Saint - Venant ' s Principle for Linear Elastic Porous Materials

Toupin's version of the Saint-Venant's principle in linear elasticity is generalized to the case of linear elastic porous materials. That is, it is shown that, for a straight prismatic bar made of a linear elastic material with voids and loaded by a self-equilibrated system of forces, at one end only, the internal energy stored in the portion of the bar which is beyond a distance s from the loaded end decreases exponentially with the distance s. Introduction Mathematical versions of SaintVenant's principle in linear elasticity due to Sternberg, Knowles, Zanaboni, Robinson and Toupin have been discussed by Gurtin [1] in his monograph. Later developments of the principle for Laplace's equation, isotropic, anisotropic, and composite plane elasticity, three-dimensional problems, nonlinear problems, and time-dependent problems are summarized in the review articles by Horgan and Knowles [2] and by Horgan [3]. For a linear elastic homogeneous prismatic body of arbitrary length and cross-section loaded on one end only by an arbitrary system of self-equilibrated forces, Toupin [4] showed that the elastic energy U ( s) stored in the part of the body which is beyond a distance s from the loaded end satisfies the inequality ~ U(s) ~ U(O)exp[-(s -l)jsc(l)]. (1) ;J~ The characteristic decay length sc( 1) depends upon the maximum and the minimum elastic moduli of the material and the smallest nonzero characteristic frequency of the free vibration of a slice of the cylinder of length 1. By using Ericksen's [5] estimate for the norm of the stress tensor in terms of the strain energy density, one can show that sc( 1) depends on the maximum elastic modulus and not on the minimum elastic m~dulus. Inequalities similar to (1) have been obtained by Berglund [6] for ,linear elastic micropolar prismatic bodies, by Batra [7-9] for non-polar and micropolar lin266 R. C. BATRA AND J. S. YANG ear elastic helical bodies and prismatic bodies of linear elastic materials with microstructure, and by Batra and Yang [10] for linear piezoelectric materials. Herein we prove a similar result for a straight prismatic body made of a linear elastic material with voids. We assume that the cross-sections are materially uniform in the sense that one cross-section can be obtained from the other by a rigid body motion. Thus the material properties are independent of the axial coordinate of the point. Ericksen . [5] has discussed material uniformity in more general terms. " Governing Equations for Linear Elastic Porous Materials Let the finite spatial region occupied by the linear elastic porous body with voids be V, the boundary surface of V be S, the unit outward normal of S be ni, and S be partitioned as Su U ST = S, Su n ST = 0. (2) The governing equations without body sources and boundary conditions for quasi static deformations of the body in rectangular Cartesian coordinates are [11, 12] Tij,i = 0, hi,i + 9 = 0 in V, Tij = ~ = CijklSkl + Bij + Dijk,k in V, 9 = -~=-B.'S.._l:"'-d."" in V 8<1> '3 '3 <. 'f' "f',' , 8W h. = -=D kl ,S kl+ d."' + A.."'. in V , 8'" . ' "f' '3'f',3 , 'f',' Sij = !(Uj,i + Ui,j) in V, Ui = Ui on Su, niTij = fj on ST, nihi = h on Sh, (3) where Ui is the displacement, Tij the stress tensor, Sij the strain tensor, the change in volume fraction, hi the equilibrated stress vector, and 9 the intrinsic ;; equilibrated body force. Su and ST are parts of the boundary S on which mechanical displacement and traction are prescribed as Ui and fi, respectively; and the normal component of the equilibrated stress vectornihi is prescribed ash on S. Throughout ;.) this paper, a repeated index implies summation over the range of the index, and a comma followed by an index j stands for partial differentiation with respect to x j. W(Sij, <1>, ,i) is the internal energy density function given by W = .!CijklSijSkl + !~2 + !Aij,i,j +BijSij + DijkSij,k + di,i, (4) SAINT-VENANT'S PRINCIPLE 267 which is assumed to be a positive definite, homogeneous quadratic function of the ten variables Sij, , ,i [10,11]. To save some writing we denote the ordered triplet (Sij, , ,i) by r and write W as W = ! r . Er . (5) Thus E is a linear transformation from a 1 O-dimensionallinear space into a 10-dimensionallinear space. Because of the positive definiteness of W 8W 8W 2 . ar . ar = Er . Er = r . E r ~ aMr . Er = 2aMW, (6) where aM is the supremum of the eigenvalues ofE. Fonnulation of the Problem Consider an unstressed prismatic bar with materially uniform cross-sections and made of a linear elastic porous material. Introduce a fixed rectangular Cartesian coordinate system so that in the unstressed reference configuration the x3-axis coincides with the axis of the bar, one end is contained in the plane X3 = 0 and for points in the bar X3 ~ O. Since the cross-sections of the bar are assumed to be materially