Finite Element Method - Pseudo - rigid and rigid - flexible bodis _12

Finite Element Method - Pseudo - rigid and rigid - flexible bodis _12 This monograph presents in detail the novel "wave" approach to finite element modeling of transient processes in solids. Strong discontinuities of stress, deformation, and velocity wave fronts as well as a finite magnitude of wave propagation speed over elements are considered. These phenomena, such as explosions, shocks, and seismic waves, involve problems with a time scale near the wave propagation time. Software packages for 1D and 2D problems yield significantly better results than classical FEA, so some FORTRAN programs with the necessary comments are given in the appendix. The book is written for researchers, lecturers, and advanced students interested in problems of numerical modeling of non-stationary dynamic processes in deformable bodies and continua, and also for engineers and researchers involved designing machines and structures, in which shock, vibro-impact, and other unsteady dynamics and waves processes play a significant role.

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 second- ary 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 deforma-

tion 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 o ] = r 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 ~ l o 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).22 The DDA form, while widely used in the geotechnical community, is

usually combined with a simple linear elastic constitutive model and linear strain-dis-

placement 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-Venant-

Kirchhoff 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^.^^,*^

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, and Ailhi, = 6, (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

UIJ simply becomes an identity Thus, a rigid body motion may be specified as

4i(x1i t ) = ri(t) + Ail(t> 1x1 - &I ( 12.4)

or, in matrix form, as

+(XI t ) = r(t) + A ( t ) [ X - R] (12.5) 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 con- straints 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

@ = S r + G A [ X - R ]

d+ = dr + &A[X - R]

Using Eq (12.5) these may be simplified to

4) = 6r - 960 where y x - r

where d+ and 60 are incremental and variational rotation vectors, respectively

In a similar manner we obtain the velocity for the rigid motion as

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

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

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

skew symmetric form

rl [ -:: w1 71

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

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

or by transforming in the opposite way

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(xu - r) x f u = m; IC = IO ( 12.17)

where IC is the angular momentum of the rigid body, m is a resultant couple and II is the

spatial inertia tensor

ll

The spatial inertia tensor (matrix) Ti is computed from

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

J = po [(YTY)I - YYT] d V where Y = X - R (12.19)

J O 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 (1 2.16) and

(12.19) together with the relation to determine the centre of mass given by

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)

-

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

where Re is the reference volume of each element e Use of Eq (12.21) in Eq (12.20) to determine the centre of mass now gives

(12.23)

and finally the reference inertia tensor (matrix) as

9 = Map [(Y;fYp)I - Yay$]; Y, = X, - R (12.24)

e

where

Rigid motions 401

The above definition of Y, tacitly assumes that Ea N , = 1 If other interpolations are

used to define the shape functions (e.g hierarchical shape functions) it is necessary to

modify the above procedure to determine the mass and inertia matrix

12.3.3 Transient solutions

The integration of the translational rigid term r may be performed using any of the

methods described in Chapter 18 of Volume 1 or indeed by other methods described

in the literature The integration of the rotational part can also be performed by many

schemes, however, it is important that updates of the rotation produce discrete time

values for rigid rotations which retain an orthonormal character, that is, the A, must

satisfy the orthogonality condition given by Eq (12.12) One procedure to obtain this

is to assume that the angular velocity within a time increment is constant, being

measured as

(12.26)

1

At

a(t) M a,+, = -e

in which At is the time increment between t , and t,+ I , 8 is the increment of rotation

during the time step, and 0 < Q < 1 The approximation

an+, = (1 - @ ) a n + aan+l (12.27)

is used to define intermediate values in terms of those at t, and t,, Equation (12.8)

now becomes a constant coefficient ordinary differential equation which may be

integrated exactly, yielding the solution

A(t) = exp[6(t - t , ) / A t ] A , t, d t d t n + l (12.28)

In particular at t n f a we obtain

A,+, = e x p [ d ] A , This may also be performed using the material angular velocity Q.' Many algorithms

exist to construct the exponential of a matrix, and the closed-form expression given by

the classical formula of Euler and Rodrigues (e.g see Wittaker2') is quite popular

This is given by

(12.29) sin lei A 1 sin2 A 2

exp[i]= I + - e + - 8 where 101 = [f3TO]1'2

le1 2 [pi2/2]

This update may also be given in terms of quaternions and has been used for

integration of both rigid body motions as well as for the integration of the rotations

appearing in three-dimensional beam formulations (see Chapter 1 l).8,29'30 Another

alternative to the direct use of the exponential update is to use the Cayley transform

to perform updates for A which remain orthonormal

Once the form for the update of the rigid rotation is defined any of the integration

procedures defined in Chapter 18 of Volume 1 may be used to advance the incre-

mental rotation by noting that 8 or 0 (the material counterpart) are in fact the

change from time t , to f,, 1 The reader also is referred to reference 8 for additional

algorithms directly based on the GNI 1 and GN22 methods presented in Chapter 18

of Volume 1 Here forms for conservation of linear and angular momentum are of particular importance

12.4 Connecting a rigid body to a flexible body

In some analyses the rigid body is directly attached to flexible body parts of the problem [Fig 12.2(a)] Consider a rigid body that occupies the part of the domain denoted as R, and is ‘bonded’ to a flexible body with domain Rf In such a case the formulation to ‘bond’ the surface may be performed in a concise manner using Lagrange multiplier constraints We shall find that these multiplier constraints can

be easily eliminated from the analysis by a local solution process, as opposed to the need to carry them to the global solution arrays as was the case in their use in contact problems (see Sec 10.8)

