Physically nonlinear shell deformation based on three-dimensional finite elements

The aim of the study is to determine the stress-strain state of the shell under step loading beyond the elastic limit. At the loading step, relations between strain increments and stress increments are obtained without accepting the hypothesis of a straight normal. For the numerical implementation of the algorithm in the calculation of the shell without using a straight normal, a prismatic finite element has been developed. We consider the physically nonlinear deformation of the shell, an arbitrary point of which is represented by a radius vector defined by the curvilinear coordinates of the reference surface and the distance from the reference surface to the point under consideration. By differentiating the radius-vector function, the basis vectors of the point under consideration are determined, the scalar products of which are the components of its metric tensor. The increments of deformations at the loading step are defined as the differences of the corresponding components of the metric tensors. The defining equations at the loading step are obtained in two versions. In the first version, they are determined by differentiating the stress functions of the deformation theory of plasticity, which are obtained on the basis of dividing deformations into elastic and plastic parts using the hypothesis of material incompressibility during plastic deformation. In the second version, they are obtained without using the operation of dividing the strain increments into elastic and plastic parts on the basis of the proposed hypothesis that the components of the deviator of the stress increments are proportional to the components of the deviator of the increments of deformations. For the numerical implementation, a three-dimensional prismatic finite element with a triangular base was used for nodal unknowns in the form of displacements and their first derivatives with respect to curved coordinates. The correctness of the proposed variant of obtaining physically nonlinear defining equations at the loading step of the deformable shell is confirmed by a numerical example. On the example of calculating a cylindrical shell under the action of internal pressure, clamped at one end and free at the other, the values of normal stresses in the embedment turned out to be approximately 14% lower in the case of using the proposed variant of obtaining the constitutive equations at the loading step. The developed algorithm for determining the stress-strain state in the physically nonlinear deformation of elements of technospheric objects can be used in the practice of engineering calculations.