COMPLETE SOLUTIONS IN THE THEORY OF ELASTIC MATERIALS WITH VOIDSsweers/papers/Chandrasekharai... · 2007. 3. 9. · elastic materials consisting of a distribution of small pores (voids)

COMPLETE SOLUTIONS IN THETHEORY OF ELASTIC MATERIALS

WITH VOIDS

By D. S. CHANDRASEKHARAIAH

(Department of Mathematics, Bangalore University, Central CollegeCampus, Bangalore-560 001, India)

[Received 17 April 1986]

SUMMARYThree complete solutions of the field equations of the linear theory of homogen-

eous and isotropic elastic materials containing a distribution of vacuous pores (voids)are obtained. These solutions are analogous to and include as special cases theGreen-Lame', Boussinesq-Papkovitch-Neuber and Cauchy-Kovalevski-Somiglianasolutions in classical elastodynamics. A connection among the solutions is exhibited.The uncoupled versions of the field equations are deduced.

1. Introduction

THE theory of elastic materials with voids is one of the recent generaliza-tions of the classical theory of elasticity. This theory is concerned withelastic materials consisting of a distribution of small pores (voids) whichcontain nothing of mechanical or energetic significance. The general versionof this theory was proposed by Nunziato and Cowin (1) and thelinearized version was deduced by Cowin and Nunziato (2). The new theoryis intended to be of practical utility in investigating various types ofgeological, biological and synthetic porous materials for which the classicaltheory is inadequate. Some problems revealing interesting characterizationsof the theory are contained in (2 to 9); some basic theorems and propertiesof solutions are obtained in (10). The inter-relationships between this theoryand various other elasticity theories are analysed in (6).

The object of this paper is to present three complete solutions of the fieldequations of the theory formulated in (2), for homogeneous and isotropicmaterials. These solutions are analogous to the well-known Green-Lame",Boussinesq-Papkovitch-Neuber and Cauchy-Kovalevski-Somigliana solu-tions in classical elastodynamics (11). Indeed, these classical solutionsemerge as particular cases of our solutions. A connection among thesolutions is also exhibited. The uncoupled versions of the field equations arededuced.

(Q. J1 Mecta. appl. Math., Vol. 40, PI. 3, 19871 © Oxford Unhtntty Pros 1987

402 D. S. CHANDRASEKHARAIAH

2. Basic equations

In the context of the theory presented in (2) the complete system of fieldequations for a homogeneous and isotropic material are given as follows:

a" (21)

(2.2)

In these equations u is the displacement vector, cp is the so-called volumefraction field (defined in (2)), b is the body force and / is the so-calledextrinsic equilibrated body force (2), each measured per unit volume, A, nare the usual Lam6 constants, p is the mass density, a, ft, £, a> and k arenew material constants characterizing the presence of voids, and t is time.The usual vector notation is adopted; A denotes the Laplacian operator. Ifwe set /3 = / = ) + - b = 0, (2.3)P1 = 0. (2.4)

Here we have put

Lj(u, <p) = D2n + 2c2a2V div a + (/3/p)V^>, (2.5)

with

"5' (2-7)I - + P * T J , (2.8)dt dt2/

(2.9)

We also need the operators

• , = c?A- |J , (2.10)

D2 = D,a,+?- A, (2.11)P

(2.12)2pa2

ELASTIC MATERIALS WITH VOIDS 403

The following relations among the operators are easily verified:

(2.13)

(2.14)2pa2 "

D2-Dla2 = 2a2D3A. (2.15)

Throughout our analysis, it is assumed that all functions appearing in thediscussion are continuous and differentiable up to the required order onD X T, where D is a regular region in the Euclidean 3-dimensional spaceand T is a time interval. Also, all the differential operators are taken to becommutative.

3. Green-Lam6 type solution

In classical elastodynamics, it is known that u admits a representation ofthe form

a = Vcp + curl i|>, (3.1)

where cp and TJ> obey the equations

•i<P = X, n2y = y, (3-2)

the functions x and y being defined through the relation

b = - p ( V * + curly). (3.3)

The representation (3.1) is known as the Green-Lam6 solution (11, p. 233).In the context of the theory presented in (2) also, we seek a repre-

sentation of the type (3.1) for u. Substituting for u from (3.1) into theright-hand side of (2.5) and using (2.13) we obtain

L,(u, 4>) = V[n,(p + (ft/p)4>] + curl D ^ . (3.4)

If b is represented as in (3.3), we verify that (2.1) is satisfied if we set

(0/p)0 = Of " Di<P) (3-5)

and assume that t|> obeys the equation

Y- (3-6)

Substituting for a and and <p are arbitrary functions obeying equations (3.6) and(3.8), then (3.1) and (3.5) constitute a solution of the field equations (2.1)and (2.2).

