The dynamic response of a stretched circular hyperelastic membrane subjected to normal impact

Wave Motion 16 (1992) 137-150 Elsevier

137

The dynamic response of a stretched circular hyperelastic membrane subjected to normal impact J.B. Haddow, J.L. Wegner Department of Mechanical Engineering, Universio, of Victoria. Victoria. B, C.. Canada

L. Jiang Martec Ltd., Halifax, Nova Scotia, Canada Received 16 September 1991

Finite amplitude wave propagation in a circular isotropic hyperelastic membrane, initially subjected to an equibiaxial stretch which is followed by a suddenly applied pressure or normal impact of a projectile, is considered. Axially symmetric deformation is assumed, and the governing equations, in Lagrangian form, are expressed in terms of the principal components of Biot stress and the principal stretches. These equations are a system of five first order quasi-linear partial differential equations in conservation form with a source term. Solutions, obtained by the method of characteristics and a finite difference scheme, are presented 3raphically for particular cases of the Mooney-Rivlin strain energy function.

1. Introduction

The governing equations for dynamic plane deformation of a stretched hyperelastic string are a system of four first order quasi-linear partial differential equations [ 1] and, tbr certain problems, similarity solutions can be obtained which are valid until the first wave reflection occurs at a fixed end. This paper considers a related problem, namely the propagation of longitudinal and transverse waves which arise in axially symmetric dynamic deformation of a hyperelastie membrane initially subjected to an equibiaxial stretch. It is assumed that the membrane is incompressible, homogeneous and isotropic in the undeformed natural reference configuration. The usual membrane approximation of zero bending stiffness is adopted and the equations of motion along with three compatibility equations are a system of five first order quasi-linear partial differential equations which is hyperbolic for realistic strain energy functions. This system has a source term, consequently similarity solutions, of the type obtained in [1] and [2] for the string problem, do not exist. The equations of motion, in Lagrangian form, which are obtained from a variational principle, and the constitutive relations, are expressed in terms of the principal components of Biot stress and principal stretches. A detailed discussion of the Biot stress tensor is given by Ogden [3]. Two axially symmetric problems are considered, which are the sudden application of a spatially uniform pressure to one surface of a diaphragm and the normal impact of a diaphragm by a flat nosed projectile. The first problem has been considered by Hallquist and Feng [4] who used a different procedure and did not consider wave propagation. 0165-2125/92/$05.00 ~.'~"1992

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J.B. Haddow et al. / Response of circular membrane

138

2. Formulation of problems A thin homogeneous hyperelastic sheet, of thickness H in the natural reference configuration, is subjected to an equibiaxial stretch Ao~>1, and is then clamped so that a diaphragm of radius b >>H is formed. Cylindrical polar coordinates of a particle in the reference configuration are denoted by (R, 19, Z) and the diaphragm occupies the region O<<,R<<,B=b/Xo,

O<~O<~2n,

-H/2~Z<.NH/2.

Cylindrical polar coordinates of a particle of the material surface Z = 0, in the current deformed configuration, are denoted by (r, 0, z). It has been noted by Haughton and Ogden [51 that, in general, the middle surface in the deformed configuration is not the same material surface as Z = 0. This distinction is neglected since the error which results is negligible if B/H>> 1, consequently the material surface Z = 0 is taken as the middle surface in all deformed configurations. The deformation of the middle surface is given by r=r(R,t),

O=O,

z=z(R,t),

(2.1)

where t is time, and r(B, t)=XoB=b,

z(B, t)=0.

(2.2)

Since the membrane is in a state of equibiaxial stretch at time t = 0, the initial conditions are r(R, O) = XoR,

z(R, O) = O,

i'(R, O) = 2(R, O) = O,

(2.3)

where a superposed dot denotes the material time derivative. For the first problem considered, a spatially uniform application of pressure p(t) =poll(t). where H(t) is the unit step function, is applied to the material surface Z = H / 2 and the other face, Z = - H / 2 is traction free. For the second problem the membrane is subjected to a normal impacl by a cylindrical projectile of radius a < b moving with constant speed q as indicated in Fig. 1. [t is assumed that, after the projectile stokes, there ,s no slip between the fiat end of the projectile and the membrane and that the velocity of the projectile remains constant. With these assumptions, r(R, t)=itoR,

-(R, t)= - q ,

0~
(2.4)

in addition to (2.1)-(2.3).

3. Constitutive relations The principal stretches, 2, and ..~.0, of the middle surface, in the de,brined configuration, are tangential ~o~ ~ meridian and -'~ circle of !atitude~ ~,~--~,-~,,,~, ...... ~ . . . . . ;..... ~,,, ,~,=

~S - ,

~R

~v ,;~0 . . . . .

