Axisymmetric response of nonlinearly elastic cylindrical shells to dynamic axial loads

Int. J. Impact Enqng Vol. 13, No. 4, pp. 545-554, 1993 Printed in Great Britain

0734-743X/93 $6.00 + 0.00 ~i 1993 Pergamon Press Lid

AXISYMMETRIC RESPONSE OF NONLINEARLY ELASTIC CYLINDRICAL SHELLS TO DYNAMIC AXIAL LOADS R. GILAT, E. FELDMAN and J. ABOUD1 Department of Solid Mechanics, Materials and Structures, Fleishman Faculty of Engineering, Tel Aviv University, Ramat Aviv 69978, Israel

(Received I September 1992; in revised form 23 December 1992)

S u m m a r y - - T h e axisymmetric response of nonlinearly elastic cylindrical shells subjected to dynamic axial loads is analysed by using an incremental formulation. The material elastic nonlinearity is modeled by the generalized Ramberg-Osgood representation. The time-dependent displacements of the shell are assumed to be governed by nonlinear equations of motion based on the von Karman-Donnell kinematic relations; moreover, both in-surface and out-of-surface inertia terms are included. The finite difference method with respect to the spatial coordinate and the R u n g e - K u t t a method with respect to time are employed to derive a solution. Numerical results demonstrate the effect of the material nonlinearity on the deflections, stiffness matrices and dynamic buckling behavior of cylindrical shells.

1. I N T R O D U C T I O N

The analyses of the response and of the critical conditions for stability of thin-walled structures subjected to time-dependent in-surface compressive loads have been the subject of many studies. A list of references was presented in a review and in a recently published book by Simitses[1,2-1. Some of those studies were concerned with nonperiodic time-dependent axial loads applied to cylindrical shells made of elastic isotropic, anisotropic and composite materials. Material nonlinearity which is usually associated with inelastic behavior was considered by Lindberg and Florence [3] and Jones [4] who provided some experimental and analytical results, and by Liaw and Yang [5] who used the finite element method. The aim of the present work is to investigate the effect of elastic material nonlinearity, which might arise due to nonlinearly elastic behavior of materials, on the axisymmetric response of an initially isotropic cylindrical shell subjected to dynamic axial load. The generalized Ramberg-Osgood representation of the strain-stress relations is used to model the elastic material nonlinearity. The analysis is confined however to small strains and moderate deflections. An incremental approach which is commonly used in plasticity analysis is offered for the present nonlinearly elastic analysis. The incremental formulation enables a linearization of the constitutive relations within each small time increment. The resulting linearized strain-stress relations involve a stress-dependent compliance matrix, which is inverted in order to establish the constitutive stress-strain relations. Thus, as a result of this formulation it is possible to establish explicit stress-strain relations without the need to solve at each time increment a system of nonlinear equations. In the framework of the classical shell theory the time-dependent deformation within each time increment is governed by the incremental nonlinear equations of motion based on the yon Karman-Donnell kinematic relations. In the present investigation both in-surface and out-of-surface inertia terms are included. The solution to the problem is obtained progressively by employing the finite difference method with respect to the spatial coordinate and the Runge-Kutta method with respect to time for each one of the time increments. An essential feature of the procedure, based on the incremental formulation, is the re-evaluation of the stiffness coefficients at every point of the shell for each time increment. Numerical results are presented for a cylindrical shell made of a nonlinearly elastic material subjected to two types of time-dependent axial loads. For the case of loads 545

R. GILATet al.

546

proportional to time which are applied to both edges of the shell, two types of boundary conditions, namely simply supported and clamped, are considered. The dependence of the deflection and the stiffness matrices of the cylindrical shell on the spatial coordinate and on time is illustrated. For the case of an impulsive load impacting one edge, the dynamic buckling behavior of the shell is analysed. Simitses [2] summarized the concepts and methodologies used in estimating critical conditions for dynamic buckling of suddenly loaded structures, and classified them in groups. In the present investigation the equations of motion approach (established by Budiansky-Roth) is adopted for the determination of the dynamic buckling state. The effect of the nonlinear elastic behavior is demonstrated by a comparison with corresponding results obtained by neglecting the material nonlinearity.

