Aeta Mechanica 84, 185--190 (1990) A C T A M E C H A N I C A | by Springer-Verlag 1990

A complete solution in elastodynamics D. S. Chandrasekharaiah, Bangalore, India

(Received October 24, 1989)

Summary. In the context of classical elastodynamics, complete representations for the displacement vector and the stress tensor are obtained in terms of a single vector obeying a wave equation. The connection between these representations and other known representations is exhibited.

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

In a recent paper [1], the author obtained complete solutions for the Navier equation and the Beltrami-Michell equation in elastodynamics, by generalizing the Naghdi-t tsu solu- tion in elastostatics. In the present Note, we continue our studies on complete solutions in elastodynamics by presenting one more solution not found in the literature. This new solution is motivated by the Korn 's solution in elastostatics (see [2, p. 143]). Like the Naghdi-Hsu type solution [1], this solution also determines the displacement vector and the stress tensor in terms of just one vector function tha t obeys a non-homogeneous wave equation. The connection between this solution and other known elastodynamic solutions is exhibited. As in [1], the notation of [2] is employed.

2 Korn ' s so lu t ion in a l t ernat ive f o r m

In the linear theory of homogeneous and isotropic elastic solids, the Navier equation of equilibrium reads as follows:

1 1 Au + - - V d i v u + - - b = O . (2.l)

1 - - 2v

When b = O, Kern has given the following complete solution for this equation (see [2, p. 143]):

1 u = ~ 1 - - 2------v curl h (2.2.1)

A h = curl x (2.2.2)

Au = 0. (2.2.3)

This solution determines u in terms of two vector functions ~ and h which are not it/dependent.

186 D.S. Ch~ndrasekharaiah

I t is possible to rewrite the above solution entirely in terms of u. To this end, let us recall the identi ty:

f dv~ =--4~/. /(Y) A Ix-yl

B

By virtue of this identity, Eq. (2.2.2) yields:

1 / curl z(y) dvy. (2.2.4) h -- 47~ ]x -- y]

B

Consequently, the expression (2.2.1) becomes

1 f curl u(y) u = u -~ 4~(1 -- 2~) curt J Ix - - y[ dvu (2.2.5)

B

which determines u entirely in terms of u. Expressions (2.2.5) and (2.2.3) together serve as an alternative version of the Korn's

solution.

3 Dynamic generalization

We now proceed to generalize the solution for u given by (2.2.5) and (2.2.3) to the dynamic case, and establish its completeness. We suppose that b is not necessarily zero.

The dynamic counterpart of Eq. (2.1) is given by [1]

1 L u -~ - - b ~- O, (3.1)

where

1 L u = []2u ~- - - Cl2V(div u). (3.2)

2(1 -- v)

We seek the following representation for u, which is a dynamic generalization of ex- pression (2.2.5):

1 ; curl u(y , t - - ix - - yj/c2) dvy. (3.3) u = u ~ 4 ~ ( l _ 2 v ) c u r l ~ I x - y ]

B

Substituting for u from (3.3) into the right-hand side of (3.2), and using the identities:

( A c 2 1 ~ 2 ) f / ( y ' t - [ x - y l / c ) d v y ~ - 4 ~ / ' ~ - ~ Jx - - Yl (3.4)

B

C12 C2 2

2(1 --~,) 1 - -2v ' (3.5)

curl curly = V div v - - A v , (3.6)

C12 D1 -- [~2 -- - - A, (3.7)

2(1 -- r)

A complete solution in elastodynamics 187

we ob ta in

Lu = D l u . (3.8)

Clearly, if we assume t h a t u obeys the equat ion

1 f--lit. = - - - - b , ( 3 . 9 )

t hen E q (3.1) is readily satisfied.

Thus, i] u is an arbitrary vector/unction obeying the Eq. (3.9), then the representation (3.3) is a solution o/Eq. (3.1).

We now show t h a t this solution is complete in the sense t ha t every solution of Eq. (3.1) admi t s a representa t ion as given b y (3.3) with u obeying Eq. (3.9).

