Three-dimensional exact elasticity solutions for antisymmetric angle-ply laminated composite plates

International Journal of Mechanics and Materials in Design - This paper is concerned with the derivation of the benchmark three-dimensional exact solutions for the static analysis of rectangular...     

Pure bending in non-linear micropolar elasticity

International Journal of Mechanics and Materials in Design - Large-deformation pure cylindrical bending of a homogeneous strip of uniform finite thickness with micropolar (Cosserats’)...     

On new symplectic elasticity approach for exact free vibration solutions of rectangular Kirchhoff plates

In the classical approach, it has been common to treat free vibration of rectangular Kirchhoff or thin plates in the Euclidian space using the Lagrange system such as the Timoshenko’s method or Lévy’s method and such methods are the semi-inverse methods. Because of various shortcomings of the classical approach leading to unavailability of analytical solutions in certain basic plate vibration problems, it is now proposed here a new symplectic elasticity approach based on the conservative energy principle and constructed within a new symplectic space. Employing the Hamiltonian variational principle with Legendre’s transformation, exact analytical solutions within the framework of the classical Kirchhoff plate theory are established here by eigenvalue analysis and expansion of eigenfunctions in both perpendicular in-plane directions. Unlike the classical semi-inverse methods where a trial shape function required to satisfy the geometric boundary conditions is pre-determined at the outset, this symplectic approach proceeds without any shape functions and it is rigorously rational to facilitate analytical solutions which are not completely covered by the semi-inverse counterparts. Exact frequency equations for Lévy-type thin plates are presented as a special case. Numerical results are calculated and excellent agreement with the classical solutions is presented. As derivation of the formulation is independent on the assumption of displacement field, the present method is applicable not only for other types of boundary conditions, but also for thick plates based on various higher-order plate theories, as well as buckling, wave propagation, and forced vibration, etc.     

Singular solutions in elasticity

Summary General solutions of the time-independent equations of motion for linear elasticity with couplestresses are studied. The completeness of a displacement potential similar to theGalerkin-Somigliana representation is proved. The fundamental singular solution of the field equations is defined with the aid of the reciprocal work theorem.

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Zusammenfassung Allgemeine Lösungen der zeitunabhängigen Bewegungsgleichungen der linearen Elastizität mit Momentenspannungen werden studiert. Die Vollständigkeit eines Verschiebungspotentials, ähnlich dem vonGalerkin-Somigliana, wird bewiesen. Die singuläre Grundlösung der Feldgleichungen wird mit Hilfe des verallgemeinertenBettischen Reziprozitätssatzes definiert.

     

A numerical method for homogenization in non-linear elasticity

In this work, homogenization of heterogeneous materials in the context of elasticity is addressed, where the effective constitutive behavior of a heterogeneous material is sought. Both linear and non-linear elastic regimes are considered. Central to the homogenization process is the identification of a statistically representative volume element (RVE) for the heterogeneous material. In the linear regime, aspects of this identification is investigated and a numerical scheme is introduced to determine the RVE size. The approach followed in the linear regime is extended to the non-linear regime by introducing stress–strain state characterization parameters. Next, the concept of a material map, where one identifies the constitutive behavior of a material in a discrete sense, is discussed together with its implementation in the finite element method. The homogenization of the non-linearly elastic heterogeneous material is then realized through the computation of its effective material map using a numerically identified RVE. It is shown that the use of material maps for the macroscopic analysis of heterogeneous structures leads to significant reductions in computation time.     

Fundamental solutions in micropolar elasticity

In this paper the general form of the fundamental solutions and fundamental matrices associated with the equations of the micropolar elastostatics as well as the integral representation by means of these matrices are determined. For the dynamic problem the time-periodic fundamental solutions and general fundamental solutions are determined. In this last case the explicit solution is given for $\text{ρ(γ + ?) = jμ}$.     

New exact solutions for free vibrations of thin orthotropic rectangular plates

In this paper, a novel separation of variables is presented for solving the exact solutions for the free vibrations of thin orthotropic rectangular plates with all combinations of simply supported (S) and clamped (C) boundary conditions, and the correctness of the exact solutions are proved mathematically. The exact solutions for the three cases SSCC, SCCC, and CCCC are successfully obtained for the first time, although it was believed that they are unable to be obtained. The new exact solutions are further validated by extensive numerical comparisons with the solutions of FEM and those available in the literature.     

On exact unsteady navier-stokes solutions

Exact “universal” unsteady rotational flow solutions are compared to an earlier class of exact unsteady flows. They apply to different flow situations.     