R

,

13.1)

where s is the distance, measured from the pole, along a meridian. It then follows that Or OR ;t, cos a,

0z - ...... a, sin ct, OR

(3.2~

~39

J.B. Haddow et al. / Response o f ciradar membrane

b ____21

[-

1 1 l

,,,

b

I

Fig. I. Impact problem.

Fig. 2. Meridian of middle surface.

where a is the angle between the tangent to a meridian and the radial direction as indicated in Fig. 2. It follows from (3.2) that ;L,= (r '2 + z,-,)t-~

(3.3)

where a prime denotes partial differentiation with re, p,.ct to R. We neglect the variation in stretch and membrane stresses across the thickness and, s;nce an isowopic incompressible membrane is considered, the strain energy ;V is expressed as a symmetric function, W= bY(2,, Ao),

(3.4)

o f ~, and ,t,o. Results are obtained for the Mooney-Rivlin strain energy function which, in the form (3.4), is given by 2 ~ y(2tF + 2~ + ,~, ~;to" - 3) + ( 1 - y) (2~-" + A0" + Z;;I.~ - 3) },

0.5)

where 0~<~,~<1 and p is the shear modulus for infinitesimal deformation from the natural reference configuration. The membrane stresses are averaged across the thickness and the normal stress, due to the pressure on the material surface Z = H / 2 , is negligible compared with the membrane stresses. Consequently, it is assumed that the principal Cauchy membrane stresses act in directions tangential to a meridian and a circle of latitude of the middle surface, and the third principal stress is assumed to be negtigiNe. A Lagrangian formulation of the governing equations is adopled and the Blot stresses [3] are introduc~:d rather than the Cauchy or nominal stresses. For the present problem the principal components of Bio~. stress (TI, T2) arc conjugate to the principal components of stretch (,~,, Ao), and 0g"

~W'

J.B. Haddow et al. / Response of circular membrane

140

Constitutive equations (3.6) are valid only for an incompressible membrane with the approximation T~ = 0. The principal c o m p o n e n t s of membrane force per unit length, referred to the reference configuration, are given by T~fl and T ~ H in the meridional and latitudinal directions, respectively.

4. Derivation of equations of motion T h e concepts of virtual work and inertia force are used to obtain a form o f Hamilton's Principle for a pressurized membrane and the equations r,: m o t i o n are obtained from this principle. A pressure, applied to one surface of a membrane, is a nonco ~servative follower loading, consequently the derivation o f Hamilton's Principle for a conservative contir~'aous system is not applicable. An annular region o f inner radius a and outer ~adius b in the, initially stretched configuration is considered, with a = 0 as a special case. For the pressure app[i,:ation problem a = 0. The virtual w o r k of the pressure p for an axially symmetric virtual displacement is f'

8 Ur = 2 ~ ( - p )

r ----

n . 8 x dr,

(4.1)

COS

where 6X is the virtual displacement with components ( r and Sz and n is the unit normal to the middle surface, as indicated in Fig. 2, with r and z c o m p o n e n t s , . -sin a and cos a, respectively. The form of (4.1) referred to the reference configuration, ~ U~= 2rtp where ,4 = a ; ( , ,

( r z ' ~ r - rr'~z) d R ,

is obtained from the relation d r = r ' dR, (3.2) and (4.1), apd can be simplified to give

6 [;,--- 2~p~

_ ,.:R.

J~

since 6z = 0 at R = B, r(0, t)----0 if A = 0, and ~z = 0 at R = .4 if A ~ 0, Equating the sum of the virtual work of the pressure and the inertia forces to the virtual change in strain energy gives 8 Up- 2~pH

(i:6r + 5 ~ z ) R d R = 27tHS

VCR d R ,

(4.2)

where p is the density. The following variational principle is obtained from (4.2) by considering the solution path in configuration space and varied paths, between times t~ and ~_, such that c ~ r ( h ) = ~ r ( t , ) = ~ z ( ¢ ~ ) = 6 : ( t 2 ) =0, ~ t, i i't

d~3~. 04 edRdt.

(4.3)

where e = H R /)

+ _ , - W ( r ",

~ .....

z

"

(4.4)

J.B. Haddowet al. / Responseof circular membrane

141

The equations of motion ~(T1 cos a) ~R

(T~ cos a - T,) pXo2t,sin a 0u - + R H P ~t'

O(TI sin a) ~-(T1 sin a)_p~,oZt cos a

OR

R

H~

(4.5)

0w -p St'

(4.6)

where u = ? and w = z:, are obtained from the Euler-Lagrange equations associated with (4.4), along with (3.2), (3.3) and (3.6).