We now show that this solution is complete in the sense that everysolution {u, <f>) of the system (2.1), (2.2) admits a representation asdescribed by (3.1), (3.5), (3.6) and (3.8).

Suppose that {u, (p) is an arbitrary solution of the system (2.1), (2.2)corresponding to b given by (3.3). Then we have

• 2 u + 2c\a2V div u + (filp)V<p - V* - curl y = 0, (3.9)

(3.10)

In view of the Helmholtz representation of a vector field, there existfunctions p and q such that

u = Vp + curl q. (3.11)

Substituting (3.11) into (3.9), and using (2.13), we obtain

V[CV + ifilp)<t> ~ X] + curl [Djq - Y] = 0. (3.12)

Taking the divergence of this equation we get

A[nlP + (J5/p)<p-X] = 0. (3.13)

This equation admits the following representation for p (see the Appendix,Theorem 1):

P = o + ^ i , (3.18)with

curlAt|>0 = 0, (3.19)

(3.20)

Substituting forp and q from (3.14) and (3.18) in (3.12) and using (3.15),(3.16), (3.19) and (3.20), we obtain

a2

(V + l jo) = 0, (3.21)


from which it follows that

(3.22)

where ty2 and % are independent of t.Taking divergence of (3.22) and using (3.15), we get

t div ij)2 + div t|>3 = 0. (3.23)

This equation holds for all t if and only if

div H>2 = 0, divt|>3 = 0, (3.24)

from which we get the representations

\l>2 = curl ij>2, T|>3 = curl ij>3, (325)

where ty2 a°d % are independent of /.

Taking the Laplacian of (3.22) and noting (3.15) and (3.19), we get

f Aii>2 + At|>3 = 0. (3.26)

For this to hold for all t, we should have, in view of (3.25),

curl AiJ>2 = 0, curl Atj>3 = 0. (3.27)

These equations yield the representations

^ = V<p2, Aij>3 = Vq>3, (3.28)

where q>2 and q>3 are independent of t.We now define the function ij> = t|>(P, t) by

^f%, (3.29)

where

<* = tq>2 + <p3 (3.30)

and R is the distance from the field point P to a point Q, the integrationbeing with respect to Q. Then we have

curl \|) = curl (t^ + ty2 + %)• (3.31)

Also, noting that

we obtain

D2tp = D2H., + D2(rti»2 + %) - clV<D. (3.33)

Since tji2 and tj>3 are independent of t, equations (3.28) and (3.30) reduce


the last two terms in (3.33) to zero; from equation (3.20) it then follows thatij> obeys the equation (3.6).

It may be verified that, in view of (3.14), (3.18), (3.23), (3.25) and (3.31),the representation (3.11) reduces to (3.1). We note that (3.16) is identicalwith (3.5). Consequently, (3.7) holds. From (3.10) it now follows that q>obeys (3.8). This proves the completeness of the solution described by (3.1),(3.5), (3.6) and (3.8).

In the classical case {fi = / = 0 = 0), equations (3.5) and (3.6) becomeidentical with (3.2); the Green-Lam6 solution of classical elastodynamics isthus recovered. It may be mentioned that the completeness proofs givenin several works (on classical elastodynamics), including the one given in(11, p. 233), assume that divy = 0 and demand that div\|> = 0. No suchrestrictions are imposed in our analysis. Our proof of completeness isanalogous to that given by Long (12) (in classical elastodynamics) in theabsence of body forces.

