are validated by comparison with the solution of nonlinear finite element method. In numerical calculations, the effects of interfacial damage on the stress in the interface and the nonlinear dynamic response of laminated plates are discussed.     

A semi-analytical approach for the non-linear large deflection analysis of laminated rectangular plates under general out-of-plane loading

A semi-analytical approach for the geometrically non-linear analysis of rectangular laminated plates with general inplane and out-of-plane boundary conditions under a general distribution of out-of-plane loads is developed. The analysis is based on the elastic thin plate theory with geometrically non-linear von Kármán strains. The solution of the non-linear partial differential equations is reduced to an iterative sequential solution of non-linear ordinary differential equations using the multi-term extended Kantorovich method. The efficiency, accuracy, and convergence of the proposed method are examined through a comparison with other semi-analytical methods and with finite element analyses. The capabilities of the approach and its applicability to the non-linear large deflection analysis of plate structures are demonstrated through various numerical examples. Emphasis is placed on combinations of lamination, boundary, and loading conditions that cannot be analyzed using alternative semi-analytical methods.     

An efficient higher order zigzag theory for composite and sandwich beams subjected to thermal loading

A new efficient higher order zigzag theory is presented for thermal stress analysis of laminated beams under thermal loads, with modification of the third order zigzag model by inclusion of the explicit contribution of the thermal expansion coefficient α₃ in the approximation of the transverse displacement w. The thermal field is approximated as piecewise linear across the thickness. The displacement field is expressed in terms of the thermal field and only three primary displacement variables by satisfying exactly the conditions of zero transverse shear stress at the top and the bottom and its continuity at the layer interfaces. The governing equations are derived using the principle of virtual work. Fourier series solutions are obtained for simply-supported beams. Comparison with the exact thermo-elasticity solution for thermal stress analysis under two kinds of thermal loads establishes that the present zigzag theory is generally very accurate and superior to the existing zigzag theory for composite and sandwich beams.     

A refined theory for laminated beams: Part II—An iterative variational approach

This paper presents a displacement-based one-dimensional model for the analysis of laminated composite beams, based on the assumption of cross sections rigid in their own planes. The proposed model is mainly focused on the boundary layer analysis. The representation of the axial displacements is given as products between line functions and warping modes of the cross section. Both the sets of unknown functions are determined by means of a variational formulation in order to obtain the ‘best choice’ for the thickness coordinate functions. The minimization of the total potential energy functional is reduced to a sequence of linear problems by means of a gradient technique. Various examples referring to simply supported and cantilever beams, subjected to distributed or concentrated loads, are solved. The results for stress distributions are found to be in excellent agreement with exact plane strain and finite element plane stress solutions even at very low distances from the end sections.

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Sommario In questo lavoro viene presentato un modello monodimens onale agli spostamenti per l'analisi di travi in laminato multistrato fondato sull'ipotesi di sezioni indeformabili nel proprio piano. Il modello è finalizzato principalmente allo studio degli effetti di bordo. Gli spostamenti assiali vengono espressi attraverso prodotti fra funzioni di linea e modi di ingobbamento della sezione. Entrambi gli insiemi di funzioni incognite sono determinati attraverso una formulazione variazionale allo scopo di ottimizzare le forme di ingobbamento della sezione. La minimizzazione del funzionale dell'energia potenziale totale viene ridotta ad una sequenza di problemi lineari mediante una tecnica tipo gradiente operando alternativamente le varizioni rispetto ai due insiemi di funzioni incognite. Negli esempi risolti, gli stati tensionali ottenuti risultano in eccellente accordo con soluzioni esatte (in stato piano di deformazione) e con soluzioni agli elementi finiti (in stato piano di tensione) anche in prossimità delle sezioni estreme libere o vincolate.

     

A finite wavelet domain method for wave propagation analysis in thick laminated composite and sandwich plates

An efficient numerical method is developed for the simulation of three dimensional transient dynamic response in thick laminated composite and sandwich plate structures involving very high frequencies and wave numbers. The proposed method incorporates Daubechies wavelet scaling functions for the interpolation of the in-plane displacements with a Galerkin formulation. It further explores the orthonormality and compact support of wavelet scaling functions to produce near diagonal consistent mass matrices and banded stiffness matrices. Hence, an uncoupled equivalent discrete spatial dynamic system is formulated, synthesized and rapidly solved in the wavelet domain using an explicit time integration scheme. The in-plane wavelet interpolation is further combined with an efficient high order layerwise laminate plate theory, that implements Hermite cubic splines for the through-the-thickness approximation of displacement fields. Numerical results are presented on the prediction of guided waves in laminated and thick sandwich composite plates and compared with respective solutions obtained by analytical, semi-analytical and time domain spectral element models. The method yielded higher convergence rates and substantial reductions in computational effort compared to respective time domain spectral finite elements.     

基于B-样条小波的复合材料层合板灵敏度分析

基于BSWI样条小波有限元方法,将响应和响应的灵敏度系数同时看作状态变量,在Hamilton体系下推导了复合材料层合板响应和响应灵敏度系数的混合控制方程。基于该混合控制方程研究了对称铺层四边固支复合材料层合板的位移响应灵敏度系数在z方向上的分布情况,并将计算结果与有限差分法进行比较。结果表明:材料参数E₁,E₂和G₁₂对位移响应影响明显,其中以E₁对位移响应的影响最大,E₂次之。与经验法相比,半解析法计算结果相对误差小于10^-4,这说明本文的计算方法是正确的,且降低了计算成本和程序实现难度,提高了计算精度。     

Non-linearities in rotation and thickness deformation in a new third-order thickness deformation theory for static and dynamic analysis of isotropic and laminated doubly curved shells

A geometrically non-linear theory is developed for shells of generic shape allowing for third-order thickness and shear deformation and rotary inertia by using eight parameters; geometric imperfections are also taken into account. The geometrically non-linear strain–displacement relationships are derived retaining full non-linear terms in all the 8 parameters, i.e. in-plane and transverse displacements, rotations of the normal and thickness deformation parameters; these relationships are presented in curvilinear coordinates, ready to be implemented in computer codes. Higher order terms in the transverse coordinate are retained in the derivation so that the theory is suitable also for thick laminated shells. Three-dimensional constitutive equations are used for linear elasticity. The theory is applied to circular cylindrical shells complete around the circumference and simply supported at both ends to study initially static finite deformation. Both radially distributed forces and displacement-dependent pressure are used as load and results for different shell theories are compared. Results show that a 6 parameter non-linear shell theory is quite accurate for isotropic shells. Finally, large-amplitude forced vibrations under harmonic excitation are investigated by using the new theory and results are compared to other available theories. The new theory with non-linearity in all the 8 parameters is the only one to predict correctly the thickness deformation; it works accurately for both static and dynamics loads.     

复合材料层合板的二次屈曲和二次分枝点分析

为了研究复合材料层合板的二次分叉特性 ,利用能量变分原理和非线性几何方程建立了具有弹性约束的复合材料层合板在面内载荷作用下的非线性稳定性控制方程组。控制方程组用广义傅立叶级数法进行求解 ,并得到载荷 -挠度曲线。基于分叉理论中的 Lerray-Schaulder定理 ,采用小挠动法 ,直接导出了复合材料层合板的二次失稳方程。研究结果表明 ,非对称层板也可能存在分叉 ,弹性转动支持系数和铺层等因素对二次分叉有很重要的影响。随着弹性系数的增大 ,二次失稳载荷值与初次失稳载荷值之比下降