Evaluating the effect of whiskerization of the fibers of composites on the fundamental frequency of a laminated cylindrical shell

Conclusion The calculations showed that whiskerization of the reinforcement of the structural material of multilaminate shells makes it possible in some cases to increase the fundamental vibration frequency of the structure up to 15–20%. In combination with the well-known [1] effect of improved strength characteristics for a whiskerized composite in the transverse and shear directions, this finding allows us to conclude that whiskerized structural materials are more efficient than ordinary laminated composites in shell-type load-bearing structures. Here, the greatest benefit can be expected in the case of whiskers which have higher elastic moduli than the main reinforcement. Since considerably higher reinforcement intensities can be achieved in whiskerized laminated composites than in composites with a reinforcement characterized by an arbitrary three-dimensional structure, it can be concluded on the basis of the results obtained here that, at least for shells of moderate thickness (10 < R/h 50), whiskerized composites are the optimum structural material for load-bearing shells.Translated from Mekhanika Kompozitnykh Materialov, No. 6, pp. 1022–1027, November–December, 1987.     

Strength and deformation properties of fiberglass-plastic in the flat stressed state as a function of reinforcement structure

Conclusion Limiting strength values have been ascertained in the flat stressed state as a function of reinforcement structure. The change in each strength surface tensor component as a function of reinforcement intensity has been approximated by the piecewise-linear approximation method. A strength condition has been derived which can be used in optimization problems. The problem of the optimum reinforcement structure of a composite at various ratios of the stresses ₁₁, ₂₂, and ₁₂ has been examined. By using the strength condition, one can predict strength values for structures which appear in the class of materials in question with various reinforcement intensities. The procedure developed can be used in designing composite materials.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 848–859, September–October, 1978.     

Nonaxisymmetric thermal stress state of laminated rotational bodies of orthotropic materials under nonisothermic loading

A procedure for numerical investigation of nonaxisymmetric temperature fields and the elastic stress-strain state of laminated rotational bodies of cylindrically and rectilinearly orthotropic materials under nonisothermal loading is proposed. The deformation of orthotropic materials is described by the equations of anisotropic elasticity theory. The equations of state are written in the form of Hookes law for homogeneous materials, with additional terms which take into account the thermal deformation, changes in the mechanical properties of materials in the circumferential direction, and their dependence on temperature. A semianalytic finite-element method in combination with the method of successive approximations is used. An algorithm for numerical solution of the corresponding nonlinear boundary problem is elaborated, which is realized as a package of applied FORTRAN programs. Some numerical results are presented.Translated from Mekhanika Kompozitnykh Materialov, Vol. 40, No. 6, pp. 731–752, November–December, 2004.     

Deformation and strength of laminated quasiisotropic carbon-reinforced plastics

Conclusions The stress-strain state and strength of the quasiisotropic carbon-reinforced plastics depend strongly on the lay-up system of the adjacent plies because the magnitude of the interply stresses and their distribution greatly change when the reinforcement system is changed. The highest stress intensity is recorded at the free edge of the specimen. The magnitude of the residual thermal stresses is comparable with that of the mechanical stresses at the instant of formation of cracks in the weak plies of the composite. The first microcracks form inside the plies with the orientation 90 and ±45° in the direction of reinforcement. The IACs start to form at the free edges of the specimen and propagate into the material at strains from 12 to 39% of limiting strain _x ^l . Delamination of the laminated composite starts later than the formation of IACs (40–88% of _x ^l ) because the normal stress _x is higher than _z. The optimum lay-up system of the composite is [0/45/90/-45]_s because this system ensures higher stiffness of the material. The ±45 and 90° plies are not suitable for external surfaces of the composite since they start to crack very early. The quasiisotropic composites are less sensitive to the value of G_{2 3} than the composites with the ply lay-up [±]_s.Translated from Mekhanika Kompozitnykh Materialov, No. 3, pp. 449–454, May–June, 1985.     

Experimental investigation of the anisotropy of strength of bolted joints of laminated composites upon variation of geometric and design parameters

Conclusion In the present work we investigated the anisotropy of the characteristics of tensile (buckling) strength and elasticity of plates made of KMU-4L in a bolted joint in a wide range of parameters of the material (stacking, number of layers), geometric parameters of the joint, parameters of the fastening element and of design parameters. An analysis of the results showed that the strength of the bolted joint under static loading can be increased for thin plates with h = 1–2 mm by local reinforcement of the zone of the hole by layers with = ±45 °, and also by increasing the moment of preliminary tightening; for plates with h = 3–4 mm the only feasible measure is increasing the moment of preliminary tightening to 20–40 Nm. All the results of the experiment were graphically generalized in the form of surfaces of maximal stress; this made it possible to predict the load bearing capacity of a joint for intermediate values of the parameters by calculating the safety factors.Translated from Mekhanika Kompozitnykh Materialov, No. 2, pp. 268–272, March–April, 1990.     

