A geometrically non-linear tensorial formulation of a skewed quadrilateral thin shell finite element

A 48-degree-of-freedom (d.o.f.) skewed quadrilateral thin shell finite element, including the effect of geometrical non-linearity, is formulated and appropriate numerical procedures are adopted for the development of an efficient approach for the static and dynamic analysis of general thin shell structures. The element surface is described by a variable-order polynomial in curvilinear co-ordinates. The displacement functions are described by bicubic Hermitian polynomials in curvilinear co-ordinates. The directions of the curvilinear co-ordinates at each nodal point are uniquely defined to coincide with the directions of the boundaries of the element. In the present case of a skewed quadrilateral with non-orthogonal curvilinear coordinates, the coupling terms of the metric tensor and curvature tensor of the surface no longer vanish, such as in the case of orthogonal co-ordinates. The tensor form is used in the setup of the shape functions, geometric derivatives, stiffness matrix and computer code. This allows for the treatment of shells with irregular shapes and variable curvatures. To evaluate the efficiency and accuracy of this formulation, a systematic list of examples is chosen: (i) linear and non-linear static analysis of square and rhombic plates, cylindrical and spherical shells; (ii) linear vibrations of trapezoidal flat and curved plates; (iii) large amplitude vibrations of a rhombic plate. For the square plate and cylindrical and spherical shell, shewed element meshes with various distortion angles are used to study the effect of the distortion angles on the accuracy of the results and to demonstrate the versatility of the present element. All results are compared with alternative available solutions including those obtained using regular rectangular meshes. Pinched thin cylindrical and spherical shells are studied using different skewed meshes and various Gauss integration meshes, and no membrane locking phenomenon is observed.     

曲线齿线圆柱齿轮啮合数学模型

参照圆弧齿线圆柱齿轮齿面方程的推导方法以及齿面组成特点,介绍了一类曲线齿线圆柱齿轮的齿面组成特点.采用齿轮啮合原理中的单参数曲面包络方法,运用坐标变换推导了曲线齿线圆柱齿轮的数学方程和共轭齿面方程的一般形式.以圆弧齿线圆柱齿轮的齿面方程推导为例,推导了圆弧齿线圆柱齿轮的空间啮合数学模型,并采用UG建模软件建立了圆弧齿线圆柱齿轮的3D模型,验证了该方法的正确性.研究结果为进一步开展新型齿线的设计、新型曲线齿线圆柱齿轮传动研究提供了理论基础.     

Phase and Dispersion of Cylindrical Surface Waves

Most theoretical and experimental work on surface waves does not take into account dispersion. When propagating along a flat planar half space, surface waves are known as Rayleigh waves and are not dispersive. When the radii of curvature are large, surface waves behave like Rayleigh waves. However, when the radii are small, dispersion becomes a contributing factor. Experimental measurements indicate that along with dispersion, there appears to be a strong phase shift effect as the wave propagates along the circumferential path of cylindrical specimens. The phase shift effect is observed even under conditions where dispersion is not detected. Classical theories provide the velocity-frequency equations, which represent the dispersion relationships, for surface waves. An alternate theoretical approach is discussed in this article that demonstrates the phase-dispersion relationship for cylindrical surface waves. Experimental data support the theoretical conclusions and indicate phase shift is directly related to the radius of curvature to an extent much more sensitive than dispersion.     

基于Pro/E机构模块的齿轮三维实体建模研究

齿廓曲线尤其是齿根过渡曲线的构建是渐开线圆柱齿轮三维实体建模的难点,也是其关键问题。现在所采用的构建方法操作复杂,计算繁琐,且不适合不同刀具加工的齿轮模型的构建。以范成法加工齿轮的原理为理论基础,介绍在Pro/E平台下利用其机构模块的“轨迹曲线”(trace curve)功能精确构建各种刀具加工出来的渐开线圆柱齿轮齿廓曲线的方法,并以磨前滚刀加工直齿圆柱齿轮为例,详细介绍该方法的使用步骤,构建其精确的三维实体模型。该方法无需推导渐开线和过渡曲线的方程就能够方便、快捷地构建出任何刀具加工出来的齿轮模型,为基于啮合原理的复杂曲面零件的有限元建模提供了参考,特别是为复杂曲面零件的CAD/CAE建模与分析奠定了基础。     

Energy Release Rates for the Penny-Shaped Crack in a Linear Piezoelectric Solid

A domain independent integral expression that is derived from the principle of virtual work and holds in curvilinear coordinates is used to derive the energy release rate for a penny-shaped crack in a linear piezoelectric solid. The virtual mechanical and electric displacement fields are chosen to allow variation of the crack tip position. Results for the energy release rate for a finite crack segment and point-wise show the effect of crack front curvature in a piezoelectric material.     

