УТОЧНЁННАЯ ЛИНЕЙНАЯ ТЕОРИЯ УПРУГОСТИ С УЧЁТОМ КВАДРАТИЧНЫХ СЛАГАЕМЫХ

Abstract

A variant of the linear theory is proposed, based on taking into account in Taylor expansions for stresses and in power series and the expansion of the radical into a binomial series for deformations of not only linear, but also quadratic terms. In this case, static equilibrium equations in stresses are written in the form of second-order partial differential equations. The law of pairing of tangential stresses is observed. The relationship between deformations and displacements is described by relations similar to Cauchy relations, but taking into account the second derivatives of displacements along spatial coordinates. The equations of continuity of deformations for this version of the theory of elasticity are written in the form of two groups of relations. The first group of relations establishes the continuity equations for the linear terms of the deformation components; The second group of relations establishes the continuity equations for the quadratic terms of the deformation components. The physical equations for this variant of the linear theory of elasticity are written in the same way as for the classical linear theory of elasticity. Equilibrium equations in stresses are represented by second-order partial differential equations. The equilibrium equations in displacements are written in the form of fourth-order partial differential equations. The equations of equilibrium, as well as the equations of continuity of deformations, along with other parameters – the physical constants of the medium – contain small parameters dx, dy, dz, the magnitude of which, as shown by numerical modeling, has little effect on the nature of the stress-strain state. To determine them, it is necessary, apparently, to use experimental data. The construction of resolving equations in both stresses and displacements for special cases of the stress-strain state of an elastic continuous medium is considered: uniaxial dressed-up state; uniaxial deformed state; plane strain; generalized plane stress state. The problem of the uniaxial deformed state of a continuous medium (solution in stresses) is numerically solved. The proposed version of the linear theory of elasticity extends the classical linear theory of elasticity and, with appropriate experimental justification, can lead to new qualitative effects in the calculation of elastic deformable bodies.