Mathematical model of the yield contour for anisotropic materials

The paper is concerned with a relevant question of processing of experimental data. Our aim is to construct yield contours for various structural materials of complicated rheology. The problem amounts to determining the minimum of a target function of the contour coefficients and belongs to the class of problems of regression analysis. Three approaches are considered for constructing a yield contour from experimental data: manual adjustment, the coordinate descent method, and the steepest descent method. On the basis of these three methods we suggest a procedure which provides a result with the least error. A program to implement this procedure is developed.     

Mathematical solution for bending of short hybrid composite beams with variable fibers spacing

In this study, the static response is presented for a simply supported functionally graded hybrid beam subjected to a transverse uniform load. Material properties of the beam are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. By varying the fiber volume fraction within a symmetric laminated beam and combining two fiber types to create a hybrid functionally graded material (FGM) can offer desirable increases in axial and bending stiffness. The equations governing the hybrid FGM beams are determined using the principle of virtual work (PVW) arising from the higher order shear deformation theories. Numerical results on the transverse deflection, axial and shear stresses in a moderately thick hybrid FGM beam under uniform distributed load are discussed in depth. The effect of power-law exponent on the deflection and stresses are also commented.     

Fictitious domain method for problems of bending of anisotropic plates

We consider the application of the fictitious domain method to the problem of bending of a rigidly clamped anisotropic plate of constant thickness with a complex shape . A rate-of-convergence bound is obtained for the proposed method in the form .Translated from Vychislitel'naya i Prikladnaya Matematika, No. 63, pp. 57–63, 1987.     

Nonlinear bending and buckling for strain gradient elastic beams

Nonlinear bending of strain gradient elastic thin beams is studied adopting Bernoulli–Euler principle. Simple nonlinear strain gradient elastic theory with surface energy is employed. In fact linear constitutive relations for strain gradient elastic theory with nonlinear strains are adopted. The governing beam equations with its boundary conditions are derived through a variational method. New terms are considered, already introduced for linear cases, indicating the importance of the cross-section area, in addition to moment of inertia in bending of thin beams. Those terms strongly increase the stiffness of the thin beam. The non-linear theory is applied to buckling problems of thin beams, especially in the study of the postbuckling behaviour.     

Mechanism of failure of reinforced beams in bending

The mechanism of shear failure is investigated. A theoretical analysis makes it possible to determine the limits of shear failure of reinforced beams in relation to the geometric parameters of the beam, the mechanical properties of the reinforcement and the resin, their volume content, and the loading and support conditions. The results obtained are consistent with the experimental data of [1]. It is shown that in the process of shear failure the axial displacement distribution is modified and that the shear failure mechanism depends on the type of loading and the support conditions.Institute of Hydrodynamics, Siberian Branch, Academy of Sciences of the USSR, Novosibirsk. Translated from Mekahnika Polimerov, No. 4, pp. 698–709, July–August, 1973.     

Mechanism of failure of reinforced beams in bending

The normal-stress brittle-fracture mechanism is investigated. It is shown that the stiffnesses and strengths of the reinforcement and matrix materials and their volume content determine the nature of the brittle fracture. The region of the mechanical parameters of the reinforcement and the matrix and of their volume content, for which an increase in load is possible after the onset of brittle failure of one of the phases, is determined. In this case the carrying capacity of the beam exceeds the load corresponding to the onset of failure of one of the phases.     

Long-wave oscillations and waves in anisotropic beams

The asymptotic integrating method is used to investigate long-wave oscillations and waves in an infinite heterogeneous (with respect to width) anisotropic beam-belt. A dispersion equation of the second-order accuracy with respect to the relative width of the beam-belt is constructed and additional qualitative effects related to the anisotropy are found.     

A recursive methodology for the solution of semi-analytical rectangular anisotropic thin plates in linear bending

A new recursive methodology is introduced to solve anisotropic thin plates bending problems, which is based on three concepts: (a) the plate differential operator additively decomposed obeying a material constitutive hierarchy; (b) the plate displacement field expanded into an infinite series and (c) an homotopy-like scheme applied to determine each term of the series. The pb-2 Rayleigh–Ritz Method is adopted to construct the solution space. Convergence conditions are presented and related to the differential operator decomposition and material’s anisotropy degree. Different cases of geometry, loading and boundary conditions were studied using the methodology and excellent agreement with available solutions was found.     

