Quasistatic problems for piecewise-continuously growing solids with integral force conditions on surfaces expanding due to additional material influx

Abstract

The piecewise continuous processes of additive forming of solids are studied. The being formed solids exhibit properties of deformation heredity and aging. The approaches of linear mechanics of growing solids in the framework of the theory of viscoelasticity of the homogeneously aging isotropic media are applied. Nonclassical boundary-value problems for describing the mentioned processes with the integral satisfaction of force conditions on some expanding due to the influx of additional material parts of the formed solid surface are investigated. A proposition about the commutativity of the time-derived integral operator of viscoelasticity with a limit depending on the solid point with the integration over an arbitrary, expanding due to the growth, surface inside or on the boundary of the growing solid is given. This proposition provides a way to construct the solution of the corresponding growing solids mechanics problem on the basis of Saint-Venant principle. The solution will retrace the evolution of the stress-strain state of the solid under consideration during and after the process of its additive formation. An example of applying the announced technic to modelling the processes of additive forming solids of conical shape under simultaneous action of end loads that are statically equivalent to an axial time-varying force is demonstrated.