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Finite Elements in Analysis and Design 39 (2003) 343354

www.elsevier.com/locate/nel

Collapsible shell nite element for composite materials

Ivelin Ivanov, Ala Tabiei

The Center of Excellence in DYNA3D Analysis, Department of Aerospace Engineering and Engineering Mechanics,

University of Cincinnati, Cincinnati, OH 45221-0070, USA

Received 6 February 2001; accepted 29 August 2001

Abstract

A higher order shell element for the explicit nite element method is presented. The development is intended

for simulation of composite materials crush and collapse. The element has cubic polynomials for displacement

interpolation in the direction of collapse. The proposed element oers the opportunity to capture the stress

gradient more accurately and consequently model degradation of composite material in direction of impact.

The element will also allow the usage of a coarse mesh. The developed formulation is incremental, the element

is iso-parametric and based on degeneration technique. Some validation samples are presented. The critical

integration time step for the element is also presented. In addition, the maximal natural frequency boundary

is also investigated. Finally, conclusions are drawn concerning the usage of high-order shell elements in crashsimulations of composite materials. ? 2002 Elsevier Science B.V. All rights reserved.

Keywords: High-order shell element; Explicit nite element; Composite material crushing; Crash simulations

1. Introduction

Composite materials are widely used in vehicle transportation. Besides their low-weight and

high-strength characteristics, the high-kinetic energy absorption capability during crashes makes the

composite materials desirable structural material for crashworthiness. Crash simulation is becomingan ecient tool for vehicle development. The nite element method for dynamic problems with ex-

plicit time integration is an appropriate tool for crash simulations. With the advancement of computer

technology, today engineers can model full systems for crash simulation.

The explicit nite element method is conditionally stable and the critical time step of the elements

as well as their computational eciency are very important, especially for large-scale simulations.

Corresponding author. Tel.: +1-513-556-3367=3548; fax: +1-513-556-5038.

E-mail address: [email protected] (A. Tabiei).

0168-874X/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved.P I I : S 0168- 874X (02)00077- X

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I. Ivanov, A. Tabiei / Finite Elements in Analysis and Design 39 (2003) 343 354 345

2. Finite element formulation

2.1. Geometry

The nite element formulation of the cubiclinear shell element is based on the iso-parametric

mapping of a bi-unit cube into shell geometry and degeneration [7,8,6,9] of a 16-node brick element

to 8-node shell element with 8 pseudo-normals (bers) representing the shell thickness (Fig. 1). The

isoparametric mapping is described by the following functions:

x(;;) =

8I=1

NI(; )( xI + zI() xI); (1)

where x(;;) is the position vector of shell point having local coordinates (;;) in global

coordinate system; xI the position vector of node I; I = 1; 2; : : : ; 8, xI the unit vector in the berdirection at node I; NI(; ) the surface (lamina) shape function at node I, zI() = ( I)hI=2the ber shape function at node I, hI the ber length at node I, I the position of the node I

through the thickness of the shell (1at the lower surface, 0at the middle, +1at the uppersurface).

The lamina shape functions are product of a linear polynomial with respect to and a cubic

Lagrangian polynomial with respect to as follows:

NI =( a)( b)( c)( d)

(I a)(I b)(I c)(I d)(2)

(1,1,1)

(-1,-1,-1)

x

y

z

1

2

3

4

5

6

7

8

x

I

0

+1

-1

xI

xI

hI

I

Fig. 1. Isoparametric mapping of a bi-unit cube into the lamina geometry and degeneration of a brick element to the

cubiclinear shell elements.

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346 I. Ivanov, A. Tabiei / Finite Elements in Analysis and Design 39 (2003) 343 354

e1

l

e3

l

e2

l

=const.=const.

=const. surface (lamina)

Fig. 2. Orthogonal basis vectors of lamina coordinate system.

with

I 1 2 3 4 5 6 7 8

I 1 +1 1 +1 1 +1 1 +1a +1 1 +1 1 +1 1 +1 1I 1 1 1=3 1=3 +1=3 +1=3 +1 +1b 1=3 1=3 1 1 1 1 1 1c +1=3 +1=3 +1=3 +1=3 1=3 1=3 1=3 1=3d +1 +1 +1 +1 +1 +1 +1=3 +1=3

