Kleine Yitt.eilungen 5.5

ZAMNI 62, 55 -5’1 (1982)

(‘. PERTNKIS / G. TZIVANTDIS / A. RAPTJS

On Plane Deformations of Incompres8ible Isotropic Hyperelastic Material

1. I n t r o d u c t i o n

In 1978 [l] a formulation of the governing equations for finite plane-strain cleformat ions of incompressible isotropic elastic solids was presented and two boundary-value problems for a circular annulus were examined. The main feature of that paper has been the use of an appropriate choice of deformation invariants different from those most commonly used [ Z ] .

Here we formtilate the equations for finite plane-strain of incompressible isotropic hyperelastic solids, utilizing complex variables and employing the principal stretches rather than a particular choice of strain invariants. Several new results are obtained and some known results are recovered. All these are discussed in sections 2, 3. 4 and 5.

In section 6 the circular horizontal and uniform Rhear of a tube of such a material with OGDEN’S strain-energy function is studied, not taking into account torsion. Some known results are also found, In section 7 we prove the uniquenem of the solution for this problem.

2. Kinemat ics of Deformat ion

In a fixed rectangular Cartesian coordinate system. the class of plane deformation fields with uniform transverse stretch is characterised in component form by

where Xt and pt (i = 1, 2 , 3 ) are, respectively, the reference and deformed coordinates of a typical material point and a, is a positive constant. It is assumed that the undeformed confi- gtiration is free of etress. For such deformations the polar decom- position theorem F = RU, where R is a proper orthogonal matrix and U is a positive definite symmetric matrix, implies that the two dimensional deformation gradient met.rix F can be written

xi = q(&, X,), 1, = z@,, X?) , r3 = a,X, , (1)

I n (2), the principal stretches a, and a, are the eigenvalues of the two dimensional strain matrix (P~P)W,where T denotes trans- position; w is the local angle of rotation of the material; @ =

1 = - ( y - m ) is the inclination to the &-axis of the local 2

referentid stretch axis in whose direction the principal stretch is al.

If we introauce tlir complex variables z = 9 $- iz,, Z = -= X, f iX, a i d consider z as a function of 2 and we find

These results are analogous t o t,hose of FOSDICR and SCHULER 13, section 51.

3. Analysis of S t ress

The in-plane components of the first Piola-Kkchhoff stresa tensor (t>he transpose of the nominal st,ress t,ensor S) form the matrix 141

(4)

where t is the second Piola-Kirchhoff st,ress matrix and whero use has been made of the polar decomposit,ion theorem. Tt, follows that, BT has a representation directly analogous to (2)

sin OJ cog OJ

where, in general, 8, and Sa are not the principal values of S.

referential form, are I n the absence of body forces the equilibrium equations, in

BlLl 4- 8-21,, = 0 , .Q12,1 4- 522,2 = 0. ( 6 )

Equations (6) imply the existence of a complex stress fiinction @ = @, + i0, whose components 0, and @, generate t,he com- ponents S,B (a, /? = 1, 2) through the relations

S;, = - 0 1 , s . A!,.. = -@* ,a . NZ1 = @,,,, S2, --- . (7)

If we consider @ a.s a fonction of Z nnd Z a e obtain

4. Const i t i i t iv r E q u a t i o n s

If t,he isotropic incompressible hyperelastic material under con- sideration possesses a strain-energy function W(4, a*), per unit volume in the undeformed configurat,ion, the conRtitntive equations take the form

where the pressure p iR not determined a priori by the defornia- tion and it arises as a reaction to material incompressibility. Since alas = ayl, because of the incompressibility, 1Y is ex- pressible as a function of a single strain invariant (see GREEN and ADIZINS [2])

F(T, IT, 111) = W(a: -1- a: i- a:, aiai + a:a; -t n:a& 1) =

= W(ai + a: + a!, a:(a? + a:) + aya) -- Ti’(a,, n,) =

= F(aS + aB), (10)

where I, 11, I11 are the principal invariants of both Cauchy- Green strain tensors C and B [5]. Then, equations (9) are written

8, = -pail 4- 2a,F’, h‘, = -pa;’ + 2a,F’, (11)

where a dash denotes differentiation with respect to the in- variant Il = a: + ag.

5. Governing Equat , ions

The elimination of the angles ru and y bet,ween (3) and (8) yields

(12) a 0 fi2 az a 0 . #,-a2 az a Z - a , + a , a z ’ ajj a, - a 2 a3’

-i-- -=-*--

and from these we derive the partial differential equations

for z and 0 respectively. From (13), and (11) we deduce

where

p, -= 2F’ + af l = W, -I a3p . and it is equation (14) from which we work in this paper.

(la)

56 Kleine Mitteilungen

0. Circular Shear of a Tube of Incompressible f so t rop ic Hypere las t ic Material wi th OGDEN'S St ra in -Energy Func t ion

We consider a circular cylindrical tube of radii A and B (B > A ) and height L in the undeformed configuration. We assume that the inner surface R = A is held fixed, while a shear traction t o = B is applied horizontally and uniformly t o the external surface R = B. Let I = b be the radius of the external surface and I the height of the tube in the deformed configuration.

