12

Pseudo-rigid and rigid-flexible bodies 12.1 Introduction Many situations are encountered where treatment of the entire system as deformable bodies is neither necessary nor practical. For example, the frontal impact of a vehicle against a barrier requires a detailed modelling of the front part of the vehicle but the primary function of the engine and the rear part is to provide inertia, deformation being negligible for purposes of modelling the frontal impact. A second example, from geotechnical engineering, is the modelling of rock mass landslides or interaction between rocks on a conveyor belt where deformation of individual blocks is secondary. In this chapter we consider briefly the study of such systems. The above problem classes divide themselves into two further sub-classes: one where it is necessary to include some simple mechanisms of deformation in each body (e.g. an individual rock piece) and the second in which the individual bodies have no deformation at all. The first class is called pseudo-rigid body deformation' and the second rigid-body behaviour.2 Here we wish to illustrate how such behaviour can be described and combined in a finite element system. For the modelling of pseudo-rigid body analyses we follow closely the work of Cohen and Muncaster' and the numerical implementation proposed by Solberg and Papadopo~los.~ The literature on rigid body analysis is extensive, and here we refer the reader to papers for additional details on methods and formulations beyond those covered

12.2 Pseudo-rigid motions In this section we consider the analysis of systems which are composed of many small bodies, each of which is assumed to undergo large displacements and a uniform deformation.* The individual bodies which we consider are of the types shown in Fig. 12.1. In particular, a faceted shape can be constructed directly from a finite element discretization in which the elements are designated as all belonging to a

* Higher-order approximations can be included using polynomial approximation for the deformation of each body.

Pseudo-rigid motions 397

Fig. 12.1 Shapes for pseudo-rigid and rigid body analysis: (a) ellipsoid; (b) faceted body.

single solid object or the individual bodies can be described by simple geometric forms such as discs or ellipsoids. A homogeneous motion of a body may be written as 4i(XI>t ) = ri(t) + F i ~ ( t[ )X I - RII

(12.1)

in which X I is position, tis time, RI is some reference point in the undeformed body, ri is the position of the same point in the deformed body, and FiI is a constant deformation gradient. We note immediately that at time zero the deformation gradient is the identity tensor (matrix) and Eq. (12.1) becomes q$(X,,O) = r i ( 0 ) + S i ~ [ X ~ - R=or] o ( 0 ) + S i ~ X , - S i , R ~ = S i ~ X ,

(12.2)

where ri(0)= SiI RI by definition. The behaviour of solids which obey the above description is sometimes referred to as analysis of pseudo-rigid bodies.' A treatment by finite elements has been considered by Solberg and P a p a d o p o ~ lo s ,~ and an alternative expression for motions restricted to incrementally linear behaviour has been developed by Shi, and the method is commonly called discontinuous deformation analysis (DDA).22The DDA form, while widely used in the geotechnical community, is usually combined with a simple linear elastic constitutive model and linear strain-displacement forms whlch can lead to large errors when finite rotations are encountered. Once the deformation gradient is computed, the procedures for analysis follow the methods described in Chapter 10. It is, of course, necessary to include the inertial term for each body in the analysis. No difficulties are encountered once a shape of each body is described and a constitutive model is introduced. For elastic behaviour it is not necessary to use a complicated model, and here use of the Saint-VenantKirchhoff relation is adequate indeed, if large deformations occur within an individual body the approximation of homogeneous deformation generally is not adequate to describe the solution. The primary difficulty for this class of problems is modelling the large number of interactions between bodies by contact phenomena and here the reader is referred to Chapter 10 and references on the subject for additional information on contact and other detail^.^^,*^ ~

398 Pseudo-rigid and rigid-flexible bodies

12.3 Rigid motions The pseudo-rigid body form can be directly extended to rigid bodies by using the polar decomposition on the deformation tensor. The polar decomposition of the deformation gradient may be given Fil = Aij

UJI

where

AilAij = SI,

Ailhi, = 6,

and

(12.3)

Here Ail is a rigid rotation* and UIJ is a stretch tensor (that has eigenvalues A, as defined in Chapter 10). In the case of rigid motions the stretches are all unity and UIJsimply becomes an identity. Thus, a rigid body motion may be specified as 4i(x1it ) = ri(t)