Suppose u is an a rb i t r a ry solution of Eq. (3.1). I n view of the t t e lmhol tz representa t ion of a vector , we m a y set

u : Vp + curl q . (3.10)

We consider a funct ion q0 defined b y

1 f q ( y , t - - I X - - y l / e i ) ! dvg (3.11)

q0 - - 4~cl 2 J { x - - YL B

and set

c2 2 curl curl qo = h, (3.12)

Vp d- []2 curl q0 : u. (3.13)

F r o m expressions (3.12) and (3.13), we get

c~ 2 curl u ~ [~2h. (3.14)

B y vi r tue of the iden t i ty (3.4), expressions (3.11) and (3.14) yield:

EJlqo = q, (3.15)

1 f curl u(y, t - - Ix - - y]/c2) h -- 4~.j Ix - - ~ dv.. (3.16)

B

With the aid of the identi t ies (3.6), (3.7), (3.5) and the expression (3.15), we find f rom expression (3.12) t h a t

curl h ~ (1 - - 2v) [[]2 curl q0 - - curl q] . (3.17)

Subs t i tu t ing for Vp and curl q f rom (3.13) and (3.17) in (3.10), we get

1 u = u 1 - - 2----~ curl h . (3.18)

Together with (3.16), this expression becomes the desired representat ion (3.3). Con- sequently, (3.8) holds. Since u is a solution of Eq. (3.1), it follows tha t u obeys Eq. (3.9).

This proves t h a t the solution o] Eq. (3.1) as descr~Sed by (3.3) and (3.9) is complete. We find f rom (3.18), (3.14), (3.5) to (3.7) and (3.12) t h a t

c~ 2 curl u = Ej lh .

188 D.S. Chandrasekhar~iah

B y vir tue of (3.4), this yields

h : - - c2----~-2 f curl u ( y , t - - Ix - - y t / c i ) dv u �9 (3.19)

4;7gCl 2 J I x - - Yl B

Subst i tu t ing this expression back into (3.18) and using (3.5), we get

1 f curl u ( y , t - - Ix - - Yl/Cl) ~r : U 8:z(1 - - v) c u r l d I x _ Y] dvy . (3.20)

B

Thus, for a given u, the funct ion ~ is uniquely determined. Expressions (3.3) and (3.20) therefore define a one-to-one t ransformat ion between u and ~; this t r ans format ion is analogous to the one defined b y (2.3) and (2.19) of [1].

I n the static case, expressions (3.3) and (3.18) and Eq. (3.14) reduce to expressions (2.2.5) and (2.2.1) and Eq. (2.2.2) respectively, and Eq. (3.9) becomes

1 An -- - - b . (3.21)

@cl 2

For b ~ 0, this equat ion becomes Eq. (2.2.3). Thus, the Korn ' s solution in elastostat ics is recovered.

4 Connec t ion w i t h o t h e r solu t ions

The connections which the solution given by (3.3) and (3.9) has with the Green-Lam5 solution, the Cauchy-Kovalevski-Somigl iana solution and the Bouss inesq-Papkovi tch- ~Ncuber t ype solution (recorded in [1]) and the Naghdi -Hsu type solution (obtained in [1]) m a y be established if the various functions occurring in these solutions are related as

follows:

= [ ~ g , h = c2 2 curl g , [~lh -= c2 ~ curl ~ ,

1 1 u - - curl h q- - - V/~ --~ ~b ~ curl r -b V F , (4.1)

1 - - 2v 2(1 - - v)

1 ~2 = 2(1 - - v) (Y - - ~0) = ci 2 div g = -~- (q~ -~ p . ~b).

Here, the func t ions /2 and Y are as defined in [1]. Recall ing f rom [1] t ha t ~2 obeys the equat ion V~,~2 = cl 2 div u and noting f rom the

representat ion (3.3) above t ha t div u = div u, we find t h a t $2 is re lated with z through

the equat ion:

E ] ~ = cl 2 div u . (4.2)

I f we subst i tu te for ~ in te rms u, h and ~(~ f rom (4.1) into the Naghdi -Hsu type solu- t ion obta ined in [1], and use the identities (3.5) to (3.7) and the expression (4.2), we arr ive a t the solution given by (3.3) and (3.9). [Since the Naghdi -Hsu type solution is complete, this procedure serves as an a l ternat ive proof for the completeness of the solution given by (3.3) and (3.9)]. Revers ing the steps, we recover the Naghdi -Hsu type solution f rom the one given by (3.3) and (3.9). The Green-Lamd solution, the Cauchy-Kovalevski-Somigl iana solution and the Boussinesq-Papkovitch-57euber type solution can also be deduced f rom the solution given by (3.3) and (3.9), and vice-versa, in an analogous way, b y using (4.1).

