Coupled thermoelasticity in laminated composite plates based on Green–Lindsay model

A new mixed finite element formulation is proposed to analyze transient coupled thermoelastic problems. The non-classical (Green–Lindsay) coupled model of dynamic thermoelasticity is selected for a laminated composite and a homogeneous isotropic plate. For the particular finite element developed here, there are 15 degrees of freedom at each node. Two simply supported plates are considered subjected to a half-sine mechanical and thermal loading. It is seen from a comparison of the present study with NISA II FEM code that the present method has good agreement and accuracy.     

Classical coupled thermoelasticity in laminated composite plates based on third-order shear deformation theory

A new mixed finite element formulation is proposed to analyze transient coupled thermoelastic problems. Coupled model of dynamic thermoelasticity is selected for a laminated composite and a homogeneous isotropic plate. For the particular finite element developed here, there are 15 degrees of freedom at each node. Two simply supported plates are considered subjected to sinusoidally distributed mechanical and thermal loading. It is seen, by comparing the present results with results from the NISA II FEM code, that they are in good agreement.     

Generalized thermoelastic waves in spherical curved plates without energy dissipation

In this article, the propagation of thermoelastic waves in orthotropic spherical curved plates subjected to stress-free, isothermal boundary conditions is investigated in the context of the Green–Naghdi (GN) generalized thermoelastic theory (without energy dissipation). The theoretical formulation is based on the linear GN thermoelastic theory. The coupled wave equation and heat conduction equation expressed by the displacement and temperature are obtained. By the Legendre orthogonal polynomial series expansion approach, the coupled controlling equations are solved. The convergence of the method is demonstrated through a numerical example. The dispersion curves of thermal modes and elastic modes are illustrated simultaneously. Dispersion curves of the corresponding purely elastic spherical plate are also shown to analyze the influence of thermoelasticity on elastic modes. The displacement, temperature and stress distributions of both elastic modes and thermal modes are calculated to show their differences. A thermoelastic spherical plate with a different ratio of radius to thickness is considered to show the influence of the ratio on the characteristics of thermoelastic waves.     

A unified 3D‐based technique for plate bending analysis using scaled boundary finite element method

This paper presents a unified technique for solving the plate bending problems by extending the scaled boundary finite element method. The formulation is based on the three‐dimensional governing equation without enforcing the kinematics of plate theory. Only the in‐plane dimensions are discretised into finite elements. Any two‐dimensional displacement‐based elements can be employed. The solution along the thickness is expressed analytically by using a matrix function. The proposed technique is consistent with the three‐dimensional theory and applicable to both thick and thin plates without exhibiting the numerical locking phenomenon. Moreover, the use of higher order spectral elements allows the proposed technique to better represent curved boundaries and to achieve high accuracy and fast convergence. Numerical examples of various plate structures with different thickness‐to‐length ratios demonstrate the applicability and accuracy of the proposed technique. Copyright © 2012 John Wiley & Sons, Ltd.     

Dispersion of Thermoelastic Waves in a Plate With and Without Energy Dissipation

In this paper, the dispersion and energy dissipation of thermoelastic plane harmonic waves in a thin plate bounded by insulated traction-free surfaces is studied on the basis of three generalized theories of thermoelasticity. The frequency equations corresponding to the symmetric and antisymmetric modes of vibration of the plate are obtained. Some limiting and particular cases of the frequency equations are then discussed. Results obtained in three theories of generalized thermoelasticity are compared. The results for the coupled thermoelasticity can be obtained as particular cases of the results by setting thermal relaxations times equal to zero. Numerical evaluations relating to the lower modes of the symmetric and antisymmetric waves are presented for an aluminum alloy plate.     

Extended rotation‐free plate and beam elements with shear deformation effects

This paper describes a methodology for extending rotation‐free plate and beam elements to accounting for transverse shear deformation effects. The ingredients for the element formulation are a Hu–Washizu‐type mixed functional, a linear interpolation for the deflection and the shear angles over standard finite elements and a finite volume approach for computing the bending moments and the curvatures over a patch of elements. As a first application of the general procedure, we present an extension of the three‐noded rotation‐free basic plate triangle (BPT) originally developed for thin plate analysis to account for shear deformation effects of relevance for thick plates and composite‐laminated plates. The nodal deflection degrees of freedom (DOFs) of the original BPT element are enhanced with the two shear deformation angles. This allows to compute the bending and shear deformation energies leading to a simple triangular plate element with three DOFs per node (termed BPT+ element). For the thin plate case, the shear angles vanish and the element reproduces the good behaviour of the original thin BPT element. As a consequence the element is applicable to thick and thin plate situations without exhibiting shear locking effects. The numerical solution for the thick case can be found iteratively starting from the deflection values for the Kirchhoff theory using the original thin BPT element. A two‐noded rotation‐free beam element termed CCB+ applicable to slender and thick beams is derived as a particular case of the plate formulation. The examples presented show the robustness and accuracy of the BPT+ and the CCB+ elements for thick and thin plate and beam problems. Copyright © 2010 John Wiley & Sons, Ltd.     

