Finite rotation exact geometry solid-shell element for laminated composite structures through extended SaS formulation and 3D analytical integration

A nonlinear exact geometry hybrid-mixed four-node solid-shell element using the sampling surfaces (SaS) formulation is developed for the analysis of the second Piola-Kirchhoff stress that extends the authors' finite element (Int J Numer Methods Eng. 2019;117:498-522) to laminated composite shells. The SaS formulation is based on choosing inside the layers the arbitrary number of SaS parallel to the middle surface and located at Chebyshev polynomial nodes in order to introduce the displacements of these surfaces as basic shell unknowns. The external surfaces and interfaces are also included into a set of SaS. The proposed hybrid-mixed solid-shell element is based on the Hu-Washizu variational principle and is completely free of shear and membrane locking. The tangent stiffness matrix is evaluated by efficient three-dimensional (3D) analytical integration. As a result, the developed exact geometry solid-shell element exhibits a superior performance in the case of coarse meshes and allows the use of load increments, which are much larger than possible with existing displacement-based solid-shell elements. It could be useful for the 3D stress analysis of thick and thin doubly curved laminated composite shells because the SaS formulation gives the possibility to obtain the 3D solutions with a prescribed accuracy.     

Analysis of the second Piola-Kirchhoff stress in nonlinear thick and thin structures using exact geometry SaS solid-shell elements

This paper presents the finite rotation exact geometry four-node solid-shell element using the sampling surfaces (SaS) method. The SaS formulation is based on choosing inside the shell N SaS parallel to the middle surface to introduce the displacements of these surfaces as basic shell unknowns. Such choice of unknowns with the consequent use of Lagrange polynomials of degree N–1 in the through-thickness distributions of displacements, strains and stresses leads to a robust higher-order shell formulation. The SaS are located at only Chebyshev polynomial nodes that make possible to minimize uniformly the error due to Lagrange interpolation. The proposed hybrid-mixed four-node solid-shell element is based on the Hu-Washizu variational principle and is completely free of shear and membrane locking. The tangent stiffness matrix is evaluated through efficient 3D analytical integration and its explicit form is given. As a result, the proposed exact geometry solid-shell element exhibits a superior performance in the case of coarse meshes and allows the use of load increments, which are much larger than possible with existing displacement-based solid-shell elements.     

Assessment of nonlinear exact geometry sampling surfaces solid-shell elements and ANSYS solid elements for 3D stress analysis of piezoelectric shell structures

In this work, the finite rotation exact geometry four-node solid-shell element using the sampling surfaces (SaS) method is developed for the analysis of the second Piola-Kirchhoff stresses in laminated piezoelectric shells. The SaS method is based on choosing inside the layers the arbitrary number of SaS parallel to the middle surface and located at Chebyshev polynomial nodes in order to introduce the displacements and electric potentials of these surfaces as fundamental shell unknowns. The outer surfaces and interfaces are also included into a set of SaS. To circumvent shear and membrane locking, the hybrid-mixed solid-shell element on the basis of the Hu-Washizu variational principle is proposed. The tangent stiffness matrix is evaluated by 3D analytical integration throughout the finite element. This novelty provides a superior performance in the case of coarse meshes. A comparison with the SOLID226 element showed that the developed exact geometry SaS solid-shell element allows the use of load increments, which are much larger than possible with existing displacement-based finite elements. Thus, it can be recommended for the 3D stress analysis of doubly-curved laminated piezoelectric shells because the SaS formulation gives the opportunity to obtain the 3D solutions of electroelasticity with a prescribed accuracy.     

A robust,four-node,quadrilateral element for stress analysis of functionally graded plates through higher-order theories

ABSTRACT

A hybrid-mixed, four-node, quadrilateral element for the three-dimensional (3D) stress analysis of functionally graded (FG) plates using the method of sampling surfaces (SaS) is developed. The SaS formulation is based on choosing an inside the plate body N, not equally spaced SaS parallel to the middle surface, in order to introduce the displacements of these surfaces as basic plate variables. Such a choice of unknowns, with the consequent use of Lagrange polynomials of the degree N ? 1 in the assumed distributions of displacements, strains, and mechanical properties through the thickness leads to a robust FG plate formulation. All SaS are located at Chebyshev polynomial nodes that permit one to minimize uniformly the error due to the Lagrange interpolation. To avoid shear locking and spurious zero-energy modes, the assumed natural strain method is employed. The proposed four-node quadrilateral element passes 3D patch tests for FG plates and exhibits a superior performance in the case of coarse distorted meshes. It can be useful for the 3D stress analysis of thin and thick metal/ceramic plates because the SaS formulation gives an opportunity to obtain the solutions with a prescribed accuracy, which asymptotically approach the 3D exact solutions of elasticity as the number of SaS tends to infinity.     