Since the material is assumed incompressible, it follows that RI - ( I - as) A2 ,.a =

as . . where R and r are the radii of any material point of the body in the undeformed and deformed configuration respectively, and u3 =- - . If the plane polar coordinates of any material point,

before and after dcforrnation, are (R, @) and (r, 0) respectively, then

1 L

Reis , = r eb9. (10)

Sinc-c z is a function ofZ and%, and T is a fanotion of R, it fo~lows

= 1 (i, j = 1, 2, 3) tilat e = e(R, 0).

The incompressibility condition J =

can be written in the form ae a @ - ' , _ -

using the relations 2, = r COB 0, z2 = r sin 0 and (15). If we do not take into account torsion, the solution of this equation is

e = o -t F ( ~ ) , (18) where p, is an arbitrary function of R finally determined from the boundary conditions. Evidently, a boundary condition is p(A) = 0. With the help of (18) we obtain

z . R z = r -ew,

az a Z

into (14) loads t o the relations az az and the substitution of the partial derivatives - and =

(+RVlp')' = 0 , p" + where ib prime denotes differentiat>ion with respect to R,

d W l - d dR d R W' - - - - (2B') =

mation invariant [I]. Equation (20), is reduced to the equation (53) given by OGDEN [I] when a, = 1.

\Ve can express the strain-energy function W in terms of a variable quantity, say a. If we put a1 = aF1I2a, a, = a.c112a-1, where a = a( R) , then the incompressibilit,y condition is satisfied. The invariant Ia becomes

,

(21) 1 I _. ---I 2

2 -- 3 (a + a-?,

and thus W is expressed in the form If' = W(a.). Henre we have

(23)

where C is an arbitrary constant. Equation (20), determines the pressure after the deformation has been found.

We consider an incompressible isotropic hyperelastic solid for which there ie a strain-energy function of the type proposed by OGDEN [0] to describe the large elastic deformations of rubber-like solids under isothermal conditions. The strain- energy function has the form

(24)

where t,he and ttn are material constants with a,, dimen- sionlesR and ,us wit.h ciimensions of stress. This equation is written

and (23) yields

The two principal strct,chcR, except of a,, are the positive roots of the determinant

d,. ae 1' ao where, in general, f = - , g := I - and A =- - - . We can dR a& R ao

show it, by observing that the deformation gradient matrix can be written in the form

) COB 0 - sin 0 ) [ ;) (,,, @ siu 0 ( Rin 0 cog 0 - sin 0 00s 0

F&fl =.:

which in tensor form is

B = PTFP" . The left Cayty-Green strain tens? B becomea, B = PTkP, where B = F P and thus B and R have the same principal stretches.

If the positive roots of (27) are yl and yz, and we recall that the principal stretches are given by a, = a,l/'a, a - a;ll%-l, we get

y: + y\ = aT'(n2 + a-9) = f a 4- g2 + h2 . We can express the quantity as + a-, in terms of R, through the relation

A

2 -

Equations (28) and (26) determine p,' arid a, and p follows from

R p,(W =i P'(W d R . (29)

7. Uniqiicness of t h e Solu t ion

N e go over the analysis of deformation in order to see if the relation (28) between a, R and p' can be simplified. We distin- guish two cases

1) 1st a, = 1. Then I =- R and (28) yicldx

(30) rp' = a - a-1.

This is the aimplified relation brtween n , R nnd q-'. Equntion (2fl) becomes

(31)

This is a relation betwccn a and R. If ttnan > 0 (for each n) and Ian1 > 1 (for each n), then the left-hand side of (31) is a strictly increasing funct,ion of a V a > 1 and therefore with C > 0 equation (31) mmt determine a unique posit,ive a(B) and hence, via (30), a unique p ( R ) [7].

2) Let a,

I% R

I. Then equtxtion (15) is writ,tcn

a $ - = 1 +-?,

where

= (a, -- 1) A Z / R e .

Kleine Mit.teilnngen / Riichbesprechungen 57

The siniplifitd rrlation is

nlld (26) brcon:t~s

(35)

This is R relation between a and (or R). We examine if (35) has i i unique solution for a given R. We ran easily see that the Ieft- hand side, say f(a), is a strictly increasing function of a. We denote the right-hand side as g(a). Let

(39)

where sgn denotes the signum function. Then g(o) is ciefined for (I > ii (> 1) and g(a) -+ 00 as a J. d. Also, g(a) -+ Ka-' as a + m and g'(a) < 0 V (I > b. The graph of y(n ) is thus as shown

BUCHRESPRECHUNBEN

Fischer,V. / Stephan, W., Y e c h e n i s c h e Schwingun- gen. Leipzig, VER Fachbuchverlag 1980. 332 S., M 22,-. R S 546 633 8