4. Boussinesq-Papkovitch-Neuber type solution

In classical elastostatics, the following representation for u, known as theBoussinesq-Papkovitch-Neuber solution, is well known (11, p. 139):

u = ft-fl2V(A + r.n). (4.1)

Here r is the position vector of a field point P, and A and ft are arbitraryfunctions obeying the equations

b. (4.2)

In the context of the theory presented in (2) also, we seek a repre-sentation of the type (4.1) for u. Substituting for u from (4.1) into theright-hand side of (2.5), we find, on using (2.13), that

L,(in, 4>) = D2fi - V[a2{D,(A + r . SI) - 2c? div SI) - (fi/p)<p]. (4.3)

Clearly, if we set

{filp)4> = a2{D,(A + v. SI) - lc\ div SI) (4.4)

and assume that SI obeys the equation

D 2 f l = - ( l / p ) b , (4.5)

then (2.1) is readily satisfied.Substituting for u and 2(A + r.ft)-2£>3divft. (4.6)pa


We verify that if A is assumed to obey the equation

= 2D3 div fi - D2(T . ft) - fill pa2 , (4.7)

then (2.2) is also satisfied.Thus, if ft and A are arbitrary functions obeying (4.5) and (4.7), then the

representations (4.1) and (4.4) constitute a solution of the system of fieldequations (2.1), (2.2).

We now show that this solution is complete as well.Suppose that {u, (p) is an arbitrary solution of the system (2.1), (2.2). In

view of the Helmholtz resolution, there exist functions p and q such that(3.11) holds.

We consider a function AQ = AQ(P, f) defined by

( 4 8 )Anc2 JD

where the notation is as in (3.29), and set

fi, (4.9)

j ) = A. (4.10)

Substituting for curl q and p from (4.9) and (4.10) in (3.11), we get (4.1).The function AQ, defined by (4.8), obeys the equation

D2A0 = (/3/p)^ + D1p. (4.11)

Eliminating p from (4.10) and (4.11) and using (2.13), (2.15) and (4.9), weobtain (4.4).

Substituting for n and ) = D2ft. (4.12)

Since {u, <p) is a solution of the system of equations (2.1), (2.2), it followsthat ft and A obey (4.5) and (4.7). This proves that the solution describedby (4.1), (4.4), (4.5) and (4.7) is complete.

In the classical case {fi = / = <p = 0) equation (4.4) yields

DjA = 2c\ div ft - D^r . fi)

= -r.ntfi. (4.13)

Thus, in the classical case (4.1) represents a complete solution for n when fiand A obey (4.5) and (4.13). This solution, which is a dynamic generaliza-tion of the Boussinesq-Papkovitch-Neuber solution, agrees with thatrecorded in (11, p. 235).


In the time-independent case, the representation (4.4) for 0 (in thepresence of voids) reduces to

(4.14)

Also, equations governing £2 and A, viz. (4.5) and (4.7), become

(4.15)

D4AA = ^ ( 1 ~ ' ^ ) div ft - D4(r. Aft) -—,, (4.16)

pa pa

where

D4 = (*A-£)c 2 + /32/p. (4.17)Thus, in time-independent problems, (4.1) and (4.14) constitute a completesolution of the field equations (2.1), (2.2) (in the presence of voids), whenft and A obey (4.15) and (4.16).

In the classical elastostatic case (/3 = / = 0=O, 3/<9r = 0) expressions(4.14) and (4.15) yield

AA = - r . b . (4.18)

Equations (4.15) and (4.18) are precisely the equations (4.2); the classicalBoussinesq-Papkovitch-Neuber solution is thus recovered.

5. Cauchy-Kovalevski-Somigliana type solution

The following representation for u, known as the Cauchy-Kovalevski-Somigliana solution, is well known in classical elastodynamics (11, p. 235):

u = D,g-2c2a2Vdivg, (5.1)

D 2 D l g =- ( l /p )b . (5.2)

In the context of the theory presented in (2), we seek the followinggeneralized version of (5.1):

o = (l/a)[D2G - a2V{2£>3 div G + (p7p)//}]. (5.3)

Here G and H are to be determined appropriately.Substituting for a from (5.3) in the right-hand side of (2.5) we find, on

using (2.14), that

<rL,(u, <p) = IhU2G - (^/p)V(a2D,H + /3D2 div G - a<t>). (5.4)

Clearly, if we set

0 = - ( a 2 n , / / + /3n2divG) (5.5)


and assume that G obeys the equation

D2D2G = -(a/p)b, (5.6)

then (2.1) is readily satisfied.Substituting for u and <p from (5.3) and (5.5) into the right-hand side of

(2.6), and using (2.11), (2.12) and (2.14), we obtain

(5.7)

We readily see that if H is assumed to obey the equation

(5.8)

then (2.2) is also satisfied.Thus, if G and H are arbitrary functions obeying (5.6) and (5.8), then the

representations (5.3) and (5.5) constitute a solution of the system ofequations (2.1) and (2.2).

