Elastoplastic stability of glass-reinforced plastic plates with a central metal layer

The problem of the stability of a three-layer plate with a central plastic layer of metal sandwiched between elastic glass-reinforced plastic outer layers is considered. The presence of a metal layer restrains the development of creep strains in the glass-reinforced plastic and makes it possible to neglect the viscous strain components. The general equations of the problem are obtained, and the approximate Il'yushin formulation [1] is considered. An example is presented for a rectangular plate in pure shear. It is shown that the elastic anisotropic layers play the part of a load-relieving system for the central plastic layer [3], which results in an increase in the over-all critical load for the layered plate.Kalinin Polytechnic Institute. Translated from Mekhanika Polimerov, No. 5, pp. 909–915, September–October, 1969.     

Topology and material orientation optimization based on evolution equations

Many modern high-performance materials have inherent anisotropic elastic properties and its local material orientation can be considered to be an additional design variable for the topology optimization [1–3]. We extend our previous model for topology optimization with variational controlled growth [4–6] for linear elastic anisotropic materials, for which the material orientation is introduced as an additional design variable. We solve the optimization problem purely with the principles of thermodynamics by minimizing the Gibbs energy. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fundamentals of the theory of non-fabric glass-reinforced plastics

Starting from the known mechanical properties of the components (glass reinforcement and resin) the stress-strain relations for an anisotropic viscoelastic material (glass-reinforced plastic) are determined. Using Volterra's principle [2] of replacing the elastic constants in the solution of the problem of the theory of elasticity [5] by integral operators, relations for the given material are obtained and found to be in fairly good agreement with the experimental data. In a separate study the high-temperature behavior of GRP is investigated by calculating the integral operators, which are functions of the viscous properties of the resin and the temperature.Mekhanika polimerov, Vol. 1, No. 1, pp. 151–158, 1965     

Elastic constants for a material reinforced in two directions in the planar case

A simplified model is proposed for an elastic material (glass laminate) reinforced in two perpendicular directions. On the basis of an analysis of the cooperation of the reinforcement and the binder in the assumed model, approximate formulas are obtained for the reduced elastic constants of the reinforced material treated as an anisotropic and macroscopically homogeneous body. A slightly different approach to the solution of a similar problem is described in [2, 4]. The formulas obtained are in agreement with the experimental data and the recommendations of other authors.Mekhanika Polimerov, Vol. 3, No. 2, pp. 266–272, 1967     

Problems of the anisotropy of strength

The concept of a strength tensor, analogous to the tensor of elastic constants, has been introduced with a view to using its components for approximate determination of laws governing the variation of strength of anisotropic materials in uniaxial tension or pure shear as a function of the orientation of the stress state. It has been shown that the plasticity condition postulated by Mises can be used as a criterion of strength of many "weakly" anisotropic materials, if one rejects the hypothesis according to which the condition is invariant in respect to hydrostatic pressure. In the case of "strongly" anisotropic materials, wood in particular, the Mises condition is at variance with the results obtained for variously oriented specimens tested in uniaxial tension. A strength condition in the form of a fourth-degree polynomial, based on the assumed existence of a strength tensor analogous to the tensor of elastic constants, has been postulated for these materials. The validity of this criterion has been demonstrated by constructing surfaces of equicritical plane stress states from experimental data obtained for pine wood, plywood, and glass-reinforced plastic laminates.Mekhanika Polimerov, Vol 1, No. 2, pp. 79–92, 1965     

Homogenization of Fatigue and Creep Durability of Composites

The paper is an extension of the theory presented in [4] to fatigue, creep and other types of time‐dependent material behaviour. In the previous paper a multi‐phase periodic composite subjected to prescribed mechanical loads was considered. The asymptotic homogenization was applied for calculation of homogenized macro‐stresses. A nonlocal approximate macro‐strength condition, defined on homogenised stress‐field, was derived from the micro‐strength conditions and their convergence to the approximate macro‐strength condition, as the structure period tends to zero, was proved. This paper presents the analogous results for the macro‐durability of elastic and viscoelastic composites.     

