The relations of deformation theory in the quadratic approximation and the problems of constructing improved versions of the geometrically non-linear theory of laminated structures

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Two versions of the approximate relations of the deformation theory of continuous media, known as the complete version (see, for example [1]) and the incomplete version (see, for example [2, 3]) of the quadratic approximation of the non-linear theory are analysed. It is shown that the relations of the complete version, which define the elongation deformation, and the relations of the incomplete versions, which define the shear deformations, are incorrect, since, when solving specific problems, they lead to the occurrence of false bifurcation points. For small elongation deformation and medium shear deformation a non-contradictory version of the kinematic relations is constructed in the quadratic approximation, representing a combination of the relations of the complete and incomplete versions. The simplest examples of its application, connected with the reduction of the two-dimensional non-linear problem of the deformation of a strip in the form of a rod to homogenous equations and their subsequent use to detect possible forms of loss of stability for characteristic forms of loading them are considered. Essentially new results are obtained connected with the investigation of forms of loss of stability of a rod under uniform transverse compression and pure shear. In the first case the behaviour of the load turns out to be important: if it remains normal to the deformation axis of the rod, bifurcation is only possible with respect to the shear form, if it retains its direction, and then, in addition to bifurcation with respect to the shear form, a bending form of loss of stability is possible, which is identical in form with the Euler form, for which there are no shears. In the second case, i.e. when there is a load which causes pure shear of the rod, to investigate its bifurcation values, it is necessary to describe the shear deformation by non-linear kinematic relations in the complete quadratic version, whereas when there are no subcritical shear stresses one can use the simplified relations. An example of the investigation of the forms of loss of stability of a circular ring when acted upon by a uniform external pressure having zero variability in the circumferential direction is also considered.