Inverse Problems of the Mechanics of Thin-Walled Composite Structures

New formulations and methods for the solution of the inverse problems of thin-walled layered and reinforced shells and plates are discussed. Rational projects with regard for the requirements of nonflexural deformations in layered structural members, the given deformability of particular surfaces, the realization of a strictly momentless state, an equally stressed reinforcement, and the breaking strength of the binder at the interfaces are investigated. A brief review of the known solutions to these problems is given, and solutions to some new problems are described.     

A Class of 3D Problems for Layered Plates

Mechanics of Composite Materials - A class of 3D problems of elasticity theory for a layered package of orthotropic plates, which, in particular, models the behavior of the lithospheric plates of...     

Dirichlet-Neumann conditions and the orthogonality of three-dimensional guided waves in layered solids

Orthogonality relations for homogeneous waves in layered plates are obtained, and they are generalized to the case of a contact with fluid layers. For layers of infinite thickness, it is shown that the homogeneous waves of the discrete spectrum are orthogonal to each other and to the waves of the continuous spectrum. For finite-size sources, exact formulas are derived for the coefficients multiplying the modes. Based on the orthogonality relations, a nonlocal radiation principle is proposed such that the infinite domain in the numerical solution of diffraction problems for layered plates can be replaced by a virtual cylinder.     

Modeling the Dynamic Behavior of Rigid-Plastic Thin Reinforced Curvilinear Plates with a Hole on a Viscous Foundation

Mechanics of Composite Materials - A mathematical model is developed for the dynamic behavior of rigid-plastic thin reinforced layered curvilinear, hinge-supported or clamped, plates with an...     

A Generalized Nonclassical Model of Stress-Strain State in the Problems of Statics,Dynamics, and Contact Interaction of Layered Plates and Shells

A new generalized two-dimensional nonclassical model for the dynamic stress-strain state of layered anisotropic plates and shallow shells is constructed. A feature of the model is the complete account of the factors which represent the spatial character of their deformation; in particular, the transverse shear and transverse compression strains as well as the geometrical nonlinearity and dynamic action of the load are accounted for.     

On the classes of problems for deformable one-layer and multilayer thin bodies solvable by the asymptotic method

A survey of studies by the author and his disciples on the solution of some classes of problems for deformable thin bodies (strip-beams, plates, and shells) is presented. Classical and nonclassical boundary-value problems of the statics and dynamics of anisotropic and layered bodies are considered. Free and forced vibrations of one-layer and multilayer thin bodies are investigated. The coupled problems of thermoelasticity are solved.     

Buckling stability of multilayer plates and shells with interfacial structural defects under axial compression

Based on the discrete-structural theory of thin plates and shells, a variant of the equations of buckling stability, containing a parameter of critical loading, is put forward for the thin-walled elements of a layered structure with a weakened interfacial contact. It is assumed that the transverse shear and compression stresses are equal on the interfaces. Elastic slippage is allowed over the interfaces between adjacent layers. The stability equations include the components of geometrically nonlinear moment subcritical buckling conditions for the compressed thin-walled elements. The buckling of two-layer transversely isotropic plates and cylinders under axial compression is investigated numerically and experimentally. It is found that variations in the kinematic and static contact conditions on the interfaces of layered thin-walled structural members greatly affect the magnitude of critical stresses. In solving test problems, a comparative analysis of the results of stability calculations for anisotropic plates and shells is performed with account of both perfect and weakened contacts between adjacent layers. It is found that the model variant suggested adequately reflects the behavior of layered thin-walled structural elements in calculating their buckling stability. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 4, pp. 513–530, July–August, 2007.     

A method for predicting the deflection of reinforced concrete beams strengthened with carbon plates

For strengthening bent beams, plates of reinforced plastics are glued to their tensioned surface. As s result, the beam becomes layered, and it is possible to control its rigidity and deflection. Based on the methods of structural mechanics of layered media, a method is elaborated for determining the deflection of such beams on the entire range of loading up to their ultimate failure. A comparison between the theoretical and experimental results is carried out. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 42, No. 1, pp. 45–60, January–February, 2006.     