2. P R O B L E M

FORMULATION

Consider an initially isotropic material which exhibits a nonlinear material behavior. It is assumed that the response of the material can be described by using the generalized Ramberg-Osgood representation, which leads to strain-stress relations of the form 1+,,

,,

3

i=x,y,z

O'iJ-- E (~kk~ij-~ ~ (Tij

EiJ~" T

(1)

where E is Young's modulus, v is Poisson's ratio, a o and n are parameters characterizing the material nonlinearity, a u - a u - ]akki)q is deviatoric stress (6u is the Kronecker delta) and 8=x/2~f'ubu. It should be noted that the adoption of the above representation was established by neglecting the nonlinear part of the volumetric strain. The equations based on Donnelrs shell theory [6] which govern the axisymmetric motion of an anisotropic cylindrical shell subjected to time-dependent axial load are ^

__

|

-

ph Ux. . = N ..... ph Uy.,

=

Nxy.x

ph U .-.. = M ...... -- ~Nyy l + (N.~ x U:.x).x-

(2)

Here x and y are the axial and circumferential coordinates respectively and t denotes the time. The z coordinate coincides with the radial direction having its zero shifted to the midsurface of the shell (z = 0 at r = R). R and h are the radius and the thickness of the shell and p is the material density. Let Ux(x,t), Ur(x,t) and U=(x,t) denote the displacements of the midsurface in the x , y and z directions, respectively. The stress and moment resultants are given by Nij

=

ff ij dz,

M~j =

triiz dz

ij=x,y

(3)

where tro(x,z,t) are the stress components related to the strain components eo(x,z,t) through the inversion of the stress-strain relations (1). The partly nonlinear strain-displacement relations in conjunction with the Love-Kirchhoffassumptions lead to the following relations between the strains and the unknown displacements U~(x,t), i = x , y , z 2 ex~ _- U ~ . ~ - z U : , x ~ , + i2 U:.~,

eyy = - U : / R 2 e x y = U y.x.

(4)

Axisymmetric response of shells to dynamic axial loads

547

It should be noted that in the present formulation in conjunction with the material nonlinearity presented by Eqn (l), the analysis is applicable to nonlinear materials which undergo small strains and moderate deflections. The governing equations are accompanied by initial and b o u n d a r y conditions. For a shell of length L being at rest at t = 0 when loading starts, the initial conditions are

i=x,y,z

U ; = UI.,=0

at

O<~x<<.L.

t=0,

(5)

The boundary conditions for the case of a simply supported edge at x = 0 or x = L have either the S2 form [7] Uy = U.. = Mx:, = 0,

Nxx = -/V(t)

at

t/> 0

(6)

or the S1 form U:, = Uy = U z = Mxx = 0

at

t/> 0.

(7)

For the case of clamped C2 edges b o u n d a r y conditions are

Nxx = -/V(t)

Uy = U. = U__:,= 0,

at

t >/0,

x = 0,L.

(8)

The function /V(t) describes the applied load. 3. METHOD OF ANALYSIS

3.1. Incremental formulation As noted previously, in order to express the equations of motion in terms of the u n k n o w n displacements Ui, i=x,y,z, the constitutive stress-strain relations must be established by inverting Eqn (1) in order to obtain the stresses in terms of the strains. Due to the nonlinearity of relation (1), an incremental formulation is adopted to enable this inversion. Since the considered problem is a dynamic one, the increments turn out to be time intervals. Suppose that at some time t = t tl~= IAt the state of a shell is characterized by

e,j(x,z,t~') ) = e~)(x,z),

aii(x,z,t"') = al~)(x,z)

Ui(x,t ~'~)= V l'~(x) etc.

Within the next time increment t "~ ~
~,j(x,z,t) = ~l~(x,z) + a~J(x,z,t),

%(x,z,t) = ,~(x,z) + Aol~Ix,z,t),

U i(x,t) = Ultl(x) + A UIl~(x,t) etc.

(9)

Substitution of (9) into (1) and linearization with respect to the incremental variables/xalj"") yields the incremental strain-stress relations in the following matrix form

A~ ul = S A a a~.

