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JOURNALOFTHEORETICAL ANDAPPLIEDMECHANICS 41 ,2,pp.241-269,Warsaw2003 FUNDAMENTAL SOLUTIONS RELATED TO THERMAL STRESS INTENSITY FACTORS OF MODES I AND II – THE AXIALLY SYMMETRIC PROBLEM BogdanRogowski MechanicsofMaterialsDivision,TechnicalUniversityofŁódź,Poland e-mail:[email protected] This elaboration considers the crack problems for infinite thermoelastic solids subjected to steady temperature or heat flux. The crack faces are assumed to be insulated. Green’s functions are obtained for the thermal stress intensity factors of modes I and II. The Green’s functions are defi- ned as a solution to the problem of a thermoelastic transversely isotropic solid with a penny-shaped or an external crack under general axisymme- tric thermal loadings acting along a circumference on the plane parallel to the crack plane. Keywords: thermoelasticity, anisotropy, crack problems, Green’s func- tions, stress intensity factors of mode I and II 1. Introduction The penny-shaped crack in a temperature field was treated by Olesiak and Sneddon (1960); the problem was symmetrical with respect to the crack plane. The features of antisymmetry were presented by Florence and Goodier (1963) in the linear thermoelastic problem of uniform heat flow disturbed by a penny-shaped insulated crack. In this paper, we consider the steady thermal stress in a cracked solid. The problems of the crack treated here are solved by using two types of axisymme- tric ring thermal loadings as fundamental solutions: a uniform heat flux and temperature. The research is aimed at the assessing of the effect of dissimilar thermal conditions on the stress intensity factors. The stress intensity factors of modes I and II are derived in this study in terms of elementary functions. The results presented for general cases are new, but some of those related
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242 B.Rogowski to special cases of isotropic or transversely isotropic solids with crack surface thermal loadings have been already known (cf. Olesiak and Sneddon, 1960; Florence and Goodier, 1963; Rogowski, 1984). 2. Basic equations The basic equations of axisymmetric thermal stress problems for homo- geneous transversely isotropic bodies are the equilibrium equations (in the absence of body forces) σ rr,r + σ rz,z + 1 r ( σ rr σ θθ ) = 0 σ rz,r + σ zz.z + 1 r σ rz = 0 (2.1) the strain-displacement relations e rr = u r,r e θθ = u r r e zz = u z,z 2 e rz = u r,z + u z,r (2.2) the constitutive equations σ rr = c 11 e rr + c 12 e θθ + c 13 e zz β 1 T σ θθ = c 12 e rr + c 11 e θθ + c 13 e zz β 1 T σ zz = c 13 e rr + c 13 e θθ + c 33 e zz β 3 T σ rz = 2 c 44 e rz (2.3) and the heat conduction equation (steady state without heat generation) T ,rr + r 1 T ,r + s 2 0 T ,zz = 0 (2.4) where partial differentiation is indicated by the comma followed by the va- riables, c ij are the elastic constants of a transversely isotropic material, β 1 = ( c 11 + c 12 ) α r + c 13 α z , β 3 = 2 c 13 α r + c 33 α z are the thermal stress coef- ficients, α r and α z are the coefficients of the linear thermal expansion in the radial and axial direction, s 2 0 = λ r z , λ r and
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  • Winter '17
  • Emoticon, stress intensity factor, fundamental solutions, B. Rogowski, external crack

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