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**Exact results and approximate models for porous viscoplastic solids.**
*(English)*
Zbl 0821.73022

This work continues earlier studies of the void growth at the ductile rupture of metals in the cases of a plastic behaviour and a viscoplastic one. At first, the constitutive law is established in special cases of Newtonian (linear) viscous and ideal-plastic materials. The problem is reduced to investigation of two macroscopic potentials. The Gurson yield criterion is used. It is noted that none of the approximate potentials which have been used earlier does simultaneously satisfy three following natural requirements: (i) to reproduce the exact solution for a hollow cylinder or sphere loaded in hydrostatic tension or compression; (ii) to be a quadratic form of the overall stress tensor in the extreme case of a Newtonian behaviour, and (iii) to yield the currently accepted Gurson criterion for an ideal-plastic matrix.

The aims of the present paper are: first, to complete available exact results for porous viscoplastic materials by providing a new inequality and the analytical solution of the typical problem for a hollow cylinder loaded axisymmetrically under generalized plane strain conditions; second, to propose new approximate potentials for both cylindrical and spherical voids based on the previous results and satisfying, in particular, the three above conditions.

Subsequently, an inequality and its generalizations including the bounds obtained previously by Ponte Castañeda, Willis and Talbot are established. Then, applications to the special case of ideal-plastic materials and to hollow cylinder problem are considered. Further, the approximate potentials are derived from a heuristic approach based on conditions (i), (ii), (iii) and on the previous inequalities. The accuracy of the proposal for cylindrical voids is assessed by comparing it with the exact potential. Finally, a similar comparison is presented for spherical voids using an upper bound estimate of the strain rate potential based on the Rice-Tracey-Gurson velocity fields.

The aims of the present paper are: first, to complete available exact results for porous viscoplastic materials by providing a new inequality and the analytical solution of the typical problem for a hollow cylinder loaded axisymmetrically under generalized plane strain conditions; second, to propose new approximate potentials for both cylindrical and spherical voids based on the previous results and satisfying, in particular, the three above conditions.

Subsequently, an inequality and its generalizations including the bounds obtained previously by Ponte Castañeda, Willis and Talbot are established. Then, applications to the special case of ideal-plastic materials and to hollow cylinder problem are considered. Further, the approximate potentials are derived from a heuristic approach based on conditions (i), (ii), (iii) and on the previous inequalities. The accuracy of the proposal for cylindrical voids is assessed by comparing it with the exact potential. Finally, a similar comparison is presented for spherical voids using an upper bound estimate of the strain rate potential based on the Rice-Tracey-Gurson velocity fields.

Reviewer: I.A.Parinov (Rostov-na-Donu)

### MSC:

74C10 | Small-strain, rate-dependent theories of plasticity (including theories of viscoplasticity) |

74E05 | Inhomogeneity in solid mechanics |