Effective high-order approximations of layered coatings and linings of anisotropic elastic,viscoelastic and nematic materials

Beginning at low frequencies, asymptotically exact models of anisotropic coatings and linings with a small ratio of the half-thickness to the longitudinal deformation scale are constructed. The requirement on the conditions of contact with the substrate, where at least one of the boundary conditions must contain a displacement component of the strain in explicit form is “non-classical” here. The action of the coating/lining on a thicker body is approximated by impedance boundary conditions at the interface. The error of the model is reduced to the third order for layered packets and to the sixth order for a single layer. The physical limit of the applicability is the frequency of the first quasi-resonance in the corresponding deformed system. A comparison with the propagator matrix and numerical testing for partial waves shows satisfactory accuracy, comparable with the accuracy of the theory of classical plates of similar order. The results can be used in contact problems and for rapid algorithms for calculating the spectrum of the eigenwaves in half-spaces and thick layered plates with any number of coatings and linings. An extension to the case of viscoelastic materials and nematic elastomers is given.     

对称铺设各向异性叠层矩形板的非线性弯曲

本文研究了对称铺设各向异性矩形叠层板在各种支承条件下的非线性弯曲.应用奇异摄动方法导出了挠度和应力函数的一致有效的N阶渐近解.对承受边缘张力和侧向载荷的简支矩形叠层板应用奇摄动方法和改进了的迦辽金方法(一种加权残数法)进行了分析和计算.     

Stress State of Multilayer Thin-Walled Elements with Interfacial Defects of Structure

Based on the discrete-structural theory of thin plates and shells, a calculation model for thin-walled elements consisting of a number of rigid anisotropic layers is put forward. It is assumed that the transverse shear and compression stresses are equal on the interfaces. Elastic slippage is allowed over the interfaces between adjacent layers. The solution to the problem is obtained in a geometrically nonlinear statement with account of the influence of transverse shear and compression strains. The stress-strain state of circular two-layer transversely isotropic plates, both without defects and with a local area of adhesion failure at their center, is investigated numerically and experimentally. It is found that the kinematic and static contact conditions on the interfaces of layered thin-walled structural members greatly affect the magnitude of stresses and strains. With the use of three variants of calculation models, in the cases of perfect and weakened contact conditions between layers, the calculation results for circular plates are compared. It is revealed that the variant suggested in this paper adequately reflects the behavior of layered thin-walled structural elements under large deformations. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 41, No. 6, pp. 761–772, November–December, 2005.     

An Iterative Analytical Theory in the Mechanics of Layered Composite Systems

An iterative analytical theory in the mechanics of layered composite systems is developed. The prehistory of the nonclassical theory of layered systems is presented. The division of this theory into two principal directions - discrete-structural and continuous-structural - is mentioned. The basic iterative Ambartsumyan theory, which belongs to the second direction, is described. The formation of the generalized iteration theory of first approximation is shown. In this theory, the disagreement between the kinematic and static models is removed, i.e., a generalization of these models is realized. The theory of second approximation is described. An iterative principle is presented for the formation of a higher-approximation nonclassical theory. Based on this principle, theories of anisotropic composite shallow shells, plates, and beams are formulated. Comparative calculation results for different layered composite systems are presented.     

The property of orthogonality and energy transfer by three-dimensional eigenwaves in transversely isotropic laminated plates with and without contact with a liquid

The properties of eigenwaves in laminated plates with anisotropy in a transforse directon properties are studied. The most general form of the solutions, dispersion relations, power flows and generalized orthogonality relations are analysed. The similarity and difference in the properties of the waves as compared with isotropic media and ideal fluids, as well as the extension to the case of layered spaces and half-spaces, is investigated. A method of determining the coefficients for the eigenwaves radiated in the plate is proposed in the case of a problem with dynamical sources of finite size. A method of summing the series over eigenwaves is suggested.     