Temperature dependence of the strength of a woven composite in a plane state of stress

Conclusions On the basis of an analysis of test results obtained by typical patterns of simple momentary quasistatic loading at various temperatures within the 20–150°C range, we have found a family of strength surfaces for a woven organic plastic material in a plane state of stress in the reinforcement plane. These experimentally found strength surfaces can be approximated by the equation of a second-degree surface. It has been established that the range of safe states of stress in the stress space narrows nonuniformly with rising temperature; namely, this narrowing is accompanied by a shift of the center and a reorientation of the axes of the strength ellipsoid. We have revealed and described the temperature dependence of the components of the strength surface tensors involved in the strength criterion. The data can be used for predicting the strength of a composite material under consideration when the latter is subject to simple quasistatic loading patterns in the three-dimensional (₁₁, ₂₂, ₁₂) stress space in the reinforcement plane within a given test range of temperatures.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 3, pp. 452–457, May–June, 1978.     

Failure of laminar fiberglass-plastics in compression

Conclusions 1. The most general form of failure of laminar fiberglass-plastics in compression, as revealed on the basis of experimental studies, is chipping along an inclined area.2. A numerical comparison of the theoretical and experimental values of the chip angles for the materials studied supports the suggestion of interpreting the chip bands as slippage bands in an ideal elastoplastic anisotropic body.3. On the basis of this suggestion, valuable equations (11)–(13) have been obtained for calculating the strength of fiberglass-plastics in compression as a function of the strength properties of the matrix and of the reinforcement, the percent reinforcement, and the cutting angle .M. V. Lomonosov Moscow State University. Translated from Mekhanika Polimerov, No. 2, pp. 240–243, March–April, 1978.     

Structure and properties of ceramic composites based on ZrO2

Monodisperse fine powders of high purity obtained by sol-gel method are used for production of high technical data ceramics. The fiber reinforcement is used for hardening of composite materials. It was of interest to study production possibility of reinforced composite material based on ZrO₂ obtained by sol-gel method with filler from fibers of partially stabilized zirconia. ZrO₂ powders were obtained by precipitation of its hydrated gel from aqueous zirconium oxychloride solution by ammonium hydroxide followed by thermal treatment. For composite reinforcement ceramic partially stabilized (8 mole Y₂O₃) ZrO₂ fibers 0.16–0.67 mm in length and 5–7 m in diameter were used. Content of the fibers in composite was 20 wt.%. From powders and their mixtures with fibers, the samples were pressed as disks, beams and cylinders, and anneal in air at 1100–1600°C temperature range. The investigation has shown that the fibers of partially stabilized zirconia change the composite structure, increase the content of tetragonal modification that promotes its hardening. Treatment temperature of precursor determines physical chemical properties of compositions with fibers. Their high specific surface and reaction ability provides a workability of forming and sintering processes into strong composite material. The ceramics was increased by 2.5–3 times as strength after fibrous filler introduction into ZrO₂ hydrogel matrix.Presented at the Ninth International Conference on the Mechanics of Composite Materials (Riga, October 1995).Presented at the Ninth International conference on the Mechanics of Composite Materials (Riga, October 1995). Institute of General and Inorganic Chemistry of the Academy Sciences of the Belarus. Minsk. Translated from Mekhanika Kompozitnykh Materialov, Vol. 32, No. 3, pp. 418–427, May–June. 1996.     

Study of microbuckling of fiber-reinforced composite materials

On the basis of the theory of microbuckling of lamina-reinforced composites and formula predicting the critical composite stress for microbuckling in the shear mode _cs published in the literature, a FORTRAN program for study of the behavior of microbuckling of fiber reinforced composites has been developed. Some types of composite materials (reinforcement of different fibers and epoxy matrix) have been studied. Graphics and curves, accounting for the dependences of the compressive stress at failure _cs from the reinforcement volume k, specimen length L, and shear modulus of resin G_r have been obtained. The comparison of the theoretical diagrams presented here and experimental and theoretical results, published in the literature shows good agreement. The basic conclusion of the work presented here is that the study could be used for other fiber reinforced composites (with different mechanical properties of matrices and fibers).Institute of Mechanics, Bulgarian Academy of Sciences, Sofia, Bulgaria. Published in Mekhanika Kompozitnykh Materialov, Vol. 32, No. 4, pp. 531–538, July–August, 1996.     

