The stress-strain state of a plate with a multilayer inclusion

The existence of two correction coefficients traditionally introduced to account for the effect of the distribution of tangential stresses over the thickness of a plate is discussed. The virtual-work principle is used to generalize the expressions for the coefficients to the case of arbitrary loading. These expressions and hypotheses for displacements help to derive equations for orthotropic rectangular plates subject to tangential surface loads. These equations account for the effect of the distribution of tangential stresses over the thickness of the plate. Numerical examples are given. The results obtained are compared with those produced by other theories Keywords: orthotropic rectangular plate, correction coefficient, distribution of tangential stresses over plate thickness, virtual-work principle, tangential surface loadsIntroduction. The shear theories of plates based on hypotheses for displacements use correction coefficients K x and K y to account for the effect of the distribution of the tangential stresses t xz and t yz over the thickness of the plate [6, 14, etc.]. Contrastingly, the theories based on hypotheses for t xz and t yz [2,8] are constructed making direct use of the distribution of these stresses, which makes the correction coefficients unnecessary.This paper discusses the existence of K x and K y . The virtual-work principle is used to generalize the expressions for K x and K y to the case of arbitrary loading. These expressions and hypotheses for displacements are used to derive equations for orthotropic rectangular plates subject to tangential surface loading. These equations account for the effect of the distribution of t xz and t yz . Numerical examples are given. The results are compared with those produced by other theories [2,4,8,[15][16][17][18].

“…Numerical examples are given. The results are compared with those produced by other theories [2,4,8,[15][16][17][18].…”

mentioning

confidence: 98%

Refined theory of orthotropic plates subject to tangential surface loads

Kirakosyan

2008

“…Other approaches to solving problems for bodies with stress concentrators such as thin inclusions and cavities by exact and approximate methods are proposed in [13][14][15][16].…”

mentioning

confidence: 99%

Convergence of solutions for noncanonical bodies found by the boundary-shape perturbation method

Chernopiskii¹

2009

The exact solution of the problem of the torsion of a solid cylindrical shaft with an ellipsoidal cavity is expanded into a series in powers of an eccentricity parameter. At the points of the cavity where the stress concentration is maximum (minimum), the expansion coefficients for the exact and approximate solutions coincide. The approximate solution is obtained by the first version of the boundary-shape perturbation method, which is used to solve three-dimensional problems of elasticity for bodies of revolution with nearly canonical shape. Successive approximations as an iteration process are shown to converge to the exact solutions Keywords: theory of elasticity, boundary-shape perturbation method, solid cylindrical shaft, ellipsoidal cavity, series expansion of exact solution, approximate solutionIntroduction. The perturbation method is used to solve some problems of continuum mechanics and outlined in [7]. The method of perturbation of elastic properties was used in [11] to solve problems of stress concentration around holes in physically nonlinear plates [4]. In [10], this method was applied to orthotropic materials using the elastic properties of a transversely isotropic material as a zero approximation. The boundary-shape perturbation method was used in [2, 3, 6] to study the stress concentration in elastic plates and shells with curvilinear holes.Three-dimensional problems of elasticity were solved in [2, 3] using two versions of the boundary-shape perturbation method for noncanonical bodies under static forces: (i) bodies with nearly canonical inclusions and cavities bounded by orthogonal surfaces and (ii) bodies bounded by nonorthogonal surfaces [8,9].The former method [3] shows practical convergence of approximate solutions of stress-concentration problems for elastic bodies with ellipsoidal cavities (a shaft under torsion or a medium under uniform tension-compression), which permit exact solutions [5,12]. Other approaches to solving problems for bodies with stress concentrators such as thin inclusions and cavities by exact and approximate methods are proposed in [13][14][15][16].In the present paper, we will expand the exact analytic solutions of stress-concentration problems for a shaft with an ellipsoidal cavity under torsion in powers of the eccentricity parameter e. We will compare the coefficients of the expressions of the exact [5] and approximate [3] solutions.1. Exact solution for a shaft with an ellipsoidal cavity under torsion is as follows [5]:

“…Similar problems for inclusions of low stiffness were addressed in [2,10]. Note also that static magnetoelastic problems for bodies with inclusions and an elastic orthotropic medium with an ellipsoidal inclusion were solved in [7,9]; the stress state of viscoelastic plates with elliptic inclusions and plates with a multilayer inclusion was studied in [8,11].1. Problem Formulation.…”

mentioning

confidence: 99%

Flexural vibrations of a thin circular elastic inclusion in an unbounded body under the action of a plane harmonic wave

Vakhonina

Попов

2008

The interaction of a plane harmonic longitudinal wave with a thin circular elastic inclusion is considered. The wave front is assumed to be parallel to the inclusion plane. Since the inclusion is thin, the matrix-inclusion interface conditions (perfect bonding) are formulated on the mid-plane of the inclusion. The bending displacements of the inclusion are determined from the bending equation for a thin plate. The problem is solved using discontinuous Lamé solutions for harmonic vibrations. Therefore, the problem can be reduced to the Fredholm equation of the second kind for a function associated with the discontinuity of normal stresses on the inclusion. The equation obtained is solved by the method of mechanical quadratures using Gaussian quadrature formulas. Approximate formulas for the stress intensity factors are derived. Results from a numerical analysis of the dependence of the SIFs on the dimensionless wave number and the stiffness of the inclusion are presented Keywords harmonic wave, elastic inclusion, discontinuous solution, stress intensity factor Introduction. In studying the vibrations of elastic bodies containing a thin inclusion, it is often assumed perfectly rigid. Though such an assumption greatly facilitates the mathematical solution, the effect of the elastic properties of the inclusion on the stress concentration near it cannot be examined. That this effect can be significant was shown in [3] where the vibrations of an unbounded body with strip inclusions were studied. In the present paper, we will use an approach based on discontinuous solutions to study the effect of plane elastic waves on an inclusion in the form of an elastic circular plate. Similar problems for inclusions of low stiffness were addressed in [2,10]. Note also that static magnetoelastic problems for bodies with inclusions and an elastic orthotropic medium with an ellipsoidal inclusion were solved in [7,9]; the stress state of viscoelastic plates with elliptic inclusions and plates with a multilayer inclusion was studied in [8,11].1. Problem Formulation. Consider an unbounded body (matrix) with an elastic inclusion in the form of a thin circular plate of thickness h. The inclusion is represented by a circle of radius a in the plane z = 0 in a cylindrical coordinate system with the origin at the center of the inclusion. The inclusion is acted upon by a plane expansion-compression wave with front parallel to its plane. The potential of and the displacements in this wave are given by