engleski

Advanced non-linear shell modelling of the Hamiltonian problems

Some aspects of continuum and numerical formulation for geometrically non-linear dynamics of shells and its temporal discretisation are considered. The main features of the approach: (i) the finite rotation shell theory, (ii) the energy and momentum conserving time-integration algorithm, (iii) the finite element implementation, and (iv) the consistent linearisation of the weak form of the equation of motio - are very briefly presented and discussed. The shell theory is based on derived approach. It is developed from the three-dimensional continuum theory by employing standard assumptions on the distribution of the displacement field in a shell body. A single director model for thin shells is obtained by approximation of terms describing the shell geometry. Using this description, a non-linear shell model, which is equivalent to the two-dimensional Cosserat continuum, is derived. The kinematics fields are exactly described. In particular, finite rotations of the directorfield are described by a rotation vector formulation, which is free of singularities. Temporal discetisation of the weak form of the equations of motion is performed using an implicit numerical time-integration scheme. In fact, an algorithm, which may be regarded as a special form of the mid-point rule, is employed. It preserves the fundamental constants of the autonomous motion: the total linear and angular momentum as well as, for Hamiltonian case, the total energy. A finite element formulation is based on the four noded isoparametric elements. To avoid locking, a so-called assumed natural approach is adopted. Particular atention is devoted to the consistent linearisation of the time discretised weak form of the equation of motion, in order to achieve a quadratic rate of asymptotic convergence typical for the Newton-Raphson based solution procedure. A linearisation is performed after the spatial element discretisation. The performance of the model is here illustrated with a numerical example.

non-linear shell modelling; Hamiltonian problems; finite rotation shell theory; consistent linearisation

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