Dynamics of unidirectional glass-fiber-reinforced plastic

The motion problem of unidirectional glass-fiber-reinforced plastic is formulated under the assumption that the fibers are under stress-strain only, while the binder is under shear stress only. The binder and fiber inertia is calculated along a direction parallel to the fibers. The system of equations in partial derivatives obtained is reduced by Laplace transformation with respect to time to a system of ordinary differential equations in which only the fiber displacements occur. As illustration, the effect of a normal stress wave on a half space is solved. The solution is obtained in the form of an infinite series provided with an explicit law by which the terms are obtained. Curves are presented for the distribution of the normal and shearing stresses at different moments of time. The binder inertia reduces to the appearance of tangential stresses at the fiber-binder boundary, which can explain the tendency towards stratification in constructions made of glass fiber-reinforced plastic.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 139–145, July–August, 1974.     

Large deflections of heterogeneous anisotropic rectangular plates

An approximate solution is presented for large deflections of clamped, uniformly loaded, unsymmetrically laminated, anisotropic, rectangular plates. Expressing the load and displacements in the form of series, the von Karman-type nonlinear differential equations and immovable boundary conditions are reduced to a series of linear partial differential equations and boundary conditions. The solution obtained by successive approximations can reduce to some existing solutions for large deflections of homogeneous plates. Numerical results based on the first three terms of the truncated series are graphically presented for unsymmetrical cross-ply and angle-ply plates having various values of fiber-reinforced material, number of layers, and aspect ratio. The results in small deflections of coupled laminates are compared with available data.     

Reducing three-dimensional elasticity problems to two-dimensional problems by approximating stresses and displacements by legendre polynomials

Shell equations are constructed in orthogonal curvilinear coordinates using approximations of stresses and displacements by Legendre polynomials. The order of the constructed system of differential equations is independent of whether stresses and displacements or their combination are specified on the shell surfaces, which provides the correct formulation of the surface conditions in terms of both displacements and stresses. This allows the system of differential equations of laminated shells to be constructed using matching conditions for displacements and stresses on the contact surfaces. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 179–190, May–June, 2007.     

Dynamic stress intensity factors around a cylindrical crack in an infinite elastic medium subject to impact load

This paper investigates transient stresses around a cylindrical crack in an infinite elastic medium subject to impact loads. Incoming stress waves resulting from the impact load impinge on the crack in a direction perpendicular to the crack axis. In the Laplace transform domain, by means of the Fourier transform technique, the mixed boundary value equations with respect to stresses and displacements were reduced to two sets of dual integral equations. To solve the equations, the differences in the crack surface displacements were expanded in a series of functions that are zero outside the crack. The boundary conditions for the crack were satisfied by means of the Schmidt method. Stress intensity factors were defined in the Laplace transform domain and were numerically inverted to physical space. Numerical calculations were carried out for the dynamic stress intensity factors corresponding to some typical shapes assumed for the cylindrical crack.     

Two-dimensional elasticity solution for bending of functionally graded beams with variable thickness

This paper studies the stress and displacement distributions of functionally graded beam with continuously varying thickness, which is simply supported at two ends. The Young’s modulus is graded through the thickness following the exponential-law and the Poisson’s ratio keeps constant. On the basis of two-dimensional elasticity theory, the general expressions for the displacements and stresses of the beam under static loads, which exactly satisfy the governing differential equations and the simply supported boundary conditions at two ends, are analytically derived out. The unknown coefficients in the solutions are approximately determined by using the Fourier sinusoidal series expansions to the boundary conditions on the upper and lower surfaces of the beams. The effect of Young’s modulus varying rules on the displacements and stresses of functionally graded beams is investigated in detail. The two-dimensional elasticity solution obtained can be used to assess the validity of various approximate solutions and numerical methods for the aforementioned functionally graded beams.     

