Vol. 17, 1966 Kurze Mitteilungen - Brief Reports - Communications br~ves 449

b z w .

OK* = 0 , ( 4 - 1,2 . . . . . l). (40) @

Jedoch sind die vera l lgemeiner ten Formen (20), (37) besonders passend dann, wenn

die Funk t ionen der Different ia lb indungen Beschleunigungen h6herer 0 r d n u n g ent-

ha l ten oder in ihnen nicht- l inear sind.

L I T E R A T U R V E R Z E I C H N I S

[1] J. NIELSEN, Vorlesungen iiber elementare Mechanik, deutsch y o n FENCHEL, pp. 345-354, Springer-Verlag, Berhn 1935.

[23 I. TZENOFF, Uber eine neue Form der Gleichungen tier analytischen Dynamik (rnssisch), Doklady Akad. Nauk UdSSR, 89 (1953) ; Nouvelles/ormes des dquations di//drentielles du mouvement des syst~mes matdriels holonomes et nonholonomes, X l I I t h Intern. Astro- naut. Congr., Varna 1962.

[33 D. MANGERON und S. DELEANU, Sur une classe d'dquations de la mdcanique analytique au sens de I. TZENOFF, C.R. Acad. bulg. Sci. 75 (1962).

[4] D. RASKOVIC, A Contribution to the Physical Reality o/the Second Acceleration (JERK), ZAMM, GAMM-Tagung 1965.

[5 3 A. WASSMUTH, Studien iiber Jourdain's Prinzip der Mechanik, S. B. Akad. Wiss. Wien, 128, 3 (1919).

[6] BE. DOLAPTSCHIEW, Uber die Nielsensche Form der Gleichungen yon Lagrange und deren Zusammenhang mit dem Prinzip yon Jourdain und mit den nichtholonomen mechanischen Systemen, ZAMM (im Druck).

Summary

A generalized form of the equations of motion of a rheonomic-holonomic mechanical system is proved. As special cases the equations of NIELSEN and TZENOFF are obtained. By a method of the last author the form of APPELL'S equations is derived, in which the kinetic energy, but not the energy of acceleration, appears. The generalized equations can be extended to the case of non-holonomical systems.

(Eingegangen: 5. Juli 1965.)

Kurze Mitteilungen - Brief Reports - Communications brSves

Control lable States in Compr e s s ib l e Elast ic Die lectr ics

By MANOHAR SINGFI, North Carolina State University, Raleigh, North Carolina, USA

1. Introduction

There are certain finite deformations which can be produced in every homogeneous, isotropic, elastic solid by surface tractions alone. These problems, known as exact de- formations, are solved by the inverse method in which the deformation is prescribed to begin with and it is then shown that the deformation can be supported without body force and without the knowledge of the form of functional dependence of the strain-energy function on the strain invariants. This observation, which revived the theory of large

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450 K u r z e M i t t e i l u n g e n - B r i e f R e p o r t s - C o m m u n i c a t i o n s b r ~ v e s ZAMP

elast ic de format ions , was m a d e b y RIVLIN [11 1), w ho also sugges ted t h a t since t he so lu t ions to e x a c t d e f o r m a t i o n s d id n o t requ i re b e f o r e h a n d t he fo rm of t he s t r a i n - e n e r g y func t ion for t h e mate r ia l , these d e f o r m a t i o n s could t h e n be employed in t he e x p e r i m e n t a l de ter - m i n a t i o n of t he s t r a i n ene rgy func t ion E2]. Whi l e t h e r e is a m o d e r a t e l y large class of p rob - lems wh ich can be solved in th i s m a n n e r for incompress ib le mate r i a l s [1, 3-81, ERICKSEN [91 ha s p r o v e d t h a t if t h e ma te r i a l is compressible , t h e n on ly pu re h o m o g e n e o u s d e f o r m a t i o n s are possible;