At each integration points, one needs to transform the strain increment and stress components from

global to local (lamina) coordinate system (Fig. 2) for anisotropic materials, because the constitutive

law is known in this co-ordinate system. The lamina reference frame is dened by its orthogonal basis vectors el1; e

l2; e

l3. The vector e

l3 perpendicular to the lamina could be easily determined by the

vectors tangent to the - and -directions as follows:

e =x;

x; ; e =

x;

x; ; el3 =

e ee e

: (3)

The other two vectors are chosen to be equally close to the tangent vectors e and e. Using

complementary vectors e and e, the basis vectors are determined [8]:

e = e + e; e = el3 e; (4)

el1 =e e

e e

; el2 =e + e

e + e

: (5)

Consequently the transformation matrix

Q = [el1 el2 e

l3]

T (6)

is used to transform quantities from global to the lamina coordinate system.

2.2. Kinematics

The same parametric representation used to describe the geometry is used to interpolate the

nodal displacements, i.e. isoparametric representation. An incremental formulation of displacements

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is given, because the shell formulation is purposed for non-linear solution using the explicit nite

element method:

u(;;) =

8I=1

NI(; )(uI + zI()uI); (7)

where u(;;) is the incremental displacement vector at any point of the shell, uI the incremental

translational displacement vector of node I, uI the displacement vector of the unit ber vector at

node I.

The shape functions are the same and the current deformed conguration at step n + 1 of time

integration is updated using the following:

xn+1 = xn + un; (8)

xn+1

I = xn

I + un

I; (9)

xn+1I = xn

I + un

I: (10)

The incremental translation displacement vector uI is updated from the translation equation of

motion. The displacement vector of the unit ber vector uI is updated from the rotational equa-

tion of motion, by applying the nodal incremental rotations, 1I, 2I and 3I. A second-order

technique referred as a HighesWinget formula is used for updating the unit ber vectors [6,9]:

xn+1I = RT

I xn

I; (11)

where

RI = I3 (2SI + SIST

I )DI; (12)

SI =

0 3I 2I

3I 0 1I

2I 1I 0

; (13)

DI =2

4 + 21I + 22I +

23I

; (14)

where I3 is the 3 3 identity matrix.

2.3. Stress update and internal forces

The strain and spin increments are calculated from the displacement increment gradient as follows:

G =@u

@x; (15)

U = 12

(G + GT); (16)

G = 12

(G GT): (17)

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The stress tensor is updated in rotated current conguration n + 1 by using the Jaumann rate [6]

from previous conguration n as follows:

A

n+1

=A

n

+ GA

n

+A

n

G

T

: (18)Then the stress tensor is transformed in the lamina coordinate system as well as strain increment

tensor using the following transformation:

Al = QAn+1QT; (19)

UT = QUQT: (20)

The increment of stress vector is calculated by using the constitutive model of the material and the

stresses in the lamina coordinate system are nally updated by the following equation:

Al = Al + CUl: (21)

Updated stress tensor is transformed back to the global coordinate system by the following transfor-mation:

A = QTAlQ: (22)

The stress vector in global coordinate system is now used to update the internal force vector

below:

fint =

V

BTA dV =

11

11

11

BTA|J| d d d; (23)

where B is straindisplacement matrix determined by the following equation:

U

= B u

u

(24)

and |J| is the determinant of the Jacobian matrix given by J = [@x=@@x=@@x=@]T.We apply numerical integration to obtain the internal forces from (23). The integration rule is

selectively reduced integration at Gauss quadrature integration points. We have reduced integration

for shear strain components of straindisplacement matrix as well as for shear stress components.

3. Maximum natural frequency

The maximal natural frequency of high-order nite elements is higher than the low-order elements

with the same distance between the nodes. We can show the dierence in one-dimensional problemcomparing 4-node cubic bar element to 2-node linear bar element. Both elements have equal distance,

l, between the nodes, cross-sectional area A, Youngs modulus E and mass density . The critical

time step for linear bar element with lumped mass matrix is [2]

tlin = l

E: (25)

The shape functions of the cubic bar element are

NI =(x xa)(x xb)(x xc)

(xI xa)(xI xb)(xI xc); (26)

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I 1 2 3 4

xI 0 l 2l 3lxa l 0 0 0

xb 2l 2l l l

xc 3l 3l 3l 2l

Calculating the stiness matrix K from shape function derivatives, we obtain

KIJ = AE

3l0

NI;xNJ;x dx = K =AE

120l

148 189 54 13

189 432 297 54

54 297 432 189

13 54 189 148

: (27)

