Wrinkle-free solutions in the theory of curved circular membranes

Rotationally symmetric deformations of a curved circular elastic membrane under a vertical surface load are studied, with prescribed radial stresses or radial displacements at the edge. Considering Reissner's theory of thin shells of revolution suffering small strains but arbitrarily large deflections and rotations, the determination of the principal stresses in the membrane is shown to be equivalent to the solution of a single, second-order ODE, expressed in terms of a geodesic variable. Analytical techniques are applied in order to determine a limit curve of those boundary data, which subdivide the parameter range into complementary domains of existence and non-existence of tensile solutions. Finally, the more restricted subdomain of those boundary parameters is determined which admit wrinkle-free solutions, i.e. solutions governed by a nonnegative radial and circumferential stress component.     

Geometrically nonlinear analysis of elastic membranes with embedded rigid inclusions

In this paper the deformation of membranes containing rigid inclusions is analyzed. These rigid inclusions can significantly change the entire stress distribution in the membrane and therefore create major difficulties for the design. The initially flat membrane, which may be prestretched by boundary in-plane tractions or displacements, is subjected to externally applied loads and to the weight of the rigid inclusions. The composite system is examined in cases where its deformation reaches a state for which the undeformed and deformed shapes are substantially different. In such cases large deflections of membranes are considered, which result from nonlinear kinematic relations. The three coupled nonlinear equations in terms of the displacements governing the response of the membrane are solved using the analog equation method, which reduces the problem to the solution of three uncoupled Poisson's equations with fictitious domain source densities. The problem is strongly nonlinear [Katsikadelis JT, Nerantzaki MS. The boundary element method for nonlinear problems. Eng Anal Boundary Elements 1999;23:365–73]. In addition to the geometrical nonlinearity, the problem is itself nonlinear, since the membrane's reactions on the boundary of the rigid inclusions are not a priori known as they depend on the produced deflection surface. Iterative schemes are developed for calculation of deformed membrane's configuration, which converge to the final equilibrium state of the membrane with the given external applied loads. Several example problems are presented, which illustrate the method and demonstrate its accuracy and efficiency. The method employed for the solution is boundary only with all the advantages of the pure BEM.     

On the boundary value problem in the nonlinear theory of dipolar elastic materials

Abstract

In our study, we formulate the boundary value problem in the context of the nonlinear theory of dipolar, porous, and elastic materials. For this problem, some existence and uniqueness results are proven. The results are natural generalizations of the results obtained by Langenbach for the classical elastic bodies.     

A BEM based domain decomposition method for nonlinear analysis of elastic space membranes

In this paper the Domain Decomposition Method (DDM) is developed for nonlinear analysis of both flat and space elastic membranes of complicated geometry which may have holes. The domain of the projection of the membrane on the xy plane is decomposed into non-overlapping subdomains and the membrane problem is solved sequentially in each subdomain starting from zero displacements on the virtual boundaries. The procedure is repeated until the traction continuity conditions are also satisfied on the virtual boundaries. The membrane problem in each subdomain is solved using the Analog Equation Method (AEM). According to this method the three coupled strongly nonlinear partial differential equations, governing the response of the membrane, are replaced by three uncoupled linear membrane equations (Poisson's equations) subjected to fictitious sources under the same boundary conditions. The fictitious sources are established using a meshless BEM procedure. Example problems are presented, for both flat and space membranes, which illustrate the method and demonstrate its efficiency and accuracy.     

Extensions of the U* theory for applications on orthotropic composites and nonlinear elastic materials

To improve the performance and the efficiency of a structure, it is essential to understand how the applied loads are transferred through the structure. The U* index was introduced as one of the methods for identifying the load transfer paths within structures. It has increasingly been used by industry due to its unique capability in structural analysis and design. However, the feasibility of the U* theory for composite materials has not been demonstrated yet. Meanwhile, the U* index was only arisen for linear elastic materials, as a result it is invalid for materials with nonlinear elasticity. In this paper, two modified load transfer indexes for orthotropic composites (\(U_{O}^{*}\)) and nonlinear elastic materials (\(U_{NL}^{*}\)) are proposed based on the basic U* theory, respectively. The computational process of both the \(U_{O}^{*}\) and the \(U_{NL}^{*}\) indexes is presented. The effectiveness of the proposed indexes on load transfer analysis is demonstrated through four case studies.     