Analysis of effective elastic properties of unidirectional boron fiber-reinforced plastic by a generalized self-consistent method

A new generalized self-consisrtent method is developed for the statistical mechanics of composites which makes it possible to reduce the problem of predicting the effective elastic properties of composites with random structures to solution of a simpler averaged problem of an inclusion with a transitional layer in a material with the effective elastic properties sought. The typical size of the transition layer is determined by the correlation radius of the random structure, and its elastic properties are considered as both the close order of the mutual position and the variation of inclusion dimensions in terms of a special averaged indicator function of the structure. A numerical calculation is presented by the generalized self-consistent method for the average indicator function and the transversely-isotropic tensor for effective elastic properties of unidirectional boron fiber-reinforced plastic based on different models for actual random structure in the plane of isotropy. Analysis of the numerical results compared with experimental data and known solutions of other authors demonstrates the high accuracy of the generalized self-consistent method for a broad class of random composite structures.Perm State Mechanical University, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 32, No. 6, pp. 747–758, November–December, 1996.     

Multidimensional Dirichlet series in the problem of the asymptotics of spectral series of nonlinear elliptic operators

Series of asymptotic solutions of nonlinear elliptic boundary-value problems in compact domains with a spectral parameter contained in the boundary condition are constructed, and the connection of these solutions with the trajectories of classical Hamiltonian systems defined on the boundary of the domains considered is established. The asymptotic solutions indicated are expressed in terms of multidimensional Dirichlet series, and a superposition law is established for them which, as it turns out, does not depend either on the number of independent variables in the original problem or on the form of the nonlinearity.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Vol. 23, pp. 137–222, 1983.     

Some properties of solutions of linear homogeneous differential equations with continuous coefficients

An order-reducing formula is obtained for the Wronskian W[y₁,..., y_n] of a system of (n–l)-times continuously differentiable functions. This formula is used to prove that the condition W[y₁,..., y_n] 0 (x 1) is necessary and sufficient for linear independence on I of the solutions y₁,..., y_m of the equation ..... of order n m. A formula isderived which, given a system of n – 1 linearly independent solutions Y_1,..., Y_n–1 of Eq. (1) constructs in explicit forma solution Y₂ linearly independent with Y_1,..., Y_n–1 (a generalization of Abel's formula).Translated from Vychislitel'naya i Prikladnaya Matematika, No. 55, pp. 51–56, 1985.     

Some exact velocity profiles for granular flow in converging hoppers

Gravity flow of granular materials through hoppers occurs in many industrial processes. For an ideal cohesionless granular material, which satisfies the Coulomb-Mohr yield condition, the number of known analytical solutions is limited. However, for the special case of the angle of internal friction equal to ninety degrees, there exist exact parametric solutions for the governing coupled ordinary differential equations for both two-dimensional wedges and three-dimensional cones, both of which involve two arbitrary constants of integration. These solutions are the only known analytical solutions of this generality. Here, we utilize the double-shearing theory of granular materials to determine the velocity field corresponding to these exact parametric solutions for the two problems of gravity flow through converging wedge and conical hoppers. An independent numerical solution for other angles of internal friction is shown to coincide with the analytical solution.Received: July 24, 2002     

Calculation of characteristics of the elastic and thermophysical properties of a multiphase composite containing composite or hollow spherical inclusions

Conclusion We generalized the self-consistent method of effective media to the case of a four-phase model consisting of a core, a shell, a binder, and the effective medium. We obtained analytic solutions for the elastic characteristics, coefficient of linear expansion, heat capacity, and thermal conductivity of a multiphase composite containing several types of composite (or hollow) spherical inclusions. In the special case of a composite containing inclusions of just one type, the solutions obtained for the bulk modulus of elasticity K, coefficient of linear expansion a, heat capacity c, and thermal conductivity agree (within the framework of the two-stage approach) with the values found using known solutions for a three-phase model [8]. The first stage entails calculation of the effective characteristics of a spherical composite inclusion, while the second stage involves calculating the analogous characteristics for the composite as a whole.The possibilities of the solutions that were found were illustrated in a calculation of the shear modulus of a composite containing spherical hollow inclusions. It was shown that by assuming a nonaxisymmetric Weibull distribution of the parameter (the ratio of the thickness of the wall of a particle to its radius) it is possible to reach better agreement between the calculations and the experimental data in [4] than when calculations are performed using only the mean value of .The solutions obtained here can be used to find optimum combinations of volume fractions of different types of fillers in multiphase composites.The work was sponsored at the University Iberoamericana in 1994 by the Mexican National Council of Science and Technology (CONACYT).Translated from Mekhanika Kompozitnykh Materialov, Vol. 30, No. 4, pp. 512–519, July–August, 1994.     

