Exact three-dimensional analysis for static torsion of piezoelectric rods

Based on the theory of elasticity, exact analytical and numerical solutions of piezoelectric rods under static torsion are studied. In this paper, direct solution method is used. The main scope is to check the extension of validity of assumptions in previous papers that had been made based on linear distribution of electric potential through the cross section and their influences on deflection and the angle of rotation. Stress and electric induction functions are employed to obtain the exact solution of the static and electrostatic equilibrium equations under torsional loading. It is shown that previous assumptions are valid only in some types of piezoelectric materials, while in other types these assumptions lead to considerable deviations from accurate modeling. The present analytical solutions are compared with three-dimensional finite element analysis results and absolute agreements are found. At the end of this article, torsional rigidity, shape-effects on induced piezoelectric deformation and the range of valid region for linear distribution of electric potential assumption have been studied.     

圆截面弹性细杆的平面振动

基于Kirchhoff理论讨论圆截面弹性细杆的平面振动．以杆中心线的Frenet坐标系为参考系建立动力学方程．杆作平面运动时,其扭转振动与弯曲振动解耦．讨论任意形状杆的扭转振动和轴向受压直杆在无扭转条件下的弯曲振动,证明直杆平衡的静态Lyapunov稳定性与欧拉稳定性条件为动态稳定性的必要条件．考虑轴向力和截面转动惯性效应的影响,导出弯曲振动的固有频率．     

Characterizing Torsional Properties of Microwires Using an Automated Torsion Balance

In order to characterize the torsional behavior of microwires, an automated torsion tester is established based on the principle of torsion balance. The main challenges in developing a torsion tester at small scales are addressed. An in-situ torsional vibration method for precisely calibrating the torque meter is developed. The torsion tester permits the measurement of torque to nN m, as a function of surface shear strain to a sensitivity of sub-microstrain. Using this technique, we performed (monotonic and/or cyclic) torsion tests on polycrystalline copper and gold wires. It is found that (i) a size effect appears in both the initial yielding and the plastic flow of torsional response; (ii) a reverse plasticity occurs upon unloading in cyclic torsion response; and (iii) the Hall-Petch effect and the strain gradient effect are synergistic. We also performed cyclic torsion tests on human hairs and spider silk which are natural protein fibers with a different morphological structure to metallic wires. It is shown that the single hair exhibits torsional recovery, and that the spider silk displays torsionally superelastic behavior whereby it is able to withstand great shear strain.     

Extension,inflation and torsion of a residually stressed circular cylindrical tube

In this paper, we provide a new example of the solution of a finite deformation boundary-value problem for a residually stressed elastic body. Specifically, we analyse the problem of the combined extension, inflation and torsion of a circular cylindrical tube subject to radial and circumferential residual stresses and governed by a residual-stress dependent nonlinear elastic constitutive law. The problem is first of all formulated for a general elastic strain-energy function, and compact expressions in the form of integrals are obtained for the pressure, axial load and torsional moment required to maintain the given deformation. For two specific simple prototype strain-energy functions that include residual stress, the integrals are evaluated to give explicit closed-form expressions for the pressure, axial load and torsional moment. The dependence of these quantities on a measure of the radial strain is illustrated graphically for different values of the parameters (in dimensionless form) involved, in particular the tube thickness, the amount of torsion and the strength of the residual stress. The results for the two strain-energy functions are compared and also compared with results when there is no residual stress.     

TORSIONAL CENTER OF BEAMS OF ARBITRARY COMPLEX CROSS-SECTION 1)

为了计算任意复杂非圆截面梁横截面扭转中心的位置，用节线法将其约束受扭后所有横截面面外变形的形状用一族包含节线未知函数的曲面表示，建立梁约束受扭时的控制方程后，再用常微分方程求解器分别求出单纯扭矩与横向载荷单独作用时节线未知函数的数值解，最后用刚度等效原理导出复杂截面梁横截面扭转中心的位置。算例计算结果表明：该方法是合理的、有效的，是计算任意复杂非圆截面梁横截面扭转中心位置的可靠方法。     

Mullins Effect for a Cylinder Subjected to Combined Extension and Torsion

We study the Mullins effect for a circular cylinder of incompressible, isotropic material under loading cycles of combined extension and torsion. The analysis is based on the constitutive model recently proposed in De Tommasi et al. (J. Rheol. 50: 495–512, 2006). This model assumes that the mechanical response at each material point results as a homogenized effect of a mixture of different materials with variable activation and breaking thresholds. We show the feasibility of this approach to treat complex, inhomogeneous deformations. In particular, we obtain for the generic loading path the analytical expressions of the stress field, of the axial force, and of the twisting moment. The proposed model exhibits the Mullins stress softening effect in the case of simple extension, simple torsion, and combined extension and torsion. We analyze in detail the path dependent behavior and the preconditioning effects.      