Algebraic model of turbulent boundary layer on a convex curvilinear surface

A simple algebraic model of turbulent boundary layer on convex curvilinear surfaces is suggested that is based on the generalization of the two-layer one-parameter algebraic model for a flat plate [ 1 ]. The model is tested in a wide range of variation of the curvature parameter (0.01 ≤ δ₀/R _w ≤ 0.09, where δ₀ is the thickness of the boundary layer at the initial cross section of the curvilinear region andR _w is the curvature radius of the surface), the results of which are indicative of a good agreement between the experimental and calculated data on the integral characteristics of the boundary layer, namely, the friction coefficientC _f , the displacement thickness δ^* and momentum thickness δ^**, and the form parameterH = δ^*/δ^**. Based on the comparison between the calculated and experimental data on the distribution of tangential turbulent stresses, a conclusion is made that the model predicts a much lower effect of the curvature on the suppression of turbulence in the outer region of boundary layers at a mild curvature of the surface (δ ₀ /R _w = 0.01) than in experiments. However, this difference has a tendency to decrease as the surface curvature increases. An analysis of the calculated and experimental velocity profiles plotted in the variables of the wall law leads to a conclusion that the generalized Townsend wall law is partially realized on a curvilinear surface.     

Curved PVDF airborne transducer

In the application of airborne ultrasonic ranging measurement, a partially cylindrical (curved) PVDF transducer can effectively couple ultrasound into the air and generate strong sound pressure. Because of its geometrical features, the ultrasound beam angles of a curved PVDF transducer can be unsymmetrical (i.e., broad horizontally and narrow vertically). This feature is desired in some applications. In this work, a curved PVDF air transducer is investigated both theoretically and experimentally. Two resonances were observed in this transducer. They are length extensional mode and flexural bending mode. Surface vibration profiles of these two modes were measured by a laser vibrometer. It was found from the experiment that the surface vibration was not uniform along the curvature direction for both vibration modes. Theoretical calculations based on a model developed in this work confirmed the experimental results. Two displacement peaks were found in the piezoelectric active direction of PVDF film for the length extensional mode; three peaks were found for the flexural bending mode. The observed peak positions were in good agreement with the calculation results. Transient surface displacement measurements revealed that vibration peaks were in phase for the length extensional mode and out of phase for the flexural bending mode. Therefore, the length extensional mode can generate a stronger ultrasound wave than the flexural bending mode. The resonance frequencies and vibration amplitudes of the two modes strongly depend on the structure parameters as well as the material properties. For the transducer design, the theoretical model developed in this work can be used to optimize the ultrasound performance.     

An assumed strain formulation of efficient solid triangular element for general shell analysis

An efficient assumed strain triangular solid element is developed for the analysis of plate and shell structures. The finite element formulation is based on the two‐field assumed strain formulation with two independent fields of assumed displacement and assumed strain. The assumed strain field is carefully selected to alleviate the shear locking effect without triggering undesirable spurious kinematic modes. The curvilinear surface of shell structures is modelled with flat facet elements to obviate the membrane locking effect. The patch tests are successfully passed, and numerical test involving various example problems demonstrates the validity and efficiency of the present element. Copyright © 2000 John Wiley & Sons, Ltd.     

低能原电子下形状因子对金属曲面二次发射系数的影响

提出金属曲面二次电子发射的形状因子概念,并从理论上论述了形状因子对曲面二次电子发射系数的影响。从半经验理论出发,推导出低能原电子入射下,金属曲面二次发射系数与入射角的近似关系,并结合曲面形状,得到形状因子一般表达式,进而求出金属曲面的二次发射系数。以圆柱曲面为例,计算其形状因子和低能入射下其二次发射系数,并用实验数据证明了该理论的正确性,最后对结果进行了讨论,得到如下结论:在低能原电子入射情况下,金属曲面二次发射系数为金属材料二次发射系数与形状因子相乘所得。     

A facet-like shell theory

A linear theory for facet-like thin elastic shells is derived where strain/displacement, curvature change/displacement and constitutive relations appear the same as for flat plates. Application of Koiter's arguments shows that the theory is a valid first approximation. The theory is of interest for limiting cases of faceted finite element analysis of smooth shells. Although the final equations of facet-like shell theory do not have quite as simple a form as more conventional equations it is possible that their derivation from equations for flat plates may appeal to engineers. A specialization of the equations is given to circular cylindrical shells where four simple illustrative examples show no essential differences with results from more conventional theory.     