Influence of edge effects on the state of stress and strength of carbon composites in tension

Conclusions A difference has been found between the theoretical and experimental strength values. The theoretical strength of flat strips with straight edges, obtained on the basis of the two-dimensional solution within the elastic range for a composite with fiber angle < 30°, is higher than the actual strength. The difference is due to the failure to take account of interlaminar shear. The strength tensor is applied to a monolayer with respect to which the internal state of stress and the nature of the interrelationship between the layers are not taken into account. In this connection, it is desirable to supplement the given criterion with a condition which would take account of the difference between the stresses _xy in adjacent layers. If the directions of reinforcement > 30°, then the agreement between the theoretical strength of flat strips of variable width, obtained on the basis of the two-dimensional solution within the elastic range using the tensor strength criterion, is in good agreement with the actual strength obtained experimentally. The big differences between the actual and theoretical strengths for the three-dimensional solution are attributable to the physical and geometric nonlinearity in the boundary zone between the elementary layers of the composite.Translated from Mekhanika Kompozitnykh Materialov, No. 2, pp. 263–270, March–April, 1985.     

A necessary condition for the regularity of a boundary point for degenerating parabolic equations with measurable coefficients

We prove a necessary condition for the regularity of a point on a cylindrical boundary for solutions of second-order quasilinear parabolic equations of divergent form whose coefficients have a superlinear growth relative to derivatives with respect to space variables. This condition coincides with the sufficient condition proved earlier by the author. Thus, we establish a criterion for the regularity of a boundary point similar to the well-known Wiener criterion for the Laplace equation.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 6, pp. 818–836, June, 2004.     

Long-term failure of a layered viscoelastic composite material with a crack under a time-dependent load

The long-term failure of a layered viscoelastic composite caused by precritical propagation of a coin-shaped crack is studied. It is assumed that the crack is located inside a viscoelastic layer (the layer of binder) parallel to the layer orientation. The crack development due to stretching of the composite massive by uniformly distributed external forces increasing with time is described. It is assumed that these forces act perpendicularly to the plane of crack propagation. The investigation is carried out within the framework of Boltzmann-Volterra linear theory for resolving integral operators with difference kernels describing the deformation of a material with time-dependent rheological properties. An irrational function of the viscoelastic integral operator is presented in the form of a proper continued fraction and transformed using the method of operator continued fractions. Numerical solutions are obtained for resolving integral operators with the kernel in the form of Rabotnov exponential-fractional function. The kinetics of crack growth with a prefailure zone commensurable with the crack length is described. A comparison with the results obtained in terms of the concept of thin structure of the crak tip is given.Timoshenko Institute of Mechanics, Ukrainian National Academy of Sciences, Kiev, Ukraine. Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 4, pp. 545–558, July–August, 2000.     

The Riemann–Hilbert Problem for Analytic Description of the DM Solitons

A simple exact formula is derived for the profile of the optical pulse propagating over a DM fiber with zero mean dispersion. The dissipation is neglected, and dispersion is assumed to be constant along the adjacent legs of the waveguide, thus providing the applicability of the integrable NLS models within each leg. The formula describes a class of solutions called dispersion-managed solitons (DM solitons), which are periodic along the waveguide and exponentially localized in time. The DM solitons are parameterized by a certain class of spectral data, specified from numerical simulations. Using a related Riemann–Hilbert problem, we reconstruct a profile of the DM soliton from the given spectral data. For sufficiently long legs, the leading term of DM soliton is found in explicit form by asymptotic undressing of the Riemann–Hilbert problem. The analytic results are compared with numerical simulations.     

Oscillations of a membrane that is orthogonal to a flow: Asymptotics of large stresses

The problem of small free oscillations of a membrane that is orthogonal to an inflowing stream is considered. A quasi-cavitation numerical hydrodynamic scheme is used that combines the assumptions on the potential character of the flow before the membrane and the assumption of a constant bottom pressure. The problem is reduced to a homogeneous system of integral-differential and singular integral equations with respect to variables that give the form of the membrane and density of associated whirls. For a small velocity pressure an analytic solution to the problem in the first order of the perturbation theory is constructed. Analytic and numerical solutions to the problem are compared, which are obtained by using the methods of finite differences and discrete whirls.Translated from Dinamicheskie Sistemy, No. 7, pp. 41–47, 1988.