6 T H E F I N I T E - E L E M E N T M E T H O D - - P A R T I

6.1 Introduction

The concept of the finite-element procedure may be dated back to 1943 when Courant [1] approximated the warping function linearly in each of an assemblage of triangular elements to the St. Venant torsion problem and proceeded to formulate the problem using the principle of minimum potential energy. Similar ideas were used later by several investigators to obtain the approximate solutions to certain boundary-value problems. It was Clough [2] who first introduced the term "finite elements" in the study of plane elasticity problems. The equivalence of this method with the well-known Ritz method was established at a later date, which made it possible to extend the applications to a broad spectrum of problems for which a variational formulation is possible. Since then numerous studies have been reported on the theory and applications of the finite-element method.

In this and next chapters the finite-element formulations necessary for the deformation analysis of metal-forming processes are presented. For hot forming processes, heat transfer analysis should also be carried out as well as deformation analysis. Discretization for temperature calculations and coupling of heat transfer and deformation are discussed in Chap. 12. More detailed descriptions of the method in general and the solution techniques can be found in References [3-5], in addition to the books on the finite-element method listed in Chap. 1.

6.2 Finite-Element Procedures

The path to the solution of a problem formulated in finite-element form is described in Chap. 1 (Section 1.2). Discretization of a problem consists of the following steps: (1) describing the element, (2) setting up the element equation, and (3) assembling the element equations. Numerical analysis techniques are then applied for obtaining the solution of the global equations. The basis of the element equations and the assembling into global equations is derived in Chap. 5 (eq. (5.20) and eqs. (5.21) or (5.22)).

The solution satisfying eq. (5.20) is obtained from the admissible

90

The Finite-Element Method---Part I 91

velocity fields that are constructed by introducing the shape function in such a way that a continuous velocity field over each element can be defined uniquely in terms of velocities of associated nodal points. In the deformation process shown in Fig. 6.1, the workpiece is divided into elements, without gaps or overlaps between elements. In order to ensure continuity of the velocities over the whole workpiece, the shape function is defined such that the velocities along any shared, element-side are expressed in terms of velocity values at the same shared set of nodes (compatibility requirement). Then a continuous velocity field over the whole workpiece can be uniquely defined in terms of velocity values at nodal points specified globally.

We define a set of nodal point velocities in a vector form as

V T = (131, L~ 2 . . . . , vN} (6.1)

where the superscript T denotes transposition and N = (total number of nodes) x (degrees of freedom per node).

An admissibility requirement for the velocity field is that the velocity boundary condition prescribed on surface Su (essential boundary condi- tion) must be satisfied. This condition can be imposed at nodes on Su by

DIE

3q

G l o b a l n o d a l p o i n t n u m b e r i n g

Element and e l e m e n t a l noda l p o i n t n u m b e r i n g

FIG. 6.1 Finite-element mesh and nodal point specifications in a forming process.

92 Metal Forming and the Finite-Element Method

assigning known values to the corresponding variables. It is to be noted that the incompressibility condition is not required for defining a velocity field in the formulation of eq. (5.21) or (5.22).

Equations (5.20) and (5.21) or (5.22) are now expressed in terms of nodal point velocities v and their variations 6v. From arbitrariness of 6Vl, a set of algebraic equations (stiffness equations) are obtained as

a~r ~ ( a ~ r ) = 0 (6.2) aVl=_. ~ (i)

where (j) indicates the quantity at the jth element. The capital-letter suffix signifies that it refers to the nodal point number.

Equation (6.2) is obtained by evaluating the (O~r/Ovl) at the elemental level and assembling them into the global equation under appropriate constraints.

In metal-forming, the stiffness equation (6.2) is nonlinear and the solution is obtained iteratively by using the Newton-Raphson method, The method consists of linearization and application of convergence criteria to obtain the final solution. Linearization is achieved by Taylor expansion near an assumed solution point v = v0 (initial guess), namely,

ra ' l ,,,,,--o + LaVl avjJ,=,o ¥5¥ 0

where Avs is the first-order correction of the velocity vo. Equation (6.3) can be written in the form

K A v = f (6 .4)

where K is called the stiffness matrix and f is the residual of the nodal point force vector.

