A Technique for Determining the Elastic and Dissipative Properties of Orthotropic Composites in Shear

A new method is presented for the characterization of three principal complex shear moduli of linear viscoelastic orthotropic materials, which is based on the measurement of complex torsional vibration frequencies of three rods of rectangular cross section. The rod-type test specimens are cut out from a composite plate along the principal material axes in the reinforcement plane. It is shown that the torsional stiffness of an elastic rod can be calculated not only by means of the Saint-Venant torsion theory, but also using a relationship obtained from the Reissner-Mindlin theory of plates. The transfer to a viscoelastic model of the material with complex moduli is realized with the help of the correspondence principle. By applying a numerical sensitivity analysis of natural frequencies to the shear moduli, the advisable width-to-thickness ratios of the specimens are found. As an illustration of data processing, the dynamic shear moduli and the loss factors for a GFRP fabric with an epoxy matrix are calculated. A comparison of the method offered with some known static and dynamic methods for determining the shear moduli of orthotropic materials is given.     

Singular integral equation method for 2D fracture analysis of orthotropic solids containing doubly periodic strip-like cracks on rectangular lattice arrays under longitudinal shear loading

The doubly periodic arrays of cracks represent an important mesoscopic model for analysis of the damage and fracture mechanics behaviors of materials. Here, in the framework of a continuously distributed dislocation model and singular integral equation approach, a highly accurate solution is proposed to describe the fracture behavior of orthotropic solids weakened by doubly periodic strip-like cracks on rectangular lattice arrays under a far-field longitudinal shear load. By fully comparing the current numerical results with known analytical and boundary element solutions, the high precision of the proposed solution is verified. Furthermore, the effects of periodic parameters and orthotropic parameter ratio on the stress intensity factor, crack tearing displacement, and effective shear modulus are studied, and an analytically polynomial estimation for the equivalent shear modulus is proposed in a certain range. The interaction distances among the vertical and horizontal periodic cracks are quite different, and their effects vary with the orthotropic parameter ratio. In addition, the dynamic problem is discussed briefly in the case where the material is subjected to harmonic longitudinal shear stress waves. Further work will continue the in-depth study of the dynamics problem of the doubly periodic arrays of cracks.     

Application of the resonance method for determining the elastic and dissipative properties of an orthotropic polymer composite

Experimentally determined resonant frequencies and damping of flexural and torsional vibrations of rod-type rectangular test specimens made of an orthotropic GFRP fabric with different ratios of cross-sectional sizes are used for calculating six principal complex elastic and shear moduli. The application of the classical theories of flexural and torsional vibrations, the theory of flexural vibrations of a Timoshenko beam, and a refined theory of torsional vibrations of free-free orthotropic rods is analyzed.     

A symplectic analytical wave propagation model for damping and steady state forced vibration of orthotropic composite plate structure

An analytical wave propagation model is proposed in this paper for damping and steady state forced vibration of orthotropic composite plate structure by using the symplectic method. By solving an eigen-problem derived in the symplectic dual system of free bending vibration of orthotropic rectangular thin plates, the wave shape of plate is obtained in symplectic analytical form for any combination of simple boundary conditions along the plate edges. And then the specific damping capacity of wave mode is obtained symplectic analytically by using the strain energy theory. The steady state forced vibration of built-up plates structure is calculated by combining the wave propagation model and the finite element method. The vibration of the uniform plate domain of the built-up plates structure is described using symplectic analytical waves and the connector with discontinuous geometry or material is modeled using finite elements. In the numerical examples, the specific damping capacity of orthotropic rectangular thin plate with three different combinations of boundary condition is first calculated and analyzed. Comparisons of the present method results with respect to the results from the finite element method and from the Rayleigh–Ritz method validate the effectiveness of the present method. The relationship between the specific damping capacity of wave mode and that of modal mode is expounded. At last, the damped steady state forced vibration of a two plates system with a connector is calculated using the hybrid solution technique. The availability of the symplectic analytical wave propagation model is further validated by comparing the forced response from the present method with the results obtained using the finite element method.     

Torsional vibrations and dispersion of torsional waves in orthotropic composite rods

The Barr’s refined theory of torsional vibrations of isotropic rods of noncircular cross section is generalized for an orthotropic material. An analysis of natural frequencies of torsional vibration of free-free orthotropic prismatic rods of rectangular cross section is carried out with the help of an exact solution of the frequency equation. For orthotropic CFRP and GFRP rods, the improved theory, which takes into account the normal stresses and inertia forces in the axial direction, in some cases, predicts a noticeable raise in the natural frequencies compared with those following from the Saint-Venant classical theory. A good agreement is obtained between the experimental and calculated values of natural frequencies of torsional vibrations of rectangular quartz and fiber glass rods. The dispersion of torsional waves in an orthotropic quasi-homogeneous rod is considered. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 44, No. 2, pp. 165–182, March–April, 2008.     

