涉及肌肉作用影响的髓内钉屈曲分析

在骨折的修复过程中，常发生髓内钉断骨内的情况。其主要原因是涉及肌肉影响后，髓内钉不是单钝的受压杆件，而是多种组合变形的屈曲问题。文中利用有限元法对目前较新结构的髓内钉在骨折治疗中涉及肌肉作用影响进行多方面的压、弯曲与扭转组合变形屈曲分析。得出了三叶瓣型髓内钉在涉及肌肉影响时的屈曲临界压力比单纯受压时髓内钉的屈曲临界压力小51％。并得出髓内钉只能额外承受15Nm左右的外来扭力。这些结果可供髓内钉进一步发展及临床应用参考。通过有限元的分析还可以对髓内钉受力状态的薄弱环节有一个全面的认识，为髓内钉合理设计、材料的选择等进行优化设计提供依据。     

空间杆系与叠层板组合结构的非线性稳定分析

本文用非线性有限元法研究一种空间杆系与复合材料叠层板组合结构的非线性稳定问题。将杆系离散成空间梁柱单元，将复合材料叠层板离散成复合材料叠层板单元，并考虑了几何非线性和板的成层正交各向异性性质。为了获得较高的计算效率并顺利通过稳定临界点，求解非线性平衡方程采用了变增量步长和柱面弧长法迭代策略。在考证了程序的正确性之后，具体分析了几个工程模型，得出了于工程设计有益的结论。     

Die freie Oberfläche einer Flüssigkeit über einer rotierenden Scheibe

Zusammenfassung Es wird eine modifizierte Form des Weissenberg-Effekts untersucht, wobei sich die viskoelastische Flüssigkeit in einem kreiszylindrischen Gefäß befindet, an dessen Boden eine Scheibe rotiert. Normalspannungsdifferenzen rufen in der Flüssigkeit eine Strömung hervor, die auf der Drehachse von unten nach oben gerichtet ist, und die freie Oberfläche wölbt sich nahe der Achse nach außen. Unter der Voraussetzung hinreichend langsamer Strömung wird eine Theorie zweiter Ordnung entwickelt. Sie führt auf elliptische Randwertaufgaben zweiter bzw. vierter Ordnung für das Geschwindigkeitsfeld der Primärströmung in Umfangsrichtung und für die Stromfunktion der Sekundärströmung in der Meridianebene. Ihnen werden äquivalente Variationsaufgaben zugeordnet und mit der Methode der Finiten Elemente numerisch gelöst. Die Gestalt der freien Oberfläche setzt sich bei geeigneter Normierung aus drei universellen Formfunktionen zusammen, die für verschiedene Füllhöhen berechnet werden. Im experimentellen Teil wird nachgewiesen, daß durch entsprechende Messungen der Auslenkung des Flüssigkeitsspiegels die unteren Grenzwerte der beiden Normalspannungskoeffizienten bestimmt werden können. Das Rheometer besitzt den Vorzug, daß die Oberflächenspannung der Flüssigkeit die Meßgröße nur unwesentlich beeinflußt.

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Some kind of Weissenberg effect is considered where the viscoelastic fluid, being within a cylindrical vessel, is set in motion by a rotating disc near the tank bottom. Because of normal-stress differences within the fluid a secondary flow arises which is directed upwards near the axis of symmetry, and thus the free surface is deformed. Under the assumption of sufficiently slow flow a second-order theory is developed. It leads to second-order and fourth-order elliptic boundary value problems for the velocity field in azimuthal direction and for the stream function of the secondary flow, respectively. Equivalent variational problems are formulated and solved by the method of finite elements. When normalized appropriately, the shape of the free surface consists of three shape functions, which are independent of any material constants. It is shown by corresponding experiments, that the zero-shear-rate normal-stress coefficients can be determined by measuring the displacement of the free surface. In this rheometer, the surface tension of the fluid causes only insignificant influence on the quantity to be measured.

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Symbole C _H [—] Verhältnis der FormfunktionenF ₂/F₁ - f [—] die Sekundärströmung treibende radiale Volumenkraft, dimensionslos - F ₀, F₁, F₂ [—] universelle Formfunktionen - Fr [—] Froude-Zahl - g [m s^–2] Erdbeschleunigung - h [—] Auslenkung der Oberfläche, aufr ₀ bezogen - H [—] dimensionslose Füllhöhe - K [—] Kennzahl der Kapillarität - r,z [m] Zylinderkoordinaten - r, z [—] dimensionslose Koordinaten - r ₀ [m] Radius des Meßbehälters - Re [—] Reynolds-Zahl - v _r, v, v_z [m s^–1] Geschwindigkeitskomponenten - We ₁, We₂ [—] Weissenberg-Zahlen - [Pa s] Nullviskosität der Flüssigkeit - [°C] Temperatur - [m] Kapillarlänge - v ₁, v₂ [Pa s²] untere Grenzwerte der Normalspannungskoeffizienten - [kg m^–3] Dichte der Flüssigkeit - [N m^–1] Oberflächenspannung - [—] Zylinderkoordinate - [—] Dissipationsfunktion der Sekundärströmung, dimensionslos - [—] Stromfunktion, dimensionslos - [—] örtliche Winkelgeschwindigkeit, dimensionslos - [s^–1] Winkelgeschwindigkeit der Scheibe     