Once the solution of eq. (6.4) for the velocity correction term Av is obtained, the assumed velocity v0 is updated according to v0 + a~ Av, where a~ is a constant between 0 and 1 called the deceleration coefficient. Iteration is continued until the velocity correction terms become negligibly small. The Newton-Raphson iteration process is shown schematically in Fig. 6.2. It is seen from the figure that convergence of Newton-Raphson iterations, the initial guess velocity should be close to the actual solution. When a deformation process is relatively simple, the initial guess velocity can be provided, for instance, by the upper-bound method. However, if the process is complex and obtaining a good initial guess solution is difficult, then the use of the direct iteration method discussed in Section 7.4 of Chap. 7 may be appropriate.

Two convergence critera may be used. One measures the error norm of the velocities, IIAvll/ l lvl l , where the Euclidean vector is defined as IIv[[ = (VTV) u2, and requires such an error norm to decrease from iteration to iteration. The other criterion requires the norm of the residual equations, Ilasr/av[I, to decrease.

In general terms, the first criterion is most useful in the early stages of

The Finite-Element Method--Part I 93

o

<

o z

J

I ! |

Solution point

Applied nodal f o r c e

NODE V E L O C I T Y

(a)

/ NODE V E L O C I T Y

(b)

r

FIG. 6.2 Schematic representation of the Newton-Raphson method: (a) convergence; (b) divergence.

iteration, when the velocity field is still far from the solution. The second test is most useful when slightly ill-conditioned systems reach the final stage of iterations. The final solution is considered to be achieved when the error norm reaches a specified small value, say 5 x 10 -5.

The finite-element method procedures outlined above are implemented in a computer program in the following way.

1. Generate an assumed solution velocity. 2. Evaluate the elemental stiffness matrix for the velocity correction

term Av in eq. (6.4). 3. Impose velocity conditions to the elemental stiffness matrix, and

repeat step 2 over all elements defined in the workpiece. 4. Assemble elemental stiffness matrix to form a global stiffness

equation.

94 Metal Forming and the Finite-Element Method

5. Obtain the velocity correction terms by solving the global stiffness equation.

6. Update the assumed solution velocity by adding the correctional term to the assumed velocity. Repeat steps 2 through 6 until the velocity solution converges.

7. When the converged velocity solution is obtained, update the geometry of the workpiece using the velocity of nodes during a time increment. Steps 2 through 7 are repeated until the desired degree of deformation is achieved.

The order of the certain steps mentioned above may change depending upon the computer programming practice.

The above procedure applies to the analysis of nonsteady-state- processes. For steady-state processes, updating the geometry of the workpiece is not necessary. The procedure for steady-state processes is described in Chap. 10 for the analysis of axisymmetric extrusion and drawing.

6.3 Elements and Shape Function

The order of a boundary-value problem is defined as that of the highest derivative of the field variable. It is 2 in our problem, and the integrand in eq. (5.20) is of order 1. Thus, the admissibility requirements are that the velocity must have a continuous first-order derivative within the element (completeness requirement) and the velocity should be continuous at the subdomain interfaces (compatibility requirement). Symbolically, a function of class C r has continuous derivatives of all orders up to and including r. Then shape functions for the velocity field within the element must be of class C 1 for completeness, and the compatibility of elements requires them to be of class C O globally.

The geometry of an element, in general, is uniquely defined by a finite number of nodal points (nodes). The nodes are located on the boundary of the element or within the element, and the shape function defines an admissible velocity field locally in terms of velocities of associated nodes. Thus, elements are characterized by the shape and the order of shape functions.

In the finite-element method, interpolation of a scalar function f (x , y) defined over an element is introduced in a form

f (x , y) = ~ q,~(x, y) . f~ (6.5) t l t

where f,, is a function value associated with a~th node, and q,~(x, y) is the shape function. It is, in general, a polynomial function of x and y defined over the element in such a way that

q,,(x~, Yt~) = 6~t~ (6.6)

The Finite-Element Method---Part I

1

L3=O

FIG. 6.3 Area coordinate system of the triangular element.

95

where (x~, y~) is the coordinates of flth node and 6 ~ is the Kronecker delta. Owing to the property of the shape functions given by eq. (6.6), f~ in eq. (6.5) has the value of the function f at (x~, y~) and the f~ are independent of each other.

There are various types of elements, depending upon the shape of the element and the polynomial order of shape funtions. In the following, the elements used in this book are discussed.

