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双语推荐：Mindlin板理论

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基于复合材料一阶剪切理论---Mindlin理论的板理论和Rayleigh表面积分方程，对嵌在无限障板上的复合材料层合板声传递问题进行分析。在此基础上，建立复合材料的结构传声损失优化设计模型。最后，基于传声损失理论，应用遗传算法进行对称型复合材料板结构的铺层几何优化分析。算例验证了优化方法的有效性。
Based on the Mindlin first order shear effect plate theory and Rayleigh integral function ,the sound transmission of laminated composite plates on an infinite baffle is investigated herein . T he sound transmission loss optimization model of composite structure was established .Finally ,with the sound transmission loss theory ,the stacking sequence problem of symmetric composite laminate opti-mization is conducted .By introducing genetic algorithm (GA ) ,the optimization problem is solved . The numerical example shows the effectiveness of the calculating ways .

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基于三维压电理论,利用有限元软件ANSYS对考虑封装结构的石英晶体谐振器进行了研究。在添加导电胶、封装基座和封盖后,获得了清晰的石英晶体板厚度剪切振动的位移云图和在长度、宽度方向的位移图。数值分析结果表明与自由振动相比,封装基座和封盖对石英晶体板厚度剪切振动的影响不大。结果验证了Mindlin板理论在分析石英晶体板高频振动的有效性。
Based on the three-dimensional theory of piezoelectricity, the quartz crystal resonator with consideration of package structures have been investigated by the finite element software ANSYS. With adding conductive adhesive support, sealing base and cover, the paper obtained clear cloud pictures of displacement of the thickness-shear vibrations of quartz crystal plate and displacement pictures on the length and width direction. Numerical results showed that compared with free vibration, the effect of sealing base and cover on the thickness-shear vibration of quartz crystal plate is little. In the paper, the obtained results have verified efficiency of Mindlin plate theory on analysis of high frequency vibration of quartz crystal plate.

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基于Mindlin板理论提出了一种高阶八节点杂交应力四边形单元.该单元不仅能通过零剪力分片检验,而且能通过非零常剪力增强型分片检验.单元边界位移插值采用任意阶Timoshenko梁函数,对不同厚跨比的四边简支、固支方板,以及圆板进行了弯曲和自由振动分析,数值结果表明无论对薄板还是中厚板,该单元均是准确和有效的,并且具有几何不变性.
Based on the Mindlin plate theory,a high-order 8-node hybrid stress quadrilateral element is presented.The proposed element can pass not only the zero shear stress patch test,but also the non-zero constant shear stress enhanced patch test.The element boundary displacement is interpolated by arbitrary order Timoshenko beam function.The bending and free vibration of the simply-supported and clamped square and circular plate with different thickness and span ratios are investigated. Numerical results show that the element can be used to analyze both moderately thick and thin plates and it is efficient,accurate and geometric invariant.

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为了解决中厚板与粘弹性地基共同作用下的非线性振动问题,在Reissner-Mindlin一阶剪切变形板的理论基础上,运用pb-2瑞利-里兹法分析双参数粘弹性地基上四边自由矩形中厚板的非线性自由振动,探讨了板的尺寸参数、横向剪切因子、粘滞系数以及地基反应模量对板的振动特性产生的影响及其变化规律。并对数值算例进行编程求解,与文献实测数据进行对比,结果十分接近,证明了该方法的可行性。
Based on the theory of Reissner-Mindlin first order shear deformation plate,the vibration method of pb-2 Rayleigh-Ritz analysis the vibration problems of moderate thick-ness rectangular plates with four free edges resting on a biparametric viscoelastic foundation is used to solve the question of free rectangular middle thick plate with biparametric viscoe-lastic foundation ,the influence and change laws of the vibration characteristics of the plates caused by the size parameters,transverse shear factor,reaction modulus of foundation are discussed.The numerical examples are programmed,the solution obtained is compared with the measured date from the literature.The solutions are almost identical,the presented method is proved to be feasible.