Finite-difference method of solving the bending problem for glass-reinforced plastic plates with allowance for creep

A new finite-difference method is proposed for solving the bending problem as it applies to thin rectangular orthotropic plates of linear hereditary material. The proposed method makes it possible to obtain a closed approximate solution in explicit form for a series of finite-difference systems of equations corresponding to the problem of the bending of plates of orthotropic glass-reinforced plastic with allowance for the rheological properties of the material under various boundary conditions.Institute of Cybernetics and Computer Center, Academy of Sciences of the Uzbek SSR, Tashkent. Translated from Mekhanika Polimerov, No. 6, pp. 1075–1082, November–December, 1969.     

Determination of the transverse strain coefficients of orthotropic materials from bending tests

A method is proposed for determining the elastic constants of orthotropic materials by testing rectangular specimens in pure bending. The method is based on the fact that in bending the tensile zone undergoes transverse contraction and the compressive zone transverse expansion.Kiev Structural Engineering Institute. Translated from Mekhanika Polimerov, No. 6, pp. 1115–1117, November–December, 1969.     

Using finite difference and differential transformation method to analyze of large deflections of orthotropic rectangular plate problem

This paper analyses the large deflections of an orthotropic rectangular clamped and simply supported thin plate. A hybrid method which combines the finite difference method and the differential transformation method is employed to reduce the partial differential equations describing the large deflections of the orthotropic plate to a set of algebraic equations. The simulation results indicate that significant errors are present in the numerical results obtained for the deflections of the orthotropic plate in the transient state when a step force is applied. The magnitude of the numerical error is found to reduce, and the deflection of the orthotropic plate to converge, as the number of sub-domains considered in the solution procedure increases. The deflection of the simply supported orthotropic plate is great than the clamped orthotropic plate. The current modeling results confirm the applicability of the proposed hybrid method to the solution of the large deflections of a rectangular orthotropic plate.     

An asymptotic approach to the problem of the free oscillations of a beam

Equations describing the free small longitudinal and transverse oscillations of a straight elastic beam of rectangular cross section are obtained using the plane linear theory of elasticity and the method of integrodifferential relations. The initial system of partial differential equations is reduced to a system of ordinary linear differential equations with constant coefficients. The effect of the geometrical and elastic characteristics of the beam on the frequency and form of the natural oscillations is investigated. For longitudinal motions it is shown that different types of natural displacements and internal stresses of the beam exist. For transverse oscillations, it is found that there are frequency zones corresponding to different forms of the solutions of the characteristic equation obtained using the proposed model.     

正交异性矩形板后屈曲摄动分析

本文从正交异性板Kármán型大挠度方程出发,以挠度为摄动参数,采用直接摄动法,研究了正交异性矩形板在面内压缩作用下的后屈曲性态.本文讨论了两种面内边界条件,同时考虑了初始挠度的影响.本文给出了多种复合材料板的计算结果.所得结果与实验结果的比较表明二者是一致的.     

A type of mechanically anisotropic,optically sensitive material

The technology for obtaining a mechanically anisotropic, optically sensitive material that can be used in photoelastic stress analysis is described. The mechanically anisotropic, photoelastic materials obtained by this method (method 3) are free of initial birefringence, have a sufficiently high degree of mechanical anisotropy, and possess high optical sensitivity. The elastic and piezooptic properties of specimens of orthotropic material have been investigated and it has been found that the isochromatics (lines of equal path differences ) are lines of equal principal strain differences. Experimental results are presented for the stress distribution in an orthotropic disk compressed by a pair of opposing concentrated forces.Translated from Mekhanika Polimerov, No. 5, pp. 835–839, September–October, 1971.     

Forced vibration of an initially stressed thick rectangular plate made of an orthotropic material with a cylindrical hole

The forced vibration of an initially statically stressed rectangular plate made of an orthotropic material is studied. The plate is simply supported along all its edges and contains an internal across-the-width cylindrical hole of rectangular cross section with rounded corners. The initial stresses are created by uniformly distributed normal forces applied to opposite end faces of the plate. Because of the hole, these stresses are not uniform in the plate and significantly affect the stress field caused by additional time-harmonic dynamical forces acting on the upper face of the plate. Hence, for solving the boundary-value problem considered, the superposition principle is unsuitable. Therefore, our investigations are carried out within the framework of the three-dimensional linearized theory of elastic waves in initially stressed bodies. The corresponding boundary- value problems on determining the initial and additional, dynamical stress states are solved by using the three-dimensional finite-element method. Numerical results on stress concentrations around the cylindrical hole and the fundamental frequencies, and on the influence of the initial stresses on the frequencies are presented.     