Recen t ly , SINGH a n d PIPKIN [101 h a v e descr ibed a s imi la r se t of exac t solut ions, wh ich t h e y call controllable states, in t he t h e o r y of incompress ib le e last ic dielectr ics. The de- f o r m a t i o n a n d e i the r t he electr ic field or t h e dielectr ic d i s p l a c e m e n t field are p rescr ibed a t t h e o u t s e t a n d i t is ver i f ied t h a t such a s t a t e can be s u p p o r t e d w i t h o u t b o d y force or d i s t r i b u t e d cha rge in eve ry homogeneous , isotropic, incompress ib le , e last ic dielectr ic. SINGH a n d PIPKIN also p rove t h a t no con t ro l l ab le s ta te , o t h e r t h a n those t h e y h a v e descr ibed, is possible in incompress ib le dielectr ics .

I n t h e p r e sen t paper , we show t h a t t he on ly con t ro l l ab le s t a t e s in compress ible elast ic dielectr ics are t he pu re h o m o g e n e o u s d e f o r m a t i o n s c o m b i n e d w i t h e i the r a u n i f o r m electr ic field or a u n i f o r m dielectr ic d i s p l a c e m e n t field. I n t he incompress ib le case, t he cond i t ions are less res t r i c t ive due to t he presence of a n a r b i t r a r y pressure p in t he c o n s t i t u t i v e re la t ions a n d the re fo re a fa i r ly large n u m b e r of con t ro l l ab le s t a t e s are o b t a i n e d [101. A n o t h e r d e p a r t u r e is t h a t while con t ro l l ab le s t a t e s cons i s t ing of h o m o g e n e o u s de fo rma t ions w i t h n o n - u n i f o r m fields and vice ve r sa exis t in incompress ib le dielectr ics, t h e y c a n n o t in compress ib le dielectr ics.

W e ou t l ine in Sect ion 2 t he t h e o r y of elast ic dielectr ics deve loped in [101. Th i s t h e o r y is fo rma l ly e q u i v a l e n t to TouPIN'S [111 theory , excep t t h a t t he stress, t h e electr ic field, and t h e dielectr ic d i s p l a c e m e n t field are no t decomposed in to sums of va r ious par t s , a n d also we do n o t use t h e express ions for response coeff icients in t e r m s of de r i va t i ve s of t h e s to red -ene rgy dens i ty . Next , we seek con t ro l l ab le s ta tes . Accord ing to ERICKEN'S t h e o r e m [9], d e f o r m a t i o n for a con t ro l l ab le s t a t e m u s t be h o m o g e n e o u s if t he elast ic dielectr ic is compress ib le a n d if e lectr ical effects are a b s e n t ; t h i s appl ies also in t he p r e sen t case since no e lect r ica l effects is a p a r t i c u l a r case. W e p r o v e t h a t r e s t r i c t ions on t h e field requi re i ts un i fo rmi ty . A s t a t e in which b o t h d e f o r m a t i o n and field are h o m o g e n e o u s is shown to be cont ro l lab le .

2 . C o n t i n u u m E l e c t r o e l a s t o s t a t i c s

W e reproduce in th i s sec t ion f rom [101 some of t he e q u a t i o n s gove rn ing t he t h e o r y of elast ic dielectrics.

T h e macroscopic electr ic field s t r e n g t h E i is c o n s e r v a t i v e :

E i , j = E : , i . (2.1)

W i t h t he absence of d i s t r i bu t ed charge, t he f lux D i is solenoidal :

Di, i ~ O. (2.2)

Ins ide t he dielectr ic b o d y if we assume to a c c o u n t for t he e l ec t romechan ica l forces, o the r t h a n g r a v i t a t i o n a l or iner t ia l , in t e r m s of a s t ress d i s t r i b u t i o n t i alone, t h e n i t follows exac t ly as in c o n t i n u u m mechan i c s t h a t

ti = a j l n j , (2.3) t h a t t he s t ress m a t r i x is symmet r i c ,

aO = aj i , (2.4)

a n d t h a t t he equa t i ons of equ i l i b r ium w i t h o u t b o d y or ine r t i a l forces are :

a i j , j = O. (2.5)

