Co-rotational Technique

In any large displacement formulation, the goal is to separate the deformation displacements from the rigid body displacements, as only the deformation displacements give rise to strains and the associated generation of strain energy. This separation is usually accomplished by comparing the current configuration with a reference configuration.

The current configuration is a complete description of the deformed body in its current spatial location and orientation, giving locations of all points (nodes) comprising the body. The reference configuration can be either the initial configuration of the body, i.e., nodal locations at time zero, or the configuration of the body at some other state (time). Often the reference configuration is chosen to be the previous configuration, say at time tⁿ = tⁿ⁺¹− ∆t.

The choice of the reference configuration determines the type of deformations that will be computed: total deformations result from comparing the current configuration with the initial configuration, while incremental deformations result from comparing with the previous configuration. In most time stepping (numerical) Lagrangian formulations, incremental deformations are used because they result in significant simplifications of other algorithms, chiefly constitutive models.

A direct comparison of the current configuration with the reference configuration does not result in a determination of the deformation, but rather provides the total (or incremental) displacements. We will use the unqualified term displacements to mean either the total displacements or the incremental displacements, depending on the choice of the reference configuration as the initial or the last state. This is perhaps most obvious if the reference configuration is the initial configuration. The direct comparison of the current configuration with the reference configuration yields displacements, which contain components due to deformations and rigid body motions. The task remains of separating the deformation and rigid body displacements. The deformations are usually found by subtracting from the displacements an estimate of the rigid body displacements. Exact rigid body displacements are usually only known for trivial cases where they are prescribed a priori as part of a displacement field. The co-rotational formulations provide one such estimate of the rigid body displacements.

The co-rotational formulation uses two types of coordinate systems: one system associated with each element, i.e., element coordinates which deform with the element, and another associated with each node, i.e., body coordinates embedded in the nodes. (The term

‘body’ is used to avoid possible confusion from referring to these coordinates as ‘nodal’

coordinates. Also, in the more general formulation presented in [Belytschko et al., 1977], the nodes could optionally be attached to rigid bodies. Thus the term ‘body coordinates’ refers to a system of coordinates in a rigid body, of which a node is a special case.) These two coordinate systems are shown in the upper portion of Figure 4.1(a).

The element coordinate system is defined to have the local x-axis ˆx originating at node I and terminating at node J; the local y-axis ˆy and, in three dimension, the local z-axis ˆz, are

constructed normal to ˆx. The element coordinate system

(

^{x y zˆ ˆ ˆ, ,}

)

and associated unit vector triad (e₁, e₂, e₃) are updated at every time step by the same technique used to construct the initial system; thus the unit vector e₁ deforms with the element since it always points from node I to node J.

The embedded body coordinate system is initially oriented along the principal inertial axes; either the assembled nodal mass or associated rigid body inertial tensor is used in determining the inertial principal values and directions. Although the initial orientation of the body axes is arbitrary, the selection of a principal inertia coordinate system simplifies the rotational equations of motion, i.e., no inertial cross product terms are present in the rotational equations of motion. Because the body coordinates are fixed in the node, or rigid body, they rotate and translate with the node and are updated by integrating the rotational equations of motion, as will be described subsequently.

The unit vectors of the two coordinate systems define rotational transformations between the global coordinate system and each respective coordinate system. These transformations operate on vectors with global components A= (A A A_x, _y, _z), body coordinates components

where b_ix, b_iy, b_iz are the global components of the body coordinate unit vectors. Similarly for the element coordinate system:

[ ] ^{{ }}

where e_ix, e_iy, e_iz are the global components of the element coordinate unit vectors. The inverse transformations are defined by the matrix transpose, i.e.,

{ }^{A =}

[ ]

^{λ T{ }A (4.3)}

{ }

^{Aˆ =}

[ ]

^{µ T{ }A (4.4)}

since these are proper rotational transformations.

The following two examples illustrate how the element and body coordinate system are used to separate the deformations and rigid body displacements from the displacements:

Figure 4.1. Co-rotational coordinate system: (a) initial configuration, (b) rigid rotational configuration and (c) deformed configuration.

Rigid Rotation. First, consider a rigid body rotation of the beam element about node I , as shown in the center of Figure 4.1(b), i.e., consider node I to be a pinned connection. Because the beam does not deform during the rigid rotation, the orientation of the unit vector e in the initial and rotated configuration will be ₁ the same with respect to the body coordinates. If the body coordinate components of the initial element unit vector e were stored, they would be identical to the ₁⁰ body coordinate components of the current element unit vector e . ₁

Deformation Rotation. Next, consider node I to be constrained against rotation, i.e., a clamped connection. Now node J is moved, as shown in the lower portion of Figure 4.1(c), causing the beam element to deform. The updated element unit vector e is constructed and its body coordinate components are compared to the ₁ body coordinate components of the original element unit vector e . Because the ₁⁰ body coordinate system did not rotate, as node I was constrained, the original element unit vector and the current element unit vector are not colinear. Indeed, the angle between these two unit vectors is the amount of rotational deformation at node I, i.e.,

0

1 1 3

e × =e θAe (4.5)

Thus the co-rotational formulation separates the deformation and rigid body deformations by using:

• a coordinate system that deforms with the element, i.e., the element coordinates; or

• a coordinate system that rigidly rotates with the nodes, i.e., the body coordinates;

Then it compares the current orientation of the element coordinate system with the initial element coordinate system, using the rigidly rotated body coordinate system, to determine the deformations.

Dans le document LS-DYNA (Page 73-76)