Triangular Element Family

In the triangular element family, it is convenient to define shape functions in the area coordinate system, Lt, L2, L3. The area coordinates for a triangle, as shown in Fig. 6.3, are defined by the following linear relations:

x = Llxt + L2x2 + L3x3

y = Ltyl + L2y2 + L3Y3 (6.7)

Ll + L E + L 3 = l

where (xo,, y~) are the coordinates of a corner of the triangle. It can be readily shown that an alternative definition of the coordinate of a point P can be given by the ratio of the area of the shaded triangle to that of the total triangle as

area P23 area P13 area P12 L1 L2 = - - and L3 = - -

area 123' area 123 area 123

Solving eq. (6.7) for Lt, L2 and L3 gives

Lt = (at + btx + cly)/2A

L2 = (a2 + b2x + c2y)/2A (6.8)

L3 = (aa + b3x + c3y)/2A

96

where

and

Metal Forming and the Finite-Element Method

A = a r e a of (123)

a l = x 2 y a - x 3 Y 2 , a2-- -x3Yl-Xlya , a 3 = x l Y 2 - x 2 y 1

bx = Y2 - Y3, b2 = Y3 - Yl, b3 = Yl - )'2 (6.9)

C I- '~X 3 ~ X 2 ~ C 2 ~ X l ~ X 3 , C 3 ~ X 2 - X 1

With a linear triangular element with nodes at its corners (see Fig. 6.4a), the shape functions q~ are linear and are given by

ql ---- L1, q2 -- L2, q3 -- L3 (6.10)

A quadratic triangular element has primary nodes at the corners and secondary nodes at the mid-sides (see Fig. 6.4b) and the shape functions are

ql = (2Lt - 1)L1, q2 = (2L2 - 1)L2, q3 = (2L3 - 1)L3 (6.11)

q4 = 4L1L2, q5 = 4LzL3, q6 = 4L3L1

The admissible velocity field within the element can be represented, for both linear and quadratic elements, by

u , (L , , L2, L3) = ~_j q~(L, , L2, L3)u(s °0

(6.12) uy(L1, L2, L3) = E q~(L1, L2, L3)u~ ~)

o/

where u~ '*), uy ~) are the velocity components of the octh node. Elements that use the same shape functions as the coordinate transfor-

mation are called isoparametric elements. Linear triangular elements, owing to eqs. (6.7) and (6.10), are isoparametric. However, with quadratic shape functions, isoparametric elements are curved triangular elements.

3 3

2 4 2

(a) Ib)

FIG. 6.4 (a) Linear triangular element; (b) quadratic triangular element.

The Finite-Element Method---Part I 97

4

I -1

-1

x

(a) (b)

4 ~ 3

o s ~ ~

1 2

(c)

FIG. 6.5 (a) Natural coordinate and rectangular parent element; (b) isoparametric element (mapped on Cartesian coordinate; quadrilateral element); (c) shape function.

In this book, the linear triangular element is used for the analysis of plate-bending under plane-strain conditions in Chap. 8. It is also used for the analysis of sheet-metal forming in Chap. 11.

Rectangular Element Family The shape functions of rectangular elements are, in general, defined in a parametric form over a domain - 1 -< ~ -< 1, - 1 -< 1/< 1 in a natural coordinate system (~, ~/). The element defined in the natural coordinate system is sometimes called the parent element. The simplest of the rectangular elements is the 4-node linear element shown in Fig. 6.5a. The shape functions, q~, which are bilinear in ~ and T/, are defined as

q~(~, r/) -- ~(1 + ~o,~)(1 + T/o,~/) (6.13)

where ( ~ , T/~) are the natural coordinates of a node at one of its corners.

98 Metal Forming and the Finite-Element Method

The value of the shape function, given by eq. (6.13), is shown schemati- cally in Fig. 6.5c. Admissible velocity fields can be defined uniquely over the rectangular element by the nodal velocity components as

ux(~, tl) = ~_~ qo:(~, tl)U~ '~) Ot

uy(~, r/) = '~, q,~(~, r/)Uty ~) (6.14) t g

where (ut~ '~), u~ '~)) is the velocity at the a~th node and summation is over all four nodes.

Coordinate transformation from the natural coordinate (~, t/) to the global coordinate (x, y) is defined by

x(~, tl) = ~ q~,(~, Tl)X,~ tit

y(~, t/) = ~ q~(~, t/)y~ (6.15) ¢I¢

where (x~, y~) are the global coordinates of the ath node. Since the coordinate transformation (6.15) uses the same shape functions (6.14), the linear element is isoparametric and takes quadrilateral shape in the Cartesian map, as shown in Fig. 6.5b.

