COMPLETE SOLUTIONS IN THE THEORY OF ELASTIC MATERIALS WITH VOIDS

Transcription

1 COMPLETE SOLUTIONS IN THE THEORY OF ELASTIC MATERIALS WITH VOIDS By D. S. CHANDRASEKHARAIAH (Department of Mathematics, Bangalore University, Central College Campus, Bangalore , India) [Received 17 April 1986] SUMMARY Three complete solutions of the field equations of the linear theory of homogeneous and isotropic elastic materials containing a distribution of vacuous pores (voids) are obtained. These solutions are analogous to and include as special cases the Green-Lame', Boussinesq-Papkovitch-Neuber and Cauchy-Kovalevski-Somigliana solutions 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 generalizations of the classical theory of elasticity. This theory is concerned with elastic materials consisting of a distribution of small pores (voids) which contain nothing of mechanical or energetic significance. The general version of this theory was proposed by Nunziato and Cowin (1) and the linearized version was deduced by Cowin and Nunziato (2). The new theory is intended to be of practical utility in investigating various types of geological, biological and synthetic porous materials for which the classical theory is inadequate. Some problems revealing interesting characterizations of the theory are contained in (2 to 9); some basic theorems and properties of solutions are obtained in (10). The inter-relationships between this theory and various other elasticity theories are analysed in (6). The object of this paper is to present three complete solutions of the field equations of the theory formulated in (2), for homogeneous and isotropic materials. These solutions are analogous to the well-known Green-Lame", Boussinesq-Papkovitch-Neuber and Cauchy-Kovalevski-Somigliana solutions in classical elastodynamics (11). Indeed, these classical solutions emerge as particular cases of our solutions. A connection among the solutions is also exhibited. The uncoupled versions of the field equations are deduced. (Q. J1 Mecta. appl. Math., Vol. 40, PI. 3, Oxford Unhtntty Pros 1987

2 402 D. S. CHANDRASEKHARAIAH 2. Basic equations In the context of the theory presented in (2) the complete system of field equations 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 volume fraction field (defined in (2)), b is the body force and / is the so-called extrinsic equilibrated body force (2), each measured per unit volume, A, n are the usual Lam6 constants, p is the mass density, a, ft,, a> and k are new material constants characterizing the presence of voids, and t is time. The usual vector notation is adopted; A denotes the Laplacian operator. If we set /3 = / = <p = 0, equation (2.2) is identically satisfied and (2.1) reduces to the classical Navier equation. For convenience, we rewrite (2.1) and (2.2) in the following form: Here we have put Li(u, d>) +-b = 0, (2.3) P 1 = 0. (2.4) Lj(u, <p) = D 2 n + 2c 2 a 2 V div a + (/3/p)V^>, (2.5) with We also need the operators "5' (2-7) I- + P*TJ, (2.8) dt dt 2 / (2.9), = c?a- J, (2.10) D 2 = D,a,+?- A, (2.11) P 2pa 2 (2.12)

3 ELASTIC MATERIALS WITH VOIDS 403 The following relations among the operators are easily verified: (2.13) 2pa 2 " (2.14) D 2 -D l a 2 = 2a 2 D 3 A. (2.15) Throughout our analysis, it is assumed that all functions appearing in the discussion are continuous and differentiable up to the required order on D X T, where D is a regular region in the Euclidean 3-dimensional space and T is a time interval. Also, all the differential operators are taken to be commutative. 3. Green-Lam6 type solution In classical elastodynamics, it is known that u admits a representation of the form a = Vcp + curl i >, (3.1) where cp andtj> obey the equations i<p = X, n 2 y = 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 representation of the type (3.1) for u. Substituting for u from (3.1) into the right-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 <p from (3.1) and (3.5) into the right-hand side of (2.6), we get (3.7) It follows that (2.2) is also satisfied if <p is assumed to obey the equation (3.8)

4 404 D. S. CHANDRASEKHARAIAH Thus, if tj> 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 every solution {u, <f>) of the system (2.1), (2.2) admits a representation as described 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\a 2 V div u + (filp)v<p - V* - curl y = 0, (3.9) (3.10) In view of the Helmholtz representation of a vector field, there exist functions 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[n lp + (J5/p)<p- X ] = 0. (3.13) This equation admits the following representation for p (see the Appendix, Theorem 1): P = <P + <Po, (3-14) with Taking curl curl of (3.12) we obtain (3.15) (3.16) curl A(D 2 q - Y) = - (3.17) This equation admits the following representation for q (see the Appendix, Theorem 2): q =M>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 a 2 (V + ljo) = 0, (3.21)