( 1O)

For a plane-stress state

A p t l y = /I}AA~a} • t A~tl} ~A~l)~ A•/) =/At-r{/) Art(1) A,.r(l) --(~ Art(I)] Sl 1 S=

SI2

S13

S22

$23 S33

\symm.

S16~ S26 /

$36/ $66 ]

(11)

R. GILATet al.

548

and the elements of the compliance matrix S are s ~ l = [ 1 + k l + 0-25kz(2°'~xo, ayy) (o2 ]e±

s12=[_v_O.5k

1 + 0.25k2(2trxx (I) - o'ry)(2o'xr (I) (1)- axx)] (I) lE

st 3 = [ - - v-- 0.5k t + 0.25k2(2trx~ (0 - ayr)( (1) - _(1) ~(I)~1_t Vyy/j E _ (l) (I) (I) I S16-[1.5k2(2a~--tryy)a~r]E

s 2 2 = [ 1 + k x + 0.25k2(2a~,r(I) a~x) (I) 2]Et_

s23=[-v-O.5kl

+ 0.25k2(2aty~ - ~luox~J~- o~'l')_ "x~Ja~(I)~l±E

$2 6 ~----["1.5k2(2a(~!y) -- O~xmxra~(I)/~(I)-Ile

J-~

s 3 3 = [ l + k x+O.25k2(__(I) _(11~211 Oxx-- O yyJ $36 =

(I)

(I)

(I)

1

[1.5k2( - ax.~ - trxr)axy] ~

S66 = [ 2 ( 1

(12)

+ v ) + 3 k 1 + 9k2ax~,(0"-] .rE

where kx -

(0(0-')"%'A - ~ -- ~ '

k2

(n -- 1)(0(I)")~ G~-

-

6(t):_ _ . ) : .T _(,): - - OXX O),y - - -O(x0 x O- .y )y ± T 3o'~,:.

Establishment of the constitutive stress-strain relations in an incremental form is now straightforward yielding Ao"") = CAn (a)

(13)

C = S -1.

(14)

with

Note that the elements of the compliance matrix S and C depend on the state of stresses al~J(x,z); consequently, Using (13) and the incremental form of (3) and (4), the of the nonlinearly elastic cylindrical shell (for the case can be written

AN(Ix)y/-~AI6 A26

hence those of the stiffness matrix they are functions of x,z and t. incremental constitutive relations of an axisymmetric deformation)

(15) A66

816

AU(yll x

"

The extension, coupling and bending stiffness matrices A;j, B~i and D~ are defined as usual

(Aij, Bij,Dij) =

Qo'(l,z,z2) dz

(16)

where Qo = C~j- Ci3Cj3/C33 , iJ = 1,2,6. Since Cij and hence Aij, Bij and Dij depend on the state of stresses a~)(x,z), they should be re-evaluated for each time increment at every point of the shell. Having established the incremental constitutive relations (15), we now return to the

Axisymmetricresponse of shells to dynamic axial loads

549

equations of motion (2) and present them in an incremental form. At each time increment t")<.~t<~.t tz+ ~) the response of the shell, namely the change in the displacements AU~)(x,t), i=x,y,z, is governed by the following equations p h A U ~ ! , , = ( N ~ + AN_~x),x'° phA U'~,),,= tY tt1-4-ANtlh x ""

Xy

--

--

""

x y / , x

(1) it) it) phAU=.-(M.~+AM~),xx-~N.+AN.,)+[(N~+AN~x)(U~,~t'~ (l}

__

(I)

") +AU:,~)],~.t')

(17)

O<~x<~L

(18)

The dependent variables have to satisfy the initial conditions AUI'I =0,

AU~.~= UI~.~

i=x,y,z

at

t = t "),

and one of the boundary conditions (6), (7) or (8) in their incremental form ill .0 . . AN _y = A U :t~)=AMx~= . "1 = - [ ~ ( t ) - / ~ ( t " l ) ] AUtt} AU~)=AUtrt)-AUtt}-AMtO_ _ --~ _ - . . - ~ _-0~ AU~rtI- A U I:t -) - A U :tl~. ~--0,

at

A"l"'") Vxx=--[N(t)--N(tttl)]

at

t")<~t<~t "+1)

t~t)<...t<...t"+11 at

tu)<<.t~t tl+l)

