Application of a genetic algorithm to optimal lay-up of shells made of composite laminates

The orientation of fiber direction in layers and the number of layers of composites play the major role in determining the strength and stiffness. Thus, the basic design problem is to determine the optimum stacking sequence of the composite laminate. Many methods are available at present for the design optimization of structural systems. However, these methods are based on mathematical programming techniques involving the gradient search and the direct search. These methods assume that the variables are continuous. In this paper, a different search and optimization algorithm, known as a Genetic Algorithm (GA), has been successfully applied to obtain the optimum fiber orientation of multilayered shells, which considers the angle of fiber orientation as a discrete variable. The principle of GA is applied to obtain optimum layers and the orientation of fibers of stiffened shells for both the symmetric and antisymmetric orientations of fibers for dynamic analysis. Shells composed of two to nine layers without stiffeners, with one stiffener, and with two stiffeners for a single as well as different materials are analyzed and the maximum frequency for each population is computed using the FEAST-C software. Submitted to the 11th International Conference on Mechanics of Composite Materials (Riga, June 11–15, 2000). Published in Mekhanika Kompozitnykh Materialov, Vol. 36, No. 2, pp. 271–278, March–April, 2000.     

Application of the method of conditional moments to investigating the deformation properties of orthotropic composites with fiber microdamages

A model of deformation of stochastic composites subjected to microdamage is developed for the case of orthotropic materials with microdamages accumulating in the fibers. The composite is treated as a matrix strengthened with elliptic fibers with orthotropic elastic properties. The fractured microvolumes are modeled by a system of randomly distributed quasi-spherical pores. The porosity balance equation and relations for determining the effective elastic moduli for the case of a fibrous composite with orthotropic components are used as the fundamental relations. The fracture criterion is given as a limit value of the intensity of average shear stresses occurring in the undamaged part of the material, which is assumed to be a random function of coordinates and is described by the Weibull distribution. Based on an analytical and numerical approach, the algorithm for determining the nonlinear deformation properties of such a material is constructed. The nonlinearity of composite deformations is caused by the accumulation of microdamages in the fibers. By using a numerical solution, the nonlinear stress–strain diagrams for an orthotropic composite in uniaxial tension are obtained. Translated from Mekhanika Kompozitnykh Materialov, Vol. 45, No. 1, pp. 17–30, January–February, 2009.     

Strength of biodegradable polypropylene tapes filled with a modified starch

The possibility of creating composite materials with high deformation and strength characteristics based on polypropylene (PP) and a natural polysaccharide in the form of a modified starch (MS) has been studied. The modified starch is shown to interact chemically with functional groups of PP, thereby positively affecting the physicomechanical properties, structure, and water absorption properties of films and oriented flat fibers based on starch-filled PP. The strength characteristics of both oriented and unoriented composites are 1.5–2.0 times as high as those of the initial PP. The water absorption ability of the materials varies symbatically with content of MS, which points to the dominant contribution of interactions at the PP-MS interface. The introduction of MS into synthetic polymers offers a possibility of producing new ecologically safe materials with high strength characteristics. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 42, No. 3, pp. 389–400, May–June, 2006.     

Explicit Reciprocity Law for Lubin-Tate Formal Groups

In this article, using Fontaine＇s ФГ-module theory, we give a new proof of Coleman＇s explicit reciprocity law, which generalizes that of Artin-Hasse, Iwasawa and Wiles, by giving a complete formula for the norm residue symbol on Lubin-Tate groups. The method used here is different from the classical ones and can be used to study the Iwasawa theory of crystalline representations.     

Hyperbolic conservation laws with vanishing nonlinear diffusion and linear dispersion in heterogeneous media

We analyze family of solutions to multidimensional scalar conservation law, with flux depending on the time and space explicitly, regularized with vanishing diffusion and dispersion terms. Under a condition on the balance between diffusion and dispersion parameters, we prove that the family of solutions is precompact in L¹_loc{L^1_{\rm loc}}. Our proof is based on the methodology developed in Sazhenkov (Sibirsk Math Zh 47(2):431–454, 2006), which is in turn based on Panov’s extension (Panov and Yu in Mat Sb 185(2):87–106, 1994) of Tartar’s H-measures (Tartar in Proc R Soc Edinb Sect A 115(3–4):193–230, 1990), or Gerard’s micro-local defect measures (Gerard Commun Partial Differ Equ 16(11):1761–1794, 1991). This is new approach for the diffusion–dispersion limit problems. Previous results were restricted to scalar conservation laws with flux depending only on the state variable.     