Torsional static and vibration analysis of functionally graded nanotube with bi-Helmholtz kernel based stress-driven nonlocal integral model

A torsional static and free vibration analysis of the functionally graded nanotube(FGNT)composed of two materials varying continuously according to the power-law along the radial direction is performed using the bi-Helmholtz kernel based stress-driven nonlocal integral model.The differential governing equation and boundary conditions are deduced on the basis of Hamilton’s principle,and the constitutive relationship is expressed as an integral equation with the bi-Helmholtz kernel.Several nominal variables are introduced to simplify the differential governing equation,integral constitutive equation,and boundary conditions.Rather than transforming the constitutive equation from integral to differential forms,the Laplace transformation is used directly to solve the integro-differential equations.The explicit expression for nominal torsional rotation and torque contains four unknown constants,which can be determined with the help of two boundary conditions and two extra constraints from the integral constitutive relation.A few benchmarked examples are solved to illustrate the nonlocal influence on the static torsion of a clamped-clamped(CC)FGNT under torsional constraints and a clamped-free(CF)FGNT under concentrated and uniformly distributed torques as well as the torsional free vibration of an FGNT under different boundary conditions.     

Torsional,flexural, and torsional-flexural buckling modes of a cylindrical shell under combined loading

Stability problems for cylindrical shells under various loading modes were considered in numerous papers. A detailed analysis of such problems can be found, e.g., in the monograph [1]. We refer to the solutions presented in this monograph as classical.For long cylindrical shells in axial compression, one of the buckling modes is the purely beam flexural mode similar to the classical buckling mode of a straight rod. It is well known that it can be studied by using the nonlinear or linearized equations of the membrane theory of shells. In [2], it was shown that, on the basis of such equations constructed starting from the noncontradictory version of geometrically nonlinear elasticity relations in the quadratic approximation [3], under the separate action of the axial compression, external pressure, and torsion, there are also previously unknown nonclassical buckling modes, most of which are shear ones.In the present paper, we show that the use of the above equations for cylindrical shells under compression and external pressure with simultaneous pure torsion or bending permits revealing the earlier unknown torsional, beam flexural, and beam torsional-flexural buckling modes, which are nonclassical, just as those found in [2]. The second of these buckling modes is realized when axially compressing forces are formed in the shell with simultaneous torsion, and the third of them is realized under compression combined with pure bending.It was found that, earlier than the classical buckling modes, the torsional buckling modes can be realized for relatively short shells with small shear rigidity in the tangent plane, while the second and third buckling modes can be realized for relatively long shells.     

The analysis of thin walled composite laminated helicopter rotor with hierarchical warping functions and finite element method

In the present paper, a series of hierarchical warping functions is developed to analyze the static and dynamic problems of thin walled composite laminated helicopter rotors composed of several layers with single closed cell. This method is the development and extension of the traditional constrained warping theory of thin walled metallic beams, which had been proved very successful since 1940s. The warping distribution along the perimeter of each layer is expanded into a series of successively corrective warping functions with the traditional warping function caused by free torsion or free bending as the first term, and is assumed to be piecewise linear along the thickness direction of layers. The governing equations are derived based upon the variational principle of minimum potential energy for static analysis and Rayleigh Quotient for free vibration analysis. Then the hierarchical finite element method is introduced to form a numerical algorithm. Both static and natural vibration problems of sample box beams are analyzed with the present method to show the main mechanical behavior of the thin walled composite laminated helicopter rotor. The project supported by the National Natural Science Foundation of China (19932030)     

Some analytical solutions for Saint-Venant torsion of non-homogeneous cylindrical bars

The Saint-Venant torsion problem of linearly elastic cylindrical bars with solid and hollow cross-section is treated. The shear modulus of the non-homogeneous bar is a given function of the Prandtl's stress function of considered cylindrical bar when its material is homogeneous. The solution of the torsional problem of non-homogeneous bar is expressed in terms of the torsional and Prandtl's stress functions of homogeneous bar having the same cross-section as the non-homogeneous bar.     

A universal relation in torsion for a mixture of solid and fluid

In his study of combined finite extension and torsion of a nonlinear, incompressible, isotropic elastic circular cylinder, Rivlin [1] established a relation for the torsional stiffness which depends only on the axial force, the axial extension ratio and the radius of the undeformed cylinder, in the case of small twist. The relationship did not depend on the structure of the stored energy function and is hence a universal relation. In this paper, we extend Rivlin's result to the case of combined extension and torsion of a cylindrical mixture of a nonlinear elastic solid and fluid.     