(19)

(20) (21)

at each of the edges x = 0,L. Note that for initially isotropic material and under the considered boundary conditions the shear coupling terms of the compliance matrix $16,$26,s36 [Eqn (12)] remain zero throughout the shell during the deformation. Hence the second of Eqns (17) is exactly satisfied and only the other two equations are to be solved. 3.2. Numerical solution By substituting relations (15) into the first and third of Eqns (17), two nonlinear partial differential equations in terms of AU~°(x,t), i = x , z are obtained. These equations are to be solved successively for each of the time increments t")...
550

R. GILAT et al.

with respect to m.... Pmax and At was ensured by choosing sufficiently small spatial and time increments. The validity of the present approach was verified in the special case of linear material behavior. This was performed by a comparison with the numerical results presented by Bogdanovich and Feldman [8-]. Excellent agreement between the present incremental formulation and the results of [8] was obtained. 4. A P P L I C A T I O N S

The previously presented analysis was applied to investigate the dynamic response of a cylindrical shell characterized by L/R = 2, R/h = 100. The shell was considered to be made ofa nonlinearly elastic material whose properties are [9-]: E = 6 GPa, v = 0.35, tro = 106 MPa, n = 3, p = 1400 K# m-3. Two loading cases were examined. (a) Loads proportional to time applied to both edges of the shell such that/V(t) = VpPst with Vp = 1500 s- 1 (if not otherwise stated) ruling the loading rate and Pscr= 0.37 MN m (Ps, is the static buckling load of the linearly elastic simply supported shell). (b) Symmetric pulse applied to one edge of the shell, i.e. N(t)= P(t) at x =0, while at x = L, U~(L,t)= 0. The form of the applied impact is chosen as

P(t) = K[ Po(t ) - Po(t- 3q)-] Po(t) = [t3H(t)- 3t3H(tt)+ 3t3H(t2)- t3H(t3),]/(6q 3) t l = t - - q,

t2=t--2q,

t3=t--3r/.

Here H(t) is the Heaviside unit function and K is an amplitude factor. It can be verified that the function P(t) is symmetric with respect to t = 3q. It rises smoothly from zero at t = 0 to P(3q)=K, and then decreases smoothly to P(t>_-6r/)=0. The duration of the pulse 6=6r/which is chosen as 6=krL/x/E/(1-v2)p is ruled by ka' In order to demonstrate the effect of the nonlinearity of the elastic material, results obtained by using the nonlinear constitutive relations (shown by solid lines in the figures), were compared with the corresponding ones obtained by neglecting the material nonlinearity (shown by dashed lines). (a) Loads proportional to time The distribution of the nondimensional time-dependent radial and axial displacements

Uz/h and Ux/h along the x axis of the simply supported shell are shown in Figs la and lb respectively, at several nondimensional times r=tVp. Due to the symmetry of the problem with respect to x/L=0.5, only half of the picture is given. It is observed that the deformed shape of the nonlinearly elastic shell is similar to that of the linearly elastic one; however, there is a difference between the responses of these two cases which is notable near the loaded edges. In both cases U~ has local extrema in the regions near the loaded boundaries and the extrema nearest to the edges turn out to be the maximum deflection maxU:. Yet, maxU. of the nonlinearly elastic shell becomes greater than the corresponding X

x

maximum deflection of the linearly elastic shell as time elapses. The buckling wavelength of the nonlinearly elastic shell is smaller than the buckling wavelength of the linearly elastic shell. This relation is in agreement with the prediction of Jones 1-4] that the critical mode number of plastic buckling is greater than that of linearly elastic buckling. While the difference between the radial displacements U_. resulting from the different (linear and nonlinear) analyses is mostly pronounced in the regions near the loaded edges, the axial displacement Ux of the nonlinearly elastic shell is greater than that of the elastic shell throughout the shell length. Since the maximum deflection varies with time, it is interesting to examine this variation for two types of shells namely simply supported and clamped at the edges. In Fig. 2 the nondimensional maximum deflection maxlU,.I/h as a function of the nondimensional time

Axisymmetric response of shells to dynamic axial loads

551

o

/

0

r.075

\r-15

r,2.25

iI

-I0 Uz

-2.0

-30

-40 0

I

I

I

I

I

OI

02

03

04

05

X L

FIG. la. Distribution of the deflection U. along the x axis of the simply supported shell at various time points. Both edges are subjected to axisymmetric axial loads proportional to time. ( - - Nonlinearly elastic analysis, - - - linearly elastic analysis.)