Guided ultrasonic waves in composite cylinders

The ultrasonic nondestructive evaluation of composite cylinders requires a thorough understanding of the propagation of waves in these materials. In this paper, the propagation of flexural and longitudinal guided waves in fiber-reinforced composite (FRC) rods with transversely isotropic symmetry properties is studied. The frequency equations obtained for free cylinders and the effect of increased fiber volume fraction (increased anisotropy) on the dispersion characteristics of the rod are explored. The numerical results reveal a number of previously unnoticed characteristics of dispersion curves for composite cylinders. The mode shapes of longitudinal waves propagating in glass/epoxy cylinders are also plotted. These plots can be used to choose an appropriate strategy for inspecting composite cylinders by ultrasonic nondestructive evaluation techniques. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 43, No. 3, pp. 411–426, May–June, 2007.     

Strength and fracture properties of advanced SiC-based fibers

The tensile strength and the fracture properties of advanced SiC-based fibers were characterized, and an extensive fractographic analysis was conducted to correlate their mechanical behavior and microstructure. Tensile tests re vealed that the strength of Hi-Nicalon™ and Hi-Nicalon™ Type S fibers was sensitive to a critical flaw. The inspection of fracture surfaces revealed that the fracture of these fibers originated mainly at the critical flaw, which was surrounded by an obvious mirror zone. The Tyranno™-SA fiber showed a transcrystalline fracture behavior. The different fracture behavior observed in this work could be related to different fabrication processes and compositions at the grain boundary. For the Hi-Nicalon™ and Hi-Nicalon™ Type S fibers, the critical flaw size was linearly related to the mirror size. By using the linear fracture mechanics, the fracture toughness and the critical fracture energy of the fibers were estimated. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 42, No. 6, pp. 759–770, November–December, 2006.     

Some generalizations of the two-dimensional Toda chain and Sinh-Gordon equations

We investigate discrete transformations of the solutions and potentials of a scalar differential equation of the second order with two independent variables. The transformations are implemented by the operator D=V₁ϱ_x+V₂ϱ_y+V₃ and simple cases of the closure of these transformations are considered. Inverse scattering problem methods are used to prove the integrability of the nonlinear equations obtained. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 2, pp. 233–241, February, 1997.     

Investigation of contact corrosion of anodic coatings depending on the type of polarization of materials of the base and coating

We propose a refined mathematical model for the fracture mechanism and prediction of durability of anodic coatings in seawater. Various types of polarization of materials of a coating and a base in electrolyte are considered. We investigate the influence of the thickness of electrolyte on fracture of anodic coatings. The results of the quantitative investigations performed enable us to propose certain recommendations concerning the optimal choice of the material of a coating depending on the geometric sizes of possible through damages in it. For noninteracting small strips-damages, we established the existence of a critical thickness of the electrolyte, for which the density of the corrosion current increased by 30–35%. Pidstryhach Institute of Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences, L'viv. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 90–95, April–June, 1998.     

Finite-element investigation of the stress-strain state of an inhomogeneous rectangular plate

With the use of the finite-element method, the generalized plane stressed state of a rectangle of isotropic functionally gradient materials under the action of normal load is investigated. A finite-element model is constructed by the Bubnov–Galerkin method. The domain of the body is split into rectangular gradient elements that take into account dependences of Young’s modulus and Poisson’s ratios on coordinates. Numerical calculations are performed for the case where Young’s modulus is a polynomial function. The influence of the material gradientness and the sizes of the rectangle on its stress-strain state is analyzed.     

Linear energy-preserving integrators for Poisson systems

For Hamiltonian systems with non-canonical structure matrix a new class of numerical integrators is proposed. The methods exactly preserve energy, are invariant with respect to linear transformations, and have arbitrarily high order. Those of optimal order also preserve quadratic Casimir functions. The discussion of the order is based on an interpretation as partitioned Runge–Kutta method with infinitely many stages.     

Predicting the elastoplastic response of fiber-reinforced metal matrix composites

This paper aims to investigate the effect of microstructure parameters (such as the cross-sectional shape of fibers and fiber volume fraction) on the stress–strain behavior of unidirectional composites subjected to off-axis loadings. A micromechanical model with a periodic microstructure is used to analyze a representative volume element. The fiber is linearly elastic, but the matrix is nonlinear. The Bodner–Partom model is used to characterize the nonlinear response of the fiber-reinforced composites. The analytical results obtained show that the flow stress of composites with square fibers is higher than with circular or elliptic ones. The difference in the elastoplastic response, which is affected by the fiber shape, can be disregarded if the fiber volume fraction is smaller than 0.15. Furthermore, the effect of fiber shape on the stress–strain behavior of the composite can be ignored if the off-axis loading angle is smaller than 30°.     

