Engineering Transactions,

**4**, 1, pp. 25-41, 1956### Sprężysto-Plastyczna Grubościenna Powłoka Kulista z Materiału Niejednorodnego, Poddana Działaniu Ciśnienia Wewnętrznego Zewnętrznego

A thick-walled spherical shell whose internal and external radii are a and b respectively, subjected to internal and external pressure p and q, is considered with the assumption that the difference of pressures Π = p - q is an increasing function of time t.

Introducing the functions €o (mean elongation) and %, defined by Eqs. (2.2) and (2.3) (where u denotes the radial displacement and * the radial coordinate of the considered point in a spherical system of coordinates) the components of the strain deviator, (2.5), and those of the stress deviator, (3.6), are expressed in terms of these functions, the moduli G (r) and v (r) being functions of the radius r. If we assume that the non-ho- mogeneity is of a type for which we can take y = const, the unknown functions εo and S, constitute the solution of the system of equations (4.8). The stresses in the elastic (external) region are determined with the simplifying assumption of ε = 0 and v =½. Introducing the boundary conditions for that region, (5.9), formulae for stresses in the elastic region are obtained in the form (5.15). Taking the yield condition in the form (6.1), which means that the material of the shell is also plastically non-homogeneous, the stress components in the plastic region are found in the form (6.14). The unknown value of the pressure D between both regions in Eqs. (5.15) is found from the condition that J, should not «jump» when passing from the elastic to the plastic zone. The corresponding expression is (7.2). The last unknown value, that of the radius n of the sphere constituting the boundary between these two zones, is obtained from the condition that the mean normal stress should not «jump» when crossing the boundary between the zones. This condition leads to a transcendental equation, (7.4), which can be solved for n. From Eq. (7.4) the following are determined: the value of the pressure difference, (8.1), corresponding to the first plastic deformations at the internal surface of the shell; the value (8.3), corresponding to the plastification of the whole shell. The condition (9.3) is established which should be satisfied by the functions G(r) and Q(r) in order that the plastic state should occur simultaneously in the whole shell. An analogous condition in the case of elastic and plastic non-homogeneity of the conjugate type takes the form (10.1).

Introducing the functions €o (mean elongation) and %, defined by Eqs. (2.2) and (2.3) (where u denotes the radial displacement and * the radial coordinate of the considered point in a spherical system of coordinates) the components of the strain deviator, (2.5), and those of the stress deviator, (3.6), are expressed in terms of these functions, the moduli G (r) and v (r) being functions of the radius r. If we assume that the non-ho- mogeneity is of a type for which we can take y = const, the unknown functions εo and S, constitute the solution of the system of equations (4.8). The stresses in the elastic (external) region are determined with the simplifying assumption of ε = 0 and v =½. Introducing the boundary conditions for that region, (5.9), formulae for stresses in the elastic region are obtained in the form (5.15). Taking the yield condition in the form (6.1), which means that the material of the shell is also plastically non-homogeneous, the stress components in the plastic region are found in the form (6.14). The unknown value of the pressure D between both regions in Eqs. (5.15) is found from the condition that J, should not «jump» when passing from the elastic to the plastic zone. The corresponding expression is (7.2). The last unknown value, that of the radius n of the sphere constituting the boundary between these two zones, is obtained from the condition that the mean normal stress should not «jump» when crossing the boundary between the zones. This condition leads to a transcendental equation, (7.4), which can be solved for n. From Eq. (7.4) the following are determined: the value of the pressure difference, (8.1), corresponding to the first plastic deformations at the internal surface of the shell; the value (8.3), corresponding to the plastification of the whole shell. The condition (9.3) is established which should be satisfied by the functions G(r) and Q(r) in order that the plastic state should occur simultaneously in the whole shell. An analogous condition in the case of elastic and plastic non-homogeneity of the conjugate type takes the form (10.1).

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Copyright © Polish Academy of Sciences & Institute of Fundamental Technological Research (IPPT PAN).

#### References

J. Nowiński i W. Olszak, O podstawach teorii ciał sprężystych fizykalnie nieliniowych, Arch. Mech. Stos. 1 (1954).

W. Olszak, O podstawach teorii ciał elasto-plastycznych niejednorodnych (I), Arch. Mech. Stos. 3 (1954).

W Olszak i W. Urbanowski, Sprężysto-plastyczny grubościenny walec niejednorodny pod działaniem parcia wewnętrznego i sity podłużnej, i Arch. Mech. Stos. 3 (1955).

W. Prager I P.G. Hodge, Theory of Perfectly Plastic Solids,

New York-Londyn 1951.