4~3-

30

2.O Ux --fipO

-iO 0

I Oi

I 0.2

I 03

I 04

I 0.5

X L

FIG. lb. Distribution of the axial displacement Ux along the x axis of the simply supported shell at various time points. Both edges are subjected to axisymmetric axial loads proportional to time. (-Nonlinearly elastic analysis, - - - linearly elastic analysis.)

r is shown. In this figure, bold lines describe the behavior of the $2 simply supported shell, while the behavior of the C2 clamped one is given by fine lines. It can be seen that as time elapses, during which the stresses increase, the deflections resulting from the nonlinear analysis are greater than those predicted using the linear Hooke's constitutive law. This deviation from a specific value of maximum deflection predicted assuming linearly elastic

R. GILAT et al.

552

50

,/

40

simply supporte~/~ 30

tlIt

/

~'luzl h

2O

ciomped

IO

,

~ I0

,

1

~

,

,

L

I

20

30

T F I G . 2.

Temporal variation of the maximum deflection maxlU:l/h of simply supported (bold

x

lines)

and clamped (fine lines) shells. Both edges are subjected to axisymmetric axial loads proportional to time. (Nonlinearly elastic analysis. - - - linearly elastic analysis.)

50

/

/,," /t /~llltllll/~/i/llllllll / /I //I/ it//I ii1III /////

4.0

3O

~'lUzl h

2.0

1.0

---~-"~ 0

,

,

,

I0

I

,

20

30

-g F I G . 3.

Temporal variation of the maximum deflection maxlU:[,'/ of a simply supported shell for

various values of loading rate. ( - -

x

Nonlinearly elastic analysis,

- --

linearly elastic analysis.)

material is greater under the simply supported boundary conditions. Hence the effect of the nonlinearity is more significant for the simply supported boundary conditions under which deflections increase faster. Figure 3 illustrates the effect of the material nonlinearity under simply supported boundary conditions for various values of loading rate. It shows that the effect of the materal nonlinearity becomes more significant with decreasing loading rate. However, the effect of material nonlinearity is weakly influenced by the considered boundary conditions and loading rates. Since the stiffness coefficients A, B, D depend, due to the nonlinear behavior of the material, on the current state of stress, it is interesting to examine the change of these coefficients with time. To this end, Table 1 shows these three matrices at point x =0.04L of an $2 simply supported shell at two times, namely, T=0.0 (which corresponds to a linear elastic behavior), and z=2.25. A comparison between these two states leads to the observation that both initial isotropy (which is characterized by the values of A, B, D at z=0.0), and initial symmetry with respect to z = 0 (according to which B = 0 ) disappear. Figure 4 illustrates the spatial and temporal changes of stiffness ratio D 1JD°I (D°t is the element Dla of the bending stiffness matrix of the corresponding linearly elastic shell). It can be seen that as loading proceeds, Dll/D°l decreases, and as a function of x it undergoes relatively sharp changes in the same region as the deflection does.

553

Axisymmetric response of shells to dynamic axial loads T A B L E 1. S T I F F N E S S MATRICES OF A NONLINEARLY ELASTIC SIMPLY S U P P O R T E D SHELL. B O T H EDGES ARE SUBJECTED TO AXISYMMETRICAL AXIAL LOADS PROPORTIONAL T O T I M E

(,,.43.990)(!00)(i5 A~-t~:"

at

at

r =0.0

r =2.25 x = O.04L

I0 [

Da-~es

B-I-~6

3.99 11.4

0

0

0

0

0

3.7

0

0

3.32

6.62 0

0

0

-

.9

2.11]

/

0

0

1.49 0

3.32 0 .32 9.5

3.09

.68 5.33 0 1.12

0

1.79

"r =075

OB "r = I ','5

__L O6

/ r-~-2 2 5

DI, O4

02

O

I

I

1

I

I

OI

02

03

04

05

X L

FIG. 4. Distribution of the bending stiffness D~~ along the x axis of the nonlinearly elastic simply supported shell at various time points. Both edges are subjected to axisymmetric axial loads proportional to time.