Fig 12.2 Lagrange multiplier constraint between flexible and rigid bodies: (a) rigid-flexible body; (b) Lagrange multipliers

12.4.1 Lagrange multiplier constraints

A simple two-dimensional rigid-flexible body problem is shown in Fig 12.2(a) in

which the interface will involve only three-nodal points In Fig 12.2(b) we show an exploded view between the rigid body and one of the elements which lies along the rigid-flexible interface Here we need to enforce that the position of the two interface nodes for the element will have the same deformed position as the corresponding

point on the rigid body Such a constraint can easily be written using Eq (12.4) as

C, = r ( t ) + A(t)[X, - R] - x,(t) = 0 (12.30)

in which the subscript a denotes a node number We can now modify a functional to

include the constraint using a classical Lagrange multiplier approach in which we add the term

IIg = lac, = l, [x,(t) - r(t) - A(t)[X, - R]] (12.31) Taking the variation we obtain

611g = SA, [x, - r - A[X, - R]] + A, [Sx, - Sr - SOA[X, - R]] (12.32)

Connecting a rigid body to a flexible body 403

From this we immediately obtain the constraint equation and a modification to the

equilibrium equations for each flexible node and the rigid body Accordingly, the

modified variational principle may now be written for a typical node (Y on the inter-

face of the rigid body as

sn + snd = [SX, SX, br 68 SA,]

MpUV” + M , p i p + P, - f,

M a u V u + M a p i p + P, - f, + h,

p - f - h ,

T

fi - m - y,h,

x, - r - h[X, - R]

(12.33)

where y, = x, - r are the nodal values of y, /3 are any other rigid body nodes

connected to node Q and p, u are flexible nodes connected to node a

Since the parameters x, enter the equations in a linear manner we can use the

constraint equation to eliminate their appearance in the equations Accordingly,

from the variation of the constraint equation we may write

which permits the remaining equations in Eq (12.33) to be rewritten as

M , , i , + M p p i p + P, - f,

Sn + 6nrf = [ SX, Sr SO J p - fM,,V, + Mapip + Pa - f,

,i - m - j : ( ~ , ~ i ~ + ~ ,+ P, ~- fa) i} = o ~

(12.35) For use in a Newton-Raphson solution scheme it is necessary to linearize Eq

(12.35) This is easily achieved

d ( S n ) + d ( 6 n r f ) = [ Sx, 6r SO]

{ e ]

(12.37)

Combining all the steps we obtain the set of equations for each rigid body as

in which e, and ep are the residuals from the finite element calculation at node a and

p, respectively We recall from Chapter 10 that each is given by a form

which is now not zero since total balance of momentum includes the addition of the I ,

The above steps to compute the residual and the tangent can be performed in each element separately by noting that

e

where I: denotes the contribution from element e Thus, the steps to constrain a flexible body to a rigid body are once again a standard finite element assembly process and may easily be incorporated into a solution system

The above discussion has considered the connection between a rigid body and a body which is modelled using solid finite elements (e.g quadrilateral and hexahedral elements in two and three dimensions, respectively) It is also possible directly to connect beam elements which have nodal parameters of translation and rotation This is easily performed if the rotation parameters of the beam are also defined in terms of the rigid rotation A In this case one merely transforms the rotation to

be defined relative to the reference description of the rigid body rotation and assembles the result directly into the rotation terms of the rigid body If one uses

a rotation for both the beam and the rigid body which is defined in terms of the global Cartesian reference configuration no transformation is required Shells can

be similarly treated; however, it is best then to define the shell directly in terms of three rotation parameters instead of only two at points where connection is to be performed 1,32

12.5 Multibody coupling by joints

Often it is desirable to have two (or more) rigid bodies connected in some specified manner For example, in Fig 12.3 we show a disc connected to an arm Both are treated as rigid bodies but it is desired to have the disc connected to the arm in such a way that it can rotate freely about the axis normal to the page This type of motion is characteristic of many rotating machine connections and it as well as many other types of connections are encountered in the study of rigid body

motion^.^'^^ This type of interconnection is commonly referred to as a joint In quite general terms joints may be constructed by a combination of two types of simple constraints: translational constraints and rotational constraints

Multibody coupling by joints 405

12.5.1 Translation constraints

The simplest type of joint is a spherical connection in which one body may freely

rotate around the other but relative translation is prevented Such a situation is

shown in Fig 12.3 where it is evident the spinning disc must stay attached to the

rigid arm at its axle Thus it may not translate relative to the arm in any direction

(additional constraints are necessary to ensure it rotates only about the one axis -

these are discussed in Sec 12.5.2) If a full translation constraint is imposed a

simple relation may be introduced as

where a and b denote two rigid bodies Thus, addition of the Lagrange multiplier

constraint

imposes the spherical joint condition It is necessary only to define the location for the

spherical joint in the reference configuration Denoting this as Xi (which is common

to the two bodies) and introducing the rigid motion yields a constraint in terms of the

rigid body positions as

n 1 - - LT j [ r [.I + A(")(X J - R(")) - r(b) - A ( b ) ( X j - R ( b ) ) ] ( 12.43)

The variation and subsequent linearization of this relation yields the contribution to

the residual and tangent matrix for each body, respectively This is easily performed

using relations given above and is left as an exercise for the reader

If the translation constraint is restricted to be in one direction with respect to, say,

body a it is necessary to track this direction and write the constraint accordingly To

accomplish this the specific direction of the body a in the reference configuration is

required This may be computed by defining two points in space X I and X2 from

Fig 12.3 Spinning disc constrained by a joint to a rigid arm