A complete solution in elastodynamics 189

5 Representation for the stress tensor

The stress tensor T associated with u is given by the Hooke's law:

T = 2 / ~ ~ ( d i v u ) l + l ~ u . (5.1)

Substituting for u from (3.3) into this expression, we get

[ ~ 1 ~curlfCurlx(y,t-,x--yl/C~)dvy]. (5.2) T = 2 / ~ ~ ( d i v x ) l + ~ - ~ 4~(1--2v) I x - y l

B

Since u given by (3.3) and (3.9) obeys the Navier equation (3.1), it follows that T given by (5.2) automatically obeys the Beltrami-Michell equation [1, Eq. (1.6)]. This fact may be verified directly also. Tile completeness of the representation for u given by (3.3) and (3.9), established in Section 3, asserts the completeness of the representation for T given by (5.2) and (3.9).

In order to see the connection which the representation for T given by (5.2) and (3.9) has with the Teodorescu's complete solution for the Beltrami-Miehell equation, given by [1, Eq. (1.7)], we first rewrite the expression (5.2) in the following form, on using (3.16), (4.2), (3.5) and the definition of c22:

T = 1--~-- ~ v ([]2D) 1 + 2~oc22~ ~ - - 1 -- 2v curl h . (5.3)

Suppose we set

[ 1 ] O~ 1 curl h + ~ V~ ~7 -- �9 (5.4)

= --~c~ ~ u 1 -- 2----~ 2(1 -- v) ' 2(1 -- v)

Then the expression (5.3) becomes

T = 2 [ c ~ F ~ - ~ j - ~g]. (5.5)

Also, together with (3.5), (3.6), (3.14) and (5.4), expressions (3.9) and (4.2) yield

1 [ ~ 2 g --" C22b , [~lV = 1 -- 2----------~ divg. (5.6)

Expressions (5.5) and (5.6) constitute the Teodoreseu solution (see [1]). Similarly, if we eliminate g and ~] from (5.5) and (5.6) with the aid of (5.4), (3.5), (3.6), (3.14) and (4.2), we re- cover (5.2) and (3.9). Thus, the representation for T given by (5.2) and (3.9) implies and is implied by the Teodorescu representation given by (5.5) and (5.6). The completeness of one of these two representations therefore ensures the completeness of the other.

6 Concluding remarks

I t has to be emphasised that like the solution obtained in [1], the solution developed in the present Note also determines u and T in terms of a single vector function that obeys a non-homogeneous wave equation. Whereas the vector ~b appearing in the solution obtained in [1] obeys the equation []2~b = - -b /e , the vector ~t appearing in the solution obtained here obeys the equation DI~ ~ --b/Q. Further, whereas ~b and u have the same curl, :r

and u have the same divergence.

190 D.S. Chandrasekharaiah: A complete solution in elastodynamics

Finally, we remark t h a t the expressions (3.3) and (3.9) can be convenient ly used in all or thogonal curvil inear coordinates. As such, like the Green-Lam6 solution and the solution obta ined in [1], the solution presented in this Note is also useful in pract ical applications.

References

[1] Chandrasckharaiah, D. S.: Naghdi-Hsu type solution in elas~odynamics. Acta Mechanica 76, 235--241 (1989).

[2] Gurtin, 5I. E. : The linear theory of elasticity. In: Encyclopedia of physics, Vol. VI a/2 (FIiigge, S., ed.), p. 1. Berlin--Heidelberg--New York: Springer 1972.

Author's address: Prof. Dr. D. S. Chandrasekharaiah, Department of Mathematics, Bangalore Uni- versity, Central College Campus, Bangalore-560001, India