uniform, E depends only on Xl and X2. Hence W = W(Sij, , ,i, XA), A = 1, 2 (7) in which W is a homogeneous quadratic function of the indicated variables except XA. An infinitesimal rigid body displacement is described by a uniform translation Ci and a rotation bji = -bij. The displacements associated with a rigid body displacement are Wi = Ci + bjiXj. (8) Thus if Vi = Ui + Wi, (9) then . .. , Sij(V) = Sij(U), (10) and W(Sij, , ,i, XA) is unchanged. '. The equations and boundary conditions for quasi static deformations of the prismatic bar are (~ ) = 0, (~ ) ~ = 0 in V, 8Sij ,i 8,i,i 8 (11) niTij = tj, nihi = 0 on S, 268 R. C. BATRA AND J. S. YANG where we have assumed that ST = S. We are interested in the case when the part X3 = 0 of the boundary S is loaded and the remainder of the boundary is traction free; hence tj is nonzero only at X3 = O. In order that there exists a solution to (11), the applied loads must be self-equilibrated and must satisfy [ ti dS = 0, [ilkjXjtk dS = O. (12) Jco Jco Here moments are taken with respect to the origin, £ijk is the alternating tensor, and C s is the cross-section of the body contained in the plane X3 = s. With the definition U(s) = [ WdV, (13) JX3~S we state and prove below the , THEOREM. If a prismatic body made of a linear elastic porous material and with materially uniform cross-sections is loaded on Co by a self-equilibratedforce system, then U(s) ~ U(O) exp[-(s -l)/sc(l)], (14) where sc(l) = 2(aM/Ao(I»1/2, (15) AO( 1) is the smallest nonzero eigenvalue of the following eigenvalue problem ( 8W ) (8W) 8W. ~ = AUj, ..aT + {iJ: = A 10 V, tJ ,i 'I',t ,i 'I' (16) niTij = 0, nihi = 0 on S, fora slice of the prismatic body of axial length 1. In (16) V is the region of the slice between X3 = sand X3 = s + 1, S is the total boundary surface ofV. Proof of the Theorem. Recalling (13) and W is a homogeneous quadratic function of the indicated variables except XA, we have by Euler's theorem 11 ( 8W 8W 8W ) U(s) = 2 ~Sij + {iJ: + aT,i dV ~ X3~S tJ 'I' 'I',t 11 ( 8W 8W 8W ) = 2 ~Uj,i + -8'" + _8'" .,i dV " X3~S tJ 'I' 'I',t 1 [ ( 8W 8W ) = 2Jr ni~Uj + niaT dS c. tJ 'I',t 1 1 ( 8W 8W ) = -2 ~Uj + ~ dB, (17) c. 3J '1',3 SAINT-VENANT'S PRINCIPLE 269 where we have used the strain-displacement relation (3)6, the divergence theorem and nk = -O3k on Cs. Using the inequality 2 f fh dV ::;; a f r dV + ~ f h2 dV, (18) Jv Jv a Jv which holds for a > 0 and is a consequence of the Schwarz and the geometricarithmetic mean inequalities (e.g. see Toupin [4]), we obtain 1l aw 1 ( l aw aw 1 1 ) -2 -as.ujdS::;;"4 aI -as .-as . dS + ujujdS. (19) C. 33 C. 33 33 at C. Similarly 1 l aw 1( l aw aw 1 1 2 ) -2 ~dS::;;"4 a2 ~~dS + dS, (20) C. '1-',3 C. '1-',3 '1-'.3 a2 C. and hence 1 [ 1 (aw aw aw aw aw aw) U(s) ::;; "4 {3 ~~ + BTBT + a-;:a-;: dB, C. '3 '3 '1-'.' '1-'.' 'I-' 'I-' + ~ fa. (UjUj + <1>2) dS] , (21) where we have set at = a2 = {3. Substituting from (6) into (21) results in U(s) ::;; l [{3 fa. 2aMW dS + ~ fa. (UjUj + <1>2) dB] . (22) Integration of both sides of (22) with respect to X3 from X3 = s to X3 = s + 1 for some 1 > 0 and setting 1 fs+1 IJs U(y)dy=Q(s,/) (23) gives {3aM f 1 f 2 , Q(s, I)::;; ~ J( WdV + 4 {31 J( (UjUj + )dS, (24) c.,! c.,! in which ~ Cs.I = {X:XE V,s::;; x3::;;s+/} = portion of the prismatic body between the planes X3 = s and X3 = S + I. (25) In order to bound the last integral on the right-hand side of (24) by an integral of W, we consider the eigenvalue problem (16) on C s.l. By taking the inner product 270 R. C. BATRA ND J. S. YANG of (16)1 with Uj and (16)2 with c/J, adding the respective sides of the resulting equations and integrating them over Cs,l, using the divergence theorem and the boundary conditions (16)3,4, we obtain . 2.1; WdV ,\ = C.,I . (26) fc (UjUj + c/Jc/J) dV ',1 Since W = 0 for a rigid body displacement, the smallest eigenvalue is zero. In order to eliminate the rigid body displacement and thereby the possibility of zero eigenvalue we consider smooth fields Vi and c/J that satisfy . 1 (VjVj + c/J2) dV i 0, 1 Vj dV = 0, 1 fijkX jVk dV = O. (27) C.,I C.,I C.,I As shown by Toupin [4], for a given Ui we can choose Wi in (9) such that Vi satisfies (27). Thus the lowest eigenvalue '\o( I) will satisfy the inequality 2fc WdV 0 < '\0(1) ~ fc (Vj~~ + c/Jc/J)dV. (28) ,,1 Substitution from (28) into (24) results in the following: Q(s,l) ~ ~ [ WdV, (29) 1c.,1 in which 1 2 sc(l) =2f3aM + ~. (30) We choose /3 = 2/(aM'\o)I/2 so that sc(l) takes on the minimum value sc(l) = 2(aM/'\o)I/2. (31) Differentiating (23) with respect o s yields ~ = ![U(s + I) U(s)] == _! [ WdV. (32):'-ds I I 1c.,1 This when combined with (29) results in i Sc(l)~ + Q ~ O. (33) Integrating (33) and using U(s + I) ~ Q(s, I) ~ U(s), (34) SAINT-VENANT'S PRINCIPLE 271 which follows from the observation that U( 8) is a nonincreasing function of 8, we arrive at U(S2 + 1) ( ) ~ exp[-(s2 sI)jsc(1)]. (35)