Elements with shape functions of higher-order polynomials can be defined in a similar manner. Figure 6.6 shows a quadratic rectangular element with 8 nodes in the natural coordinate and global coordinate systems. The shape functions are defined by

corner nodes:

q,~(~, r/) = 1(1 + ~,~)(1 + t / ~ t / ) ( ~ + ~/~r/- 1)

mid-side nodes: (6.16)

q~(~, t/) = ½(1 - ~2)(1 + r/~/), ~ = 0

q~(~, n) = ½(1 + ~ ) ( 1 - T/2), r/~ = 0

Rectangular quadratic elements can also be defined by 9 nodes by adding an internal node to an 8-node quadratic rectangular element. The shape functions of the 9-node quadratic element can be derived from those of 8-node element. It may be of interest to note that the internal node does not affect the velocity distributions at the element boundary and the element shape. The rectangular element with 9 nodes is said to be of the Lagrangian family, since the shape functions are derived from Lagrange polynomials [6].

The isoparametric element can be distorted, for instance as shown in Fig. 6.6b, for a quadratic rectangular element by mapping according to eq. (6.15) with the shape functions given by eq. (6.16). The distorted shapes of

The Finite-Element Method---Part I 99

1

F-l 1

-1

5

(a)

I y

2

6

3 7 5

x

(b)

FIG. 6.6 Eight-node quadratic rectangular element: (a) natural coordinate and parent element; (b) isoparametric element after Cartesian mapping.

the isoparametric elements are sometimes useful, since they allow more flexibility and convenience in mesh generation. On the other hand, there are some limitations on the element geometry. To be a valid element, the coordinate transformation given by eq. (6.15) should be one-to-one, or the Jacobian of the coordinate transformation (see eq. (6.29)) should be positive at all points within the element. The validity of the element geometry is important, since elements deform considerably in metal- forming simulation.

Elements used for axisymmetric deformation are the same as two- dimensional elements in terms of shape functions and coordinate transfor- mations of the element. However, the axisymmetric element represents the cross section of a torus (ring element), while the other two dimensional elements represent the cross section of a straight bar. The strain-rate definition and the volume integration procedure are therefore different.

100 Metal Forming and the Finite-Element Method

The 4-node linear isoparametric element (quadrilateral element) is used extensively in this book for the analyses of two-dimensional and axisym- metric deformation processes.

Three-Dimensional Brick Element

The three-dimensional brick element is a natural extension of the two-dimensional linear element. The parent element is defined over a domain - 1 -< ~-< 1, - 1 < r/--- 1, - 1 -< ~-< 1, in natural coordinate system with a node at each corner. The shape functions are defined as

q~(~, t/, ~) = 18(1 + ~ ,~) (1 + r/~,q)(1 + ~ ) (6.17)

In eq. (6.17), ( ~ , t/r, ~,) are the coordinates of the ath node in the natural coordinate system. Figure 6.7 shows the brick element and its

8 7 A

3

(a)

z 5

8 7

Sb. . . . . 3

I 2 (b)

FIG. 6.7 Three-dimensional brick element in natural and Cartesian coordinate systems.

The Finite-Element Method--Part I 101

node-ordering convention. The velocity field can be expressed by

ux(~, T/, ~ )= ~ q~(~, r/, ¢)u(~ ~)

ur(~, T/, ¢ ) = ~ q~(~, r/, ¢)u(/~) (6.18) o¢

Uz(~, 17, ~ )= ~ q~(~, ~/, ¢)u(~ ")

where (u(~ "), u(r °°, u~ ~')) is the velocity of the a'th node. The coordinate transformation is given by

x(~, rl, ¢) = ~, q~x~ tY

Y(~, tl, ¢) = E q~Y~ (6.19) t l t

z(~, r I, ¢) = ~ q,,z,~ t l t

where (x,,, y,,, z,,) are the coordinates of the trth node. The use of the brick element is shown in Chap. 14 for the analysis of

three-dimensional deformation.

Note on No ta t ion

In expressing elemental functions, we treat each velocity component as a scalar. Sometimes it is more convenient to express a velocity field in a vector form as

u = NTv

where superscript T denotes transposition. The use of the transpose is only for the convenience of expressing formulations in matrix forms.