Antiplane periodic contact problems fora layer non-uniform along its thickness

Antiplane periodic contact problems for an elastic layer with a shear modulus which variesexponentially along its thickness are considered. The problems are reduced to an integral equation of the first kind with an irregular, periodic, difference kernel. A method which has been described previously [1,2] is used for the approximate solution of this equation.     

静止平面应力裂纹尖端的各向异性塑性应力场的补充研究(Ⅰ)

文献[1]的结果对α≥2情形不适用。为此,我们用文献[1]的方法导出了α=2和α>2两者的静止平面应力裂纹尖端的各向异性塑性应力场的一般表达式。作为实例,我们给出了α=2的静止平面应力Ⅰ型和Ⅱ型裂纹尖端的各向异性塑性应力场的解析表达式。     

Ausbreitung von Stosswellen in einem Bodenmaterial mit linearem Zustandsdiagramm

Summary Going out from two earlier reports [1, 2] on one-dimensional shock waves in the ground, the special case of the shock wave propagation in a material with a linear pressure-deformation relation is discussed. The pressure distribution along the shock front as well as the velocity distribution on the surface can explicitly be given by means of an infinite sum, for which in the case of a strongly plastic ground an integral gives a good approximation. For completely elastic or plastic ground material the sum is explicitly evaluated. The theory is applied to a shock loading followed by a triangular pressure fall-off. Also the case of the exponentially decreasing shock loading is discussed.     

Sensitivity analysis of weakened beams on elastic foundation with Green's functions

The approach of Sensitivity Analysis with Green's Functions (SAGF) [1,2] was developed to predict changes in deformations, stresses or eigenfrequencies of structures resulting from stiffness modifications or cracking by considering only the weakened or damaged parts of the structures. This approach results in a local analysis instead of a global analysis by recalculating the whole structure. Consequently, it is computationally less time-consuming than the conventional methods based on a global analysis. The key idea of the SAGF approach is based on the comparison between the elastic strain energies of the original and the weakened structures and the substitution of the virtual displacements by the corresponding Green's functions [1, 2]. Furthermore, an approximate approach for the sensitivity analysis was suggested which is described in [1, 3] in details. This approach enables us to predict the changes in the structural responses due to the stiffness weakening in the beam or in the elastic Winkler foundation by considering only the internal forces or the deflections of the original unweakened system. In addition, an iterative method was developed to enhance the accuracy of the SAGF approach. In this paper, the local SAGF method for sensitivity analysis of elastic beams on Winker foundation with stiffness weakening is presented. The accuracy and the efficiency of the proposed method are verified by using a numerical example. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Formulation of Additive Finite Anisotropic Thermo-Plasticity in Logarithmic Lagrangean Strain-Entropy Space

A phenomenological model of rate-independent thermo-plasticity at finite strains is discussed. The formulation is based on an additive decomposition of the strain measure into an elastic and plastic part as proposed by Green and Naghdi. A constitutive model in the logarithmic Lagrangean strain-entropy space is developed capable of modelling isotropic elastic and anisotropic plastic material behaviour. The staggered solution scheme for coupled thermo-mechanical problems employs an isentropic phase for the deformation and an iso-geometrical phase for the thermal field. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Refined linear theory of elastic anisotropic multilayer shells. Part 2

On the basis of the refined linear theory of elastic anisotropic multilayer shells of arbitrary shape derived in [1] it is established that a number of theorems of the linear theory of elasticity have analogues in the theory of multilayer anisotropic shells.For Part 1 see [1].Institute of Fluid Mechanics, Bucharest. Translated from Mekhanika Polimerov, No. 1, pp. 100–109, January–February, 1976.     