We now show that this solution is also complete.Suppose that {u, <p} is an arbitrary solution of (2.1), (2.2). In view of

Helmholtz resolution there exist p and q such that (3.11) holds.We set

where p0 and q0

Then we have

G = Vpo +

{fia2lp)H = D]

are such that

D2Q2P0 = ^ [D i j

£>2qo =

divG =

curlqo,

D + {filp)4>\,

orq.

Apo»

+ a curl q.

(5.9)

(5.10)

(5.11)

(5.12)

(5.13)

(5.14)

Substituting for p and curl q from (5.10) and (5.14) in (3.11) and takingnote of (2.15) and (5.13), we obtain (5.3). Together with (5.10) and (5.13)as well as (2.11), equation (5.11) yields (5.5). Substituting for n and <p fr°m

(5.3) and (5.5) in the right-hand sides of (2.5) and (2.6), and using (2.11),(2.12) and (2.14), we obtain (5.7) and

aLx(n, <p) = D ^ G . (5.15)

Since {u, (p) is a solution of the system (2.3), (2.4), it follows that G and Hobey (5.6) and (5.8). This proves the completeness of the solution describedby (5.3), (5.5), (5.6) and (5.8).

In the classical case (fi = l = 2 = ADj and


(l/a)D3 = c2A; see (2.11) and (2.12). Consequently, (5.3) and (5.5) become

u = Dtg - 2c2a2V div g, (5.16)

• 2 D l g = - ( l / p ) b . (5.17)

Here we have set AG = g.The representation (5.16) and equation (5.17) are identical with (5.1) and

(5.2) respectively; the Cauchy-Kovalevski-Somigliana solution of classicalelastodynamics is thus recovered.

In the time-independent case, (5.3), (5.5), (5.6) and (5.8) become

a = (l/a)[D4AG - a2V{D5 div G + {filp)H}}, (5.18)

4> = {U a)A[a2c\H + j8c! div G], (5.19)

(5.20)

(5.21)

Here, we have set2 (5.22)

Thus, in time-independent problems, (5.18) and (5.19) constitute a com-plete solution (in the presence of voids) when G and H obey (5.20) and(5.21).

In the classical elastostatic case (0 = % = / = <f> = 0, 9/dt = 0), (5.18) and(5.20) become

u = Ag - 2a2V div g, (5.23)

AAg=-(l / / i )b , (5.24)

where we have set cfAG = g. We note that (5.23) and (5.24) correspond tothe Boussinesq-Somigliana-Galerkin solution in classical elastostatics (11,p. 141).

6. Connections between solutions

A connection between the three solutions presented in sections 3 to 5 maybe established easily if the functions q>, t|», A, ft and H, G are relatedthrough the equations

ft = curl t|i + VI\ 1

A = (l/a2)(r-<p)-(r.ft), j ( 6 1 )

where T = T{P, t) is given by

1r = _ _


and

I^ I (6.3)ifilp)H = o-(A + r . ft) - 2Z>j div G. J

It is straightforward to verify that if we substitute for q> and ty fromrelations (6.1) in (3.1), (3.5), (3.6) and (3.8), taking note of (2.13), (2.15)and (6.2), we obtain (4.1), (4.4), (4.5) and (4.7). Similarly, if we substitutefor ft and A from (6.3) in (4.1), (4.4), (4.5) and (4.7), taking note of (2.14),we obtain (5.3), (5.5), (5.6) and (5.8). Other interrelationships among thesolutions also follow from relations (6.1) to (6.3).

Note that whereas the system of equations governing ty and <p, viz. (3.6)and (3.8), and the system of equations governing G and H, viz. (5.6) and(5.8), are completely uncoupled, the system of equations governing ft andA, viz., (4.5) and (4.7), are not completely uncoupled. While determining ftand A in a given problem, first (4.5) has to be solved for ft and then (afterdetermining SI) equation (4.7) has to be solved for A.

7. Uncoupled field equationsIn what follows we deduce uncoupled versions of the coupled field

equations (2.1), (2.2).Eliminating q> from (3.5) and (3.8) and using (2.11) and (3.3) we obtain

the following equation that contains div u + Dj div b - 0 A/= 0. (7.2)

From (3.8), (7.1) and (7.2) we note that in the absence of b and /, thefunctions q>, <fi and divu satisfy one and the same equation: D^F = 0.