The Use of 9-Parameter Shell Theory for Development of Exact Geometry 12-Node Quadrilateral Piezoelectric Laminated Solid-Shell Elements

The present work focuses on the development of the exact geometry (EG) 12-node piezoelectric solid-shell element with three translational degrees of freedom per node. The term “EG” reflects the fact that coefficients of the first and second fundamental forms of the reference surface are taken exactly at each element node. The finite element formulation developed is based on the higher-order 9-parameter equivalent single-layer shell theory accounting for thickness stretching, which permits the use of 3D constitutive equations. In this theory, we introduce three sampling surfaces, namely, bottom, middle, and top, and choose nine displacements of these surfaces as basic shell unknowns. Such a way allows one to represent the EG piezoelectric solid-shell element formulation in a very compact form and to derive the strain-displacement equations, which describe exactly all rigid-body shell motions in any convected curvilinear coordinate system. The element matrices are evaluated through the use of 3D analytical integration by employing the extended ANS method. To avoid shear and membrane locking and have no spurious zero energy modes, the assumed displacement-independent strains and stress resultants fields are invoked.     

Refined shell elements for the analysis of functionally graded structures

The present paper considers the static analysis of plates and shells made of Functionally Graded Material (FGM), subjected to mechanical loads. Refined models based on the Carrera’s Unified Formulation (CUF) are employed to account for grading material variation in the thickness direction. The governing equations are derived from the Principle of Virtual Displacement (PVD) in order to apply the Finite Element Method (FEM). A nine-nodes shell element with exact cylindrical geometry is considered. The shell can degenerate in the plate element by imposing an infinite radius of curvature. The Mixed Interpolation of Tensorial Components (MITC) technique is extended to the CUF in order to contrast the membrane and shear locking phenomenon. Different thickness ratios and orders of expansion for the displacement field are analyzed. The FEM results are compared with both benchmark solutions from literature and the results obtained using the Navier method that provides the analytical solution for simply-supported structures subjected to sinusoidal pressure loads. The shell element based on refined theories of the CUF turns out to be very efficient and its use is mandatory with respect to the classical models in the study of FGM structures.     

A hybrid‐mixed four‐node quadrilateral plate element based on sampling surfaces method for 3D stress analysis

The hybrid‐mixed assumed natural strain four‐node quadrilateral element using the sampling surfaces (SaS) technique is developed. The SaS formulation is based on choosing inside the plate body N not equally spaced SaS parallel to the middle surface in order to introduce the displacements of these surfaces as basic plate variables. Such choice of unknowns with the consequent use of Lagrange polynomials of degree N–1 in the thickness direction permits the presentation of the plate formulation in a very compact form. The SaS are located at Chebyshev polynomial nodes that allow one to minimize uniformly the error due to the Lagrange interpolation. To avoid shear locking and have no spurious zero energy modes, the assumed natural strain concept is employed. The developed hybrid‐mixed four‐node quadrilateral plate element passes patch tests and exhibits a superior performance in the case of coarse distorted mesh configurations. It can be useful for the 3D stress analysis of thin and thick plates because the SaS formulation gives the possibility to obtain solutions with a prescribed accuracy, which asymptotically approach the 3D exact solutions of elasticity as the number of SaS tends to infinity. Copyright © 2016 John Wiley & Sons, Ltd.     

Exact 3D stress analysis of laminated composite plates by sampling surfaces method

A paper focuses on the use of the efficient approach to exact 3D elasticity solutions of cross-ply and angle-ply laminated composite plates. This approach is based on the new method of sampling surfaces (SaS) developed recently by the authors. We introduce inside the nth layer I_n not equally spaced SaS parallel to the midsurface of the plate and choose displacements of these surfaces as fundamental plate unknowns. Such an idea permits the representation of the proposed higher order layer-wise plate theory in a very compact form. This fact gives in turn the opportunity to derive the exact 3D solutions of elasticity for thick and thin laminated composite plates with a prescribed accuracy by utilizing a sufficiently large number of SaS, which are located at interfaces and Chebyshev polynomial nodes.     