+ Ail(t>1x1 &I -

( 12.4)

or, in matrix form, as

+

(12.5) +(XI t ) = r(t) A ( t ) [X - R] Alternatively, we can express the rigid motion using Eq. (12.1) and impose constraints to make the stretches unity. For example, in two dimensions we can represent the motion in terms of the displacements of the vertices of a triangle and apply constraints that the lengths of the triangle sides are unchanged during deformation. The constraints may be added as Lagrange multipliers or other constraint methods and the analysis may proceed directly from a standard finite element representation of the triangle. Such an approach has been used in reference 27 with a penalty method used to impose the constraints. Here we do not pursue this approach further and instead consider direct use of rigid body motions to construct the formulation. For subsequent use we note the form of the variation of a rigid motion and its incremental part. These may be expressed as

@=Sr+GA[X-R]

d+ = dr

+ &A[X

- R]

Using Eq. (12.5) these may be simplified to

4)= 6r - 960

where y

d+ = dr - jrd0

x

-r

(12.6)

where d+ and 60 are incremental and variational rotation vectors, respectively. In a similar manner we obtain the velocity for the rigid motion as +=r-yo

(12.7)

in which i is translational velocity and o angular velocity, both at the centre of mass. The angular velocity is obtained by solving A=bA

(12.8)

li=ASt

(12.9)

or

* Often literature denotes this rotation as R,; however, here we use R, as a position of a point in the body and to avoid confusion use Am to denote rotation.

Rigid motions 399

where fl is the reference configuration angular velocity.' This is clearer by writing the equations in indicia1 form given by hi1

(12.10)

= wghi, = hij RjI

where the velocity matrices are defined in terms of vector components and give the skew symmetric form

0

w.. = rl

-w3

[ -:

0 w1

w2

71

(12.11)

and similarly for a,. The above form allows for the use of either the material angular velocity or the spatial one. Transformation between the two is easily performed since the rigid rotation must satisfy the orthogonality conditions

PA= AP= I

( 12.12)

at all times. Using Eqs (12.8) and (12.9) we obtain

c;, = A

~

A

~

(12.13)

or by transforming in the opposite way

h = A%A

( 12.14)

12.3.1 Equations of motion for a rigid body If we consider a single rigid body subjected to concentrated loads f, applied at points whose current position is xu and locate the reference position for R at the centre of mass, the equations of equilibrium are given by conservation of linear momentum (12.15) ll

where p defines a linear momentum, f is a resultant force and total mass of the body is computed from

( 12.16) and conservation of angular momentum

i=

C(xur) x -

fu

= m;

IC = IO

( 12.17)

ll

where IC is the angular momentum of the rigid body, m is a resultant couple and II is the spatial inertia tensor. The spatial inertia tensor (matrix) Ti is computed from

II = AJP

(12.18)

400 Pseudo-rigid and rigid-flexible bodies

where 9 is the inertia tensor (matrix) computed from an integral on the reference configuration and is given by

J=

po [(YTY)I - YYT] dV

where Y = X - R

(12.19)

JO

Thus, description of an individual rigid body requires locating the centre of mass R and computing the total mass m and inertia matrix J. It is then necessary to integrate the equilibrium equations to define the position r and the orientation of the body A.

12.3.2 Construction from a finite element model If we model a body by finite elements, as described throughout the volumes of this book, we can define individual bodies or parts of bodies as being rigid. For each such body (or part of a body) it is then necessary to define the total mass, inertia matrix, and location of the centre of mass. This may be accomplished by computing the integrals given by Eqs (12.16) and (12.19) together with the relation to determine the centre of mass given by

mR

=

jOpoXdV

(12.20)

In these expressions it is necessary only to define each point in the volume of an element by its reference position interpolation X. For solid (e.g., brick or tetrahedral) elements such interpolation is given by Eq. (10.55) which in matrix form becomes (omitting the summation symbol)

-

X = N,X,

(12.21)

This interpolation may be used to determine the volume element necessary to carry out all the integrals numerically (see Chapter 9 of Volume 1). The total mass may now be computed as =

C(jO