Non-linear coupled multi-field mechanics and finite element for active multi-stable thermal piezoelectric shells

In this paper, a coupled multi-field mechanics framework is presented for analyzing the non-linear response of shallow doubly curved adaptive laminated piezoelectric shells undergoing large displacements and rotations in thermal environments. The mechanics incorporate coupling between mechanical, electric and thermal fields and encompass geometric non-linearity effects due to large displacements and rotations. The governing equations are formulated explicitly in orthogonal curvilinear coordinates and are combined with the kinematic assumptions of a mixed-field shear-layerwise shell laminate theory. A finite element methodology and an eight-node coupled non-linear shell element are developed. The discrete coupled non-linear equations of motion are linearized and solved, using an extended cylindrical arc-length method together with a Newton–Raphson technique, to enable robust numerical predictions of non-linear active shells transitioning between multiple stable equilibrium paths. Validation and evaluation cases on laminated cylindrical strips and cylindrical panels demonstrate the accuracy of the method and its robust capability to predict non-linear response under thermal and piezoelectric actuator loads. Moreover, the results illustrate the capability of the method to model piezoelectric shells undergoing large shape changes by actively jumping between stable equilibrium states and quantify the strong relationship between shell curvature, applied electric potential, applied temperature differential and induced shape change. Copyright © 2008 John Wiley & Sons, Ltd.     

半圆孔板承受拉伸和弯曲时的三维应力集中

根据Hellinger-Reissner原理建立了具有一个无外力圆柱表面三维8节点杂交应力元,元内假定应力场满足以柱坐标表示的三维平衡方程及无外力圆柱面上的外力边界条件,当元退化为二维时也满足协调方程。单元位移场选取与相邻元协调。用这种特殊杂交应力元,在相当粗的网格下即能准确地分析具有半圆孔厚(薄)板的三维(二维)应力集中。     

STUDIES OF STRESS CONCENTRATION BY USING SPECIAL HYBRID STRESS ELEMENTS

The stresses around rectangular holes of rounded corners and rounded notches are analysed by using the combination of three kinds of three-dimensional special hybrid stress finite elements. One kind of three-dimensional hybrid stress finite element with a traction-free inclined surface has been developed here based on an extended Hellinger–Reissner principle by a rational approach. Other two kinds of 12-node three-dimensional hybrid stress elements with a traction-free cylindrical surface and a traction-free perpendicular surface, respectively, were derived based on the Hellinger–Reissner principle. Examples have indicated that the combination of these special solid elements is far superior in predicting the stress concentration factors, the distributions of circumferential stresses and normal stresses for solids with holes and notches. © 1997 by John Wiley & Sons, Ltd.     

Coupled mechanics and finite element for non‐linear laminated piezoelectric shallow shells undergoing large displacements and rotations

A theoretical framework is presented for analysing the coupled non‐linear response of shallow doubly curved adaptive laminated piezoelectric shells undergoing large displacements and rotations. The formulated mechanics incorporate coupling between in‐plane and flexural stiffness terms due to geometric curvature, coupling between mechanical and electric fields, and encompass geometric non‐linearity effects due to large displacements and rotations. The governing equations are formulated explicitly in orthogonal curvilinear co‐ordinates and are combined with the kinematic assumptions of a mixed‐field shear‐layerwise shell laminate theory. Based on the above formulation, a finite element methodology together with an incremental‐iterative technique, based on Newton–Raphson method is formulated. An eight‐node coupled non‐linear shell element is also developed. Various evaluation cases on laminated curved beams and cylindrical panels illustrate the capability of the shell finite element to predict the complex non‐linear behaviour of active shell structures including buckling, which is not captured by linear shell models. The numerical results also show the inherent capability of piezoelectric shell structures to actively induce large displacements through piezoelectric actuators, by jumping between multiple equilibrium states. Copyright © 2005 John Wiley & Sons, Ltd.     