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基于Mindlin平板理论,采用波函数展开法和局部坐标系方法,对含双圆孔平板结构中弹性波散射与动应力集中问题进行了研究.文中对Mindlin板含双圆孔时的开孔动弯矩集中系数做了数值计算,并分析了开孔间距对动弯矩分布的影响.结果表明:与单圆孔情况相比,由于开孔之间的相互影响,双圆孔间的动弯矩分布会发生比较复杂的变化.孔间距相互作用有时会使动应力集中得到缓解,而有时会使动应力集中加剧.在低频和平板较薄的情况下,平板开孔动弯矩互不影响间距较小;在较高频率和平板较厚的情况下,平板开孔动弯矩互不影响间距较大.所有这些现象都与入射波波长与孔径等特征尺度有关.因此,在工程结构动力学分析与强度设计中,应对不同波长和特征尺度下的动应力作具体的分析计算,而不是简单地套用静载强度设计标准或规范.
Based on Mindlin''s theory of plates,elastic wave scattering and dynamic stress concentrations in plates with two circular cutouts were studied by using the wave function expansion method and local coordinate system. Numerical results of dynamic moment concentration factors around the cutouts were obtained by setting different parameters and the influence of the parameters on dynamic moment distributions were discussed.It is shown that unlike single-cutout situation,the complex changes take place on dynamic moment concentration factors at two-cutout situation due to the interaction between two openings.Sometimes the dynamic moment was increased,or sometimes the one was reduced.When the plate is thicker at the high-frequency,the opening spacing of the non-significant influence is larger.Therefore,in the dynamic design the comprehensive dynamic stress analysis would be done instead of directly applying the static standard and code for the structure design.

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根据Mindlin理论,考虑宏观尺度非线性效应、二次和三次微尺度非线性效应以及微尺度频散效应,建立了描述一维微结构固体中纵波传播的一种新模型.用动力系统定性分析理论,分析了微结构固体中孤立波的存在条件及其几何特征,证明了当介质参数和孤立波传播速度满足适当条件时,在二次微尺度非线性效应的影响下微结构固体中可以形成一种非对称孤立波,在三次微尺度非线性效应的影响下微结构固体中可以形成一种对称孤立波.最后,用数值方法进一步验证了上述结论.
By taking into account the macroscale nonlinear effect, quadratic and cubic microscale nonlinear effects, and microscale dispersion effect, a new model capable to describe the longitudinal wave propagation in one-dimensional microstructured solid is established based on the Mindlin theory. Using the qualitative analysis theory in the planar dynamical systems, we have analyzed the existence conditions and geometrical characteristics of solitary waves, and proved that the faces under the influence of quadratic microscale nonlinear effect, can form an asymmetric solitary wave in the microstructured solid; and under the influence of cubic microscale nonlinear effect, they can form a symmetric solitary wave in the microstructured solid, when the medium parameters and the propagation speeds of solitary waves satisfy certain appropriate conditions. Finally, the above results are further verified using a numerical method.

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在Toupin-Mindlin应变梯度理论框架内,建立了一种关于厚壁筒的应变梯度模型,并利用打靶法和拟牛顿迭代法实现了两点复杂边界条件和四阶常微分平衡方程的求解,得到了厚壁筒应力场和位移场精确数值解。为了降低该模型的复杂性和提高该模型的适用性,通过对高阶应力做出简化假设,得到了简化的平衡方程和相应的厚壁筒应力、应变的解析解。通过对厚壁筒经典弹性理论解析解、梯度理论简化解和梯度理论精确数值解的比较,检验了简化模型解析解的可行性和精确性,并得到了梯度理论和经典理论在厚壁筒应用上的差异性。
The simplified equilibrium equation and analytical solutions to strains and stresses in thick hollow cylin-der are obtained in the framework of Toupin-Mindlin′s strain gradient theory by adopting simplifying assumptions with respet to the higher-order stresses. To verify the feasibility of the analytical solutions of the simplified model, Quasi-Newton method, together with a shooting method,was used to obtain accurate numerical solutions of the origi-nal fourth-order ordinary differential equilibrium equation with complex boundary conditions. The analytical solu-tions of conventional elastic theory, analytical solutions of the simplified gradient model and the accurate numerical solutions of the original gradient model were compared to test the feasibility and accuracy of the simplified gradient model and to acquire the differences between the original gradient theory and the conventional elastic theory in the thick hollow cylinder.