The vectors n and v are defined by their components according to uT= {u.(~, ,1), u.(~, n))

vT= {u~', u~'), .~2~, ~2~ . . . . } = {o , , o , , o~ . . . . )

respectively, for the ease of two-dimensional deformation. The shape functions are also arranged in the matrix form as

. . __[q , 0 0 0 :] ql 0 q2 0 q3

for the two-dimensional case.

6.4 Element Strain-Rate Matrix

In Chap. 4, the strain-rate components in Cartesian coordinate system are defined by

1 ( a u i + a u j ~ (6.20) ~ = 2 \ axj axd

102 Metal Forming and the Finite-Element Method

It was also shown, in Section 6.3, that the admissible velocity for all type of elements can be expressed by

u, = ~'~ q~,u~ ~) (6.21)

Substituting eq. (6.21) into eq. (6.20), we have

(6.22)

It is seen from eq. (6.22) that strain rate components can be evaluated if aq,Jaxi is known.

For the Cartesian coordinate system, we denote the coordinate xi by (x, y, z) for three-dimensional deformation, by (r, z, 0) for axisymmetric deformation, and by (x, y) for two-dimensional deformation.

Let X~,, Y~, and Z~ be defined as

aq,~ aq,~ aq~, (6.23a) x '~=ax ' Y~'=ay' z '~=az

Then the strain-rate components given by eq. (6.22) are expressed by

~. = X x..~ ~, *. = X v~.~;), ~. = X z~.:

~.. = ½ E (Y~,u~ ~ + x~,,,~.")) (6.23b)

~,, = ½ ~ (Z.u~ ~ + V~u~*))

gz~ = ½ E (x~"~. ~) + z~"~ ~))

It is convenient to arrange the strain-rate components in a vector form. For two-dimensional elements and axially symmetric deformation, the strain-rate components can be written as

~= ~ %

au~

0x

0uy ay

auy au~

for plane-stress (6.24a)

The Finite-Element Method--Part I

L;~y ) ] aoY for plane-strain

|a, , , a_, ~.ax+a!J

103

(6.24b)

and

kz ~= ko =

aUr Or OUz

Oz Ur

r

L guz Ou

for axisymmetric deformation (6.24c)

Substituting eqs. (6.23) into vectors are represented in a unified form, as

eqs. (6.24a, 6.24b and 6.24c), the strain-rate

~4 Z (X~,u(~=) + Y=.<,:))J

(6.25)

In eq. (6.25), ul and u2 correspond to ux and ur, respectively, for two-dimensional deformation, and P,~ is zero for plane-strain and the row of k3 is deleted for plane-stress deformation. For the axially symmetric case, ul and u2 represent Ur and uz, respectively, and P, becomes q J r .

Equation (6.25) can be written in the matrix form as

~= By

where B is called the strain-rate matrix and written as

(6.26)

L

YI 0 112 0 Y3 0 Y4 (6.27) B=/PI o P2 o P3 o P4 o

Y1 X1 ]I2 X2 Y3 X3 Y4 X4

The number of columns of the B matrix is determined by the number of degrees of freedom allowed to the element.

The evaluation of the strain-rate matrix, or of X~, Y,,, and Z,,, requires the differentiation of shape functions with respect to the global coordinate.

104 Metal Forming and the Finite-Element Method

Since the shape functions are expressed in the natural coordinate system, it is necessary to express the global derivatives in terms of the derivatives with respect to the natural coordinate. Consider a coordinate transforma- tion given by eq. (6.19), where shape functions are defined in the natural coordinate system. Then the derivatives of the shape functions with respect to the natural coordinate system can be expressed, using the chain rule, as

( aq,Jc~ I f Oq,~/ ax" aq~l a~l ~ = dJ aq,/ ay (6.28)

aq~/acJ Laq~laz. where J is the Jacobian matrix of the coordinate transformation, given by

r o.ao ay/a~ az/ae 1 J=|ax/a,7 ay/a~ az/a,7| (6.29)

L ax/a~ ay/a¢ az/a~j Then the derivatives in eq. (6.23a) can be obtained as {x,~} faqo/ax] faq,~/a~"

vo : ! aq:l ay t : J- '! aqol a,~ Z~ I.aqo~lazJ I. aqo~la¢~

(6.30)

iaqo [ ay ay ]faq~] ax~ 1 aE ~,/~1~ aq./=~ ax ax //aqo/ (6.31)

where IJI is the determinant of the Jacobian matrix and expressed by

axay axay I d l - - - - ( 6 . 3 2 )

aLl aL2 aL2 aLl

The strain-rate matrix of a linear triangular element, shown in Fig. 6.4a, can be obtained in a closed form by substituting q~=L~, q2=L2,

where j -1 is the inverse of J. It may be mentioned that in plane-strain deformation, the strain rate ~3

is not necessary, since it is always zero. However, it is convenient to include e3 in eq. (6.25) so that the strain rate matrix B of the plane strain deformation has the same form as that of the axisymmetric deformation as shown in eq. (6.27).