均布荷载作用下功能梯度简支梁弯曲的解析解

利用应力函数半逆解法,研究了均布载荷作用下、材料属性在厚度上任意变化的功能梯度简支梁弯曲的解析解,给出了各向应力应变与位移的解析显式表达式.首先根据平面应力状态的基本方程,得出了功能梯度梁的应力函数应满足的偏微分方程,并根据应力边界条件得出了各应力分布的表达式;进而根据功能梯度材料的本构方程和位移边界条件,得出了应变和位移的分布.最后,通过将本文的解退化到均质各向同性梁并与经典弹性解比较,证明了本文理论的正确性,并求解了材料组分呈幂律分布的功能梯度梁的应力和位移分布,分析了上下表层材料的弹性模量比λ与组分材料体积分数指数n对应力和位移分布的影响.     

THEORY OF THICK-WALLED SHELLS AND ITS APPLICATION IN CYLINDRICAL SHELL

In this paper, a theory of thick-walled shells is established by means of Hellinger-Reissner's variational principle, with displacement and stress assumptions. The displacements are expanded into power series of the thickness coordinate. Only the first four and the first three terms are used for the displacements parallel and normal to the middle surface respectively. The normal extruding and transverse shear stresses are assumed to bt, cubic polynomials and to satLyfy the boundary stress conditions on the outer and inner surfaces of the shell. The governing equations and boundary conditions are derived by means of variational principle. As an example, a thick-walled cylindrical.shell is disscussed with the theory proposed. Furthermore, a photoelastic experiment has been carried out, and the results are in fair agreement with the computations.     

Dynamic compactness of coupled thermoelastic waves in continuously nonuniform anisotropic media

In the present work, the dynamic problem of coupled thermoelasticity with the most general type of nonuniformity and anisotropy is analyzed. The hyperbolic nature of the system of equations of coupled thermoelasticity is demonstrated, effects of extinction of separate waves by superposition of elastic and thermoelastic wave fronts are investigated, and the interrelationship of different orders of discontinuity of stresses, displacements, and temperature is determined. The case of the uncoupled problem of thermoelasticity is especially analyzed. Sufficient conditions are obtained for the dynamic density for wave processes in thermoelasticity, previously investigated for boundary value problems of hyperbolic systems of second order differential equations [1], andelastic stress waves [2] are obtained. The generally accepted system of tensor notation for the theory of thermoelasticity is used [3].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 154–163, May–June, 1981.     

3D dynamic stress intensity factors around two parallel square cracks in an infinite elastic medium under impact load

Summary  Transient stresses around two parallel cracks in an infinite elastic medium are investigated in the present paper. The shape of the cracks is assumed to be square. Incoming shock stress waves impinge upon the two cracks normal to tzheir surfaces. The mixed boundary value equations with respect to stresses and displacements are reduced to two sets of dual integral equations in the Laplace transform domain using the Fourier transform technique. These equations are solved by expanding the differences in the crack surface displacements in a double series of a function that is equal to zero outside the cracks. Unknown coefficients in the series are calculated using the Schmidt method. Stress intensity factors defined in the Laplace transform domain are inverted numerically to the physical space. Numerical calculations are carried out for transient dynamic stress intensity factors under the assumption that the shape of the upper crack is identical to that of the lower crack. Received 2 February 2000; accepted for publication 10 May 2000     

New Statement of the elasticity problem for a layer

The problem on the equilibrium of an inhomogeneous anisotropic elastic layer is considered. The classical statement of the problem in displacements consists of three partial differential equations with variable coefficients for the three displacements and of three boundary conditions posed at each point of the boundary surface. Sometimes, instead of the statement in displacements, it is convenient to use the classical statement of the problem in stresses [1] or the new statement of the problem in stresses proposed by B. E. Pobedrya [2]. In the case of the problem in stresses, it is necessary to find six components of the stress tensor, which are functions of three coordinates. The choice of the statement of the problem depends on the researcher and, of course, on the specific problem. The fact that there are several statements of the problem makes for a wider choice of the method for solving the problem. In the present paper, for a layer with plane boundary surfaces, we propose a new statement of the problem, which, in contrast to the other two statements indicated above, can be called a mixed statement. The problem for a layer in the new statement consists of a system of three partial differential equations for the three components of the displacement vector of the midplane points. The system is coupled with three integro-differential equations for the three longitudinal components of the stress tensor. Thus, in the new statement, just as in the other statements in stresses, one should find six functions. In the new statement, three of these functions (the displacements of the midplane points) are functions of two coordinates, and the other three functions (the longitudinal components of the stress tensor) are functions of three coordinates. It is shown that all equations in the new statement are the Euler equations for the Reissner functional with additional constraints. After the problem is solved in the new statement, three components of the displacement vector and three transverse components of the stress tensor are determined at each point of the layer. The new statement of the problem can be used to construct various engineering theories of plates made of composite materials.     