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基坑开挖卸荷导致工程桩桩侧极限阻力降低.建立单桩模型,用Mindlin应力解考虑开挖引起的竖向有效应力变化,分别计算开挖前后的桩侧极限阻力.通过某工程案例,将理论结果与其他结果相对比.最后分析了桩侧阻力降低系数随基坑开挖深度、边长、长宽比及桩长的变化规律.结果表明:桩侧阻力降低系数随开挖深度增加先减小而后缓慢增大,存在谷值临界深度,随开挖边长、长宽比增加先减小而后趋于稳定;增加桩长导致桩侧阻力降低系数增大.
The effect of excavation-related unloading on the underlying soil will reduce the ultimate pile shaft resistance.In this study,Mindlin''s stress solution was used to allow for the variation of vertical ef-fective stresses induced by excavation around a single pile.The ultimate resistances of the pile shaft before and after the excavation were calculated.The results of theoretical calculation were compared with others through a case study.Furthermore,the variations of reduction factors for shaft resistance with excavation depth,excavation length,ratio of length to width and pile length were analyzed.The results showed that reduction factors for shaft resistance decreased firstly and then increased with the increase of excavation depth,and it decreased first and then tended to be stable with the increase of excavation length and ratio of length to width.With the increase of pile length,the reduction factors for shaft resistance increased.

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疏桩在软土地区的应用可以分为两个方面:疏桩复合地基和疏桩基坑支护.本文针对疏桩基础中的摩擦桩,采用弹性力学中的Mindlin-Geddes解和Boussinesq解联合求解的方法,分析了疏桩基础的附加应力场.通过对比不同桩间距下的桩土荷载分担比的变化规律,分析了软土地区疏桩间距的合理范围.同时对疏桩水平方向上存在的土拱效应,采用三维有限元数值方法模拟了桩间距对疏桩成拱的影响,得出了产生土拱的最大桩间距范围在6~8d之间.针对疏桩复合地基的沉降控制计算方法,本文总结出应用Poulos弹性理论法计算单桩沉降和复合地基方法计算群桩沉降.
The application of scattered piles in soft soil can be divided into two aspects:composite founda-tion and excavation pit supporting .This article analyzed the additional stress field of friction piles in sparse pile foundation with Mindlin and Boussinesq solutions .By comparing the load sharing percent of soil with different pile spacing, the reasonable range of pile spacing was recommended in soft soil region .Meanwhile, 3D finite el-ement method was adopted to analyze the horizontal soil arching effect of scattered piles influenced by pile spac -ing, concluding that the maximum pile spacing can reach 6~8d.While elastic theory methods were used in sin-gle pile settlement calculation , the designing method of composite foundation was adopted in pile group calcula-tion.

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针对颗粒流法中颗粒几何参数等对土体宏观力学性质的影响进行了数值模拟,并将数值模拟结果与室内模型试验验证进行了对比。对二维、三维下土体级配及孔隙率进行了理论分析,并提出二维数值试样颗粒数目及孔隙率转换公式。采用PFC2D软件进行了6组参数验证及24组交叉对比(含4组级配、3组孔隙率及2类接触模型)双轴压缩试验。结果表明:孔隙率对土体力学性质的影响大于级配;随着孔隙率增加,土体强度峰值逐渐降低,对应轴向应变增大,体应变则由剪胀性逐渐向剪缩性过渡。当孔隙率较低时,采用线性及Hertz-Mindlin(H-M)接触模型得到的模拟结果均较接近;而当孔隙率升高时,建议采用H-M接触模型,以更好地反映土体到达强度峰值后产生的后续剪胀效应。
To analyze the impact of geometrical particle parameters on macromechanical properties of the soil in particle flow method, the conversion formula of the granule number and modified porosity of numerical sample group in the 2D condition was proposed;and the numerical simulation and the laboratory model tests were carried out. It conducted 6 groups of parameter validation tests and 24 groups of variables cross-comparison biaxial compression tests, and included 4 types of gradations, 3 types of porosities and 2 kinds of contact models. The results show that the effect of porosity is larger than that of gradation on soil mechanical properties. With the increase of porosity, the peak value of soil strength gradually decreases; and the axial strains of corresponding points increase accordingly. Compared with the results of the Hertz-Mindlin (H-M) contact model which is focused on the nonlinear relation between the contact force and relative displacement;the results of the linear contact mod

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