The d e f o r m a t i o n of a b o d y can be descr ibed b y speci fying t he Car te s i an coord ina tes x d X A ) af te r d e f o r m a t i o n of t he m a t e r i a l pa r t i c le or ig inal ly loca ted a t X A in t he same

x) N u m b e r s in b r a c k e t s r e f e r to R e f e r e n c e s , p a g e 453.

Vol. 17, i966 Kurze Mitteilungen - Brief Reports - Comnlunications br~ves 451

Cartesian system. Finger ' s s train tensor g = I I gij I I is defined in te rms of the deformat ion gradients xi, A (= OXi/OXA) by

gij = xi, A Xj, ,4 �9 (2.6)

We use the no ta t ion g~ to indicate the i / ' - componen t of the ma t r ix gN, and h(~ r and/~r) to denote the i / ' - components of the matr ices defined by

h~) : ~,%~ ~,) + (g~ E~) E~ (2.7) and

/!~:~ = vi(gj~ D~) + (g;~"~ Dk) Dj (2.8) 31 The mater ia ls which we consider are those for which the dielectric d isplacement D i and

the total stress ai j a r e determined by E i and the quant i t ies xi, A t h a t describe the s ta te of deformat ion of the medium. If the dielectric med ium is isotropic in its undeformed, f ie ld free state, then the cons t i tu t ive relat ions take the form

Di = (Ao ~ij + A1 gii + A2 gi 2") Ei ' (2.9)

% : ~o ~j + r g,j + r g~,j + ~ h~ + r h~}~ + ~ h~ , (2.1o)

where the scalar coefficients A and ~ are funct ions of the following or thogonal invar ian ts :

I = g i ~ i ( # = 1, 2, 3), h + , = E i g i ~ i E i ( /*= 0, 1, 2) . (2.11)

On the o ther hand, if D i a r e considered as independen t var iables instead of E l , then cons t i tu t ive relat ions tha t replace (2.9) and (2.10) are:

Ei = (Do 6ij + Ol gij + De g~i) D i ' (2.12) and

tg) + ~0 5 t (z) (2.13) 2 i(~ + to4 -,~ . i i a i i = ~~ dii + ~01 gii + ~P2 gii + ~P3 -,~

The coefficients ~2 and ~ are now funct ions of the following or thogonal invar ian ts :

J~ = gi~i (# = 1, 2, 3), J4+ , = Di gii " Di (a = 0, 1, 2) . (2.14)

3 . C o n t r o l l a b l e S t a t e s w i t h S p e c i f i e d E l e c t r i c F i e l d

To de te rmine all the control lable s tates in the p r e s e n t case, we follow a procedure similar to t h a t adopted for incompressible dielectrics [10?.

Suppose we are g iven a symmet r i c and posi t ive defini te tensor field gi j which is twice cont inuously differentiable.

In order t h a t gij be der ived from a possible deformat ion xi(XA ) of the body, it is necessary and sufficient t h a t gij meet the compat ib i l i ty condi t ions:

g-1 ~-1 -~ _g-1 = (3.!) 4 R i i k l = 2 ( i l , k i + ik, i l - - g i k i l il, i ~ ) + g m n ( A i k m A i l n • O.

