Exact resultant equilibrium conditions in the non-linear theory of branching and self-intersecting shells

We formulate the exact, resultant equilibrium conditions for the non-linear theory of branching and self-intersecting shells. The conditions are derived by performing direct through-the-thickness integration in the global equilibrium conditions of continuum mechanics. At each regular internal and boundary point of the base surface our exact, local equilibrium equations and dynamic boundary conditions are equivalent, as expected, to the ones known in the literature. As the new equilibrium relations we derive the exact, resultant dynamic continuity conditions along the singular surface curve modelling the branching and self-intersection as well as the dynamic conditions at singular points of the surface boundary. All the results do not depend on the size of shell thicknesses, internal through-the-thickness shell structure, material properties, and are valid for an arbitrary deformation of the shell material elements.     

Existence Theorems in the Geometrically Non-linear 6-Parameter Theory of Elastic Plates

In this paper we show the existence of global minimizers for the geometrically non-linear equations of elastic plates, in the framework of the general 6-parameter shell theory. A characteristic feature of this model for shells is the appearance of two independent kinematic fields: the translation vector field and the rotation tensor field (representing in total 6 independent scalar kinematic variables). For isotropic plates, we prove the existence theorem by applying the direct methods of the calculus of variations. Then, we generalize our existence result to the case of anisotropic plates.     

Determination of the midsurface of a deformed shell from prescribed surface strains and bendings via the polar decomposition

We show how to determine the midsurface of a deformed thin shell from known geometry of the undeformed midsurface as well as the surface strains and bendings. The latter two fields are assumed to have been found independently and beforehand by solving the so-called intrinsic field equations of the non-linear theory of thin shells. By the polar decomposition theorem the midsurface deformation gradient is represented as composition of the surface stretch and 3D finite rotation fields. Right and left polar decomposition theorems are discussed. For each decomposition the problem is solved in three steps: (a) the stretch field is found by pure algebra, (b) the rotation field is obtained by solving a system of first-order PDEs, and (c) position of the deformed midsurface follows then by quadratures. The integrability conditions for the rotation field are proved to be equivalent to the compatibility conditions of the non-linear theory of thin shells. Along any path on the undeformed shell midsurface the system of PDEs for the rotation field reduces to the system of linear tensor ODEs identical to the one that describes spherical motion of a rigid body about a fixed point. This allows one to use analytical and numerical methods developed in analytical mechanics that in special cases may lead to closed-form solutions.     

功能梯度截顶圆锥壳的热弹性弯曲精确解

研究了功能梯度材料截顶圆锥壳在横向机械载荷与非均匀热载荷同时作用下的变形问题. 基于经典线性壳体理论推导出了以横向剪力和中面转角为基本未知量的功能梯度薄圆锥壳轴对称变形的混合型控制方程. 假设功能梯度圆锥壳的材料性质为沿厚度方向按照幂函数连续变化的形式. 然后采用解析方法求解,得到了问题的精确解. 分别就两端简支和两端固支边界条件,给出了圆锥壳的变形随其载荷、材料参数等变化的特征关系曲线,重点分析和讨论了载荷参数与材料梯度变化参数对变形的影响.      

Numerical Analysis of Axisymmetric Buckling of a Conical Shell under Radial Compression

The nonlinear boundary-value problem of the axisymmetric buckling of a simply supported conical shell (dome) under a radial compressive load applied to the supported edge is formulated for a system of six first-order ordinary differential equations for independent fields of finite displacements and rotations. Multivalued solutions are obtained using the shooting method with specified accuracy. For various values of the loading parameter, bifurcation of the solutions of the problem is studied and a parametric branching diagram is constructed. The buckling modes are obtained for three branches of the solution. Curves of the buckling modes corresponding to three isolated branches of the solution are given.     

压电材料中切口/接头端部平面电弹性场奇异性有限元分析

为了对平面载荷作用下压电材料中切口或接头端部附近电弹性场奇异性问题进行分析，首先以应力平衡方程、Maxwell方程和和边界条件为基础，得到一种求解压电材料特征问题的弱式方程；其次，假定楔形切口或接头端部附近单元内位移和电势沿径向分布为指数形式，而周向方向分布则采用泡函数插值，将其代入弱式方程，建立一种只需对楔形切口或接头端部附近周边进行离散的一维简单有限元方法．压电材料的极化轴可以是任意方向．利用该有限元模型讨论了楔形切口角度、极化轴方向和边界条件对奇性场的影响．通过和其它特定情况下的现有解相比，证实了该文有限元数值方法的有效性，而且精度很高．     