Triangular Element Family The strain-rate matrix of the triangular family of elements can be derived by applying eq. (6.30) to the shape functions given in eqs. (6.10) and (6.11). Since the area coordinates are not independent of each other, we can eliminate L3 from the expressions of qo, by using L3 = 1 - L ~ - L2. Equation (6.30) can be written for triangular element as

The Finite-Element Method---Part I

q3 = 1 - L1 - L2, and is written as follows:

where

1 X, = I-~ (Y2 - Y3),

- 1 Y, = ] ~ (x2 - x3),

X 3 = - X 1 - X2,

- 1 X2 = I - ~ (Yl - Y3)

1 Y2 = I-~ (X, - x3)

Y3 = - Y, - Y2

105

(6.33a)

IJI = (x l - x3) (yz - y3) - (x2 - x g ( y , - y3) ( 6 . 33b )

Note that I,!1 is twice of the area of the triangle. It may also be noted that all the strain-rate components of a linear triangular element are constant over one element, since X~, and Y~ of eq. (6.33) are not functions of the area coordinates.

The strain-rate matrix of the quadratic triangular element can be derived in a similar manner to that shown above. However, the expressions for X~ and Y~ are much more complex, and it is easier to evaluate these numerically following procedures similar to those used for the linear element. It may be also mentioned that the strain-rate is not constant and varies within the quadratic element.

Rectangular Element Family

For the rectangular family of elements, X~ and Y~ in eq. (6.25) can be written as

r ay a y q f a q ~ ]

(6.34a) LY~,=-~I 91 a x | | a q ~ |

a,7 a ~ J t an ) where I,II is the determinant of the Jacobian matrix of eq. (6.15) and is expressed by

a x a y a x a y IJI - a~g aT/ a'o a~j (6.34b)

For a quadrilateral element with the numbering of nodes shown in Fig. 6.5c, X,,, Y~, and I,II can be expressed in the closed form as

)(2 = ~ ~ - y , 3 + Y34~ + y,4T/

X 3 / - Y 2 4 4- Y12~ - YI4T//

X4 k. Y13 - Y I 2 ~ +Y23 'OJ (6.35a)

II2 = x13 -- X34~ -- XI4T/~

Y3 x24 - x12~ ']- X l 4 l ~ |

Y4 . --x13 --I- x12 ~ -- x231~ j

106 Metal Forming and the Finite-Element Method

and IJI is expressed by

Idl = .X[(x13 • Y24 -- X24" YI3) -I- (X34- Y12 - Xlz" Y34)~ + (X23" Y14 -- XI4" yz3) r/] (6.35b)

where Xu = xt - xj and Yu = Y t - - Y~. For higher-order elements, it is easier to evaluate X~ and Y,, numerically

according to eq. (6.30).

Three -Dimens iona l Br ick E l e m e n t

The strain-rate matrix, or X~,, Y,,, and Z,, for a three-dimensional brick element, can be derived by extending the derivation for the rectangular family elements• Rewriting eq. (6.22) in the matrix form, we have

, o ,

Ex

~ u u , ~ Z ~

Yxy ~z

~. ~ZJc

5". x~u~ ~ Z r~u~ ~ E Z~u~ ~

E (Y~u~/0 + Xo, u~ ~'))

E (Z,,u(y °') + Y',u~)) Y. (X~u~ ~ + Z~u~ ~)

(6.36)

By using procedures similar to those of rectangular element family, we can derive the differential operator as

0 o I - - IOx I

1 - ~ ,=a- ' ,' ~ ' lay an l

l a a~ _ _ - - |

l az a~l

with ~_~ az ay az ay az ay az ay az ay az "

ax az ax az ax az ax Oz ax az ax az

1 l ax ay axay ax ay ax ay ax ay axay a,7 a¢ a¢ ao a~ a¢ + a¢ a~ a~ art art a~

(6.37)

where [JI, the determinant of the Jacobian matrix, is given by

ax ay az ax ay az ax ay az ax ay az ax ay az ax ay az I,JI - a~ a~ at; ~ at; a~ a~ ~- a~ at; a~ a¢ a~ at5 a~ a~g a¢ a~ at; a~

(6.38)

The evaluation of X~, Y~,, and Z,, in eq. (6.36) can be made by using eqs.

