Application of the finite-element method to improve the quasi-exact reinforcement theory of fibrous polymeric composites

In the paper, the WL quasi-exact reinforcement theory of fibrous polymeric composites is improved. An optimum compatibility condition related to the transverse shear problem for a unit cell, which brings solutions closest to reality, is derived. This condition is formulated in the form of a linear combination of maximum radial and circumferential displacements. Optimum coefficients of this combination are determined by comparing analytical and numerical solutions for a test specimen in the form of a rectangular thin plate, which is in a plane strain state and is subject to selected loading schemes. The analytic solutions are obtained for a homogenized material by using the WL reinforcement theory. The numerical solutions are found for an actual heterogeneous composite material by using the finite-element method, and they verify the WL reinforcement theory, in particular, the admissibility of Hills assumption. An analysis performed for two composite materials shows that the improved WL reinforcement theory gives adequate displacement fields.Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 41, No. 1, pp. 79–92, January–Febrauary, 2005.     

The stressed state and certain fracture characteristics of glass-reinforced plastics loaded in tension at an angle to the direction of reinforcement

The stressed state of glass-reinforced plastic specimens with various structures is investigated experimentally in the case of tensile loads acting at 45° to the direction of reinforcement. A relation is established between the stress distribution over the cross section and the scale factor associated with the relative width of the understressed zones along the edges of the specimen. The fracture characteristics of GRP specimens with different structures are analyzed. Recommendations are given for testing GRP with different structures in tension at an angle of 45° to the direction of reinforcement.Mekhanika Polimerov, Vol. 3, No. 6, pp. 1030–1034, 1967     

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.     

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.     

Bond of FRP Reinforcement in Concrete - a Challenge

The bond of ordinary steel reinforcement in concrete depends on many factors, such as the pullout resistance, the geometry of a concrete member, the placement of a bar in the member cross section, the cover splitting, the confinement caused by concrete and the surrounding reinforcement, the order of bond-crack appearance, and the bond-stress distribution along the bond length. The bond of FRP reinforcement depends on even a greater number of factors. Moreover, the types of FRP bars are numerous. Their surface is weaker than that of steel bars and may fracture by bond forces. The surface of FRP bars is softer and does not create as high local stress concentrations in bond contact points to concrete as the harder steel bars do. This fact often delays the appearance of cover splitting cracks along the bars. However, the load necessary for developing the crack pattern of ultimate splitting failure in concrete is then very dependent on whether the bar surface is glossy or rough. The FRP reinforcement can also be used for external shear and/or flexural strengthening of existing members. For this application, FRP bars are placed in grooves cut on the surface of the member to be strengthened and are fixed there with a cement mortar or epoxy paste. In such an application, the performance of bond between the FRP rod and the mortar or resin and then between the mortar or resin and concrete is critical for the effectiveness of the technique. The presence of two interfaces increases the number of parameters needed to characterize the global joint behavior and introduces new possible failure modes. The fundament for the bond resistance estimation should be an accepted bond philosophy linked to appropriate models. A system of bond tests should provide necessary coefficients for the models.     

Bending of prestressed glass-reinforced plastic beams

Initial waviness of the fibers and prestressing are investigated in relation to their effect on flexural stiffness and strength for beams with low shear strength. It is shown that prestressing the reinforcement increases the flexural stiffness but at the same time adds to the shear correction as a result of an increase in the modulus of elasticity E_x in the direction of reinforcement and the insensitivity of the shear modulus G_xz to prestress. It is established that prestressing increases the shear strength and the degree of anisotropy . Materials of two types are investigated: unidirectional (AG-4S) and cloth-reinforced (SKT-11).Mekhanika Polimerov, Vol. 3, No. 5, pp. 888–893, 1967     

Composite materials reinforced with a system of three straight,mutually orthogonal fibers

An extension of existing methods of calculating fibrous composites to the case of reinforcement in three mutually perpendicular directions is presented. The basic approach involves the introduction of a "modified" matrix, the properties of which are determined in terms of the elastic moduli of the binder and the reinforcement coefficient in one of the three directions. Approximate expressions are obtained for the elastic characteristics of materials with a high-modulus reinforcing phase. There is a considerable increase in the transverse modulus by comparison with the shear modulus, even for a low reinforcement coefficient in the third direction.     

Comparing reinforcement learning approaches for solving game theoretic models: a dynamic airline pricing game example

Games can be easy to construct but difficult to solve due to current methods available for finding the Nash Equilibrium. This issue is one of many that face modern game theorists and those analysts that need to model situations with multiple decision-makers. This paper explores the use of reinforcement learning, a standard artificial intelligence technique, as a means to solve a simple dynamic airline pricing game. Three different reinforcement learning approaches are compared: SARSA, Q-learning and Monte Carlo Learning. The pricing game solution is surprisingly sophisticated given the game's simplicity and this sophistication is reflected in the learning results. The paper also discusses extra analytical benefit obtained from applying reinforcement learning to these types of problems.     

Study of the residual (internal) stresses in reinforced epoxy resin

An optical polarization method has been used to study the residual (internal) stresses in reinforced ED-6 epoxy resin cured with maleic anhydride. The effects of "chemical" shrinkage, volume change relaxation during during curing, and the difference linear coefficients of thermal expansion for reinforcement and resin are elucidated. It is shown that adjacent reinforcing elements interact. The stress state is investigated with reference to a model of the elementary cell of the regular structure of a unidirectional glass-reinforced plastic. The residual stresses are found as a function of the resin/reinforcement ratio.Mekhanika Polimerov, Vol. 1, No. 4, pp. 76–80, 1965     

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.     