The same lumped masses are assumed for the nodes as for the linear bar element and the lumped

mass matrix of the cubic bar element is

M = 3Al

1=4 0 0 0

0 1=4 0 0

0 0 1=4 0

0 0 0 1=4

: (28)

Solving equation Det|K !2M| = 0 for ! we get the maximum natural frequency of the cubic barelement

!max =3:0162

l

E

(29)

and then the critical time step

tcubic =2

!max=

2l

3:0162

E= 0:6631tlin : (30)

Local buckling accompanies composite material crushing and it is related to the high geometrical

non-linearity of structures. For isoparametric shell element formulation with numerical integration, it

is not clear how the stiness matrix is changing during collapse and how this results in change of the

maximum natural frequency of the element. In order to investigate the maximum natural frequencyof a nite element model, one needs to solve the free vibration problem:

M d + Kd = 0: (31)

If the general solution of (10) is written as d = aei!t, then ! can be determined from the following:

(K !2M)a = 0: (32)

Substituting !2 = and multiplying on both sides by M1=2, one obtains:

|M1=2KM1=2 I| = 0: (33)

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which is a general eigenvalue problem for the following system matrix

R = M1=2KM1=2: (34)

The calculation of the system matrix R eigenvalues at each time steps during an explicit niteelement simulation is ambitious. Instead of system matrix R, one can use the element matrix Re,

which is based on the assertion that the maximum natural frequency of the system is less or equal

to the maximum natural frequency of any part of the system [2]:

!6

maxei ; i = 1; 2; : : : ; ne (35)

where maxei is the maximum eigenvalue of the matrix Rei for the element i:

Rei = M1=2ei KeiM

1=2ei : (36)

Here an investigation of the change of the frequency !b as a function of time is carried out:

!b = maxi

(

maxei ); i = 1; 2; : : : ; ne; (37)

The above natural frequency bounds the maximum natural frequency of the system. The following

element stiness matrix Ke is calculated by numerical integration using the same integration rule as

for the internal force calculations.

Ke =

V

BTCB dV =

11

11

11

BTCB|J| d d d: (38)

To investigate the change of the natural frequency during shell element deformation a couple of

examples are considered. These examples are presented in the next section.

4. Numerical examples

The nite element formulation of cubiclinear shell element was programmed in the Matlab soft-

ware with the central dierence explicit time integration scheme. In order to validate the program two

simple standard test samples were solved. A cantilever plate 1030 mm under tip transverse load of4 kN and dierent thicknesses were solved using the developed program. The results are compared to

the exact solutions and presented in Table 1. The deection at the free end is denoted by umax. The

stresses max and max are the maximum bending stress and the transverse shear stress, respectively.

These stresses are collected near the built-in end at the in-plane integration point (25 mm from the

free end). Good agreement with the exact solutions is observed. The HughesLiu shell formulationfor bi-linear shell element with reduced integration as well as with selective reduced integration was

also programmed for comparison of the maximum natural frequency. Viscosity hourglass control was

implemented for the bi-linear shell element with reduced integration in the program. The hourglass

control is the same as the default shell hourglass control in the public nite element code DYNA3D.

Both versions of the bi-linear shell element were validated as well.

To investigate the natural frequencies of the developed element and compare with the existing

reduced and selective=reduced integration elements two cases are considered. The plate shown in

Fig. 3 was modeled by means of 18 bi-linear shell elements with reduced and selective=reduced

integration and by means of 6 cubiclinear shell elements. The plate is clamped at edge A and a

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Table 1

Results for straight cantilever plate

Thickness Quantity Cubiclinear shell Exact solution

(mm) element

3 umax (mm) 7.7716 7.6039

max (MPa) 6343.1 6364.8

max (MPa) 132.38 133.33

5 umax (mm) 1.7566 1.7280

max (MPa) 2384.5 2400.0

max (MPa) 79.620 80.000

10 umax (mm) 0.23458 0.21600

max (MPa) 599.30 600.00

max (MPa) 39.952 40.000

x

yz

60 mm

30 mm

A

B

Young modulus = 200 GPaPoisson ratio = 0.3

Density = 7800 kg/mThickness = 3 mm

3

Fig. 3. Plate model for in-plane impact.

constant velocity in negative direction of y was prescribed for nodes of edge B, causing buckling of

the plate and simulating collapse. The maximum natural frequency calculated during the deformation

for the prescribed velocity of 10 m=s is shown in Fig. 4. A geometric imperfection in nodal coordi-

nates of edge B is imposed in order to initiate the buckling mode. The plate model is subjected to

dierent prescribed velocities 1, 10 and 20 m=s. The character of the maximum natural frequencychange is the same for the dierent values of the prescribed velocities. Other boundary conditions

are also considered for the analyzed plate. The results had the same trend as the one depicted

in Fig. 4.