The Finite-Element Method--Part I

(6.37) and (6.38). The strain rate matrix B becomes

B =

"x1 o 0 x2 0 0 x3 o . . . x s 0 0

o Y, o o Y2 o o Ya . . . o Ys 0

0 0 Z 1 0 0 Z 2 0 0 . . . 0 0 Z 8

Y, X , 0 Y2 X2 0 Y3 X3 . . . Ys Xs 0

o z , Y, o z2 Y2 o z3 . . . o 7_, Y~

z~ o x , z~ o x~ z~ o . . . zs o x~

107

(6.39)

(~)2 = ~TD/~ (6.40)

The diagonal matrix D has z3 and ] as its components; corresponding to normal strain-rate and engineering shear strain-rate, respectively. Substi- tution of eq. (6.26) into eq. (6.40) gives

(~)2 ~__ vTBTDBv = vTpv (6.41)

where P = BTDB. The matrix D in eq. (6.40) takes different forms depending upon the

expression of the effective strain-rate, in terms of strain-rate components. For example, the effective strain-rate in plane-stress problems is expressed in a different form from that of plane-strain problems, although the definition of the effective strain-rate is identical in both cases. The matrix D written for plane-stress problems is not diagonal (Chap. 11). The expression of the effective strain-rate also depends on the yield criterion. Thus, the matrix D is different for anisotropic materials (Chap. 11) and for porous materials (Chap. 13), and is described in corresponding chapters.

The volumetric strain-rate ~ is given by

and expressed by k~ = CTv = Clv, (6.42)

with Ct = Bll + B21 + B31, where Bu is an element of the strain-rate matrix B.

or, in the matrix form

Matrices o f Effective Strain-Rate and Volume Strain-Rate

In the finite-element formulation for the analysis of metal forming, the effective strain-rate ~ and the volumetric strain-rate k~ are frequently used. Therefore, it is necessary to express the effective strain-rate and volumetric strain-rate in terms of the strain-rate matrix.

The effective strain-rate is defined in terms of strain-rate components, according to eq. (4.39) in Chap. 4, as

108 Metal Forming and the Finite-Element Method

6.5 Elemental Stiffness Equation

It can easily be seen from the way in which the element was introduced that the global integrals over the whole workpiece stem from the assembly of integrals over the local domain of disjoint finite elements. Therefore, it is convenient to evaluate the stiffness matrix given by eq. (6.3) at the elemental level, and to assemble into a global stiffness matrix.

Of the two variational formulations (5.21) and (5.22) in Chap. 5, we first discuss the penalty function method, eq. (5.22). Denote the first, second, and third terms (including signs) of eq. (5.22) 6~o, 6~e, 6:rsF, respectively. In metal forming, the boundary conditions along the die- workpiece interface are mixed (see Section 5.4 of Chap. 5). Therefore, along the interface Sc the treatment of traction depends on the friction representation. This aspect of the surface integral terms is discussed in Chap. 7.

Using the discrete representation of the quantities involved in 6:r that are developed in Sections 6.3 and 6.4, we can express the integrals of 6:r in terms of nodal-point velocities. Equation (6.3) then becomes

an = a~o + O~zp -I- O:rsv Ovt art art Ovt

where

Ost° Iv O = zP~jvjdV av~ e

O~te = fv KCjv~CI dV Ovt

O~sF = - f F,N,, dS av~ JsF

(6.43)

It should be noted that the term (-O~rsffOvl) is the applied nodal point force and that Oaro/OVl + O~tt,/Ovt is the reaction nodal point force.

The second derivatives of :t are expressed as

avtvj ~ PH dV + ~ Ok e 21 k PIKVKVmPMJ dV + KGC, dV

(6.44)

Evaluating stiffness matrices at the elemental level from eqs. (6.43) and (6.44), assembling them for the whole workpiece, we obtain a set of simultaneous linear equations (6.4).