5. Governingequations Quantities with dimension of length, stress, velocity and time are nondimensionalized by dividing by B, 4p, Co and B~ Co, respectively, where Co = 2(p/p)W' is the wave speed for generalized plane stress longitudinal waves propagating into an unstressed region of an incompressible elastic solid. Henceforth, unless otherwise noted, nondimensional quantities are used. The nondimensional system of governing equations consists o f the equations o f motion, which have the same form as (4.5) and (4.6) with p omitted, and the compatibility equations, 0(Z, cos a) _ 0u

St

0(,~., sin a ) _ 0w

OR"

St

8£0_ u

OR"

c3t

(5.1~

R

The matrix representation of the wstem o f governing equations is

St

OR

where G = {2, cos a, ,t, sin a, ),o, u, w} r,

~T H = - {u, w, 0, T1 cos ~, TI sin a j ,

B = - {0, 0, u/R, (Ti cos a - Tz)/R+pZo2~, sin a/H, T1 sin a/R-p2~o,~, cos a / H } x

ROT, ROL

t-1 !F~ R + ~U

,..r

~,

,

;t+l

R-"T i

R-. L

m-1 R

'

,'

'R + mT

"

mL

Fig. 3. Grid for numerical implementationof method of characteristics.

142

or B. Haddow et al. / Response o f ciradar membrane

0,5

0.0

Z

-0.5

- 1.0

t = 1.434

-I .5

m

0,0

0.5

! .0 r

t .5

2,0

Fig. 4. Deformed configurations for ?"=0.6, p H = 0.5, ,~o= 1.5 ~nd nondimensional times 0.142, 0.280, 0.427, 0.570, 0.712, 0.854, ~.017, 1.15t. 1.282, 1.434.

and a superposed T denotes the transpose. System ( 5 2 ) is in conservation form and a non-conservation fo..~n is given u3'-,

0c ~fl_+ --+A B=0,

(5.3)

where the Jacobian matrix A is given by

A-

dH(G ) dG

The eigenvalues of A are 0, :kct, :~CT where I/2

< = t ear

TI

. . . . £2~J .

(5.4)

It follows from (3.5) and (5.4) that, for the M o e n e y - R M i n strain energy function. CL C'F2~O,if~t > 1 and Z~ I> t, consequently system (5.2) is then strictly hyperbolic with five distinct families o f characteristics with

J.B. Haddow et al. / Response of circular membrane

143

2.5 ht

-

.

-

-

.

.

.

.

he

2.25

t = !.434

2.0

1.75

1.5 0.0

0.25

0.5

1.0

0.75

R Fig. 5. Distributions of stretch corresponding to configurations shown in Fig, 4.

slopes d R / d t = :t:cL, d R / d t = ± c v , O, in the (R~ ,*) plane. The Lagrangian wave speeds are CL and c-r corresponding to the propagation of discontinuities of the partial derwatives of ,~, and a, resp~tive|y. Relations along the characteristics are given by P'dG+B=O-dt

on

dR

=C,

dt

where I is the left eigenvector corresponding to the eigenvalue Ce { +cL, :kcv O} of A. These relations are du el.

- -

dt

T

COS

C~ - -

dt

T

•

Sii~

~

/ dwj_ _:ST, jdAe

~t)- ' "" o.~o \ dt

\ :~/ ~ (T,

--

T2 r.r.~ . .

r~

R/

dw+ -.Lc~ da Tsin a dU±cos a [~: Tu sin ~ I ' R ± v ; L o ~ , f ' H ] = O , • dt dt dt " '

d)'°- u=0, dt

R

on dR --=0. dt

) / ' = f ~.

.

.

.

cm

.

"

dR -=±c~. dt

dR

on - - - = + c v ,

15.6)

dt

(5.7)

144

J.B. Haddow et al.

Response of circular mt.nbrane

80

60

40 O[

t = 1.434 /

deg. 20

/ Z////¢

0 --

-20 0.0

0.25

0.5

I).75

1.0

R Fig. 6. Distributions of a, corresponding to configurations shown in Fig, 4,

The jump relations across shock fronts are obtained from the nondimensional fore1 o f the governing equations (4.5), (4.6) and (5.1) are ~'Iu] = - [T, c o s a],

l"I~., cos a] = -

[;M =

[ul,

V[w] -- - IT, sin a l ,

(5.8)

V[.~, sin e l = - [w]

(5.9)

0

(5.10)

where V is the shock velocity and the square brackets [ ] indicate the j u m p of the enclosed quantity, across the shock. J u m p relation (5.10) indicates that 2o is continuous and it follows from (5.8) and (5.9) that the shock velocities V~ and V-r which arc the Lagrangian ve!ocities of propagation of discontinuities of 2, and a. respectively, are given by IT,] I

(5.11) where the + ( - 1) sign refers to a shock travelling in the +( - 1 )R direction. It follows from (5.4/2 and (5.11 )2 that c.~ = IVrl.