5 from which it follows that ELASTIC MATERIALS WITH VOIDS 405 where ty 2 and % are independent of t. Taking divergence of (3.22) and using (3.15), we get This equation holds for all t if and only if from which we get the representations (3.22) t div ij) 2 + div t > 3 = 0. (3.23) div H> 2 = 0, divt > 3 = 0, (3.24) \l>2 = curl ij> 2, T > 3 = curl ij> 3, (325) where ty 2 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), These equations yield the representations curl AiJ>2 = 0, curl Atj> 3 = 0. (3.27) where q> 2 and q> 3 are independent of t. We now define the function ij> = t >(P, t) by where ^ = V<p 2, Aij>3 = Vq> 3, (3.28) ^f%, (3.29) <* = tq>2 + <p 3 (3.30) and R is the distance from the field point P to a point Q, the integration being with respect to Q. Then we have Also, noting that curl \ ) = curl (t^ + ty 2 + %) (3.31) we obtain D 2 tp = D 2 H., + D 2 (rti»2 + %) - clv<d. (3.33) Since tji 2 and tj> 3 are independent of t, equations (3.28) and (3.30) reduce

6 406 D. S. CHANDRASEKHARAIAH the last two terms in (3.33) to zero; from equation (3.20) it then follows that ij> 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 identical with (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) become identical with (3.2); the Green-Lam6 solution of classical elastodynamics is thus recovered. It may be mentioned that the completeness proofs given in several works (on classical elastodynamics), including the one given in (11, p. 233), assume that divy = 0 and demand that div\ > = 0. No such restrictions are imposed in our analysis. Our proof of completeness is analogous to that given by Long (12) (in classical elastodynamics) in the absence of body forces. 4. Boussinesq-Papkovitch-Neuber type solution In classical elastostatics, the following representation for u, known as the Boussinesq-Papkovitch-Neuber solution, is well known (11, p. 139): u = ft-fl 2 V(A + r.n). (4.1) Here r is the position vector of a field point P, and A and ft are arbitrary functions obeying the equations b. (4.2) In the context of the theory presented in (2) also, we seek a representation of the type (4.1) for u. Substituting for u from (4.1) into the right-hand side of (2.5), we find, on using (2.13), that L,(in, 4>) = D 2 fi - V[a 2 {D,(A + r. SI) - 2c? div SI) - (fi/p)<p]. (4.3) Clearly, if we set {filp)4> = a 2 {D,(A + v. SI) - lc\ div SI) (4.4) and assume that SI obeys the equation D 2 fl=-(l/p)b, (4.5) then (2.1) is readily satisfied. Substituting for u and <pfrom (4.1) and (4.4) into the right-hand side of (2.6), we obtain j^tf, ) = > 2 (A + r.ft)-2 > 3 divft. (4.6) pa

7 ELASTIC MATERIALS WITH VOIDS 407 We verify that if A is assumed to obey the equation = 2D 3 div fi - D 2 (T. ft) - fill pa 2, (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 field equations (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 Anc 2 JD where the notation is as in (3.29), and set fi, (4.9) (48) 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 D 2 A 0 = (/3/p)^ + D 1 p. (4.11) Eliminating p from (4.10) and (4.11) and using (2.13), (2.15) and (4.9), we obtain (4.4). Substituting for n and <p fr m (4.1) and (4.4) into the right-hand sides of (2.5) and (2.6) and using (2.13), we obtain (4.6) and L^u,tf>) = D 2 ft. (4.12) Since {u, <p) is a solution of the system of equations (2.1), (2.2), it follows that ft and A obey (4.5) and (4.7). This proves that the solution described by (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.n t fi. (4.13) Thus, in the classical case (4.1) represents a complete solution for n when fi and A obey (4.5) and (4.13). This solution, which is a dynamic generalization of the Boussinesq-Papkovitch-Neuber solution, agrees with that recorded in (11, p. 235).

8 408 D. S. CHANDRASEKHARAIAH In the time-independent case, the representation (4.4) for 0 (in the presence of voids) reduces to Also, equations governing 2 and A, viz. (4.5) and (4.7), become where (4.14) (4.15) D 4 AA = ^ (1 ~'^ ) div ft - D 4 (r. Aft) -,, (4.16) pa D 4 = (*A- )c 2 + /3 2 /p. (4.17) Thus, in time-independent problems, (4.1) and (4.14) constitute a complete solution of the field equations (2.1), (2.2) (in the presence of voids), when ft 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 classical Boussinesq-Papkovitch-Neuber solution is thus recovered. pa 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-2c 2 a 2 Vdivg, (5.1) D 2 D lg =-(l/p)b. (5.2) In the context of the theory presented in (2), we seek the following generalized version of (5.1): o = (l/a)[d 2 G - a 2 V{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 Clearly, if we set <rl,(u, <p) = IhU 2 G - (^/p)v(a 2 D,H + /3D 2 div G - a<t>). (5.4) 0=-(a 2 n,// + /3n 2 divg) (5.5)

9 ELASTIC MATERIALS WITH VOIDS 409 and assume that G obeys the equation D 2 D 2 G = -(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 We readily see that if H is assumed to obey the equation (5.7) (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 of equations (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 G = Vp o + curlqo, {fia 2 lp)h = D ] where p 0 and q 0 are such that D2Q2P0 = ^[DijD + {filp)4>\, >2qo = orq. Then we have divg = Ap o» + 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 taking note 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 al x (n, <p) = D^G. (5.15) Since {u, (p) is a solution of the system (2.3), (2.4), it follows that G and H obey (5.6) and (5.8). This proves the completeness of the solution described by (5.3), (5.5), (5.6) and (5.8). In the classical case (fi = l = <p = 0), we have (l/ar) >2 = ADj and

