On solving the plastoelastic problen in space of the commutative matrixes

Abstract

A method for calculating mechanical parameters of multilayer structural elements (MSE) and their layers in elastic and plastoelastic zones are presented and grounded in the case of axial stret­ching. The method of diagonal matrices is proposed to define the parameters of MSE's and their layers. In the elastic zone, MCE's are completely defined by two matrices: that of the modulus of elasticity and layer cross-section areas. In the plastoelastic zone, by the matrix of the layer cross­section areas and a diagonal matrix-function that defines σ - ε diagrams of the layers. In the case of stretching, the above mentioned matrices make up a commutative group with respect to product operation which makes it possible to obtain compact expressions for the required parameters that do not depend on the number of layers and are analogous to scalar ones (a single-layer case). This kind of calculation methods enables us to compute the values of axial stiffness and normal stress as well as the quantity of limiting axial load or the zones areas of elastic and plastoelastic deformation, when the diagrams σ - ε of deformation materials composing it correspond to that diagram that of plastoelastic and elastically-strengthening materials.