Non-linear viscoelastodynamic equations of three-dimensional rotating structures in finite displacement and finite element discretization

This paper deals with the non-linear viscoelastodynamics of three-dimensional rotating structure undergoing finite displacement. In addition, the non-linear dynamics is studied with respect to geometrical and mechanical perturbations. On part of the boundary of the structure, a rigid body displacement field is applied which moves the structure in a rotation motion. A time-dependent Dirichlet condition is applied to another part of the boundary. For instance, this corresponds to the cycle step of a helicopter rotor blade. A surface force field is applied to the third part of the boundary and depends on the time history of the structural displacement field. For example, this might corresponds to general unsteady aerodynamics forces applied to the structure. The objective of this paper is to model the non-linear dynamic behavior of such a rotating viscoelastic structure undergoing finite displacements, and to allow small geometrical and mechanical (mass, constitutive equations) perturbations analysis to be performed. The model is constructed by the introduction of a reference configuration which is deduced from the non-linear steady boundary value problem. A constitutive equation deduced from the Coleman and Noll theory concerning the viscoelasticity in finite displacement is used. Thereafter, the weak formulation of the boundary value problem is constructed and discretized using the finite element method. In order to simplify the mathematical study of the equations, multilinear forms are introduced in the algebraic calculation and their mathematical properties are presented.     

梁板壳的几何大变形--从近似的非线性理论到有限变形理论

对梁板壳的线性理论、近似几何非线性理论与有限变形理论作了比较,介绍了有限转动理论,指出了应用有限变形理论求解梁板壳的大变形问题的高效率、高精度的巨大优越性。     

HOMOGENIZATION—BASED TOPOLOGY DESIGN FOR PURE TORSION OF COMPOSITE SHAFTS

In conjunction with the homogenization theory and the finite element method, the mathematical models for designing the corss-section of composite shafts by maximizing the torsion rigidity are developed in this paper. To obtain the extremal torsion rigidity, both the cross-section of the macro scale shaft and the representative microstructure of the composite material are optimized using the new models. The micro scale computational model addresses the problem of finding the periodic microstructures with extreme shear moduli. The optimal microstructure obtained with the new model and the homogenization method can be used to improve and optimize natural or artificial materials. In order to be more practical for engineering applications, cellular materials rather than ranked materials are used in the optimal process in the existence of optimal bounds for the elastic properties. Moreover, the macro scale model is proposed to optimize the cross-section of the torsional shaft based on the tailared composites. The validating optimal results show that the models are very effective in obtaining composites with extreme elastic properties, and the cross-section of the composite shaft with the extremal torsion rigidity. The project supported by the National Natural Science Foundation of China (10172078 and 10102018)     

基于Hellinger-Reissner变分原理的应变梯度杂交元设计

从一般的偶应力理论出发，基于Hellinger-Reissner变分原理，通过对有限元 离散体系的位移试解引入非协调位移函数，得到了偶应力理论下有限元离散系统的能量相容 条件，并由此建立了应变梯度杂交元的应力函数优化条件. 根据该优化条件，构造了一 个C0类的平面4节点梯度杂交元，数值结果表明，该单元对可压缩和不可压缩状态的 梯度材料均可给出合理的数值结果，再现材料的尺度效应.     

有限变形弹性杆中的非线性波及周期解

利用Ham ilton变分原理,导出了计及有限变形和横向Possion效应的弹性杆中非线性纵向波动方程.利用Jacob i椭圆正弦函数展开和第三类Jacob i椭圆函数展开法,对该方程和截断的非线性方程进行求解,得到了非线性波动方程的两类准确周期解及相应的孤波解和冲击波解,讨论了这些解存在的必要条件.     

NONLINEAR WAVES AND PERIODIC SOLUTION IN FINITE DEFORMATION ELASTIC ROD

A nonlinear wave equation of elastic rod taking account of finite deformation, transverse inertia and shearing strain is derived by means of the Hamilton principle in this paper. Nonlinear wave equation and truncated nonlinear wave equation are solved by the Jacobi elliptic sine function expansion and the third kind of Jacobi elliptic function expansion method. The exact periodic solutions of these nonlinear equations are obtained, including the shock wave solution and the solitary wave solution. The necessary condition of exact periodic solutions, shock solution and solitary solution existence is discussed.     

广义协调平板型矩形壳元

本文根据修正势能原理通过广义协调方法提出了一种列式简单的平板型矩形壳元ＧＣＲ２４。它在四个角点处各有六个自由度，总共二十四个自由度。作为一种极限协调元，单元的收敛性得到保证，并且不发生薄膜闭锁现象。通过标准问题的数值检验，表明本文提出的平板型矩形薄壳元是性能可靠、计算精度高的优质单元。