A pseudo-layered, elastic-plastic, flat-shell finite element

H. Darendeliler et al. / Comput. Methods Appl. Mech. Engrg. 174 (1999) 2 1 1 - 2 1 8

,P

o'~, + ~

o'k,} 8Dk; - 2o',,;D~,, ~Dk, + o'm, 8x-~ ~

dV° =

. 7~k 8vk dA°

213

(9)

In the updated Lagrangian formulation, the Cauchy and Kirchoff stress tensors, o" and ~-, and their rates are related as /9o 7-k;= p ok; *

7-* :

+

(10a) i)V.,

,,,,

(10b)

For cases where Po/P ~ 1 and Or.,/3x k < < 1, one obtains

7"* = C~;,.,,D,,,.

( 11 )

where Ck;.... is the elastic-plastic constitutive tensor. Hence, (9) takes the form

fv(

Ck;""D"" ~Dk;

-

20"m;Dkm~Dk; +

°'m; Ox; ~ x , . J d r =

fA

7~k 3v k dA

(12)

where V and A are the volume and surface area of the configuration at time t. For convenience, the second-order tensors D O and Ov/OXi can be expressed as vectors d and w as

d T : [Dl, W

T

= [Vl, I

D22

2D12 2D23

Vl, 2

Vl, 3

V2, I

2D3j] U2, 2

V2, 3

(13a) V3,1

O3,1]

(13b)

and the fourth-order tensor C,jk; is expressed as a matrix C which can be formed by utilizing the following relation

=

.

~6q6,,

m~,]o',,

(14)

where m equals zero for unloading and elastic loading, and equals unity for plastic loading, v is the Poisson's ratio, E is the elasticity modulus, o" is the deviatoric stress tensor, ~- is the equivalent stress, h is the slope of Cauchy stress-logarithmic plastic strain curve for a simple tension test, and is Kronecker's delta. Then, (12) can be written as

fv (SdT Cd-- 2 ~dT ~ l + ~wT O w ) d V = fA ~ T f" da

(15)

where 6" and 6" are appropriately formed matrices.

3. The fiat shell finite element In this section a layered flat shell element will be developed for elastic-plastic analysis. It must be noted that the discretization of the element thickness by layers is fictitious, and it serves to assign elastic or plastic properties at a finite number of locations through the thickness of the element in an easy and systematic manner depending on the current stress states at those locations. The formulation is based on the shear-deformable theory of Mindlin. In this theory, the velocity field of a flat shell element whose midsurface coincides with the x~x2-plane can be expressed as o

v j(x~, x2) = v~(x~, x2) + x332(x,, x2)

v2(x , , x 2) = v°(xl, x2) - x 3jS~ (xj, x2)

(16)

where v; are the velocity components at a point of the shell, vl~ are the velocities at the midsurface and/3,, are the

H. Darendeliler et al. / Comput. Methods Appl. Mech. Engrg. 174 (1999) 211-218

214

rate of change of normal rotations. By employing the anisoparametric interpolation functions suggested by Tessler and Hughes [15] to avoid shear-locking, v~ and ~ can be assumed for a three-node element in terms of nodal variables as 0 i 0 UI ~--- ~i UI 0 U2=~

i 0 I U2

o

io

1

1

03 ~- ~t U3 q- 2 (bj~k(i -- bk~i~j)il~l q- 2 (aj~k~ i _ ak~i~j )i~2

(17)

~&

& = <

where sci are the triangular area coordinates, ai = xjk - X l j and bs = x2j - X2k, and the permutation is i = 1, 2, 3; j = 2, 3, 1; k = 3, 1, 2. The left superscript indicates the node number. Substituting (17) into (16), the velocity vector can be expressed in terms of the element degrees of freedom q as

(18)

v =Nq where q

T

1 0 = [ UI

I 0 U2

I 0 U3

Ifil

2 0 Ol''"

lfi2

3fl 2]

(19)

Then, the vectors d and w of (14a-b) can be expressed as n

0

m

DI. 1 -~- X3 t~2, t 0 02, 2 -- X 3 ~ I . 2

d=

O01,2 + 02,1 0 .+. X 3 ( ~ 2 2 0 03.2 -- 131

__ ~ t , i )

(20)

0 +f12 i03,l "

i

V lo, l q_x3/~2

-

U01,2 -}- X3 J~2,2

& 0

U2.1 -~- X3t~I, 1 W -----

0 U2, 2 -}- X3]~I. 2 0 ['/3,l 0 U3, 2

(21)

m

The vectors d and w can be rewritten as

d = HBq

(22a)

w = GRq

(22b)

where the matrices G and H are functions of x 3 and the matrices B and R are appropriately formed so as to express d and w in terms of element degrees of freedom vector q. Then, (14) can be written as

fz where

(l~)qT

BT-CBq - 2 NIT BT grBq + NIT RgrRq) da = fA NIT N ~ f. dA

(23)

H. Darendeliler et al. / Comput. Methods Appl. Mech. Engrg. 174 (1999) 2 1 1 - 2 1 8

0.0

215

n

-2.0 -4.0 &

E

E t..