Torsional damage of concrete beams with softening behaviour

Analysed in this paper is the torsional damage of concrete beam with softening behaviour. Change in the local stiffness and dissipated strain energy density are determined as the torsional load or rotation is increased. The idealized stress-strain curve is bilinear with a positive and negative slope. Use is made of the equations of elasticity for torsion and isoparameric mapping with finite difference. Numerical results are obtained for the pure torsion of a rectangular beam and combined torsion/compression of an I-beam. Determined are the critical torques which tend to agree well with the test data.     

Torsional rigidity of shells of revolution

In this paper, the general equations of equilibrium for axisymmetrical deformation including the torsional deformation of revolutional shells are derived. It is shown that the shearing stress distribution due to torsion is independent of other stress components including those of membrane stress and bending stress. In this paper, the torsional deformation is considered to be represented by membrane action only, and also by the combined action of bending membrane deformation. It is shown that the main contribution of torsional rigidity is that related to membrane action.     

Free torsion of thin-walled structural members of open- and closed-sections

Free torsion of thin-walled structures of open- and closed-sections is a classical elastic mechanics problem, which, in literature, is often solved by the method of membrane analogy. The method of membrane analogy, however, can be only applied to structures of a single material. If the structure consists of both open- and closed-sections, the method of membrane analogy is difficult to be applied. In this paper, a new method is presented for solving the free torsion of thin-walled structures of open- and/or closed- sections with multiple materials. By utilizing a simple statically indeterminate concept, torsional equations are derived based on the equilibrium and compatibility conditions. The method presented here not only is very simple and easy to understand but also can be applied to thin-walled structures of combined open- and closed-sections with multiple materials.     

Comparison of the rheology of polymer melts in shear,and biaxial and uniaxial extensions

The experimental properties of different polymer melts, polystyrene, high density polyethylene and low density polyethylene are compared for the first time in three different deformations: step shear, step biaxial extension and steady uniaxial extension. Properties of three other melts are also studied in step biaxial and shear experiments. For our comparative purposes some data of Laun and Winter from the literature are used, as well as new data reported here. In all the step strain experiments, the stresses can be factored into a time dependent relaxation modulus and a strain dependent damping function. The data are interpreted using a differential constitutive equation of Larson which satisfies this time-strain separability and has a single parameter that describes the strain softening character of the material. Results show that differences in the properties of the melts are most pronounced in uniaxial extension and least in biaxial extension. All melts follow the Doi-Edwards prediction relatively closely in biaxial extension. In uniaxial extension, the branched material shows a strong strain hardening effect although its shear and biaxial properties are similar to the other melts. The constitutive model gives a reasonably good fit to the data in all three deformations for unbranched materials for the same value of the adjustable parameter; the model, however, fails for the branched low density polyethylene.     

Effect of decoupled fluid loading on nonlinear vibration of a flat plate

Numerical simulations of nonlinear responses of a flat plate subject to decoupled fluid loading are carried out. Under clamped boundary conditions and subject to forced vibration at its natural frequency corresponding to the (5,1) mode, the various response modes of the plate are determined. It is found that increasing the excitation amplitude, the response changed from periodic to chaotic. In addition, the fluid-wall shear stresses are found to change the response from linear to nonlinear and vice versa depending on their magnitudes. When a static pressure load is combined with fluid-wall shear stresses and low excitation amplitude, the resulting response was chaotic.     

Characterization of the non-linear response of asphalt mixtures using a torsional rheometer

There is sparse documentation of the large normal stresses that develop in asphalt mixtures subjected to shear, a typical characteristic of non-linear materials. In this study, we continue our initial investigations of using a torsional rheometer for measuring the normal stresses developed in asphalt mixtures when subjected to torsion. Samples of sand-asphalt mixture are subjected to rotation rates as low as one revolution in ninety minutes. This study provides further clear evidence of significant development of normal stresses due to shearing and emphasizes the need for the development of models that can describe such a phenomenon.     

Nonlinear flexural-flexural-torsional interactions in beams including the effect of torsional dynamics. II: Combination resonance

In Part I of this work nonlinear coupling between torsional motion and both in-plane and out-of-plane flexural motion was examined for inextensional beams in the presence of a one-to-one internal resonance. Here the nonlinear response of the system considered in Part I is investigated for the case of an internal combination resonance involving modes associated with bending in two directions and torsion. The analysis presented is based on a consistent set of nonlinear differential equations which contain both curvature and inertia nonlinearities and account for torsional dynamics.     

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)