The second sample investigated for the maximum natural frequency analysis is the arch shown

in Fig. 5. The arch is simply supported at edges A and C or clamped. The nodes of line B move

in the negative z-direction with a prescribed constant velocity causing snapping-through of the arch.

The variation of the maximum natural frequency as a function of time for the velocity of 10 m =s is

depicted in Fig. 6. The maximum natural frequency change is observed to be the same for dierent

values of the considered velocity1, 10 and 20 m=s.

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0 1 2 3 4 5 6 7 8 9 101.2

1.25

1.3

1.35

1.4

1.45

1.5

1.55

1.6

1.65x 10

6 Impact Velocity 10 m/s

Displacement, mm

Frequency,

Hz

bi-linear reduced

bi-linear selectivecubic-linear selective

Fig. 4. Maximum natural frequency of plate buckling.

xy

z

30 mm

A

B

Young modulus = 200 GPaPoisson ratio = 0.3

Density = 7800 kg/mThickness = 3 mm

3

C

15o

R=120 mm

Fig. 5. Arch model for transverse impact.

5. Discussion and conclusion

High-order elements are widely used in most nite element codes to solve specic problems. In the

explicit nite element codes however, where computational eciency is very important, high-order

elements are not desirable. Although high-order shell elements are not proper for general use in

explicit nite element code, they could be used to model the crush front of composite material

under impact loading. The restricted usage of the high-order shell elements at the crash front could

avoid the high computational expense associated with the element. The critical time step of the

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0 1 2 3 4 5 6 71.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

1.55

1.6x 10

6 Impact Velocity 10 m/s

Displacement, mm

Frequency,

Hz

bi-linear reduced

bi-linear selectivecubic-linear selective

Fig. 6. Maximum natural frequency of snap-through arch.

developed cubiclinear shell element in collapse is not very dierent from the widely used reducedintegrated shell elements according to the presented investigation. The critical time step of the cubic

linear shell element that fuses three regular shell elements is not less than the critical time step of

the reduced integrated bi-linear shell elements for a mesh, which is rened twice.

The cubiclinear shell element proposed here oers a better distribution of strains in direction of

the impact in crash simulations. The distribution of the strain could be used for gradual degradation

of composite material stiness as the crush front progresses. This more realistic representation of the

composite crush front improves the accuracy of composite crush simulations using the nite element

method.

Acknowledgements

The nancial support for the presented work is provided by the ACC. Computing support is

provided by the Ohio Supercomputer Center. Their support is gratefully acknowledged.

References

[1] A.K. Pickett, E. Haug, J. Ruckert, A fracture damaging law suitable for anisotropic short bre=matrix materials in an

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[2] T. Belytschko, T.J.R. Hughes, Computational Methods for Transient Analysis, North-Holland, Amsterdam, 1983,

pp. 1155 (Chapters 1 and 2).

[3] E. Haug, O. Fort, A. Tramecon, M. Watanabe, I. Nakada, Numerical crashworthiness simulation of automotive

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[4] D. Kohlgruber, A. Kamoulakos, Validation of numerical simulation of composite helicopter sub-oor structures under

crash loading, Annual Forum ProceedingsAmerican Helicopter Society, Vol. 1, 1998, pp. 340349.

[5] A.F. Johnson, C.M. Kindervater, D. Kohlgruber, M. Lutzenburger, Predictive methodologies for the crashworthiness

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[6] J.O. Hallquist, LS-DYNA3D. Theoretical Manual, Rev. 2, July 1993, Livermore Software Technology Corp.

[7] T.J.R. Hughes, W.K. Liu, Nonlinear nite element analysis of shells: Part I. Three-dimensional shells Comput. Methods

Appl. Mech. Eng. 26 (1981) 331362.

[8] T.J.R. Hughes, The Finite element Method: Linear Static and Dynamic Finite Element Analysis, Prentice-Hall,

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[9] K. Bathe, Finite element Procedures in Engineering Analysis, Sec. 5, Prentice-Hall, Englewood Clis, NJ, 1982,

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