In der vorgelegten Monographie wird das Cesamtgebiet der Schwingungsmechanik fur Studierende des Maschinenbaues imd fur in der Praxis auf diesem Gebiet tatige Ingenieure in inoderner Form IehrbuchmiiOig dargestellt. Es ist didaktisch klar. anschaulich und fur den angespmchenen Leserkreis mathe- niatisch angeniessen formuliert. Allerdings beschriinkt es sich auf rein mechanische Schwingungsprobleme und spart z. R. elektmmecheniache %hmhgangssysteme vollig ails. Xach einer Einfuhrung in die mathematischen Beschreibungsmijglichkeiten 1-on Schwingungen werden einfarhe linearc Schwinger nnd die dahin fiihrenden Modellbildungen ausfuhrlich behandelt. Dabei wird eiiie methodisch und phiinomenologisch umfasaende Dar- stelliing geboten. So sind z. B. nicht-stationiire Vorgiinge, die Laplace-Transformation und fhochnstische Schwingungen darin enthalten, und auch Aufgaben werden bis zum zahlen- miiDigen Endergebnis durchgerechnet. EE schlieflt sich dann ein iimfengreicher Abschnitt iiber nicht-lineare Schwingungen mit einem Freiheitegrad an. In ihm wird ein uberblick uber die Lcieungsmethoden gegeben, der durch die Vielzahl der aufge- nornmenen Methoden besonders bemerkenswert ist. So findet man z.B. neben der Energie-Xethode imd der Methode der Anetlckeliing die Sqnivalenti~ LinrariRirriing, die Stonings-

Fig. 1. Yarktion of the fimi*tionsj(i?) mid g(a)

(Fig. 1) and it clearly intersects the graph of f(a) in preoiwly one point. The value of a at the point of intersection excerda (7.

Therefore, equation (35) must dc.termine t i iiiiiqiir N ( I ? ) and lim*e. via (34). n iiniqitr p( R).

1 Onurrxv, R. W.. On plnne deCorniation$ of Inconipresailile irotropie [4;istic

2 GEEEX. A. 2.: ADXIIS. J . B.. Larvc Elastic I>eformationc. (:lnrenfiwii Prcsi. solids, JIath. Proc. Camh. Phil. Roc. 68, pp. 127-136 (1978).

. . .. oxford (iwoj.

(1900).

3 FO'OBDICP, R. L.; SCEUL.EX, li. T., On EX~CKSEX'S prolilem for plnne dt~furina- t,ions with uniform tran~verse stret.ch, Int. J. Engng. Sei. i, pp. 217- ~ 2 3 3

4 CHADWICK, P., Continuum Xerhanir*. George Allen and U n e n LtB, London (1970).

5 !lllIXBDELL. C.; SOLL, W., The Non-Linear Field Theories of Mrrhanirr, FL~~WE'S Handbuch der Phpik, I I T / Y , Springer (11165).

(I OonEa, H. W., Large deforination isotropic elasticiby - on the corrclntion of theory and experiment for iucompressiblc rubber-like wolldx, Pror. I:. Suv. Lond. A. 846, pp. 565-581 (1972).

7 OQDEX, R. W.; CHADWICK, P., On the deformation of aolid and t.uliulur cylinders of incompresnible isotropic elastic materinl, J. Nrrli. L'hpb. Solirli 211, pp. 77-90 (1973).

Eingereicht. am 17. 12. 1980

dnwhrijt: Dr. C. PERUIKIS, Prof. G. TZTVAXIDIS and Dr. A. RAPTIS University of Ioannina, School of Phyfiics and Mathemnticfi, Ioa.nninrt, Creecv

rechnung, das Verfahren von GALERKIN und dns Vcrfehren von Booo~~umvundhZrrao~o~s9lr. In dem Kapitel iiber linearc Systeme mit mehreren Freiheitsgraden wird die Aufstellung der Schwingungsgleichungen uber die Lagrangeechen Gleichiingen, init der KraftgrijBenmethode und mit der Deformationsmethode vorgefiihrt. Im weiteren steht dann die l\latrizeneigeilwertanfga- be im Vordergmnd, und man findet uber den Rayleighschen Quotienten hinaus das Verfahren von JACOBI, die Vektoritera- tion nach V. MZBES und auch Abschatzungen der Eigenfrequen- Zen. Fiir die nichtlinearen Systemeist dieSystematik schwieriger, doch versucht das Buch erfolgreich den AnschluD an die fur einen Freiheitsgrad dargestellten Methoden. Fiir die linearen parametererregten Systeme wird in einem Abschnitt immerhin die Floquetsche Theorie vorgestellt. Die abschlieDenden Kapitel uber die Schwingungen der Kontinua und uber die Stabilitlt konnen die Fragestellungen dieser umfangreichen Qrbiri(b naturgemBf3 nur andenten.

Die vorliegende Dmstellung der mcohanischeii Schwinguiigen fur Studicrende und Ingenieure des Maschinenbaues iRt hr- sonders von der Vielzahl der vorgestellten Prage8tellungen uiitl Methoden her interessant und aktuell. Sie miissen vrrstiind- lichemeise bei der Breite ihrer Zielsetziing gelegentlich auf detaillierte Herleitungen verzichten, doch findet diis Buc11 liierbei durcheus eine fiir die Leserzielgruppe navh J nliitlt iiwl Umfang angemessene Form.

Karlsnihe F. \Vmmm,\\rmn