不对称的各向异性叠层矩形板的非线性弯曲(Ⅰ)

本文研究了不对称的各向异性叠层矩形板在多种支承条件下的非线性弯曲,利用文[1]中提出的奇异摄动方法,导出了板在横向载荷和边缘拉力的联合作用下,其挠度和应力函数的一致有效的N阶渐近解。因此,本文的研究对于这样一个复杂的问题提供了一个简单而又有效的方法。     

A finite element formulation for the analysis of interlaminar shear stresses in layered composite plates

Fiber reinforced composite plates are often highly stressed structures. For an efficient design of such plates interlaminar stresses have to be evaluated as precisely as possible. In this paper we propose a finite element formulation for the analysis of transverse shear stresses in layered composite plates. For this purpose the equilibrium equations are exploited with appropriate shape functions in weak form which leads to a local discrete boundary value problem. Two different models to compute the derivatives of the strains are considered. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solution of a thermoelasticity problem for layered plates according to a more precise model

An applied theory is constructed for layered plates taking into account deformations due to transverse shear in a stationary thermal field. Equilibrium equations and boundary conditions are derived. It is shown that for plates of average thickness, consideration of transverse shear deformations and thermal expansion in the transverse direction leads to results significantly differing from those calculated according to the classical theory.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 29, pp. 25–29, 1989.     

Second-approximation shear theory for shallow layered shells and plates

Analysis of a second-approximation refined shear model for shallow layered composite shells and plates with a substantially inhomogeneous structure over the thickness is presented. The tangential displacements and corresponding normal stresses are expressed in the form of a polynomial of the fith degree in the transverse coordinate and contain squared rigidity characteristics. In this way, the accuracy of results and practical coincidence with the 3D solutions is ensured. Based on the refined model, a theory of shallow layered shells is developed. A system of resolving equations of sixteenth power together with appropriate boundary conditions was obtained and solved analytically. It is shown that the area of application of the formed model is extended as compared with the model of the first approximation. The model proposed allows us to examine the stress-strain state of layered composite structures of substantially different thickness and physical-mechanical characteristics of the layers, including the possibility of simulating relatively large shear deformations of rigid layers separated by a low-modulus thin interlayer pliable to transverse shear.Presented at the 10th International Conference on the Mechanics of Composite Materials (Riga, April 20–23, 1998).Ukrainian Transport University, Kiev, Ukraine. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 3, pp. 363–370, May–June, 1998.     

Model of a layered plate considering nonideal contact between layers

We propose a mathematical model for doing calculations for layered plates, allowing for both rigid and sliding contact in the presence of frictional forces between the sliding layers. The model takes into account the distribution of tangential and normal displacements across the thickness of the sliding layered stack, and also the distribution of transverse normal stresses. The strain tensor is obtained using the Cauchy relations; the stress tensor is obtained based on Hooke's law. Tne Lagrange variational principle allows us to obtain the resolvent system of differential equations and the corresponding boundary conditions. The spatial model for deformation of a layered plate has a number of special features compared with familiar models. The system of differential equations has operators no higher than second order. It is described relative to displacements on the faces of the stack. This is convenient in solving problems involving sliding of layers with and without friction.Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 5, pp. 671–676, September–October 1995.     

Stress state of freely supported multilayered elliptical plates of anisotropic materials

The problem of a stressed state in elliptic plates has been considered in general for a rigid contour fixation. It is much more difficult to obtain a solution for the freely supported plates, even for isotropic materials. In this paper we suggest an approach for defining the stressed state of thin elliptic plates with layered structure under the condition of a freely supported contour. The solution is obtained in a rectangular cartesian coordinate system. The displacements, which are the fundamental unknowns, are given in the form of polynomials with unknown coefficients defined by a system of algebraic equations. The resolving equations and three out of the four boundary conditions are satisfied precisely. One boundary condition, is satisfied by means of collocation method of separate points of the contour. Estimation of the accuracy of the suggested approach is carried out by comparing the obtained results with the known ones. The problem of deformation of a twolayered plate has been discussed, in which the principal direction of elasticity does not coincide with the coordinate directions.S. P. Timoshenko Institute of Mechanics, National Academy of Science of the Ukraine, Kiev. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 4, pp. 496–504, July–August, 1997. Original article submitted March 19     

Analytical Solution of the Problem on the Thermally Stressed State of Composite Plates with Rigid and Sliding Contacts Between Layers Based on the 3D Elasticity Theory

Mechanics of Composite Materials - Two approaches to studying the thermally stressed state of layered composite plates in the 3D formulation have been developed. In the first approach, using the...