Here the ma t r ix g~ l is the inverse of the ma t r ix gii' and

g-1 g-1 -1 (3.2) Aiik Ai ik = ik, i + ik, i -- gii, k "

Next , suppose a field E i sat isfying (2.1) is given inside the dielectric body. I f the specified gi j and E~ do indeed provide a solution, then the f lux D i der ived f rom (2.9) will sat isfy (2.2) and aij der ived f rom (2.10) will sat isfy (2.5). If the s ta te specified by gij and E i happens to be such t h a t (2.2) and (2.5) are satisfied no m a t t e r wha t forms funct ions A and i~ in (2.9) and (2.10) take, then the given s ta te is defined as controllable since i t can be suppor ted wi thou t body force or d is t r ibuted charge in every homogeneous, isotropic, compressible, elastic dielectric,

452 Kurze Mitteilungen - Brief Reports - Communications brbves ZAMP

TO seek res t r ic t ions on g i j and E i for a control lable s ta te , we subs t i tu te f rom (2.10) in (2.5):

+ ~ 0~1 0~~ I,~,i + ~1 glj , j gl.i I,~,j + . . . . 0 ( 3 . 3 ) a=l 0I~ a~=l OIa '

where the dots indicate analogous t e rms arising f rom the t e rms for coefficients r to r Necessary and suff icient condi t ions t h a t (3.3) be sat isf ied for every choice of the funct ions r in (2.10) are t h a t each of the coefficients of the funct ions r and the i r der iva t ives in (3.3) should vanish :

i~, i = gN gN 13., h!. N). = h(N) (3.4) . , i = i = . , , . I~, i = 0 .

Here and elsewhere in th is paper , N = 0, 1, 2, and )t = 1 . . . . . 6. W i t h (2.9) in (2.2),

N = 0 2 = 1

In order t h a t t he above equa t ion be sat isf ied for every possible form of the funct ions A of inva r i an t s (2.11), i t is necessary and suff icient t h a t :

(g~ii E : ) , i = ILl gi N E i = O. (3.6)

The s t ra in tensor gi j and the field E i cons t i tu t e a control lable s ta te if and only if condi t ions (2.1), (3.1), (3.4), and (3.6) are all satisfied. I t is therefore qui te a p p a r e n t t h a t a homo- geneous s t a te in which gi j is a c o n s t a n t t ensor and E i a un i fo rm field is ev iden t ly con- trollable. Below we show t h a t the only control lable s ta tes in the p resen t p rob lem are the homogeneous s tates .

According to ERICKSEN'S t h e o r e m [9], de fo rma t ion for a control lable s ta te m u s t be homogeneous if electrical effects are absent . The same has to be t rue in the p resen t p rob lem since no electr ical effects ( E i = 0) is a par t i cu la r case. To f ind fields permissible for a control lable s ta te , we focus our a t t en t i on on condi t ions (2.1), (3.4a) (2 = 4), and (3.6a) (X = 0):

Ei , j - - E j , i = (E j E ) ) , i = E i , i = 0 . (3.7) W i t h (3.7b),

o = (Ej E j ) , ~ = 2 E:,~ Ei,~ + 2 E : , ~ ,

which on use of (3.7a) and (3.7c) becomes

0 = 2

imply ing therefore t h a t E i m u s t be a c o n s t a n t field.

4. Contro l lab le States w i t h Spec i f ied Die lec tr ic D i s p l a c e m e n t Fie ld

In th is sect ion we look for possible control lable s ta tes when the f lux D i r a the r t h a n the electric field E i is prescr ibed init ially. I f t he specified s t ra in gi j and f lux D i are to cons t i tu t e a control lable s ta te , t h e n in addi t ion to g l j being compat ib le and D i solenoidal, E i and aij der ived f rom (2.12) and (2.13) should sat isfy (2.1) and (2.5), respect ively, regardless of t he funct ional form of coefficients Q and ~ in t e rms of invar ian t s (2.14). A procedure similar to t h a t used in Sect ion 3 leads to the following condi t ions :

x /(N). = /:N) = 0 (4.1) jJ ,j= , , , , ,/ J ,j , and

(gN Dk) , ] - - ( g ~ D k ) ' i = (gi N Dk) J~, i - ( g ~ Dk) Ja, i -- o . (4.2)