With the aid of equations (7.1) and (7.2) as well as (2.12) and (2.13),equation (2.1) reduces to the following equation that contains u as the onlyunknown function:

pEhP2u - V{2fl2D3 div b + /3CV} + D2b = 0. (7.3)

The coupled system of field equations (2.1) and (2.2) has thus beendecoupled into two independent equations (7.1) and (7.3). Whereas each ofthe equations in the coupled system (2.1), (2.2) is of order two, in theuncoupled system equation (7.3) governing n is of order six and equation(7.1) governing 0 is of order four.

Acknowledgement

The author thanks the University Grants Commission, New Delhi, forfinancial assistance through research grant F.8-3/84SRIII.

412 D. S. C H A N D R A S E K H A R A I A H

R E F E R E N C E S

1. J. W. NUNZIATO and S. C. COWDM, Arch, ration. Mech. Analysis TL (1979)175-201.

2. S. C. COWIN and J. W. NUNZIATO, / . Elast. 13 (1983) 125-147.3. and P. PURI, ibid. 13 (1983) 157-163.4. S. L. PASSMAN, ibid. 14 (1984) 201-212.5. S. C. COWIN, ibid 14 (1984) 227-233.6. , Q. Jl Mech. appl. Math. 37 (1984) 441-465.7. P. PURI and S. C. COWIN, /. Elast. 15 (1985) 167-183.8. S. C. COWIN, ibid. 15 (1985) 185-192.9. D. S. CHANDRASEKHARAIAH, Acta Mech. 58 (1986) to appear.

10. D. IESAN, / . Elast. 15 (1985) 215-224.11. M. E. GURTIN, Encyclopedia of Physics, Vol. VI a/2 (ed. S. Flugge; Springer,

Berlin 1972).12. C. F. LONG, Acta Mech. 3 (1967) 371-375.

APPENDIX

THEOREM 1. The equation

A(D l P-9) = 0 (A.I)

admits the following representation for p:

p = q> + q?0, (A.2)

withA<po = 0, (A.3)

•i<P = q- (A.4)

Proof. Put

D,p - q = - p , (A.5)

so that

Ap,=0. (A.6)

Consider the function p2 defined by

Jo JofPi(P,to)dtodi. (A.7)o

Then we have

by (A.6), so that we may set

&p2 = q, + tq2, (A.8)

where qt and q2 are independent of t. Define <p0 by

<Po = p 2 + qi + t42, (A.9)

where <J, and Q2 are particular solutions of


Equations (A.8) to (A.10) then yield (A.3); consequently (A.5), (A.7) and (A.9)yield

If we set

<p=p-<p0 (A.12)

we get the representation (A.2) and (A.11) yields (A.4).

THEOREM 2. The equation

curlA(D2q-Y) = 0 (A.13)

admits the following representation for q:

with

= Y- (A-16)

Proof. Put

where ^ is a scalar to be chosen appropriately at a later stage. Then we have, by(A.13),

Acurlq, = 0. (A. 18)

Consider the function <fo defined by

<b(P> ° = { f qi(F>Then we have

by (A. 18), so that we may set

A curl q^ = ij>, + rtp2, (A.20)

where t)>, and t ^ are independent of t. Define y0 by

where rj), and ty2 a^e particular solutions of

curl Aij>, = -tyu curl AtJ>2 = - ^ 2 - (A.22)

Equations (A.20) to (A.22) then yield (A.15); consequently, we may write

Atl>0 = V<?2 (A.23)

for some scalar q2. If we set

i|>i = q-tl>o, (A.24)


we obtain the representation (A.14), and (A.17), (A.19), (A.21) and (A.23) yield

Djifc = V<? - c\Vq2 + y. (A.25)

If we set <J = c\q2, then (A. 16) follows.

It may be noted that the two theorems proved above are analogous to Boggio'sdecomposition theorem on the repeated wave equation (11, p. 237).

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COMPLETE SOLUTIONS IN THE THEORY OF ELASTIC MATERIALS WITH VOIDSsweers/papers/Chandrasekharai... · 2007. 3. 9. · elastic materials consisting of a distribution of small pores (voids)