J.B. Haddow et al. / R~sponse o f c.rcular membrane

145

F o r the Mooney-Rivlin strain energy function, Vr = Vv(2,. ~to), where ;:Lois any fixed value of Ao> l. has a unique inverse A., = A,( VT, ~o), V,~,> !. It can be deduced from this. and the continuity requiremen~ (5.10), that a discontinuity of ;l, and u cannot coincide.

6. Method of characteristics A numerical implementation of the m e t h o d of characteristics, based on forward differences, was used to obtain numerical results. This scheme is a modification of a procedure proposed by W h i t h a m [6]. Characteristics with slopes :I:CL, +Ca" and 0 are described as cf., c~- and Co characteristics, respectively. Arcs of the c~ and cf_ characteristics are replaced by chords as indicated in Fig. 3 which shows the grid used. The grid occupies the region # ~0 where ~b= 0 for the pressure problem, a~d ~b= A for the projectile problem. Grid points are ( R , t"), i~ {0 . . . . . m}, n ~ {0 . . . . }, where R~ = dp+ iA R, t"= n A t, A R = ( B - d p ) / in, and a grid with z~t / A R = 1 was used. Since 0 ~t 1, ~., >I 1, it follows that c,,,,.,A t / A R ~< 1 where cm,,.~is the maximum wave speed. At, t = 0, u = w = a = 0 and ~., = ~,o = 2o VRe [~b, B], consequently the numerical scheme can be started at t=0.

0 .... ::".

"-

-I

~

...............

O.0

0.25

0.5

~,V

Ll

0.75

1.0

R Fig. 7. Distributions of velocity corresponding to configurations shown in Fig. 4.

J.B. Haddow et al. / Response of circular membrane

146

0

-0.4

-0.8 Z

-1.0

0

0.4

0.8

1.2

r

Fig. 8. Deformed configurations for impact problem with ),= 0.9, B/.4 = 6, q = 0.5. 2~= 1.2 and nondimcns:o.ml times 0.230, 0.461, 0.691, 0.992, 1.152, 1.398, 1.656. 1.920. 2.192, 2.474.

Referring to Fig. 3, consider a mesh point P (R,, t "+ ~). The c~, c-~, Co characteristics through P intersect t = t" at R = R~., R ~ , R,, and iP is a segment o f tke co characteristic through P. F o r . = 0 and i = m o n l y R,~, Ri;~_, R;i and R .... R,;,L, R,,,T, respectively, need be considered. Since a forward diffe :ence discretization scheme is used, and A t t A R = I, it can be shown that R.~,,= R,-~

...... :..... i + ( c o . , - c,~,,_ j)

where .n~ .IL, T} a n d c,~.: is the value o f c ~ at R,. The matrix Q = (,1.,, A0, or, u, w) is k n o w n at the grid points tR,, ¢") from the previous step and the elemenf.s of Q at Ri~ are f o u n d by interpolation, and, for each i, are substituted, along with the elements of Q at R,, in the forward difference forms o f (5.5), (5.6), and (5.7). Foc i~ [l, 2 . . . . . m - 1 ~j these five finite difference equations arc solved for the five u n k n o w n elements o f

147

j.t3. Haddow et al. / Response of circular membrane

Q for grid point (R,, t"+~). For the grid point (Ro, t"+~), A 0 = 2 , a =0, u = 0 , for the pressure problem, and Ao = 2o, u = 0, w = - q, for the impact problem, so that for each problcm there are two unknowns which can be obtained from the finite difference forms of the relations along the CT and CL characteristics. For the grid point (R,,,, t" + ~), ,'1,o= ~ , u = w = 0, so that again there are two unknowns and these can be obtained from the finite difference forms of the relations along the c~ and CL characteristics. Numerical results were also obtained using a modification o f MacConnack's finite difference scheme. This scheme is described in [7] where it is applied to a problem goverr~ed by a system o f three first order partial differential equations with a source term. The extension to consider a system o f five equations is straightforward.