(b)

Impact loads

Let us e m p l o y the concept of Budiansky [10l which associated the d y n a m i c buckling of a structure with the state at which small changes in the m a g n i t u d e of loading lead to large changes in the structure response. T o this end, the response of the simply s u p p o r t e d shell to impact of various a m p l i t u d e s is studied. Since the response of the shell, subjected to an impact of an a m p l i t u d e K, depends both on time and space, it is necessary to characterize this response by a specific value. This value can be chosen to be max maxiU=l. !

x

N o n d i m e n s i o n a l values of max maxtU:t/h versus n o n d i m e n s i o n a l l o a d a m p l i t u d e s 0~
K/Psor

x

for various pulse d u r a t i o n s are shown in Fig. 5. A c c o r d i n g to the a b o v e m e n t i o n e d concept of d y n a m i c buckling, the steep slopes of the curves in Fig. 5 indicate a d y n a m i c buckling state. Let the value of K at which the slope of the d y n a m i c buckling curve exhibits an a b r u p t change be Kcr. It can be readily seen that the value of Kcr of the nonlinearly elastic

R. GILAT et al.

554 ao

/

7o

/

f/

60

I

50 mox mox U

t x h

z 40

30

/Jr"/

2O IO

0

v IVy7~'1'ii"~i 05

I0

15

i 20

I

25

30

K__ PSer

FIG. 5. Dynamic buckling behavior of a simply supported shell subjected to axisymmetric axial impact of various durations at one edge. ( - - Nonlinearly elastic analysis, - - - linearly elastic analysis.)

shell is lower than that of the linearly elastic shell. This implies that the nonlinearity of the elastic material reduces the amplitude of the applied impact which leads to dynamic buckling of the shell. Furthermore, at the buckled state, the slope of the dynamic buckling curve of the nonlinearly elastic shell is steeper than that of the corresponding linearly elastic curve. This means that the increase of the deflection of the nonlinearly elastic shell at the buckling state is faster. As expected, the results show the increasing of Kcr with decreasing pulse duration. 5. C O N C L U S I O N S

An incremental procedure is offered to analyse the dynamic axisymmetric response and buckling of cylindrical shells made of nonlinearly elastic materials. It is shown that the present formulation is rather simple since it does not require the solution of systems of nonlinear equations. The results exhibit the transition from linear to nonlinear material elastic behavior. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

G. J. SIMITSES, Instability of dynamically-loaded structures. Appl. Mech. Review. 40, 1403 (1987). G. J. SIMITSES, Dynamic Stability of Suddenly Loaded Structures. Springer, Berlin (1990). H. E. LINDRERG and A. L. FLORENCE, Dynamic Pulse Buckling. Martinus Nijhoff, Dordrecht (1987). N. JONES, Structural Impact. Cambridge University Press, Cambridge (1989). D. G. LIAW and T. Y. YANG, Elastic-plastic dynamic response and buckling of laminated thin shells. J. Compos. Mater. 25, 1039 (1991). J. R. VINSON, The Behavior of Thin Walled Structures: Beams, Plates, and Shells. Kluwer, Utrecht (1989). R. M. JONES, Mechanics of Composite Materials. McGraw-Hill, New York (1975). A. E. BOGDANOVICHand E. G. FELDMAN, Strength and axisymmetric deformation of laminated cylindrical shells under axial impact. Mech. Compos. Mater. 4, 449 (1982). J. ABOUDJ,The nonlinear behavior of unidirectional and laminated composites--a micromechanical approach. J. Reinforced Plastics Compos. 9, 13 (1990). B. B•D1ANsK•• Dynamic buck•ing •f e•astic structures: criteria and estimates. •n Dynamic Stability •f Structures (edited by G. HERMANN), p. 83. Pergamon Press, Oxford (1966).