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The elastoplastic problem of the half-space with a hole subjected to axially symmetric loading

The elastoplastic problem of the half-space with a hole subjected to axially symmetric loading considered in this paper is based on the elastoplastic deformation process theory. Solution of t his problem is carried out by using the modified elastic solution method and the finite element method. Some results of numerical calculation are presented here to give the picture of plastic domains enlarging in the body and t he obtained displacements on the free boundary of the half-space. | Vietnam J ournal of Mechanics, VAST, Vol. 27, No. 4 (2005), pp. 245 - 255 THE ELASTOPLASTIC PROBLEM OF THE HALF-SPACE WITH A HOLE SUBJECTED TO AXIALLY SYMMETRIC LOADING Vu Do LONG Vietnam National University, Hanoi Abstract.The elastoplastic problem of the half-space with a hole subjected to axially symmetric load ing considered in t his paper is based on the elastoplastic deformation process theory. Solution of t his problem is carried out by using the modified elastic solution method and the finite element method. Some results of numerical calculation are presented here to give the picture of plastic doma ins enlarging in the body a nd t he obtained displacements on the free boundary of the ha lf-space. 1. GOVERNING EQUATIONS Let 's consider an elastoplastic half-space with a hole subjected to axially symmetric loading, the strain state of which is determined by Cauchy relation u EO = - ; r 8u /rz = 8v az + Br , /rO = /zO = O; where u = ur(r, z), v = Uz(r , z) and uo = 0 are displacement components. The stress-strain relation for elastic state can be expressed as follows: >. >. ~1 [::1 >. + 2µ = [D]{c}, 0 µ (1.1 ) /rz where [DJ -the matrix of elastic constants >. - Ev - (1 + v)( l - 2v) ' E µ = G = 2(1 + v) Stress and strain intensity are determined in the form Eu = j~eijeij = v; V( Er - Ez) 2 + (Ez - Eo) 2 + (Eo - Er ) 2 + (3/2)1';~ , (1.3) 246 Vu Do Long and the arc-length of the strain trajectory t s j J~e.ije.ij = (1.4) dt . 0 For a plastic problem we use the elastoplastic deformation process theory [1] (1.5) dO"ijkl = DijkldEkl' or in matrix form {dO"} = dO"r dO"z ] dO"B rD1 D5 Dg D13 = rdTrz D2 D5 D10 D14 D3 D1 Du D15 4 D Ds ] D12 D 16 rdc:z dc:r] dc:B = [D]d{dc} , (1.6) drrz where [D]d is matrix of stress-strain relation of elastic-plastic behaviour, Di can be written as following D1=.\+2G -H1 ; D2 = .\-H2; D5 = D2; D5=.\ + 2G-H5 ; Dg = D3; D10 = D1; D13 = D4; D3 = .\-H3; D1=.\-H1; Du