Resonance methods for determining the complex shear moduli of orthotropic composites

A survey of various methods for determining the complex elasticity and shear moduli from the resonant frequencies of flexural and torsional vibrations of rectangular rods cut out from a plate of an orthotropic composite is presented. The errors in the computed values of dynamic shear moduli caused by inaccuracies in the experimental determination of resonance frequencies are examined. A new variant of the resonance method is developed, which permits one to find three complex shear moduli of a composite from the resonant frequencies and the damping of torsional vibrations of three rods oriented along three symmetry axes of the material. For computing the moduli in the case of an overdetermined system, an algorithm of nonlinear optimization based on the least-squares method is recommended. From the results obtained it follows that, for determining the interlaminar shear moduli with a necessary accuracy, the rods must be sufficiently thick. It is shown that a good agreement alone between calculated and experimental frequencies of flexural and torsional vibrations of rods does not ensure a reliable determination of the moduli of interlaminar shear if experiments are carried out on wide test specimens cut out from a thin plate. Recommendations are given for the choice of geometrical sizes of test specimens for resonance experiments. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 6, pp. 721–744, November–December, 2007.     

Propagation of elastic waves in certain composites

Conclusions 1. Relationships have been obtained for determining nine elastic characteristics of orthotropic composite materials from the properties of the starting components and the assigned reinforcement scheme.2. Formulas are given for calculating the propagation velocity of three types of elastic flat waves for an arbitrary direction in one of the planes of elastic symmetry of a uniform orthotropic material.3. It has been shown that the velocity of the first arrival of a packet of ultrasonic vibrations which is recorded in an experiment is equal to the velocity of motion of the wave front in a limitless medium even for rather thin (5–10 mm) fiberglass-plastic specimens with unidirectional or cross-reinforced schemes.4. The dependences of elastic properties and rates of propagation of elastic vibrations on direction which are calculated theoretically from the properties of the starting components and the reinforcement scheme agree satisfactorily with experimental results.Translated from Mekhanika Polimerov, No. 3, pp. 531–536, May–June, 1978.     

Bending of a nonlinear orthotropic cantilever beam

The method of elastic solutions is employed to investigate the plane problem of the deformation of a cantilever beam of orthotropic glass-reinforced plastic under a concentrated load with allowance for the non-linear properties of the material. The first approximation of the stress function is given and the stress distribution over the cross section is calculated for a specific GRP.Mekhanika Polimerov, Vol. 2, No. 5, pp. 773–778, 1966     

Asymptotic solutions of non-classical boundary-value problems of the natural vibrations of orthotropic shells

The natural vibrations of orthotropic shells are considered in a three-dimensional formulation for different versions of the boundary conditions on the faces: rigid clamping rigid clamping, rigid clamping free surface, and mixed conditions. Asymptotic solutions of the corresponding dynamic equations of the three-dimensional problem of the theory of elasticity are obtained. The principal values of the frequencies of natural vibrations are determined. It is shown that three types of natural vibrations occur in the shell: two shear vibrations and a longitudinal vibration, which are due solely to the boundary conditions on the faces. It is proved that each boundary layer has its own natural frequency. The boundary-layer functions are determined and the rates at which they decrease with distance from the faces inside the shell are established.     

Fundamental frequency of isotropic and orthotropic rectangular plates with linearly varying thickness by discrete singular convolution method

The subject of this article is solving free vibration problems of isotropic and orthotropic rectangular plates with linearly varying thickness along one direction. For the numerical solution to evaluate the frequencies of plates, the method of discrete singular convolution (DSC) is adopted. Frequency parameters are obtained for different types of boundary conditions, taper and aspect ratios. The effect of the mode number is also analyzed. The results obtained by the present numerical method show an excellent agreement with available published results.     

hp-Finite element for free vibration analysis of the orthotropic triangular and rectangular plates

The hp-version of the finite element method based on a triangular p-element is applied to free vibration of the orthotropic triangular and rectangular plates. The element's hierarchical shape functions, expressed in terms of shifted Legendre orthogonal polynomials, is developed for orthotropic plate analysis by taking into account shear deformation, rotary inertia, and other kinematics effects. Numerical results of frequency calculations are found for the free vibration of the orthotropic triangular and rectangular plates with the effect of the fiber orientation and plate boundary conditions. The results are very well compared to those presented in the literature.     

An efficient mixed methodology for free vibration and buckling analysis of orthotropic rectangular plates

In this paper, a simple and efficient mixed Ritz-differential quadrature (DQ) method is presented for free vibration and buckling analysis of orthotropic rectangular plates. The mixed scheme combines the simplicity of the Ritz method and high accuracy and efficiency of the DQ method. The accuracy of the proposed method is demonstrated by comparing the calculated results with those available in the literature. It is shown that highly accurate results can be obtained using a small number of Ritz terms and DQ sample points. The proposed method is suitable for the problem considered due to its simplicity and potential for further development.     

Application of the Haar wavelet-based discretization technique to problems of orthotropic plates and shells

The Haar wavelet discretization technique for solving the elastic bending problems of orthotropic plates and shells is proposed. Free transverse vibrations of orthotropic rectangular plates with a variable thickness in one direction are considered as a model problem. In the case of constant plate thickness, the numerical results are validated by comparing them with an exact solution. The results obtained are found to be in good agreement with those available in the literature.