F r o m the res t r ic t ions on g l j i n d e p e n d e n t of t he f lux D i , and ERICKSEN'S t h eo rem [9], the s t ra in g i j has to be a c o n s t a n t tensor . T h a t t he field D i m u s t also be un i form follows,

Vol. 17, 1966 Kurze Mitteilungen - Brief Reports - Communications br~ves 453

as in Section 3, in view of the condit ions (2.2), (4.1a) (4 = 4), and (4.2a) (N = 0) :

Di, i = (Dj D j ) , i = Di, j -- Dj , i = O . (4.3)

The s ta te in which a homogeneous deformat ion is accompanied by a uniform flux D i satisfies all the condit ions (2.2), (3.1), (4.1), and (4.2), of a control lable state.

A c k n o w l e d g e m e n t

The au thor is grateful to Professor A. C. PIPKIN for valuable suggestions.

R E F E R E N C E S

[1] R. S. RIVLIN, Phil. Trans. R. Soc. London A 247, 379 (1948). [2] R. S. RIVLIN and D. W. SAUNDERS, Phil. Trans. R. Soc. London A 243, 251 (1951). [3] R. S. RIVLIN, Phil. Trans. R. Soc. London A 242, 173 (1949). [4] J. L. ERICKSEN and R. S. RIVLIN, J. rat. Mech. Analysis 3, 281 (1954). [5] A. E. GREEN and R. T. SHIEL1L Proc. Roy. Soc. London A 202, 407 (1950). [6] J. L. ERICKSEN, Z. angew. Math. Phys. 5, 466 (1954). [7] W. W. KLINGBEiL and R. T. SHIELD, for thcoming. [8] M. SINGH and A. C. PIPKIN, Z. angew. Math. Phys. 76, 706 (1965). [9] J. L. ERICKSEN, J. Math. Phys. 3d, 126 (1955).

[10] M. SINGH and A. C. PIPKIN, Arch. Ra t ' l . Mech. Anal. 21, 169 1966). [11] R. A. TOUPIN, J. rat . Mech. Analysis 5, 849 (1956).

Z u s a m m e n / a s s u n g

Ein kontrol l ier ter Zustand ist definiert als einer, in welchem eme endliche stat ische Ver formung und entweder das elektrische Feld oder das dielektr ische Verschiebungsfeld anf/inglich vorgeschrieben sind. Es kann gezeigt werden, dass der resul t ierende Zustand ohne Volumkr~Lfte oder Ladungen in j edem homogenen, isotropen und zusammendrf ick- baren Die lek t r ikum hervorgerufen werden kann.

Es wird gezeigt, dass die einzig m6glichen kontrol l ier ten Zust/~nde aus reinen homogenen Verschiebungen bestehen, kombin ie r t mi t e inhei t l ichen Feldern.

(Received: January 27, 1966.)

A Note on P lane N o n l i n e a r Creep of Free B o u n d a r i e s

By CHARLES A. BERG and FRANK A. MCCLINTOCK, Dpt. of Mech. Engineering, Massachuset ts Ins t i tu te of Technology, Cambridge, Mass., U S A

Introduction

The mot ion of a free boundary in a body which undergoes creep is of great in teres t in fracture mechanics ; the c h a n g e in conf igurat ion (e.g., b lun t ing or sharpening) of cracks dur ing creep exerts a s t rong influence on subsequent f racture of the mater ia l (BERG, 1962, 1963). If a l inear viscous fluid body conta in ing a free surface undergoes creep under a g iven loading program and then the loading program is exac t ly reversed, the free surface will re turn to its original configurat ion. This p roper ty of free surfaces in l inear fluids has been called k inemat ic revers ibi l i ty (McCLINTOCK, 1962), and follows f rom the l inear i ty of the fluid ve loc i ty wi th respect to the applied loading. Exper imen t s show tha t in cer ta in mater ia ls free surface deformat ion is no t k inemat ica l ly reversible (McCLINTOCK, 1962),