Numerical solution of axisymmetric nonlinear elastic problems including shells using the theory of a Cosserat point

Starting with a recently developed three-dimensional eight node brick Cosserat element for nonlinear elastic materials, a simplified Cosserat element is developed for torsionless axisymmetric motions. The equations are developed within the context of the theory of a Cosserat point and the resulting theory is hyperelastic and is valid for dynamics of nonlinear elastic materials. The axisymmetric Cosserat element has four nodes with a total of eight degrees of freedom. As in the more general element, the constitutive equations are algebraic expressions determined by derivatives of a strain energy function and no integration is needed throughout the element region. Examples of large deformations of a nearly incompressible circular cylindrical tube and large deflections of a compressible clamped circular plate are considered to test the accuracy of the element.     

Problems of the mathematical theory of plasticity

This paper gives the results of investigations aimed at the development and elaboration of a theory for the relationships governing plastic-deformation processes in the contemporary mathematical theory of plasticity. We discuss its two main classical directions: the theory of flow and the theory of processes. The first direction is based on the concept of existence of a limiting surface and the possibility of decomposition of strains into elastic and plastic components. On the contrary, the second direction, which is progressing rapidly, does not use the concept of existence of limiting surfaces and does not allow decomposition of strains into elastic and plastic components except for the cases of simple loading and simple unloading. It is thought that in complex loading and complex unloading the deformation is elastoplastic (partially plastic or partially elastic). We believe that these two directions in the theory of plasticity in the case of complex loading will eventually approach each other because they are intended for the investigation of the mechanisms of the same physicomechanical processes of plastic deformation of various media. This work shows one possibility of such approach. Tver' State Technical University, Tver', Russia. Translated from Problemy Prochnosti, No. 1, pp. 22–41, January–February, 2000.     

Thermal effects on nonlinear vibrations of functionally graded doubly curved shells using higher order shear deformation theory

Geometrically nonlinear vibrations of functionally graded (FG) doubly curved shells subjected to thermal variations and harmonic excitation are investigated via multi-modal energy approach. Two different nonlinear higher-order shear deformation theories are considered and it is assumed that the shell is simply supported with movable edges. Using Lagrange equations of motion, the energy functional is reduced to a system of infinite nonlinear ordinary differential equations with quadratic and cubic nonlinearities which is truncated based on solution convergence. A pseudo-arclength continuation and collocation scheme is employed to obtain numerical solutions for shells subjected to static and harmonic loads. The effects of FGM power law index, thickness ratio and temperature variations on the frequency–amplitude nonlinear response are fully discussed and it is revealed that, for relatively thick and deep shells, the Amabili–Reddy theory which retains all the nonlinear terms in the in-plane displacements gives different and more accurate results.     

Gauge theory of defects in the elastic continuum

A gauge theory of defects in an elastic continuum is developed after providing the necessary background in continuum elasticity and gauge theories. The gauge group is the three-dimensional (3D) Euclidean group [semi-direct product of the translation group T (3) with the rotation group SO (3)]. We obtainboth dislocations and disclinations by breaking of the translational invariance. Breaking of the rotational invariance is shownnot to lead to any interesting effects in a linear analysis. These results are shown to be consistent with the topological analysis which is briefly discussed at the end of the paper. Any defect given by the present theory acquires acore which removes the singularity of the stress field at the origin. The stress field agrees with the continuum result asymptotically, as is expected. Geometrical aspects of the deformed state of condensed matter are also briefly touched upon.     

On the strain-displacement equations of a geometrically nonlinear curved beam

The paper is concerned with the strain-displacement equations of a curved beam undergoing arbitrarily large deflections/rotations. Two alternative procedures are used for this derivation: a mathematically consistent and an engineering-type approach. It is demonstrated that upon making certain simplifying assumptions regarding higher-order terms both approaches lead to identical results. VDI     

Experimental techniques to determine the convective heat transfer coefficients for flat and curved surfaces

This paper surveys the techniques which are available or being developed for measuring local and mean convective heat transfer coefficients, primarily for forced convection between a surface ,and an incompressible gas. After defining the heat transfer coefficient, the paper summarizes methods of surface temperature measurement, outlines the direct and indirect methods of determining the heat transfer coefficient and compares the pertinent features of three techniques suitable for use during film and multijet impingement cooling and with annular ducts.     