Thermoelastic wave characteristics in a hollow cylinder using the modified wave finite element method

This paper represents a modified formulation of the wave finite element (WFE) method for propagating analysis of thermoelastic waves in a hollow cylinder without energy dissipation. The 2D-high-order spectral element with the Gauss–Legendre–Lobatto integration is applied into the WFE method, which produces the diagonal mass matrix. Based on the assumption of harmonic displacement fields by Fourier series expansion, the general discretization wave equation is simplified from the 3D problem to 2D. Dispersion properties of elastic wave propagation in the hollow cylinder are computed considering the choice of the spectral element orders, and the results indicate the high efficiency and high accuracy of the modified formulation compared with that of the software Disperse. Then, using the modified formulation, the thermoelastic dynamic equation of the cylinder is derived from the generalized thermoelasticity theory. The propagation of the thermoelasticwave (including two kinds of wave modes) in the cylinder without energy dissipation is discussed in differentcases. Finally, wave structures along the radial direction of thermoelastic wave modes are shown at thenondimensional frequency 1.25, which can be used for the recognition of different modes.     

Meshless local Petrov–Galerkin method for coupled thermoelasticity analysis of a functionally graded thick hollow cylinder

In this article, coupled thermoelasticity (without energy dissipation) based on Green–Naghdi model is applied to functionally graded (FG) thick hollow cylinder. The meshless local Petrov–Galerkin method is developed to solve the boundary value problem. The Newmark finite difference method is used to treat the time dependence of the variables for transient problems. The FG cylinder is considered to be under axisymmetric and plane strain conditions and bounding surfaces of cylinder to be under thermal shock loading. The mechanical properties of FG cylinder are assumed to vary across thickness of cylinder in terms of volume fraction as nonlinear function. A weak formulation for the set of governing equations is transformed into local integral equations on local subdomains by using a Heaviside test function. Nodal points are regularly distributed along the radius of the cylinder and each node is surrounded by a uni-directional subdomain to which a local integral equation is applied. The Green–Naghdi coupled thermoelasticity equations are valid in each isotropic subdomain. The temperature and radial displacement distributions are obtained for some grading patterns of FGM at various time instants. The propagation of thermal and elastic waves is discussed in details. The presented method shows high capability and efficiency for coupled thermoelasticity problems.     

A hybrid stress ANS solid‐shell element and its generalization for smart structure modelling. Part I—solid‐shell element formulation

In the recent years, solid‐shell finite element models which possess no rotational degrees of freedom and applicable to thin plate/shell analyses have attracted considerable attention. Development of these elements are not straightforward. Shear, membrane, trapezoidal, thickness and dilatational lockings must been visioned. In this part of this paper, a novel eight‐node solid‐shell element is proposed. To resolve the shear and trapezoidal lockings, the assumed natural strain (ANS) method is resorted to. The hybrid‐stress formulation is employed to rectify the thickness and dilatational locking. The element is computationally more efficient than the conventional hybrid elements by adopting orthogonal‐assumed stress modes and enforcing admissible sparsity in the flexibility matrix. Popular benchmark tests are exercised to illustrate the efficacy of the elements. In Part II of the paper, the element will be generalized for smart structure modelling by including the piezoelectric effect. Copyright © 2000 John Wiley & Sons, Ltd.     

An adhesively laminated plate element for PZT smart plates

An adhesively laminated element taking into consideration peel stress is developed for a piezoelectric smart plate. In this novel finite element analysis formulation, a four node piezoelectric element is firstly derived, and an adhesive element of finite thickness with both shear and peel stiffness is sandwiched between two collocated four node plate elements to form an adhesively laminated element for a piezoelectric smart plate. In this framework of finite element analysis, because the displacement filed in this adhesively laminated element is continuous and a plate element is derived based on the Reissner–Mindlin plate theory, and thus it can be accurately applied to a thin or moderately thick host plate with bonded or debonded piezoelectric actuators and sensors. The formulation is performed for an isotropic host plate and a fiber reinforced laminate plate. Numerical results are presented to compare with those of the exact solutions for smart beams, and validate with the experimental results of the isotropic and composite host plates available in the literature. Using the present finite element analysis formulation, energy transfer stresses in the adhesive and equivalent forces induced in the host plate are investigated. The present formulation is demonstrated to allow debondings of piezoelectric patches and the debonding detection.The authors are grateful to the support of the Australian Research Council via a Discovery Projects grant (grant No: DP0346419).     