基于拟应变法的壳单元大变形分析及在冲击动力问题中的应用

为了简便有效地解决板壳结构的大变形问题，本文针对八节点相对自由度壳单元进行研究。该单元的位移场由壳的中面节点位移和上表面节点的相对位移组成，不带有转动变量。所有的研究都是基于完全的三维位移、应力、应变场。采用拟应变法，对应变场另行假设，能够改善该单元在大变形情况下的计算精度。通过引入Wilson非协调模式，构造了大变形情况下的拟应变场表达式，给出了该单元用于解决非线性动力分析问题的有限元求解方程。通过算例表明，本文针对相对自由度壳单元提出的方法及推导的公式，能够解决冲击动力问题中的大变形问题。     

BOUNDARY VALUE PROBLEMS OF TWO-DIMENSIONAL ANISOTROPIC BODY WITH A PARABOLIC BOUNDARY

Based upon Stroh formalism we derive a novel and convenient scheme for determiningthe elastic fields of a two-dimensional anisotropic body with a parabolic boundary subject to two kindsof boundary conditions, which are free surface and rigid surface, respectively. The correspondingGreen's functions are found by using the conformal mapping method. When the parabolic curve de-generates into a half-infinite crack or rigid inclusion, the singular stress fields near the tip of defectsare obtained. In particular, those Green's functions for a concentrated moment M_0 applied at a pointon the parabolic curve are also studied. It is easily found that arbitrary parabolic boundary value prob-lems can be solved by using these Green's functions and associate integrals.     

Regularization of Discontinuous Vector Fields on $${\mathbb{R}^3}$$ via Singular Perturbation

Singular perturbations problems in dimension three which are approximations of discontinuous vector fields are studied in this paper. The main result states that the regularization process developed by Sotomayor and Teixeira produces a singular problem for which the discontinuous set is a center manifold. Moreover, the definition of sliding vector field coincides with the reduced problem of the corresponding singular problem for a class of vector fields.      

A new variable-order singular boundary element for calculating stress intensity factors in three-dimensional elasticity problems

A new three-dimensional variable-order singular boundary element has been constructed for stress analysis of three-dimensional interface cracks and internal material junctions. The singular fields in the vicinity of crack front or junction have been accurately represented by the singular elements by taking account the variable order of singularities and the angular profiles of field variables. Both the singular stress fields and displacement fields are independently formulated by the element’s shape functions. Different kinds of displacement formulations are investigated. The formulation combining singular and linear terms is found to be the most accurate one. The mixed-mode stress intensity factors are treated as nodal unknowns. The variation of stress intensity factors along the line of singularity can be obtained directly from the final system of equations and thus no post processing, such as three-dimensional J-integral or domain integral, is necessary. Numerical examples involving stress singularity, such as penny-shaped cracks in homogeneous and dissimilar material interface, plates with through-thickness cracks, and a dissimilar inclusion, are investigated. The analysis results are in good agreement with those reported in the literature.     

基于非协调元特征法的奇异性问题分析

提出了一个基于位移的、分析平面尖劈尖端奇性应力场和位移场问题的非协调FE特征分析法．该方法与过去原有求解裂纹尖端近似场的有限元特征分析方法导出公式的出发点不同，并且采用的单元形式为非协调元，尖劈尖端邻域内的位移场假定没有采用奇异变换技术，运用该方法处理了若干尖劈和接头的算例，所有的计算结果表明，该方法较原有方法使用的单元少而且精度高，具有应用广泛性。     

基于真实工况下的工业机器人关节转角辨识与分析

本文基于多目视觉测量系统,对真实工况下连续运动的工业机器人进行关节转角的实时重构．该方法通过机器人运动前后的坐标集,在对刚体运动进行最优拟合的条件下,采用最小二乘法获得了各关节的旋转矩阵与平移向量．在此基础上,在考虑相邻关节牵连运动的前提下,获得了各关节的相对旋转矩阵．结合罗德里格斯变换理论通过相对旋转矩阵,确定了各关节转角．仿真与实验分析,验证了该方法的有效性与正确性．在该测量与辨识体系下,初步确定了各关节转角随机器人运动的真实状态．2与3杆臂由于物理尺寸呈细长形状,连杆挠度较大,这时变形误差与振动建立了关系,角振动幅度很大,曲线随机性较强．其余杆臂由于刚度较大,关节转角曲线呈光滑状态．     

An Experimental Study of the Rotational Effects on Separated Turbulent Flow During Stall Delay