Dynamic analysis and nonlinear simulation of aircraft flat spin

Abstract

In this paper we investigate the dynamic characteristics of flat spin of a fighter aircraft in open-loop configuration. The aircraft model under study is of the multi-role fighter configuration co-sponsored by China and Pakistan. The aerodynamic model used in the study is in the form of look-up tables that have been developed from rotary balance steady coning and oscillatory coning motion wind tunnel data. The set of all possible equilibrium spin states is numerically computed for various values of control settings using eighth-order aircraft equations of motion. Results from dynamical systems theory are applied to investigate local stability characteristics of aircraft around steady spin state. The complete set of dynamic modes of aircraft in spin is evaluated and mode content in each of the motion variable is determined using modal decomposition. This analytical technique is complemented by performing numerical simulations to investigate flat spin dynamic features. The methodology is applied to investigate dynamic behavior of two flat spins: right flat spin at 72.1° and much flatter left spin at 84.4°. The presented work provides insight into the global overview of aircraft flat spins as a function of various control settings, and their dynamic characteristics and it can facilitate the designing of flight control laws for spin recovery/prevention.     

Measuring the nonlinear elastic properties of tissue-like phantoms

A direct mechanical system simultaneously measuring external force and deformation of samples over a wide dynamic range is used to obtain force-displacement curves of tissue-like phantoms under plain strain deformation. These measurements, covering a wide deformation range, then are used to characterize the nonlinear elastic properties of the phantom materials. The model assumes incompressible media, in which several strain energy potentials are considered. Finite-element analysis is used to evaluate the performance of this material characterization procedure. The procedures developed allow calibration of nonlinear elastic phantoms for elasticity imaging experiments and finite-element simulations.     

流线型回转体外形设计综述与线型拟合

综述了流线型回转体外形设计的主要方法：(1)具有精确数学表达式的几何组合外形曲线；(2)用源汇法设计回转体外形曲线；(3)用离散外形坐标型值表示的回转体线型。给出了流线型离散型值回转体线型的拟合解析表达式。这些外形设计方法与拟合解析表达式具有广泛的工程实用价值。     

Uniqueness in the theory of bending of elastic plates

In this paper classes of functions are indicated for which Betti and Somigliana formulae hold in the exterior of a bounded domain in the problem of bending of elastic plates. Uniqueness theorems are derived and an example is discussed to illustrate the theory.     

Material-dependent stability conditions in the buckling of nonlinear elastic bars

Summary A thorough study of the critical and post-critical large displacement response of simple discrete and continuous systems made from a nonlinear elastic material, is presented. Simple material-dependent stability conditions are established, whose application does not require the solution of the intractable nonlinear differential equations of equilibrium. The predominent role of the nonlinear component of the curvature on the buckling mechanism of the foregoing systems, is revealed. Moreover, it is found that elastic systems which were considered as exhibiting post-buckling strength might be imperfection sensitive, if the effect of material nonlinearity is taken into account. An approximate but very efficient analytical approach leading to very reliable results is derived for establishing the large displacement behavior of a nonlinear elastic cantilever bar. The degree of accuracy of this approach is checked by comparing it with the exact numerical solution of Runge Kutta. The numerical results presented herein contribute to our understanding of this problem and at the same time show the efficiency, reliability and range of applicability of the proposed approach.With 4 Figures     

Convergence problems in the theory of random elastic media

A main result of the rigorous theory of random, linearly elastic media consists in the representation of the tensor of effective elastic moduli as a Neumann type infinite series which contains the infinite set of correlation functions of the distribution of the local elastic moduli. Under the restriction to statistically homogeneous and isotropic finite media it is proved that convergent series can always be obtained provided the local elastic moduli remain finite everywhere in the medium. This means that the mentioned theory cannot be applied in the above mentioned form to media with pores and/or rigid inclusions. It also means that the theory is not restricted to media with small fluctuations of the elastic parameter.