夹芯梁结构有效性能的多尺度变分渐近模型
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摘要
为有效分析夹芯梁结构性能,基于变分渐近法建立多尺度变分渐近模型。首先基于旋转张量分解概念建立三维夹芯梁几何非线性的能量方程;利用梁结构细长和非均质较小的特征,将三维夹芯梁结构各向异性、非均质问题严格分解为宏观层面的梁轴线的一维非线性分析和细观层面的单胞本构分析。基于最小势能原理,通过对单胞应变能泛函变分主导项最小化得到有效属性和波动函数解,代入梁的一维模型进行全局非线性响应分析。利用得到的全局响应、波动函数解重构局部场。由于变分特性,构建多尺度模型可以很容易通过有限元数值实现。通过三类夹芯梁结构算例表明:构建模型得到的全局位移和局部应力场与三维有限元具有很好的一致性,但计算成本和建模工作量明显减少,为结构设计人员在初始设计阶段对夹芯梁结构性能评估提供了一种简洁的途径。
In order to effectively analyze the properties of sandwich beams,a multi-scale variational asymptotic model was established based on the variational asymptotic method.Firstly,the geometrical nonlinear equations of the original 3 Dsandwich beam were established based on the concept of rotation tensor decomposition.By using the characteristics of slender and heterogeneous,the anisotropy and heterogeneity problems of sandwich beams were strictly decomposed into 1 Dnonlinear analysis along the beam reference line at the macroscopic level and unit cell constitutive analysis at the microscopic level.Based on the principle of minimum potential energy,the effective stiffness and fluctuation function solution were obtained by minimizing the variational leading items in the strain energy functional,and substituted into the 1 Dmodel of beam to perform the global response analysis.Then,the resulting global response and fluctuation function solution were used to recover the local fields.Due to the variational characteristics,the constructed multiscale model can be easily numerical implemented by finite element method.The example results of three kinds of sandwich beams show that the global displacements and local stress fields obtained by the constructed model are in good agreement with the 3 Dfinite element method,but the computational cost and modeling workload are significantly reduced,which provides a simple way for the structural designer to evaluate the performance of the sandwich beams at the initial design stage.
In order to effectively analyze the properties of sandwich beams,a multi-scale variational asymptotic model was established based on the variational asymptotic method.Firstly,the geometrical nonlinear equations of the original 3 Dsandwich beam were established based on the concept of rotation tensor decomposition.By using the characteristics of slender and heterogeneous,the anisotropy and heterogeneity problems of sandwich beams were strictly decomposed into 1 Dnonlinear analysis along the beam reference line at the macroscopic level and unit cell constitutive analysis at the microscopic level.Based on the principle of minimum potential energy,the effective stiffness and fluctuation function solution were obtained by minimizing the variational leading items in the strain energy functional,and substituted into the 1 Dmodel of beam to perform the global response analysis.Then,the resulting global response and fluctuation function solution were used to recover the local fields.Due to the variational characteristics,the constructed multiscale model can be easily numerical implemented by finite element method.The example results of three kinds of sandwich beams show that the global displacements and local stress fields obtained by the constructed model are in good agreement with the 3 Dfinite element method,but the computational cost and modeling workload are significantly reduced,which provides a simple way for the structural designer to evaluate the performance of the sandwich beams at the initial design stage.
引文
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[19]钟轶峰,余文斌.压电复合材料层合板的热压电弹性简化模型研究[J].工程力学,2012,29(6):314-319.ZHONG Y F,YU W B.Study on simplified thermopiezoelastic model for piezoelectric composite laminates[J].Engineering Mechanics,2012,29(6):314-319(in Chinese).
[20]DAI G M,ZHANG W H.Size effects of basic cell in static analysis of sandwich beams[J].International Journal of Solids&Structures,2008,45(9):2512-2533.
[2]ARVIN H,SADIGHI M,OHADI A R.A numerical study of free and forced vibration of composite sandwich beam with viscoelastic core[J].Composite Structures,2010,92(4):996-1008.
[3]KUMAR M,SHENOI R A,COX S J.Experimental validation of modal strain energies based damage identification method for a composite sandwich beam[J].Composites Science &Technology,2009,69(10):1635-1643.
[4]KHALILI S M R,DEHKORDI M B,CARRERA E,et al.Non-linear dynamic analysis of a sandwich beam with pseudoelastic SMA hybrid composite faces based on higher order finite element theory[J].Composite Structures,2013,96(4):243-255.
[5]SCIUVA M D,GHERLONE M,IURLARO L,et al.A class of higher-order C0,composite and sandwich beam elements based on the refined zigzag theory[J].Composite Structures,2015,132:784-803.
[6]ACCORSI M L,NEMAT-NASSER S.Bounds on the overall elastic and instantaneous elastoplastic moduli of periodic composites[J].Mechanics of Materials,1986,5(3):209-220.
[7]HASHIN Z,SHTRIKMAN S.A variational approach to the theory of the elastic behaviour of polycrystals[J].Journal of the Mechanics &Physics of Solids,1962,10(4):343-352.
[8]PALEY M,ABOUDI J.Micromechanical analysis of composites by the generalized cells model[J].Mechanics of Materials,1992,14(2):127-139.
[9]WILLIAMS T O.A two-dimensional,higher-order,elasticity-based micromechanics model[J].International Journal of Solids &Structures,2005,42(3):1009-1038.
[10]MURAKAMI H,TOLEDANO A.A high-order mixture homogenization of bi-laminated composites[J].Journal of Applied Mechanics,1990,57(2):388-397.
[11]SUN C T,VAIDYA R S.Prediction of composite properties from a representative volume element[J].Composites Science&Technology,1996,56(2):171-179.
[12]BUANNIC N,CARTRAUD P.Higher-order effective modeling of periodic heterogeneous beams I—Asymptotic expansion method[J].International Journal of Solids &Structures,2016,38(40):7139-7161.
[13]KOLPAKOV A G.Calculation of the characteristics of thin elastic rods with a periodic structure[J].Journal of Applied Mathematics &Mechanics,1991,55(3):358-365.
[14]BUANNIC N,CARTRAUD P.Higher-order effective modeling of periodic heterogeneous beams II—Derivation of the proper boundary conditions for the interior asymptotic solution[J].International Journal of Solids &Structures,2001,38(40-41):7163-7180.
[15]CARTRAUD P,MESSAGER T.Computational homogenization of periodic beam-like structures[J].International Journal of Solids &Structures,2006,43(3-4):686-696.
[16]SCHREFLER B A,LEFIK M.Use of homogenization theory to build a beam element with thermo-mechanical microscale properties[J].Structural Engineering &Mechanics,1996,4(6):613-630.
[17]ZHONG Y F,CHEN L,YU W.Asymptotical Construction of a Reissner-like model for multilayer functionallygraded magneto-electro-elastic plates[J].Composite Structures.2013,96(8):786-798.
[18]ZHONG Y F,CHEN L,YU W.Variational asymptoticmodeling of the multilayer functionally graded cylindrical shells[J].Composite Structures.2012,94(3):966-976.
[19]钟轶峰,余文斌.压电复合材料层合板的热压电弹性简化模型研究[J].工程力学,2012,29(6):314-319.ZHONG Y F,YU W B.Study on simplified thermopiezoelastic model for piezoelectric composite laminates[J].Engineering Mechanics,2012,29(6):314-319(in Chinese).
[20]DAI G M,ZHANG W H.Size effects of basic cell in static analysis of sandwich beams[J].International Journal of Solids&Structures,2008,45(9):2512-2533.