Enhanced mixed interpolation XFEM formulations for discontinuous Timoshenko beam and Mindlin‐Reissner plate

Shear locking is a major issue emerging in the computational formulation of beam and plate finite elements of minimal number of degrees of freedom as it leads to artificial overstiffening. In this paper, discontinuous Timoshenko beam and Mindlin‐Reissner plate elements are developed by adopting the Hellinger‐Reissner functional with the displacements and through‐thickness shear strains as degrees of freedom. Heterogeneous beams and plates with weak discontinuity are considered, and the mixed formulation has been combined with the extended finite element method (FEM); thus, mixed enrichment functions are used. Both the displacement and the shear strain fields are enriched as opposed to the traditional extended FEM where only the displacement functions are enriched. The enrichment type is restricted to extrinsic mesh‐based topological local enrichment. The results from the proposed formulation correlate well with analytical solution in the case of the beam and in the case of the Mindlin‐Reissner plate with those of a finite element package (ABAQUS) and classical FEM and show higher rates of convergence. In all cases, the proposed method captures strain discontinuity accurately. Thus, the proposed method provides an accurate and a computationally more efficient way for the formulation of beam and plate finite elements of minimal number of degrees of freedom.     

Mixed plate bending elements based on least‐squares formulation

A finite element formulation for the bending of thin and thick plates based on least‐squares variational principles is presented. Finite element models for both the classical plate theory and the first‐order shear deformation plate theory (also known as the Kirchhoff and Mindlin plate theories, respectively) are considered. High‐order nodal expansions are used to construct the discrete finite element model based on the least‐squares formulation. Exponentially fast decay of the least‐squares functional, which is constructed using the L₂ norms of the equations residuals, is verified for increasing order of the nodal expansions. Numerical examples for the bending of circular, rectangular and skew plates with various boundary conditions and plate thickness are presented to demonstrate the predictive capability and robustness of the new plate bending elements. Plate bending elements based on this formulation are shown to be insensitive to both shear‐locking and geometric distortions. Copyright © 2004 John Wiley & Sons, Ltd.     

Effect of Initial Stress on a Fiber-Reinforced Anisotropic Thermoelastic Thick Plate

The two-dimensional problem of generalized thermoelasticity for a fiber-reinforced anisotropic thick plate under initial stress is studied in the context of the Lord and Shulman theory. The upper surface of the plate is thermally insulated with prescribed surface loading while the lower surface of the plate rests on a rigid foundation and temperature. The problem is solved numerically using a finite element method. Numerical results for the temperature distribution, and the displacement and stress components are given and illustrated graphically. It is found from the graphs that the initial stress significantly influences the variations of field quantities. The results obtained in this paper may offer a theoretical basis and meaningful suggestions for the design of various fiber-reinforced anisotropic thermoelastic elements under loading to meet special engineering requirements.     

几何非线性平板的热力,动力耦合振动

本文根据由热力学第一、第二定律推出的能量守恒方程,进而导出平板几何非线性热力动力耦合振动有限元方程组,提出一种计算量少,精度高的校正显式积分格式,求解耦合方程组的时间历程问题。研究了热力动力耦合平板横向振动振幅频率的影响及热弹阻尼现象。     

A two-dimensional problem for a fibre-reinforced anisotropic thermoelastic half-space with energy dissipation

The theory of thermoelasticity with energy dissipation is employed to study plane waves in a fibre-reinforced anisotropic thermoelastic half-space. We apply a thermal shock on the surface of the half-space which is taken to be traction free. The problem is solved numerically using a finite element method. Moreover, the numerical solutions of the non-dimensional governing partial differential equations of the problem are shown graphically. Comparisons are made with the results predicted by Green–Naghdi theory of the two types (GNII without energy dissipation) and (GNIII with energy dissipation). We found that the reinforcement has great effect on the distribution of field quantities. Results carried out in this paper can be used to design various fibre-reinforced anisotropic thermoelastic elements under thermal load to meet special engineering requirements.     