-6.0

&

O []

.u0

-8.0

[]

13

* FEM []

-10.0

O Experiment 1 [] Experiment 2

-12.0

A Experiment 3 -14.0

0

I

,

,

I

,

,

,

,

=

13

26

39

52

65

78

91

104

117

130

Distance from the clamped end (mm) Fig. 1. Deflections of the plate along the span in the first case for a total load of 175 N. The value of the cut angle is 0 °, 45 ° and 90 ° for experiments 1, 2 and 3, respectively.

C. =

H x C H dx 3

(24a)

/-/TO/'/dx 3

(24b)

GT~

(24c)

n=l

o" = n=

I

Jh.

~i"= n--I

dx 3

Jh~l

where L is the number of layers and h. is the distance from the midsurface o f the shell to the bottom of the nth

Fig. 2. Plastic zones (shaded) through the thickness of the plate along the span in the first case for a tip deflection of 4.0 mm (not to scale).

H. Darendeliler et al. / Comput. Methods Appl. Mech. Engrg. 174 (1999) 2 1 1 - 2 1 8

216

layer. The layers are numbered from bottom to top. Hence, the stiffness matrix and the rate of load for an element in local coordinates are obtained as

k = fa (B TC.B - 2B r~rB + RO'R) dA

(25a)

f = fA NTT" dZ

(25b)

The element matrices and vectors are transformed and assembled to form the system equations and the solution is obtained by using Newton-Raphson method. In the integration from time t to t + At, it is assumed that the rates are constant within At and all rate quantities are multiplied by At to obtain the incremental quantities. The size of the time step, At, is controlled by using the r-min method [16] to minimize the integration errors.

4. Experimental and numerical studies In this section, the experimental and numerical results are compared for three cases. The experimental results for the displacements have been obtained by Digimar CX1 height measuring instrument. For the first two cases, three experiments have been carried out by using plates cut from a sheet in 0 °, 45 ° and 90 ° directions to compensate for the effects of material anisotropy. The stress-strain relation in the plastic range has been obtained by simple tension tests. In the first case, a rectangular plate of 120mm × 130mm is clamped along a 120mm side and loaded gradually along the opposite 120 mm side with a transverse line load. The plate thickness is 2 mm, the elastic modulus is 210 GPa, yield strength is 295 MPa and Poisson's ratio is 0.3. The stress-strain relation in the plastic range is o- = 670,~°19. In the finite element analysis, a 200-element mesh has been used for the entire plate. The

0.0

[]

-1.0

B

.... -2.0 E E o -3.0

a

• FEM

-4.0

•

o Experiment 1 -5.0

• • o

a Experiment 2 ~ Experiment 3

-6.0

0

I

I

I

I

I

i

i

i

i

26

52

78

104

130

156

182

208

234

260

Distance from a clamped end (ram) Fig. 3. Deflections of the plate along the span in the second case for a total load of 2928 N. The value of the cut angle is 0 °, 45 ° and 90 ° for experiments 1, 2 and 3, respectively.

217

H. Darendeliler et al. / Comput. Methods Appl. Mech. Engrg. 174 (1999) 2 1 1 - 2 1 8

Fig. 4. Plastic zones (shaded) through the thickness of the plate along the span in the second case for a center deflection of 4.0 mm (not to scale).

10o 100 8O

80 6O

60

z z o

40

tt. 40

20

/

20

--O-- Experiment

0

---o- Experiment

/

/ 0

/

J

i

J

L

i

i

i

i

20

40

60

80

100

120

140

160

Displacement (mm)

Fig. 5. Deflections of the loaded comer in the third case.

0 180

i

i

J

i

L

i

i

20

40

60

80

100

120

140

160

180

Displacement (mm)

Fig. 6. Deflections of the unloaded comer in the third case.

displacements obtained by experiments and finite element analysis are compared along the span in Fig. 1. Fig. 2 shows the plastic zones in the plate for a 4.0 mm tip deflection. In the second case, a rectangular plate of 120mm × 260 mm is clamped along the 120mm sides and loaded gradually with a central transverse line load parallel to clamped edges. The material properties are the same as those of the first case. Half of the plate has been discretized by 100 elements. The displacements obtained by experiments and finite element analysis are compared along the span in Fig. 3. Fig. 4 shows the plastic zones in the plate for a 4.0 mm center deflection. In the third case, a rectangular plate of 100mm × 3 0 0 m m is clamped along a 100mm side and loaded gradually at one comer of the opposite 100 mm side with a transverse point load. The plate thickness is 2 mm, the elastic modulus is 210 GPa, yield strength is 253 MPa and Poisson's ratio is 0.3. The stress-strain relation in the plastic range is o-= 617e °25. The plate has been discretized by 560 finite elements. The numerical and experimental results for the displacements of the loaded and unloaded corners of the free end are plotted with respect to the applied load in Figs. 5 and 6, respectively. It is seen that numerical and experimental results are in good agreement. The small discrepancies between the results can be attributed to using constant values for material properties that might be different throughout the plate due to anisotropy and nonhomogeneity. In the first case, the plate is stiff due to its geometry. In the second case, the stiffness is due to the boundary conditions. Hence, in both cases, the material yields before large displacements occur. However, in the third case, the plate is flexible enough to allow large displacements prior to the plastic deformations.