Similarity solutions for nonlinear diffusion — further exact solutions

Although the nonlinear diffusion equation has been extensively studied and there exists substantial literature in many diverse areas of science and technology, the number of exact concentration profiles is nevertheless limited. In a recent article in this journal (Hill [1]) a brief review of known exact results is given, as well as an elementary integration procedure which appears to be a general device for obtaining integrals associated with similarity solutions. This paper extends the results given in [1] and for particular power law diffusivitiesc ^m (such asm = –/12, –1, –/32 and –2) presents a number of new exact solutions obtained by fully integrating the ordinary differential equations derived in [1]. In addition new results are found for a general nonlinear diffusion equation which includes one-dimensional diffusion with an inhomogenouus and nonlinear diffusivityc ^mx^mas well as symmetric nonlinear diffusion in cylinders and spheres. Moreover by a separate and ad-hoc procedure a new solution is obtained of the travelling wave type but with a variable wave speed. Some of the new exact solutions obtained for one-dimensional nonlinear diffusion with power law diffusivitiesc ^mare illustrated graphically.     

Some exact wave propagation solutions in viscoelastic materials

Summary A superposition principle obtained recently by the author, reduces the viscoelastic wave problem to a static elastic problem, an elastic eigenvalue problem and an integrodifferential equation of theVolterra type involving time only. This principle is utilized to obtain exact solutions to a plane, a cylindrical, and a two-dimensional wave problem in a viscoelastic material with constantPoisson's ratio.

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Zusammenfassung Vom Verfasser wurde, kürzlich ein Überlagerungsprinzip hergeleitet, das das viskoelastische Wellenproblem auf ein statisches elastisches Problem, ein elastisches Eigenwertproblem und eineVolterrasche Integrodifferentialgleichung, die nur mehr die Zeit enthält, zurückführt. Dieses Prinzip wird nunmehr benützt, um exakte Lösungen eines ebenen, eines zylindersymmetrischen und eines zweidimensionalen Wellenproblems in einem viskoelastischen Material mit konstanterPoissonzahl zu erhalten.

     

Approximate solutions for non-linear random vibration problems

Two methods are described for obtaining approximate solutions to non-linear random vibration problems. The first method approximates the non-linear system with the linear system whose corresponding probability density function best solves the Fokker-Planck equation associated with the non-linear system. In the second method, a class of non-linear systems with known solutions to the Fokker-Planck equation is used to best approximate the non-linear system of interest. Two illustrative examples are presented and the results are compared with existing methods.     

On the fundamental solutions in micropolar elasticity with voids

Summary By means of an uncoupling representation, we derive the fundamental solution for the differential system of micropolar elasticity with voids in the steady vibration case. Reciprocity properties are also explored.     

Geometrically non-linear method of incompatible modes in application to finite elasticity with independent rotations

We discuss a geometrically non-linear method of incompatible modes. The model problem chosen for the discussion is the finite elasticity with independent rotations. The conditions which ensure the convergence of the method and the methodology to construct incompatible modes are presented. A detailed derivation of variational equations and their linearized form is given for a two-dimensional plane problem. A couple of geometrically non-linear two-dimensional elements with independent rotational freedoms are proposed based on the presented methodology. The elements exhibit a very satisfying performance over a set of problems in finite elasticity.     

A general exact transformation of body-force volume integral in BEM for 2D anisotropic elasticity

In a previous study (Zhang, Tan and Afagh, 1995), the present authors successfully transformed the body-force volume integrals in BEM for 2D anisotropic elasticity, to boundary ones. This restores the BEM as a truly boundary solution process for treating anisotropic bodies involving body forces. However, the formulation is valid only for problem domains which are geometrically convex and simply connected. This paper presents a general and exact transformation of the bodyforce volume integrals in BEM to line integrals for 2D anisotropic elasticity, in which the above-mentioned restriction on the geometry of the domain is eliminated. The successful implementation of the formulation is demonstrated by three practical examples.     

Variational principles and membrane finite elements with drilling rotations for geometrically non-linear elasticity

Variational principles for geometrically non-linear continuum with independent rotation field are constructed based on the polar factorization theorem. Their regularized forms are next discussed suitable for the finite element implementation. The considerations are specialized to a two-dimensional membrane problem, and standard isoparametric interpolations are used in order to construct membrane elements with drilling rotations. The elements are evaluated on a set of problems in geometrically non-linear elastostatics.     

A class of exact solutions to the Navier-Stokes equations

Recently, Berker established the existence of an infinite set of solutions to the flow of two infinite parallel plates rotating with the same angular velocity about an axis[4]. In this work we extend Berker's analysis by allowing the plates to be porous. We find that similar to Berker's results an infinite set of solutions is possible.