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.     

Mechanical properties of a composite with a carbonized matrix

The effect of reinforcement schemes that differ in the magnitude of the disorientation angle of neighboring filler layers on the mechanical properties of a "carbon-carbon" composite under extension, compression, or bending is shown.Translated from Mekhanika Polimerov, No. 2, pp. 235–240, March–April, 1976.     

On the complexity of the bondage and reinforcement problems

Let G=(V,E)$G=(V,E)$ be a graph. A subset D⊆V$D\subseteq V$ is a dominating set if every vertex not in D$D$ is adjacent to a vertex in D$D$. A dominating set D$D$ is called a total dominating set if every vertex in D$D$ is adjacent to a vertex in D$D$. The domination (resp. total domination) number of G$G$ is the smallest cardinality of a dominating (resp. total dominating) set of G$G$. The bondage (resp. total bondage) number of a nonempty graph G$G$ is the smallest number of edges whose removal from G$G$ results in a graph with larger domination (resp. total domination) number of G$G$. The reinforcement (resp. total reinforcement) number of G$G$ is the smallest number of edges whose addition to G$G$ results in a graph with smaller domination (resp. total domination) number. This paper shows that the decision problems for the bondage, total bondage, reinforcement and total reinforcement numbers are all NP-hard.     

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.     

Fiber waviness in textile composites and its stochastic modeling

A generic stochastic theory of composite materials with continuous, randomly curved (imperfect) fiber reinforcements, recently developed by the present authors, enables one to quantify the effect of fiber deviations from the assumed perfect paths. The theory of random functions and stochastic extension of the orientational averaging approach are utilized to evaluate the mean values and standard deviations of the full set of anisotropic stiffness characteristics. The major advantage of this novel stochastic approach is its applicability to practically any fiber reinforcement architecture, from unidirectional to multidirectional, 3-D woven, and braided composites. Importantly, the approach does not ask for exact quantification of the reinforcement imperfections, but needs only a limited knowledge of the mean path of the reinforcement and standard deviation of the local tangent. Numerical examples illustrating applications of the stochastic theory developed consider three types of composites having (i) unidirectional, (ii) biaxial, 2-D braided, and (iii) 3-D orthogonally woven reinforcements. The first example concerns validation of the model. The second example is selected due to the commonly observed significant randomness of the fiber architecture in biaxially braided composite shell elements. The third example illustrates the effect of Z-yarn waviness (illustrated by optical microscopy) in orthogonally woven composites on their elastic characteristics.     

Various methods of determining the natural frequencies and damping of composite cantilever plates. 3. The Ritz method

The Ritz method was used to determine the frequencies and forms of free vibrations of rectangular cantilever plates made of anisotropic laminated composites. Orthogonal Jacobi and Legendre polynomials were used as coordinate functions. The results of the calculations are in good agreement with the published experimental and calculated data of other authors for plates made of boron and carbon fiber reinforced plastics with different angles of reinforcement of unidirectional layers and different sequence of placing the layers, and also of isotropic plates. The dissipative characteristics in vibrations were determined on the basis of the concept of complex moduli. The solution of the frequency equation with complex coefficients yields a complex frequency; the loss factors are determined from the ratio of the imaginary component of the complex frequency to the real component. For plates of unidirectionally reinforced carbon fiber plastic with different relative length a detailed analysis of the influence of the angle of reinforcement on the interaction and frequency transformation and on the loss factor was carried out. The article shows that the loss factor of a plate depends substantially on the type of vibration mode: bending or torsional. It also examines the asymptotics of the loss factors of plates when their length is increased, and it notes that the binomial model of deformation leads to a noticeable error in the calculation of the loss factor of long plates when the angle of reinforcement lies in the range 20°<<70°.For Communication 2, see [1].Institute of Engineering Science of the Russian Academy of Sciences, St. Petersburg, Russia. St. Petersburg State University, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 2, pp. 215–225, March–April, 1997.     

Carrying capacity and deformation properties of glass-reinforced plastic pipes of various construction

Glass-reinforced plastic pipe under internal hydrostatic pressure starts to sweat at loads lower than the mechanical strength of the material. It is established that the life of such pipes is well described by an exponential law. The leakage mechanism is mainly controlled by the deformation properties of the reinforcement and resin. The period of reliable service can be estimated from the time of development of the strain corresponding to the end of the phase of steady-state creep. It is shown that for pipe made with cross-wound glass rovings, the winding angle and the number of roving ends have a considerable influence on the creep and life.Mekhanika Polimerov, Vol. 3, No. 6, pp. 1082–1088, 1967     

Stability of a transversely isotropic spherical shell coupled with an elastic foundation

The stability "in the small" of a flat spherical shell with elastic reinforcement is investigated. It is assumed that the shell is made of material (glass-reinforced plastic) with low shear resistance [7, 8], which determines the choice of calculation procedure: generalized applied shell theories of the Timoshenko and Ambartsumyan types [1, 3]. The results obtained are compared with the corresponding results of the Kirchhoff-Love theory.L'vov Polytechnic Institute. Translated from Mekhanika Polimerov, No. 1, pp. 129–131, January–February, 1970.     

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.     

Relation between the interlaminar shear modulus and thermal conductivity of oriented glass-reinforced plastics

The relation between the interlaminar shear modulus G of oriented glass-reinforced plastics and their thermal conductivity in a direction orthogonal to the plane of reinforcement is investigated. It is shown that G can be predicted from the measured values of and the bending deformations from and the propagation velocity of ultrasound.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Makhanika Polimerov, No. 6, pp. 985–989, November–December, 1970.