218

H. Darendeliler et al. / Comput. Methods Appl. Mech. Engrg. 174 (1999) 2 l l - 2 1 8

5. Conclusion A triangular flat shell finite e l e m e n t based on M I N 3 plate bending e l e m e n t has b e e n d e v e l o p e d for large strain and large deformation elastic-plastic analysis of plates and shells. The material is a s s u m e d to be isotropic and o b e y i n g the M i s e s yield criterion. The w o r k hardening characteristics o f the material are incorporated. In the formulation, the updated L a g r a n g i a n m e t h o d is used and all nonlinear terms are included. The e l e m e n t is effective and easy to use in m o d e l i n g arbitrary plate and shell g e o m e t r i e s due to its three-noded configuration and C ° - t y p e nodal variables. The layered structure o f the e l e m e n t enables easy and accurate determination o f the plastic zones through the thickness. The accuracy o f the e l e m e n t has been verified in both g e o m e t r i c and material nonlinear p r o b l e m s by c o m p a r i n g the numerical results with those obtained by experiments. A g o o d a g r e e m e n t b e t w e e n the results has been observed. Hence, the e l e m e n t can effectively be used in the numerical analysis o f sheet metal forming processes.

References [1] R.M. McMeeking and J.R. Rice, Finite-element formulations for problems of large elastic-plastic deformation, Int. J. Solids Struct. 11 (1975) 601-616. [2] M. Gotoh, A finite element analysis of general deformation of sheet metals, Int. J. Numer. Methods Engrg. 8 (1974) 731-741. [3] J.C. Nagtegaal, D.M. Parks and J.R. Rice, On numerically accurate finite element solutions in the fully plastic range, Comput. Methods Appl. Mech. Engrg. 4 (1974) 153-177. [4] J. Backlund and H. Wennerstrom, Finite element analysis of elasto-plastic shells, Int. J. Numer. Methods Engrg. 8 (1974) 415-424. [5] J.H. Argyris and M. Kleiber, Incremental formulation in nonlinear mechanics and large strain elasto-plasticity-natural approach. Part I, Comput. Methods Appl. Mech. Engrg. 11 (1977) 215-247. [6] J.H. Argyris, J. St. Doltsinis and M. Kleiber, Incremental formulation in nonlinear mechanics and large strain elasto-plasticity-natural approach, Part II, Comput. Methods Appl. Mech. Engrg. 14 (1978) 259-294. [7] J.H. Argyris and J. St. Doltsinis, On the large strain inelastic analysis in natural formulation, Part I: Quasistatic problems, Comput. Methods Appl. Mech. Engrg. 20 (1979) 213-251. [8] S.1. Oh and S. Kobayashi, Finite element analysis of plane-strain sheet bending, Int. J. Mech. Sci. 22 (1980) 583-594. [9] J.H. Argyris, H. Balmer, M. Kleiber and U. Hindenlang, Natural description of large inelastic deformations for shells of arbitrary shape-application of trump element, Comput. Methods Appl. Mech. Engrg. 22 (1980) 361-389. [10] J.C. Simo and J.G. Kennedy, On a stress resultant geometrically exact shell model. Part V. Nonlinear plasticity: formulation and integration algorithms, Comput. Methods Appl. Mech. Engrg. 96 (1992) 133-171. [11] E.N. Dvorkin, D. Pantuso and E.A. Pepetto, A finite element formulation for finite elasto-plastic analysis based on mixed interpolation of tensorial components, Comput. Methods Appl. Mech. Engrg. 114 (1994) 35-54. [12] E.N. Dvorkin, D. Pantuso and E.A. Repetto, A formulation of the MITC4 shell element for finite strain elasto-plastic analysis, Comput. Methods Appl. Mech. Engrg. 125 (1995) 17-40. [13] K.J. Bathe and H. Ozdemir, Elastic-plastic large deformation static and dynamic analysis, Comput. Struct. 6 (1976) 81-92. [14] R.K. Jain and J. Mazumdar, Research note on the elastic-plastic bending of rectangular plates: a new approach, Int. J. Plasticity 10 (1994) 749-759. [15] A. Tessler and T.J. Hughes, A three node Mindlin plate element with improved transverse shear, Comput. Methods Appl. Mech. Engrg. 50 (1985) 71-101. [ 16] Y. Yamada, N. Yoshimura and T. Sakurai, Plastic stress-strain matrix and its application for the solution of elastic-plastic problems by the finite element method, Int. J. Mech. Sci. 10 (1968) 343-354.