Measuring and modeling the thermal conductivities of three-dimensionally woven fabric composites

The effect of a three-dimensional fiber reinforcement on the out-of-plane thermal conductivity of composite materials is investigated. Composite preforms with different fibers in the thickness direction were fabricated. After in fusion by using a vacuum-assisted resin transfer molding process, their through-thickness thermal conductivities were evaluated. The measured thermal conductivities showed a significant increase compared with those of a typical laminated composite. Although the through-thickness thermal conductivity of the samples increased with through-thickness fiber volume fraction, its values did not match those predicted by the simple rule of mixtures. By using finite-element models to better under stand the behavior of the composite material, improvements in an existing analytical model were performed to predict the effective thermal conductivity as a function of material properties and in-contact thermal properties of the composite. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 45, No. 2, pp. 241–254, March–April, 2009.     

Integral representations in the theory of controlled systems

We consider methods of analysis of controlled dynamical systems based on integral representations: the transition of the system from one admissible state to another is associated with some integral transformations of the configuration space. The main focus is on transforms with respect to integration over the Euler characteristic, which make it possible to study the topological properties of systems. An analog of the Gauss-Ostrogradskii theorem is proved for integration over the Euler characteristic. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 4, pp. 59–86, 2004.     

Control of biopolymer poly-β-hydroxybutyrate characteristics by γ-irradiation

The influence of γ-irradiation on the structure and some mechanical properties of poly-β-hydroxybutyrate was studied. Specimens of PHB were γ-irradiated with various doses (1–20 Mrad) in air and vacuum. The molecular weight, structural characteristics of the crystalline and amorphous phases, characteristics of thermal degradation, and deformation of the polymer were determined. The crystallinity was found to increase with increase in the radiation dose and with the corresponding reduction in molecular weight. The increase of crystallinity was the greater the smaller the length of macromolecules (higher radiation dose). The melting temperature T _m , which characterizes the crystalline order, decreased with decrease in the molecular weight. The results of calorimetric studies suggest that radiation-caused degradation, which occurs at a temperature at which “cold” crystallization (60°C) is possible, might also affect the crystalline part of the polymer.     

Effect of surface modification of fibers with a polymer coating on the interlaminar shear strength of a composite and the translation of fiber strength in an F-12 aramid/epoxy composite vessel

The surface of aramid fibers was modified with a polymer coating — a surface treatment reagent containing epoxy resin. The resulting fibers were examined by using NOL tests, hydroburst tests, and the scanning electron microscopy. The modified fibers had a rougher surface than the untreated ones. The interlaminar shear strength of an aramid-fiber-reinforced epoxy composite was highest when the concentration of polymer coating system was 5%. The translation of fiber strength in an aramid/epoxy composite vessel was improved by 8%. The mechanism of the surface treatment of fibers in improving the mechanical properties of aramid/epoxy composites is discussed. Russian translation publeshed in Mekhanika Kompozitnykh Materialov, Vol. 42, No. 6, pp. 729–738, November–December, 2006.     

A Distributional Approach to Multiple Stochastic Integrals and Transformations of the Poisson Measure

We study a class of ‘nonpoissonian’ transformations of the configuration space and the corresponding transformations of the Poisson measure. For some class of Poisson measures we find conditions which are sufficient for the transformed measure (which in general is nonpoissonian) to be absolutely continuous with respect to the initial Poisson measure and get the expression for the corresponding Radon–Nikodym derivative. To solve this problem we use a distributional approach to Poisson multiple stochastic integrals.     

Music as Embodied Mathematics: A Study of a Mutually Informing Affinity

The argument examined in this paper is that music – when approached through making and responding to coherent musical structures,facilitated by multiple, intuitively accessible representations – can become a learning context in which basic mathematical ideas can be elicited and perceived as relevant and important. Students' inquiry into the bases for their perceptions of musical coherence provides a path into the mathematics of ratio,proportion, fractions, and common multiples. Ina similar manner, we conjecture that other topics in mathematics – patterns of change,transformations and invariants – might also expose, illuminate and account for more general organizing structures in music. Drawing on experience with 11–12 year old students working in a software music/math environment, we illustrate the role of multiple representations, multi-media, and the use of multiple sensory modalities in eliciting and developing students' initially implicit knowledge of music and its inherent mathematics. This revised version was published online in July 2006 with corrections to the Cover Date.     

Polydisperse structures of composites with random elastic properties of inclusions

A generalized self-consistent method [1, 2] is developed and applied to the boundary-value problems of composites with random elastic properties of inclusions. The approach suggested makes it possible to allow for a random mutual arrangement, statistical dispersion of elastic properties and sizes of the inclusions, and their mutual correlation in terms of special homogenized indicator functions. For comparison, the analytical solutions and those obtained from a corresponding sequence of H+1 (H=0,1,…) linked homogenized problems of the self-consistent method for the strain distribution in the inclusions and for the tensor of effective elastic properties of the composite are given. A numerical calculation of the effective transversely isotropic elastic characteristics for a unidirectional polydisperse fibrous composite is also presented. Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 1, pp. 33–58, January–February, 2000.