The semi-infinite elastic strip

Analytical solutions for determining residual stresses in two-dimensional domains using the contour method

“…Both direct and Galerkin methods result in a general eigenvalue problem with square matrix size of N × N and N unknown complex constants. The more elegant techniques are to use a set of biorthogonality approaches exploited by Johnson & Little [28] or Spence [29] and developed by Gregory [30]. By means of the biorthogonality relations, the complex expansion coefficients can be determined analytically.…”

Laminated beams of isotropic or orthotropic materials subjected to temperature change

“…Since Cry . = 0x 2 and T xyi = 0x~y ' the satisfaction of the boundary The problem of exact satisfaction of end conditions of an elastic strip has been treated by authors [11,12] in the literature. It is beyond the scope of this article, however, to modify the solution by satisfying more precisely the free end conditions of the beam.…”

The anisotropic elastic semi-infinite strip

“…Explicit solutions have not been found for the end conditions (5.2a, b) and (5.2g, h) for any materials, not even for isotropic materials [4], 7. Discussion of the end conditions.…”

“…(1.2) Under the assumption of plane stress or plane strain, if all stress and displacement components approach zero as Xj becomes large, the classical solution for the stress is of an exponential decay form, <r,j = e~"X,fu(x2) (i,j= 1,2), (1.3) where the decay exponent X is determined by the eigenequation sin2A = ±2A. (1.4) Related problems for the isotropic elastic semi-infinite strip have been considered by Papkovich [1], Fadle [2,3] and Johnson and Little [4], Extensions to certain anisotropic elastic strips have been carried out by Horgan and his coworkers [5][6][7][8][9], In this paper we consider the associated problem for the semi-infinite strip of general anisotropic materials. For anisotropic materials the in-plane and anti-plane displacements are in general coupled and we have to consider all three displacements.…”

“…where u is the displacement vector whose elements are («,, u2, m3) and the components of the 3x3 matrices Q, R, T are Qik = Ci\k\ ' ^ik = ^ilk2 ' ^ik = ^i2k2' (2)(3)(4)(5) We see that Q and T are symmetric and positive definite. Equations (2.4) can be rewritten as [15] (2.6) dy dx where Im denotes the imaginary part and the overbar stands for the complex conjugate.…”