基于耦合扩展多尺度有限元方法的功能梯度材料热应力分析

以高效模拟功能梯度材料(FGM)微观非均质性对整体热力学性能的影响为研究目的,通过随机形态描述函数(RMDF)法和体积分数的指数分布建立FGM二维微结构,在此基础上,发展了FGM热应力分析的耦合扩展多尺度有限元方法(CEMsFEM)。该方法基于扩展多尺度有限元方法(EMsFEM)的基本思想,对温度场和位移场构造数值基函数,以把微观非均质材料性质带到宏观响应中。同时为了考虑泊松效应导致的不同方向间的耦合作用,在位移场数值基函数中增加了耦合附加项。通过数值基函数建立宏微观单元信息的映射关系,在宏观尺度求解有效方程,节约计算量。为了更好地考虑微观载荷的影响,把结构的真实响应分解为宏观响应和微观扰动,进一步推导出修正的宏观载荷向量。通过不同体积分数分布的FGM在不同载荷工况下的热应力分析算例验证了本文中方法的正确性和有效性,最后讨论了微结构的尺寸效应对结构热力学响应的影响。     

圆板状SiC/C功能梯度材料残余热应力特征有限元分析

功能梯度材料残余热应力的大小及分布对其性能有效发挥及长期稳定使用有着较大的负面影响,为了尽可能充分发挥材料性能,增加材料的使用寿命,需尽可能减小残余应力以及使其合理分布.本文采用ANSYS有限元分析软件对不同叠层工艺参数的等离子体第一壁候选材料--SiC/C功能梯度材料(FGM)的残余热应力进行了数值模拟,获得了使热应力有效缓和的较适宜的工艺参数,对实际研发制备目标材料也可提供一些理论参照.相关结果表明,适量增加梯度叠层数及中间梯度层厚度可逐步有效缓和残余热应力,同时,针对本文今后应用的仍以炭材料为主体的炭基陶瓷保护层复合SiC/C FGM而言,纯SiC层厚度应取较小值,而叠层成分分布指数应取0.8～1.0为宜.     

A stabilized piezolaminated nine-nodded shell element formulation for analyzing smart structures behaviors

An explicit hybrid stabilization method is utilized together with a reduced order integration scheme to stabilize spurious zero energy modes from the sub-integrated degenerated shell element. This stabilization is achieved after employing appropriate contravariant higher order stress modes. The relevant finite element formulation of the piezolaminated nine-nodded shell element is then derived to analyze smart structures behaviors. To show the capabilities of the presented formulation, it has been implemented in a finite element code. The developed code is used to analyze some typical problems. The results are compared with those obtained from other schemes in the literature and experiments.     

A new boundary element formulation for shear deformable shells analysis

A new domain‐boundary element formulation to solve bending problems of shear deformable shallow shells having quadratic mid‐surface is presented. By regrouping all the terms containing shells curvature and external loads together in equilibrium equation, the formulation can be formed by coupling boundary element formulation of shear deformable plate and two‐dimensional plane stress elasticity. The boundary is discretized into quadratic isoparametric element and the domain is discretized using constant cells. Several examples are presented, and the results shows a good agreement with the finite element method. Copyright © 1999 John Wiley & Sons, Ltd.     

Optimization of volume fractions for functionally graded panels considering stress and critical temperature

Volume fraction optimization of Functionally Graded Materials (FGMs) is investigated considering stress and critical temperature. Material properties are assumed to be temperature dependent, and are assumed to be varied continuously in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituent materials. The effective material properties are obtained by applying linear rule of mixtures. The 3-D finite element model is adopted using an 18-node solid element to analyze more accurately the variation of material properties and temperature field in the thickness direction. For the various FGMs volume fraction distributions, mechanical stress analysis and thermo-mechanical buckling analysis are performed to get the critical conditions. Finally, the optimal designs of FGMs panels are investigated for stress reduction and improving thermo-mechanical buckling behavior.     