When effective strain-rate ~ approaches 0, or becomes less than a preassigned value eo, we have, by eq. (5.31) in Chap. 5,

= fv ° dv = f,v ~ k 6k (6.45)

The Finite-Element Method---Part I

where 0o/~o = constant. The derivatives of ~tD can be expressed by

= P , , , , , dV

109

(6.46)

av~avj Jv eo

The penalty constant K and the limiting strain rate to are introduced rather arbitrarily for computational convenience. However, proper choices of these two constants are important in successful simulation of metal- forming processes. A large value of K is, in general, preferred to keep the volume strain-rate close to zero. However, too large a value of K may cause difficulties in convergence, while too small a K results in unaccept- ably large volumetric strain. Numerical tests show that an appropriate K value can be estimated by restricting volumetric strain rate to 0.0001-0.001 times the average effective strain-rate.

The limiting strain-rate, to, under which the material is considered to be rigid, has been introduced to improve the numerical behavior of the rigid-plastic formulation [7]. Too large a value of the limiting strain-rate results in a solution in which the strain-rate of the rigid zone becomes unacceptably large. On the other hand, if we choose too small a value of limiting strain-rate, then the convergence of the Newton-Raphson method deteriorates considerably. Numerical tests show that an optimum result can be obtained by choosing the limiting strain rate as ~ of the average effective strain-rate.

Equation (5.21) is the basic formulation using a Lagrange multiplier in order to remove the incompressibility constraint. It replaces 6~re in eq. (5.22) by 6~rx = f 6 (~ i~ )dV . The Lagrange multiplier ~, is treated as an independent variable and a function of material points defined over the workpiece. The Lagrange multipliers are defined at points where the volume constancy is enforced and are introduced at the reduced integra- tion points (see Section 7.1 of Chap. 7). For example, one variable is assigned for each linear rectangular [8] and three-dimensional brick element, and four 3. are needed for a quadratic rectangular element.

Since 6~, is also arbitrary in eq. (5.21), we obtain a set of simultaneous equations given by

a~ a~ = 0 and - - = 0 (6.47)

av~ a3.

The first equation of (6.47) is nonlinear and eq. (6.3) applies for linearization. The additional terms necessary for evaluation of matrices in the stiffness equation are

a=X = fv C, dV, a=Z fv avl 03. = Civl d V

Then eqs. (6.3) and (6.47) can be arranged in the system of equations

110 Metal Forming and the Finite-Element Method

linear in Avj and ). as

aE:tx 8

(6.48) 4

O~. a v f f - \ 8~ /,o

Each coefficient appearing in eqs. (6.48) can be evaluated at the elemental level and assembled into the global stiffness equation.

It is difficult to assess which method, the penalty function method or the Lagrange multiplier method, is better in the finite-element implementa- tion. However, it can be mentioned that the Lagrange multiplier method introduces a larger number of independent variables. Also, the diagonal terms of the stiffness matrix corresponding to the ~. always become zero and special attention is required during the assembly of the stiffness matrix so that an equation corresponding to any A does not become the first one in the stiffness equations after applying the boundary conditions.

References

1. Courant, R., (1943), "Variational Methods for the Solution of Problems of Equilibrium and Vibrations," Bull. Amer. Math. Soc., Vol. 49, p. 1.

2. Ciough, R. W., (1960), "The Finite Element Method in Plane Stress Analysis," J. Struct. Div., ASCE, Proc. 2nd Conf. Electronic Computation, p. 345.

3. Bathe, K. J., and Wilson, E. L., (1976), "Numerical Methods in Finite-Element Analysis," Prentice-Hall, Englewood Cliffs, NJ.

4. Oden, J. T., (1972), "Finite Element of Nonlinear Continua," McGraw-Hill, New York.

5. Desai, C. S., and Abel, J. F., (1972), "Introduction to the Finite Element Method," Van Nostrand Reinhold, New York.

6. Hildebrand, F. B., (1974), "Introduction to Numerical Analysis," 2d Edition, McGraw-Hill, New York.

7. Chen, C. C., Oh, S. I., and Kobayashi, S., (1979), "Ductile Fracture in Axisymmetric Extrusion and Drawing, Part I: Deformation Mechanics of Extrusion and Drawing," Trans. ASME, J. Engr. Ind., Vol. 101, p. 23.

8. Lee, C. H., and Kobayashi, S., (1973), "New Solutions to Rigid-Plastic Deformation Problems Using a Matrix Method," Trans. ASME, J. Engr. Ind., Vol. 95, p. 865.