7. Numerical results 7.1. Pressure problem

Numerical results, in nondimensional form, are shown graphically in Figs. 4 - 7 for Zo = 1.5, t < 1/ cL(Ao, 2o), p / H = 0.5 and the Mooney-Rivlin strain energy function with y = 0.6.

3.5

3.0

2.5

kt 2.0

1.5

0

0.2

0.4

0.6

0.8

1.0

R Fig. 9. Distributionsof)t, corresponding to configurations shown in Fig. 8 for nondimensior~altimes 0.230. 0.461.0.691, 0.992. I. 152. 1.398, 1.656.

J.B. Haddow et al. / Response of circular membrane

|48

It is evident from the results shown in Fig. 5 that, when the pressure is applied, an acceleration (longitudinal) wave propagates radially inwards, from R = l, with speed c~ (Zo, ;to), and since CL > CT, the wave 'front is at Rf= l - cL(2o. 2o) and there is a flat central region 0 ~
~)

(7.1)

0 <~R < Rf, t <~ 1/CL(20, Z0),

z = --~.er, 2H

where nondimensional quantities are used. Equation (7.1) is a check on the numerical procedures, also results from the m e t h o d o f characteristics and MacCon'nack's scheme are in close agreement. Just behind the wave front there is a region in which a is negative as indicated in Fig. 6 and the magnitude of the axial velocity c o m p o n e n t is a m a x i m u m in this region as indicated in Fig. 7. T h e region of negative

I00

75

50 O~

deg. 25

t 0

.............

-25 0.O

0,2

0,4

0.6

{}.8

[.0

R Fig. [0. Distributions o f ~, corresponding to configuration~ :)lown in [::it~. S tor same n,,):~Jimensk)nal times as in Fig. 9.

149

J.B. Haddow et al. / Response of circular membrane

a is barely evident in the deformed shapes shown in Fig. 4 since the negative angle has a maximum value of about 2 ~' for t = 1.434. The numerical results shown in Figs. 5 and 6 indicate that the transverse wave front and the tongitudm~l wave front coincide. This appears to be paradoxical, since ca-< eL, however behind the longitudinal wave front the current thickness h decreases so that an increase in the magnitude o f the axial velocity is to be expected with a consequent negative value o f a.

7.2. Impact problem Numerical results, in nondimensional form are shown graphically in Figs. 8-1 i for Zo = 1.2, q = 0.5, B~ A = 6, and the Mooney-Rivlin strain energy function with y = 0.9. The deformed configurations are shown in Fig. 8. Results shown in Fig. 8 do not indicate discontinuities in a and those in Fig. 9 show that the longitudinal wave propagates outwards as an acceleration wave and is reflected at the boundary R = 1 as an acceleration wave. It is evident from the results shown in Figs. 9 and 10 that, before the first reflection, a = 0 behind the longitudinal wave front and there is a well defined transverse wave front behind the longitudinal wave front.

0. I ---

o.o

-0.

W

I

.......

..=-----

.

-0.2 . . . . . . . . . . . . . . . . . . . . . . . . . .

-0.3 . . . . . . . . . . . . . . . . . . . . . .

o

,0

-0.4

..................

-0.5 ~

-0.6 . . . . 0

.......................................................

0.2

0.4

0.6

0.8

1.0

R Fig. 1t. Distributions of axial velocity corresponding to configurations shown in Fig. 8 for nondimensional times 0.230, 0.461,0.691 0.992, 1.152.

150

J.B. Haddow et al. / Response of circular membrane

References [I] M.F. Beatty and J.B. Haddow, "Transverse impact of a stretched strilJg", ASME, J. Appl. Mech., 52, i 37 143 (i985). [2] J.L. Wegner, J.B. Haddow and R.J. Tait, "'Unloading waves in a plucked hyperelastic string", ASME, d. AppL Mech. 56, 459465 (1989). [3] R.W. Ogden, Non.Linear Elastic Deformations, Ellis Horwood, Chichester, UK (1984). [4] J.O. Hallquist and W.W. Feng, "Dynamic response of axisymmetric hyperelastic membranes", ASME, J. AppL Mech. 42, 890891 (1975). [5] D. Haughton and R,W. Ogden, "On the incremental equations in non-linear elasticity - I. Membrane theory", J. Mech. Phys. Solids 26, 93-110 (1978). [6] G.B. Whitham, Linear and Nonlinear Waves, Wiley, New York (1974). [7] P. Janele, J.B. Haddow and A. Mioduchowski, "Finite amplitude spherically symmetric wave propagation in a compressible hyperelastic solid", Acta Mechanica 79, 25-41 (1989).