Basic equations of the theory of reinforced media

The author derives the basic equations of the theory of composite elastic media obtained by reinforcing some elastic medium with a large number of linear or planar elastic elements with high strength and deformation resistance. The argument is based on macrostructural considerations. The stress-strain state of each of the reinforcing elements is considered with allowance for interaction with the matrix material. In addition, the "smoothing" principle introduced in [1–3] is applied. This corresponds to approximating the reinforced medium with some equivalent quasi-homogeneous anisotropic medium.The case of a fibrous medium in which the reinforcing elements are rods or filaments [4] is discussed in detail. Allowance for moment effects leads to equations analogous to the equations of the Voight-Cosserat moment theory and its later generalizations. Similar equations are obtained for the case of laminated media, where the reinforcing elements are membranes or plates. On the basis of the viscoelastic analogy [7], the equations of the theory of reinforced media are extended to include the case in which the matrix and/or reinforcing materials are linear viscoelastic.Mekhanika Polimerov, Vol 1, No. 2, pp. 27–37, 1965     

Linearkombinationen von iterierten Bernsteinoperatoren

The Bernstein polynomials B_n(f) approximate every function f which is continuous on [0, 1] uniformly on [0, 1]. Also the derivatives of the Bernstein polynomials approach the derivatives of the function f uniformly on [0, 1], if f has continuous derivatives. In this paper we shall introduce polynomial operators, namely linear combinations of iterates of Bernstein operators, which have the same properties but, under definite conditions, approximate f more closely than the Bernstein operators.     

Stability of transversally isotropic shells coupled with an elastic foundation

The stability of shells coupled with an elastic Winkler foundation is investigated. It is assumed that the shell is made of a material (glass-reinforced plastic) with low resistance to shear, as a result of which generalized theories that take transverse shear strains into account [1–4] must be used in the stability calculations. The solution obtained is compared with the corresponding solution obtained on the basis of the classical Kirchhoff-Love theory [8].Lvov Polytechnic Institute. Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 4, pp. 669–673, July–August, 1969.     

The representation of the displacement gradient of isotropic elastic body in terms of the piola stress tensor : PMM vol. 40, n 6, 1976, pp. 1070–1077

The representation of the displacement gradient of an isotropic elastic body is analyzed. It is shown on the basis of a single controlling inequality and a polar expansion of the Piola tensor that such representation has generally four branches. The mechanical meaning and the nature of that ambiguity is explained. It is established that when the angles of turn of material fibers are not excessively large, only one of the four branches is obtained. Particular cases in which the nature of ambiguity is more complex are investigated. It is noted that in many practical problems the representation of the displacement gradient by the Piola stress tensor is unambiguous.The considered problem is associated with the variational principle of complementary energy in the nonlinear theory of elasticity, where the statistically feasible fields of the asymmetric Piola stress tensor is varied [1], A method was proposed there for expressing the displacement gradient in terms of the Piola stress tensor for an isotropic elastic body. Later the concept of complementary energy and the representation of the strain gradient in terms of the Piola stress tensor were considered in [2, 3]. Examples of the use of the complementary energy concept are given in [2] and the case of an anisotropic body is considered in [3], These investigations disclosed that the considered representation of the strain tensor leads to ambiguity, but the character and nature of the ambiguity were not fully investigated.     

Analysis of the anisotropic properties of honeycomb sandwich plates

A mathematical model has been described and an approach proposed to the diagnostics of the anisotropic properties of the relative elasticity constants of three-layer (sandwich) plates with a honeycomb filler. Compact analytical relationships between the anisotropy coefficients and the geometrical dimensions of the cells have been obtained. A practical approach has been proposed for the selection of rational parameters of the honeycomb structure, at which its anisotropic properties would be in the chosen region of the functional space. The conditions have been formulated at which the constructive anisotropy, inherent to honeycomb structures, degenerates to isotropy for some relative elastic constants.Paper to be presented at the Ninth International Conference on the Mechanics of Composite Materials (Riga, October 1995).Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 4, pp. 482–487, July–August, 1995.     

Crack plane identification in anisotropic linear elastic material using the reciprocity principle

Nondestructive test methods are important for examination of elastic devices regarding existence, position and size of cracks. In the case of hidden cracks (which do not touch the boundary), a simple visual control is not sufficient. The basic idea of this paper is to examine appropriate boundary measurements under certain loads. We focus on a method presented by ANDRIEUX, BEN ABDA and BUI [1] for isotropic linear elasticity, and generalize the crack plane detection to anisotropic linear elastic material. The main idea is the use of the reciprocity principle in order to connect data from the outer boundary with the unknown crack properties. Some 2D numerical examples demonstrate, that the method is working with simulated data. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)