Variational formulation of the static Levinson beam theory

In this communication, we provide a consistent variational formulation for the static Levinson beam theory. First, the beam equations according to the vectorial formulation by Levinson are reviewed briefly. By applying the Clapeyron's theorem, it is found that the stresses on the lateral end surfaces of the beam are an integral part of the theory. The variational formulation is carried out by employing the principle of virtual displacements. As a novel contribution, the formulation includes the external virtual work done by the stresses on the end surfaces of the beam. This external virtual work contributes to the boundary conditions in such a way that artificial end effects do not appear in the theory. The obtained beam equations are the same as the vectorially derived Levinson equations. Finally, the exact Levinson beam finite element is developed.     

Linear viscoelastic analysis of a semi-infinite porous medium

This paper considers the problem of a semi-infinite, isotropic, linear viscoelastic half-plane containing multiple, non-overlapping circular holes. The sizes and the locations of the holes are arbitrary. Constant or time dependent far-field stress acts parallel to the boundary of the half-plane and the boundaries of the holes are subjected to uniform pressure. Three types of loading conditions are assumed at the boundary of the half-plane: a point force, a force uniformly distributed over a segment, a force uniformly distributed over the whole boundary of the half-plane. The solution of the problem is based on the use of the correspondence principle. The direct boundary integral method is applied to obtain the governing equation in the Laplace domain. The unknown transformed displacements on the boundaries of the holes are approximated by a truncated complex Fourier series. A system of linear equations is obtained by using a Taylor series expansion. The viscoelastic stresses and displacements at any point of the half-plane are found by using the viscoelastic analogs of Kolosov–Muskhelishvili’s potentials. The solution in the time domain is obtained by the application of the inverse Laplace transform. All the operations of space integration, the Laplace transform and its inversion are performed analytically. The method described in the paper allows one to adopt a variety of viscoelastic models. For the sake of illustration only one model in which the material responds as the standard solid in shear and elastically in bulk is considered. The accuracy and efficiency of the method are demonstrated by the comparison of selected results with the solutions obtained by using finite element software ANSYS.     

基于精细积分技术的非线性动力学方程的同伦摄动法

将精细积分技术（PIM）和同伦摄动方法（HPM）相结合,给出了一种求解非线性动力学方程的新的渐近数值方法。采用精细积分法求解非线性问题时,需要将非线性项对时间参数按Taylor级数展开,在展开项少时,计算精度对时间步长敏感;随着展开项的增加,计算格式会变得越来越复杂。采用同伦摄动法,则具有相对筒单的计算格式,但计算精度较差,应用范围也限于低维非线性微分方程。将这两种方法相结合得到的新的渐近数值方法则同时具备了两者的优点,既使同伦摄动方法的应用范围推广到高维非线性动力学方程的求解,又使精细积分方法在求解非线性问题时具有较简单的计算格式。数值算例表明,该方法具有较高的数值精度和计算效率。     

Solving the stress problem for solid cylinders with different end conditions

The paper proposes an approach to solving a spatial stress problem for solid circular cylinders under axisymmetric surface loading. Two types of boundary conditions at the ends are examined: simply supported or clamped. The circumferential variable is separated using Fourier series for the former type of boundary conditions, and spline-approximation in longitudinal coordinate is used for the latter type of boundary conditions. The resulting one-dimensional problems are solved by the stable discrete-orthogonalization method, evaluating indeterminate forms on the cylinder axis in the governing equations. Radial displacements and circumferential and longitudinal stresses are plotted __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 6, pp. 24–31, June 2006.     