An analytical approach for free vibration and transient response of functionally graded piezoelectric cylindrical panels subjected to impulsive loads

In this article, the free vibration and dynamic response of simply supported functionally graded piezoelectric cylindrical panel impacted by time-dependent blast pulses are analytically investigated. Using Hamilton’s principle, the equations of motion based on the first-order shear deformation theory are derived. Also, Maxwell’s electricity equation is taken as one of the governing equations. Three sets of electric surface conditions including closed circuit and two mixtures of closed and open circuit surface conditions are considered. By introducing an analytical approach and using the Fourier series expansions, the Laplace transform and Laplace inverse method, the solution of unknown variables are obtained in the real time domain based on a combination of system frequencies. Finally, the effects of various electric surface conditions, geometric parameters and the material power law index on the free vibration and transient response of functionally graded piezoelectric cylindrical panels subjected to various impulsive loads are examined in detail.     

提高曲面近似展开精度的方法与实现

通过对不可展曲面近似展开精度的研究，提出了根据曲面曲率的大小对曲面自动进行网格划分，通过对综合展开误差数据场进行数据拟合来选取最优展开基点，基于空间曲面与其展开图间近似的等距变换关系进行展开样形的误差矫正等3种方法，这几种方法的实现在不同程度上有效地减小了曲面的近似展开误差，提高了曲面的近似展开精度。     

Development of magnetorheological finishing process for external cylindrical surfaces

The recent increase in demand for functional and technological requirements of the component results in the development of complex geometrical shapes and that too with close tolerances and fine surface quality. To fulfill the needs for finishing the external cylindrical surfaces such as groove, taper, step surfaces, and threads, an advanced finishing process based on magnetorheological (MR) fluid has been developed. The developed process can finish the external cylindrical surfaces with controlled magnetic field as likely similar to turning operation. Fine finishing of external cylindrical surfaces is a significant requirement in many functional applications. The wide applications of this present process can be valuable in automotive, machine tool production, valves manufacturing, and aerospace. A modified new MR finishing tool with flat and curved tip surface has been made to perform finishing on external cylindrical surfaces. The present cylindrical finished workpiece is useful in macaroni manufacturing machine. The surface roughness values Ra, Rq, and Rz are reduced to 54.41%, 51.65%, and 40% with flat tool tip surface and 80.88%, 81.32%, and 82.5% with curved tool tip surface in 90 min of finishing time. The overall results reported that the present process with curved tool tip surface is comparatively more useful in finishing the external cylindrical surfaces.     

曲率参数对纤维增强树脂基复合材料构件固化压实的影响

根据有效应力准则和达西定律,结合曲线坐标系,构建了曲面复合材料构件固化过程中的树脂流动和纤维压实控制方程;建立了基于AS4/3501-6纤维树脂体系的层合板仿真模型,并将分析结果与实验数据对比,验证了模型的有效性;研究了曲率参数对复合材料构件非等温固化压实的影响规律,比较和分析了曲率参数与厚度和压力参数耦合作用下对树脂流动行为的影响机制。结果表明:构件曲率参数会对树脂分布产生一定影响,曲率越大,树脂分布的非均匀性越大;曲率参数的作用表达受到厚度与压力参数的影响,当厚度与压力较小时,对曲率的影响不明显。      

Automated print inspection for cylindrical surfaces

The objective of this research was to determine the criteria for automated inspection of a string of alphanumeric characters printed directly to the surfaces of cylindrical containers. The technical problems encountered, because a curved surface was used instead of a flat surface, were container positioning for locating surface features, character position compensation due to curvature, resolution limits defined for sensor and reflected illumination. Automated print inspection requires the use of machine vision to perform the registration of characters, template comparison for correct text being produced and the interrogation of the individual characters for correctness, position, size and type.     

Crack propagation in cylindrical shells

Results of a theoretical study of dynamic steady crack propagation in a circular cylindrical shell are presented. Equations of thin circular cylindrical shells are utilized for theoretical investigation. The semi-analytic crack tip stress solutions to shell equations are obtained by the perturbation method as well as an iterative method. Plots of internal stresses in the shell with a stationary as well as a steadily running crack are presented. These results are compared with those related to flat plates. It is concluded that the shell curvature has a pronounced effect on increasing the values of crack tip stresses. It would also qualitatively affect the variation of stress field around the crack tip. In some cases, the curvature action is shown to be even more pronounced than the inertia effects.