Modal validation of sandwich shell finite elements based on a p‐order shear deformation theory including zigzag terms

This paper describes a set of improved C⁰‐compatible composite shell finite elements for evaluating the global dynamic response (natural frequencies and mode shapes) of sandwich structures. Combining a through‐the‐thickness displacement approximation of variable high order with a first‐order zigzag function, the proposed finite elements are suited for modelling sandwich plates and doubly curved shells with a non‐uniform thickness and are more accurate than conventional models based on the first‐ and third‐order shear deformation theories, especially in sandwich panels with highly heterogeneous properties. The new finite element model is then validated by a comparison with the standard shell and 3D solid models. From these investigations, it can be concluded that adding a zigzag function even to high‐order polynomial approximations of the through‐the‐thickness displacement is a useful tool for refining the modelling of sandwich structures. In addition, the proposed formulation is sufficiently versatile to represent with the same level of accuracy the behaviour of thin‐to‐thick laminated shells as well as of strongly heterogeneous sandwich structures. Copyright © 2008 John Wiley & Sons, Ltd.     

Shock-induced thermoelastic wave propagation analysisin a thick hollow cylinder without energy dissipation using mesh-free generalized finite difference (GFD) method

The main objective of this article is the exploitation of the generalized finite difference method to study the thermoelastic wave propagation, that is, the dynamic behaviors of displacement and temperature field in a thick hollow cylinder. The thermoelasticity governing equations are derived based on Green–Naghdi coupled thermoelasticity theory (without energy dissipation). The generalized finite difference (GFD) method is used to approximate the space variables, and Newmark finite difference (NFD) is employed to obtain the behaviors of parameters in time domain. The time histories of displacement and temperature fields across the thickness of the cylinder are obtained and the propagations of thermal and elastic waves are illustrated at various times. Using the GFD method, the wave front in temperature and elastic domains can be tracked, and the comparison between results based on GFD and other numerical methods shows very good agreement. The application of GFD method in coupled thermoelasticity problems has a high capability because it does not require a mesh generation. A comparison between the presented mesh-free GFD method and meshless local Petrov–Galerkin (MLPG) method shows a good agreement of the results.     

A four‐node,shear‐deformable shell element developed via explicit Kirchhoff constraints

An efficient, four‐node quadrilateral shell element is formulated using a linear, first‐order shear deformation theory. The bending part of the formulation is constructed from a cross‐diagonal assembly of four three‐node anisoparametric triangular plate elements, referred to as MIN3. Closed‐form constraint equations, which arise from the Kirchhoff constraints in the thin‐plate limit, are derived and used to eliminate the degrees‐of‐freedom associated with the ‘internal’ node of the cross‐diagonal assembly. The membrane displacement field employs an Allman‐type, drilling degrees‐of‐freedom formulation. The result is a displacement‐based, fully integrated, four‐node quadrilateral element, MIN4T, possessing six degrees‐of‐freedom at each node. Results for a set of validation plate problems demonstrate that the four‐node MIN4T has similar robustness and accuracy characteristics as the original cross‐diagonal assembly of MIN3 elements involving five nodes. The element performs well in both moderately thick and thin regimes, and it is free of shear locking. Shell validation results demonstrate superior performance of MIN4T over MIN3, possibly as a result of its higher‐order interpolation of the membrane displacements. It is also noted that the bending formulation of MIN4T is kinematically compatible with the existing anisoparametric elements of the same order of approximation, which include a two‐node Timoshenko beam element and a three‐node plate element, MIN3. Copyright © 2000 John Wiley & Sons, Ltd.     

Consistent coupling of beam and shell models for thermo‐elastic analysis

In this paper, the finite element formulation of a transition element for consistent coupling between shell and beam finite element models of thin‐walled beam‐like structures in thermo‐elastic problems is presented. Thin‐walled beam‐like structures modelled only with beam elements cannot be used to study local stress concentrations or to provide local mechanical or thermal boundary conditions. For this purpose, the structure has to be modelled using shell elements. However, computations using shell elements are a lot more expensive as compared to beam elements. The finite element model can be more efficient when the shell elements are used only in regions where the local effects are to be studied or local boundary conditions have to be provided. The remaining part of the structure can be modelled with beam elements. To couple these two models (i.e. shell and beam models) at transitional cross‐sections, transition elements are derived here for thermo‐elastic problems. The formulation encloses large displacement and rotational behaviour, which is important in case of thin‐walled beam‐like structures. Copyright © 2004 John Wiley & Sons, Ltd.