Postbuckling behavior of rectangular moderately thick plates and sandwich plates

ＰＯＳＴＢＵＣＫＬＩＮＧＢＥＨＡＶｉＯＲＯＦＲＥＣＴＡＮＧＵＬＡＲＭＯＤＥＲＡＴＥＬＹＴＨＩＣＫＰＬＡＴＥＳＡＮＤＳＡＮＤＷＩＣＨＰＬＡＴＥＳＣｈｅｎｇＺｈｅｎ－ｑｉａｎｇ（成振强）ＷａｎｇＸｉｕ－ｘｉ（王秀喜）ＨｕａｎｇＭａｏ－ｇｕａｎｇ（黄茂光）...     

Unsymmetrical nonlinear bending problem of circular thin plate with variable thickness

Introduction Manystructuralelements(pole,plate,shell)withunevenandvariablethicknessarewidely usedinallkindsofengineeringfields.Engineerscansavematerialswhentheydesignbecause theseelementshavebetteroptimizedshapeofstructuralfeature,butthisaddsdifficultytotheanalysisoftheirmechanicalcapability.Manypreviouspapers[1-4]havesolvedtheproblemof symmetricalaxis,butnobodyhassolvedtheunsymmetricalnonlineardeformationproblemof circularthinplatewithvariablethicknessandunsymmetricalaxisuptonow,afewworkonly …     

Non-linear behavior of delaminated unidirectional sandwich panels with partial contact and a transversely flexible core

The paper presents the results of an investigation of the non-linear behavior of delaminated sandwich panels with a compressible core. The delaminated zone, at one of the face-core interfaces, consists of through-the-width crack, which is free of shear stresses but is capable of accommodating partial contact with compressive stresses only within the debonded zone. The governing non-linear equations along with the appropriate boundary conditions and the continuity conditions are derived through variational principles. The governing equations include moderate deformations type of kinematic relations, and include the high-order effects due to the transverse flexibility of the core. The governing equations along with the stress and displacements fields for the core and the appropriate continuity conditions are presented. The effects of the non-linear response and the partial contact are described through some numerical cases of three points bending typical sandwich panels with inner delaminations in the vicinity of a concentrated load, in the vicinity of a stiffened core and, finally, far from the load. Numerical results in the form of displacements, bending moments, shear stresses in the core and vertical interfacial normal stresses at the upper and lower face-core interfaces along the panel length and at the delamination crack tips are presented. Buckling curves of load versus various extreme structural parameters are included. The analyses show that a full contact type of delamination transforms into a partial contact area with buckling of the compressed face sheet, as the load is increased and it is associated with extreme large displacements and stresses.     

Scattering of harmonic anti-plane shear waves by an interface crack in magneto-electro-elastic composites

IntroductionCompositematerialconsistingofapiezoelectricphaseandapiezomagneticphasehasdrawnsignificantinterestinrecentyears,duetotherapiddevelopmentinadaptivematerialsystems .Itshowsaremarkablylargemagnetoelectriccoefficient,thecouplingcoefficientbetweenst…     

An inverse problem for a functionally graded elliptical plate with large deflection and slightly disturbed boundary

This paper deals with the inverse problem of a functionally graded material (FGM) elliptical plate with large deflection and disturbed boundary under uniform load. The properties of functionally graded material are assumed to vary continuously through the thickness of the plate, and obey a simple power law expression based on the volume fraction of the constituents. Based on the classical nonlinear von Karman plate theory, the governing equations of a thin plate with large deflection were derived. In order to solve this non-classical problem, a perturbation technique was employed on displacement terms in conjunction with Taylor series expansion of the disturbed boundary conditions. The displacements of in-plane and transverse are obtained in a non-dimensional series expansion form with respect to center deflection of the plate. The approximate solutions of displacements are solved for the first three terms, and the corresponding internal stresses can also be obtained.     

Linearized equations of nonlinear elastic deformation of thin plates

A linearized system of equations governing elastic deformation of a thin plate with arbitrary boundary conditions at its faces in an arbitrary curvilinear coordinate system is proposed. This system of equations is the first approximation of a oneparameter sequence of equations of twodimensional problems obtained from the initial threedimensional problem by approximating unknown functions by truncated series in Legendre polynomials. The stability problem of an infinite plate compressed uniaxially is solved. The results obtained are compared with the existing solutions.