icccbe 2010 Nottingham University Press Proceedings of the International Conference on Computing in Civil and Building Engineering W Tizani (Editor) Computational non-linear structural dynamics and energy-momentum integration schemes C. Sansour Department of Civil Engineering, University of Nottingham Abstract The paper deals with the nonlinear dynamics of geometrically exact shells and rods of the Cosserat type. First, the dynamical theory of shells and spatial rods is developed on the basis of the Cosserat continuum. The continuum is two-dimensional in the case of a shell and is one-dimensional in the case of a rod. A noteworthy feature of the theoretical formulation is the fact that six degrees of freedom are assigned to each point of the shell or rod, three translations and three rotations. Accordingly, so-called drilling degrees of freedom are incorporated in a shell formulation in a completely natural way. An energy/momentum conserving scheme recently developed by the authors is applied to capture the dynamics. Different applications to nonlinear dynamics of structures including chaotic and free large overall motion are presented. Keywords: structural dynamics, time integration, energy conservation, finite element methods 1 Introduction The so-called generalized continuum is defined by assigning to every point of the continuum a displacement vector and, in addition, further fields characterizing the continuum under consideration. Within Cosserat continua it is a rotation tensor which is taken as a further kinematical quantity enhancing the degrees of freedom of the system. Six degrees of freedom related to a displacement vector and a three-parametric rotation tensor, are retained in theories resting on the direct approach and the use of the so called Cosserat strain measures as formulated by Sansour & Bednarczyk [1]. Concerning the dynamics, the design of mechanical integrators which conserves momentum, angular momentum, and energy whenever the external loading and the constitutive behaviour allows for, is of special interest. The so-called of energy-momentum methods enhance the stability properties of the mechanical integrator. On the other side the stability in long term dynamics is a very important feature in order to follow the dynamics of the system for long times especially when specific phenomena like chaotic behaviour is of interest. Simo & Tarnow [5] developed a method which applies for elastic systems when the nonlinearity in the strain-displacement relation is a quadratic one. Accordingly, the method does not apply for systems when a rotation tensor is explicitly included in the formulation. As already mentioned, the later is typical for geometrically exact rod formulations as well as for shell formulations based on stretch- or Cosserat-type tensors. Motivated by the above facts, the author suggested an alternative energy-momentum method which has the most important feature of being independent of the configuration space as well as of the
kind of the non-linearities retained in the strain-displacement relations. Accordingly, the method applies equally to shell and rod formulations including a rotation tensor or not. First applications of the method was carried out in Sansour et al. [3] for a 7-paprameter shell formulation resting on the Green strain tensor with cubic non-linearities involved in the strain-displacement relations. Moreover, the method proves very stable and well designed for non-classical analysis of shell problems like detecting chaotic motion, period doubling mechanism, quasi periodic motion etc. In this paper we focus on the nonlinear dynamics of structures of the Cosserat type. Shells with six degrees of freedom as well as curved spatial rods are considered. The formulation in both cases is essentially the same. It is only the dimension of the continuum under consideration which differs. 2 The shell theory 2.1 The strain measures of the Cosserat continuum Let x and X define the placement vectors of a point of the body at the actual and reference configurations, respectively. The displacement vector is defined as u = x X and the deformation gradient F as (1) Let defines a rotation tensor given as a field over X, where SO(3) means the group of the proper orthogonal transformations. The Cosserat continuum is defined as a continuum the points of which have displacement as well as rotational degrees of freedom. Each point of the Cosserat continuum can be considered as a rigid body with six degrees of freedom where the rotations are independent of the displacements. Corresponding to the two independent kinematical fields, the displacements and the rotations, two strain measures are necessary to describe the deformation state of the body. The first one is the stretch tensor U (the first Cosserat strain measure) which is defined by (2) In general is not symmetric. The second strain measure is the second Cosserat strain tensor (the curvature tensor) which can be constructed as follows. Since the derivative with respect to the coordinate gives: and so the products must be skew-symmetric. Thus, there exist corresponding axial vectors with the help of which the second Cosserat deformation tensor is defined as (3) Here are some co- and contra base vectors defined at the reference configuration. The rotation tensor can be expressed in form of the exponential map where is an antisymmetric tensor, and in closed form the representation holds (4)
w denotes the axial vector of. Taking the transpose of and the derivative with respect to the coordinate, one can find an expression for the axial vector : 2.2 Momentum, moment of momentum, and the Hamilton For the formulation of the equations of motion of the Cosserat continuum, one needs some relations between the time derivative of the rotation vector, on the one side, and the variation of the angular velocity, on the other side. In linear vector spaces, as is the case of the displacement field, the following simple relation holds: (5) where D/Dt denotes the time derivative which is also denoted by a dot. For and one has first (6) (7) (8) Now, let be the density of the Cosserat-continuum in the Reference configuration, the surface forces, and the body forces. The momentum and angular momentum balance equations read: ((( (9) (10) where, the tensor of inertia, is constant and positive-definite, denotes the volume element and the surface element, is the boundary of. In the classical continuum the terms with and must vanish. The kinetic energy of the system is given by (11)