10 410 D. S. CHANDRASEKHARAIAH (l/a)d 3 = c 2 A; see (2.11) and (2.12). Consequently, (5.3) and (5.5) become u = D t g - 2c 2 a 2 V div g, (5.16) 2 D lg =-(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 classical elastodynamics is thus recovered. In the time-independent case, (5.3), (5.5), (5.6) and (5.8) become Here, we have set a = (l/a)[d 4 AG - a 2 V{D 5 div G + {filp)h}}, (5.18) 4> = {U a)a[a 2 c\h + j8c! div G], (5.19) 2 (5.20) (5.21) (5.22) Thus, in time-independent problems, (5.18) and (5.19) constitute a complete 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 - 2a 2 V 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 to the 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 may be established easily if the functions q>, t», A, ft and H, G are related through the equations ft = curl t i + VI\ 1 A = (l/a 2 (61) )(r-<p)-(r.ft), j where T = T{P, t) is given by r= 1

11 and ELASTIC MATERIALS WITH VOIDS 411 ^ 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 from relations (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 substitute for 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 the solutions 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 and A, viz., (4.5) and (4.7), are not completely uncoupled. While determining ft and A in a given problem, first (4.5) has to be solved for ft and then (after determining SI) equation (4.7) has to be solved for A. 7. Uncoupled field equations In 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 <p as the only unknown function: D2<p + {filp) div b + LV = 0. (7.1) Together with (3.3) and (3.8), equation (3.1) yields pa> 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 /, the functions 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 only unknown function: pehp 2 u - V{2fl 2 D 3 div b + /3CV} + D 2 b = 0. (7.3) The coupled system of field equations (2.1) and (2.2) has thus been decoupled into two independent equations (7.1) and (7.3). Whereas each of the equations in the coupled system (2.1), (2.2) is of order two, in the uncoupled 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, for financial assistance through research grant F.8-3/84SRIII.

12 412 D. S. CHANDRASEKHARAIAH REFERENCES 1. J. W. NUNZIATO and S. C. COWDM, Arch, ration. Mech. Analysis TL (1979) S. C. COWIN and J. W. NUNZIATO, /. Elast. 13 (1983) and P. PURI, ibid. 13 (1983) S. L. PASSMAN, ibid. 14 (1984) S. C. COWIN, ibid 14 (1984) , Q. Jl Mech. appl. Math. 37 (1984) P. PURI and S. C. COWIN, /. Elast. 15 (1985) S. C. COWIN, ibid. 15 (1985) D. S. CHANDRASEKHARAIAH, Acta Mech. 58 (1986) to appear. 10. D. IESAN, /. Elast. 15 (1985) M. E. GURTIN, Encyclopedia of Physics, Vol. VI a/2 (ed. S. Flugge; Springer, Berlin 1972). 12. C. F. LONG, Acta Mech. 3 (1967) APPENDIX THEOREM 1. The equation A(D lp - 9 ) = 0 admits the following representation for p: p = q> + q? 0, with A<p o = 0, i<p = q- Proof. Put D,p - q = -p, so that Ap,=0. Consider the function p 2 defined by (A.I) (A.2) (A.3) (A.4) (A.5) (A.6) Then we have Jo Joo f Pi (P,to)dt o di. (A.7) by (A.6), so that we may set &p 2 = q, + tq 2, where q t and q 2 are independent of t. Define <p 0 by <Po = p 2 + qi + t42, where <J, and Q 2 are particular solutions of (A.8) (A.9)

13 ELASTIC MATERIALS WITH VOIDS 413 Equations (A.8) to (A.10) then yield (A.3); consequently (A.5), (A.7) and (A.9) yield If we set <p=p-<p 0 we get the representation (A.2) and (A.11) yields (A.4). THEOREM 2. The equation curla(d 2 q-y) = 0 admits the following representation for q: (A.12) (A.13) with Proof. Put = Y- (A-16) where ^ is a scalar to be chosen appropriately at a later stage. Then we have, by (A.13), Consider the function <fo defined by <b(p> = { f qi(f> Then we have Acurlq, = 0. (A. 18) by (A. 18), so that we may set where t)>, and t^ are independent of t. Define y 0 by A curl q^ = ij>, + rtp 2, (A.20) where rj), and ty 2 a^e particular solutions of curl Aij>, = -ty u 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 q 2. If we set i >i = q-tl>o, (A.24)

14 414 D. S. CHANDRASEKHARAIAH we obtain the representation (A.14), and (A.17), (A.19), (A.21) and (A.23) yield If we set <J = c\q 2, then (A. 16) follows. Djifc = V<? - c\vq 2 + y. (A.25) It may be noted that the two theorems proved above are analogous to Boggio's decomposition theorem on the repeated wave equation (11, p. 237).