Residual-Stress Determination of Concentric Layers of Cylindrically Orthotropic Materials by G.Z. Voyiadjis and C.S. Hartley ABSTRACT--Equations have been obtained for determining residual stresses in the wall of a hollow, axially symmetric body consisting of concentric layers of elastically dissimilar materials, all having cylindrical elastic orthotropy. These equations permit residual normal stresses in the radial, circumferential, and axial directions and residual shear stresses on planes normal to the axis of the body to be calculated from measurements of the strains developed on the inner or outer cylindrical surface of the body as thin layers of stressed material are serially removed from the outer or inner surfaces, respectively. The equations are applied to a parametric study of stresses in an elastically isotropic, two-component body to determine the nature of the differences in stresses between the composite body and a homogeneous body as a function of the difference in elastic constants. List of Symbols am = exterior radius of mth layer am-1 = interior radius of mth layer Cm = reduced radial dimension E t'~) = modulus of elasticity of mth layer f = ratio of moduli of elasticity (in the text) km = measure of the orthotropy in the mth layer 34, = resultant moment produced by the removal of stressed material n = total number of layers qm = normal pressure acting across internal interface q, = external pressure qo = internal pressure YM = modulus of elasticity (in the figures) Or, e~, e,, % ~ = components of strain ~ml = Poisson's ratio of ruth layer at, a~, az, rzr = components of stress ar~,~), e~m) = stresses and strains of mth layer o.h, h h (r~, ~r~ = stresses for homogeneous material c c c crr, cry, az = stresses for composite ring Introduction Residual stresses in the wall of hollow, axially symmetric tubes can be calculated from measurements on a free surface of the strains resulting from the removal of a thin layer of stressed material from the opposite surface. Early solutions assumed material homogeneity, elastic isotropy, and coincidence of the principal axes of residual stress with the principal axes of the tube to obtain relationships G.Z. Voyiadjis is Associate Professor, Department of Civil Engineering, and C.S. Hartley (SEM Member) is Associate Dean and Professor, College of Engineering, Louisiana State University, Baton Rouge, LA 70803. Professor Hartley is on assignment as Director, Metallurgy Program, Division of Materials Research, National Science Foundation, Washing- ton, D.C. 20550. Original manuscript submitted: September 24, 1986. Final manuscript received: February 6, 1987. between the measured strains and the stresses in the material removed. 1-3 Recent extensions of these solutions have included elastic orthotropy ''~ and the determination of a residual shear stress in the wall of the tube.S Experi- mental studies have shown that tubing formed by non- axisymmetric forming operations may not possess an axisymmetric distribution of residual stress. 6 Consequently the analyses developed should be applied to only products which have been fabricated under conditions leading to axisymmetric distributions of residual stress. Composite tubular products consisting of concentric tubes of different materials possess a combination of features of each constituent. Such products exhibit a residual-stress distribution determined by the relative plastic deformation experienced by the constituents, the volume fraction of each, and the geometry of the forming process. In addition, if the materials are elastically aniso- tropic, fabrication can cause elastic anisotropy in the product by the development of a deformation texture. Residual stresses in such composite members can be determined by an extension of the procedure used for homogeneous, elastically anisotropic materials. In this study expressions are obtained relating residual normal stresses along the axial, radial and circumferential direc- tions and shear stresses acting on planes normal to the axis of symmetry to strain measurements made on a free surface as a thin layer of stressed material is removed from the opposite surface. Solutions are obtained for an n-layered tube. A parametric study is presented for the special case of a two-component tube of elastically iso- tropic materials examining the effect of the relative elastic properties on the stresses calculated from the equations developed in the analysis. Development of Analysis Stress Distribution in an N-Layered Composite Ring GENERAL CONSIDERATIONS The following treatment develops equations for the residual-stress distribution in an internally stressed com- posite tube having zero resultant forces and moments on the outer surfaces. Constituents of the tube are assumed to be homogeneous within each layer of the composite and elastically orthotropic with different elastic constants but the same elastic symmetry. Axial strains resulting from the removal of stressed material are assumed to be independent of radial position in the remaining material. Stresses are expressed in terms of strains developed on the inner or outer free surface when stressed material is removed from the outer or inner surface, respectively. Material is removed in thin serial sections so that the stress distribution in each section can be considered 290 September 1987