Three-dimensional velocity fields were measured using tomographic particle image velocimetry (Tomo-PIV) on a model of the blade of a small-scale horizontal axis wind turbine (HAWT) to study the effects of rotation on separated turbulent flows during stall delay at a global tip speed ratio (TSR) of 3 and a Reynolds number of 4800. The flow fields on a static airfoil were also measured at a similar angle-of-attack (AOA) and Reynolds number for comparison. It was observed that the blade’s rotation in the streamwise direction significantly affected both the mean flow and the turbulence statistics over the suction surface. The mean velocity fields revealed that, different from the airfoil flow at large AOA, the recirculation region with reversed flow did not exist on the suction surface of the blade and the flow was rather attached. Mean spanwise flow from blade’s root to its tip was also generated by the rotation. The mean vorticity vector of the blade flow was found to be tilted in the rotational direction of the blade, as well as in the wall-normal direction. Of particular effects of the rotation on Reynolds stresses were the enhancement of 〈w ²〉 and the creation of strong 〈v w〉. The production of Reynolds stresses was also affected by blade’s rotation directly through the rotational production terms and indirectly by dramatically changing the fluctuating velocity fields. The distribution of enstrophy was observed to be modified by rotation, too.     

Asymptotic Justification of the Kirchhoff–Love Assumptions for a Linearly Elastic Clamped Shell

The displacement vector of a linearly elastic shell can be computed by using the two-dimensional Koiter's model, based on the a priori Kirchhoff–Love assumptions. These hypotheses imply that the displacement of any point of the shell is an affine function of the transverse variable x ₃. The term independent of x ₃ of this approximation is equal to the displacement vector of the two-dimensional Koiter's model. The term linear in x ₃ depends on the infinitesimal rotation vector of the normal. After an appropriate scaling, we estimate here the difference between the three-dimensional displacement and this affine vector field in the case of shells clamped along their entire lateral face. Besides, in the case of shells with uniformly elliptic middle surface, taking into account the term depending of the rotation of the normal allows to improve the asymptotic estimate between the three-dimensionnal displacement and Koiter's bidimensional displacement. This revised version was published online in August 2006 with corrections to the Cover Date.     

Relationships of the Timoshenko-type theory of thin shells with arbitrary displacements and strains

A new modified version of the Timoshenko theory of thin shells is proposed to describe the process of deformation of thin shells with arbitrary displacements and strains. The new version is based on introducing an unknown function in the form of a rotation vector whose components in the basis fitted to the deformed mid-surface of the shell are the components of the transverse shear vector and the extensibility in the transverse direction according to Chernykh. For the case with the shell mid-surface fitted to an arbitrary non-orthogonal system of curvilinear coordinates, relationships based on the use of true stresses and true strains in accordance with Novozhilov are obtained for internal forces and moments. Based on these relationships, a problem of static instability of an isotropic spherical shell experiencing internal pressure is solved. The shell is considered to be made either of a linear elastic material or of an elastomer (rubber), which is described by Chernykh’s relationships.     

Higher order zig-zag theory for smart composite shells under mechanical-thermo-electric loading

A higher order zig-zag shell theory based on general tensor formulation is developed to refine the predictions of the mechanical, thermal, and electric behaviors. All the complicated curvatures of surface including twisting curvatures can be described in a geometrically exact manner in the present shell theory because the present theory is based on the geometrically exact surface representation. The in-surface displacement fields are constructed by superimposing the linear zig-zag field to the smooth globally cubic varying field through the thickness. Smooth parabolic distribution through the thickness is assumed in the out-of-plane displacement in order to consider transverse normal deformation and stress. The layer-dependent degrees of freedom of displacement fields are expressed in terms of reference primary degrees of freedom by applying interface continuity conditions as well as bounding surface free conditions of transverse shear stresses. Thus the proposed theory has only seven primary displacement unknowns and they do not depend upon the number of layers. To assess the validity of present theory, the developed theory is evaluated under the thermal and electric load as well as under the mechanical load of composite cylindrical shells. Through the numerical examples, it is demonstrated that the proposed smart shell theory is efficient because it has the minimal degrees of freedom. The present theory is suitable in the predictions of deformation and stresses of thick smart composite shells under the mechanical, thermal, and electric loads combined.     

Variable kinematic models applied to free-vibration analysis of functionally graded material shells

Closed-form solutions of free-vibration problems of simply supported multilayered shells made of Functionally Graded Material have been examined in the present paper. A variable kinematic shell model, which is based on Carrera’s Unified Formulation is extended, in this work, to dynamic shell cases. Classical shell theories are compared to refined ones as well as to layer-wise kinematics and mixed assumptions based on the Reissner mixed variational theorem. A comparison with the few results available in the open literature is presented and conclusions are drawn regarding the accuracy of classical and advanced shell modeling to evaluate lower and higher vibration modes as well as the behavior of these modes in the shell thickness direction.