Volume fraction optimization for step-formed functionally graded plates considering stress and critical temperature

Step-formed Functionally Graded Materials (FGMs) flat panels are investigated for volume fraction optimization by considering stress and critical temperature. The structure is composed of numerous layers with homogeneous and different isotropic material properties from ceramic to metal. Material properties are assumed to be temperature dependent, and remain constant in each layer. Further, the properties are assumed to be varied in the thickness direction according to a simple power law distribution in terms of the ceramic and metal volume fractions for the layer. The effective material properties of the plate are obtained by applying linear rule of mixtures for the layers. The 3-D finite element model is adopted to analyze more accurately the variation of material properties and temperature field in the thickness direction. For the various FGM volume fraction distributions and geometric parameters, mechanical stress analysis and thermo-mechanical buckling analysis are performed to get the critical conditions. Based on the results, the volume fraction optimization of the flat panels is performed for stress reduction and improving thermo-mechanical buckling behavior and compared with previous results.     

Thermomechanical advantages of functionally graded dental posts: A finite element analysis

This study aimed to fabricate dental posts with functionally graded structures comprised of zirconia, titanium, and hydroxyapatite and compare their thermomechanical behavior with homogeneous zirconia and titanium posts in simulated models of upper central incisor. The results indicated the gradual behavior of functionally graded dental posts in terms of physical and mechanical properties. The finite element analysis revealed a more efficient equilibration to the oral environment after removing the thermal stress in functionally graded dental post compared to the homogeneous counterparts. Therefore, the functionally graded structures could reduce the stress/strain concentrations and interfacial stresses in root canal and minimize the likelihood of root fracture.     

Finite element evaluation of mixed mode stress intensity factors in functionally graded materials

This paper is directed towards finite element computation of fracture parameters in functionally graded material (FGM) assemblages of arbitrary geometry with stationary cracks. Graded finite elements are developed where the elastic moduli are smooth functions of spatial co‐ordinates which are integrated into the element stiffness matrix. In particular, stress intensity factors for mode I and mixed‐mode two‐dimensional problems are evaluated and compared through three different approaches tailored for FGMs: path‐independent J^*_k‐integral, modified crack‐closure integral method, and displacement correlation technique. The accuracy of these methods is discussed based on comparison with available theoretical, experimental or numerical solutions. Copyright © 2001 John Wiley & Sons, Ltd.     

An extended element free Galerkin method for fracture analysis of functionally graded materials

An extended element free Galerkin method (XEFGM) has been adopted for fracture analysis of functionally graded materials (FGMs). Orthotropic enrichments functions are used along with the sub-triangle technique for enhancing the Gauss quadrature accuracy near the crack, and the incompatible interaction integral method is employed to calculate the stress intensity factors. Numerical simulations have proved that XEFGM provides more accurate results by less number of nodes (DOFs) in comparison with the unenriched EFGM and other conventional methods for several FGM problems with different crack locations and loadings. The results have been compared with the reference results, showing the reliability, stability, and efficiency of present XEFGM.

Received 9 June 2014 Accepted 17 September 2014.     

基于组合单元的层压复合材料三维应力分析

为了分析层压复合材料层间特性,推导了将刚性元-弹簧元相结合的离散型界面单元的刚度矩阵。建立了层压板的准三维模型,即将Mindlin板单元应用于层压板的各子层,层间作用则利用上述界面单元来模拟。通过弯曲板元计算子层面内应力,通过界面单元的弹簧力确定层间应力。对受面内拉伸的多向层压板条进行了应力分析,与使用商业软件三维实体模型计算得到的层间和面内应力对比,结果表明准三维模型的计算结果合理。这种新型界面单元的优点是可用来表征层间损伤,并且能通过对弹簧刚度的消减来模拟分层损伤的演变。     

Finite element analysis of thermoelastic contact problem in functionally graded axisymmetric brake disks

An analysis of thermoelastic contact problem of functionally graded (FG) rotating brake disk with heat source due to contact friction is presented. Finite element method (FEM) is used. The material properties of disk are assumed to be represented by power-law distributions in the radial direction. The inner and outer surfaces considered are metal and ceramic, respectively. Pure material is considered for the brake pad. Coulomb contact friction is assumed as the heat source. It is divided into two equal parts between pad and brake disk which leads to thermal stresses. Mechanical response of FG disks are compared and verified with the known results from the literatures. The results show that the maximum value of radial displacement in mounted FG brake disk is not at outer surface. It is found that the all areas between pad and brake disk is in full-contact status when the ratio of pad thickness to brake disk thickness is 0.66. It is observed that the total strain due to thermomechanical load is negative for some parts of the disks, whereas, the thermal strains are always positive. It can be concluded that gradation index of the metal-ceramic has significant effect in the thermomechanical response of FG disks.