Inference for an anisotropic diffusion model

The vector sum of a white noise in an unknown hyperspace and an Ornstein-Uhlenbeck process in an unknown line is observed through sharp linear test functions over a finite time span. The parameters associated with the white noise (including the hyperplane) are determinable with precision and index the measure-equivalence classes in the relevant sample space. An intraclass relative density provides a basis for Bayesian inference of the remaining parameters.     

Simulation model for anisotropic fibrous materials

Paper and paper–based materials such as cardboard are used in a wide variety of applications and in the development of new applications such as boxes an accurate simulation model is of major importance. Industrially made paper material typically has an orthotropic fibrous structure, due to the manufacturing process, where the fibers tend to align in the direction of motion in the machine. In this work a plasticity–based material model allowing for finite strains is developed. The model is suitable for materials with an anisotropic fibrous structure such as paper. The general framework is based on separate mappings describing the deformations of the continuum and the substructure and a multiplicative split of these mappings into elastic and plastic parts. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A study of bending waves in infinite and anisotropic plates

In this paper we present a unified approach to obtain integral representation formulas for describing the propagation of bending waves in infinite plates. The general anisotropic case is included and both new and well-known formulas are obtained in special cases (e.g. the classical Boussinesq formula). The formulas we have derived have been compared with experimental data and the coincidence is very good in all cases.     

Dynamic modeling and analysis of rotating FG beams for capturing steady bending deformation

A dynamic analysis of rotating functionally gradient (FG) beams is presented for capturing the steady bending deformation by using a novel floating frame reference (FFR) formulation. Usually, the cross section of bending beams should rotate round the point at the neutral axis while centrifugal inertial forces are supposed to act on centroid axis. Due to material inhomogeneity of FG beams, centroid and neutral axes may be in different positions, which leads to the eccentricity of centrifugal forces. Thus, centrifugal forces can be divided into three componets: transverse component, axial component and force moment acting on the points of the neutral axis, in which transverse component and force moment can make the beam produce the steady bending deformation. However, this speculation has not been presented and discussed in existing literatures. To this end, a novel FFR formulation of rotating FG beams is especially developed considering centroid and neutral axes. The FFR and its nodal coordinates are used to determine the displacement field, in which kinetic and elastic energies can be accurately formulated according to centroid and neutral axes, respectively. By using the Lagrange's equations of the second kind, the nonlinear dynamic equations are derived for transient dynamics problems of rotating FG beams. Simplifying the nonlinear dynamic equations obtains the equilibrium equations about inertial and elastic forces. The equilibrium equations can be solved to capture the steady bending deformation. Based on the steady bending state, the nonlinear dynamic equations are linearized to obtain eigen-frequency equations. Transient responses obtained from the nonlinear dynamic equations and frequencies obtained from the eigen-frequency equations are compared with available results in existing literatures. Finally, effects of material gradient index and angular speed on the steady bending deformation and vibration characteristics are investigated in detail.     

Mathematical model for cell division

The division of a living cell, such as the whitefish blastula, is modeled with the Ovals of Cassini as the basic building block. The model uses two fixed points which could correspond to the approximate center of the chromosomes about which the nuclei reforms in the divided cell. The location of these points are described in terms of the scaling parameter used to maintain a constant volume under the transformation.     

A network evolution model for the anisotropic Mullins effect

In this contribution, a micro-mechanical constitutive model for filled rubberlike materials is proposed. Rubber network is decomposed into two parts, elastic rubber network and polymer-filler network. As a consequence of filler contribution, the last one is subjected to damage. A non-Gaussian strain energy function for a single chain with a constant valence angle has been postulated. Damage in different directions is governed by the corresponding maximal micro-stretch and is considered as a result of network evolution. Directional network rearrangement as a consequence of network evolution has been employed in order to describe induced anisotropy and permanent set. The model shows good agreement with experimental data on uniaxial tension tests in two orthogonal directions. (© 2008 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Behavior of Gaussian beams in an anisotropic medium with interface

A region that consists of two parts with anisotropic Riemannian metrics is considered. The metric has a jump on the interface. Asymptotic solutions of the wave equation, reflected and transmitted from the interface, i.e., Gaussian beams (“quasiphotons”), are constructed. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 324, 2005, pp. 77–109.