Abstract
This paper presents a new nonlinear stiffness matrix of a finite element without making any simplifications. This matrix inserts the quadratic and cubic dependences of the unknown increments of generalized displacements of nodes into the initially linearized system of equations. For a bar element, the exact form of the matrix is defined and for illustrative one-dimensional problems of pure tension the full system of non-linear equations is given, which is solved by the method described in [M. J. D. Powel, Hybrid Method for Non-linear Equations. McGraw-Hill (1970)]. The calculation is evaluated as far as its convergence is concerned, and the results are compared with those of a classical approach to geometrically non-linear problems represented by the method of finite elements. Thanks to the development of numerical methods of solving the systems of non-linear equations, the new non-linear stiffness matrix can contribute significantly to a more efficient solution of non-linear problems in the mechanics of elastic bodies.
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