In case of hyper-elastic materials, there exists an internal potential (12) which is a function of both deformation tensors and. Further, the external virtual work is defined as (13) Here too we note that the virtual rotation conjugate to an external moment is given by the spatial quantity and not by the material one. For the balance equation of the moment of momentum the load of the system can be enhanced to include external fields of moments and, which act in the field and on the boundary, respectively. Attention must be given for, since it is a material vector, and the moment of momentum must be formulated with the spatial angular velocity vector. The moment of momentum is given by The equations of motion are the Euler-Lagrange equations of the following action: (14) They read ( (15) (16) The total momentum and the total angular momentum are given by (17) It can be shown that in the absence of external forces or external moments (18)
and so (19) In other words: In the absence of external forces and moments the total momentum and moment of momentum of the moving body remain conserved during the dynamical process as time evolves. In the presence of an external potential (conservative forces), one can show that the total energy of the system remains conserved during the motion. In case the internal forces and moments, as well as the external forces can be derived from potentials, we get (20) and from this we can see that (21) This means that during the dynamics the total energy of the system remains conserved. Such systems are called Hamiltonian systems. The function is called the Hamiltonian. 3 Time integration schemes 3.1 The New integration method Now, within a time integration method one generates with the help of a given transversal velocity and a given angular velocity, the latter is an antisymmetric tensor, finite quantities which are superimpoed on given displacements and rotations. Let the displacement, velocity, and acceleration fields be known at time t. For the determination of the kinematical fields at time the following updating procedure appears as a natural formula: (22) The different methods provide different expressions for u and W and their dependence on the corresponding transversal and angular velocities. Three methods are discussed in what follows. To take the velocities and angular velocities for the above updating to be in the middle of a given interval, corresponds to the midpoint-rule. Explicitly one has (23) which together with (22) define the midpoint-integration scheme. In the following we give a summary of a new integration method which we have developed recently. The method preserves the energy and momentum during integration of the conservative system. For detailed description the reader in referred to [4]:
To that end let us define first the following family of algorithms (24) According to that said in the general remarks we intend to make use of the updating formula stated above and in the same time to preserve, in a since to be made precise, the energy of the system. Let us define the following family of strain fields (25) (26) According to these definitions of strains, which are at the heart of the new integration method, the kinematical relations (2) and (5) are dropped in defining the strain fields. Instead, the strain field is defined by means of a corresponding strain-velocity field. Accordingly, the kinematical relations (2) and (5) are not used to define the strains themselves but to define the corresponding velocities. Once the quantities: and are known, the strain measures at time step are well defined according to (25) and (26). It should be clear that the resulting strain measures could differ from those calculated directly by means of the relations (2) and (5). The formulation of the midpoint rule for the strain measures (not for the kinematical fields themselves) can then achieve the desired result. 3.2 Chaotic motion of shallow arch A simply supported shallow arch having its hysteresis region of the statically calculated displacement-load diagram around zero is considered next. A point-excitation is assumed to act in the middle of the arch. The model with its corresponding data is illustrated in (Fig. 1). A chaotic motion is shown in different presentations in (Fig. 2).
Figure 1. The geometry, the data, and the load-displacement diagram of the simply supported shallow arch
Figure 2. Chaotic attractor; POINCARE-section with the corresponding (z-t)- and (v-x)-plots. F = 360, frequency = 1000
3.3 Free large overall motion The second example will demonstrate the conservation of the mechanical quantities (momentum, moment of momentum, and energy) when using our new energy/momentum integrator. It is shown in the following figures that all of the above mentioned mechanical quantities of a plate undergoing free large overall motion remain constant during integrations. Figure 3. Free motion of a plate. Problem definition Figure 4. Energy versus time steps Figure 5. Momentum versus time steps Figure 6. Angular momentum versus time steps References SANSOUR, C. and BEDNARCZYK, H., 1995., The Cosserat surface as a shell model, theory and finite-element formulation, Computer Methods Appl. Mech. Engrg. 120, 1-32. SANSOUR, C., WRIGGERS, P., and SANSOUR, J., 1997. Nonlinear dynamics of shells, theory, finite element formulation, and integration schemes. Nonlinear Dynamics 13: 279-305. SANSOUR, C., WRIGGERS, P. and SANSOUR, J., 2004. On the design of energy-momentum integration schemes for arbitrary continuum formulations. Applications to classical and chaotic motion of shells, Int. J. Numer. Meths. Engrg. 60, 2419-2440. SANSOUR, C. et al, 2010. Conmputational non-linear dynamics of the Cosserat Continuum, submitted. SIMO, J.C. and TARNOW N., 19992 The discrete energy-momentum method. Conserving algorithms for nonlinear elastodynamics. Z. angew. Math. Phy., ZAMP 1992; 43: 757-792.