Wave propagation and uniqueness in magneto-elastodynamics

InI. J. Pr int ed Engng Sci. Vol. 24 , No. 5, pp. 713-717, 0020-7225/ 86 1986 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA in Great Britain. Q WAVE PROPAGATION AND UNIQUENESS MAGNETO-ELASTODYNAMICS 1986 $3.00 Per gamon + Pr cs .OO Ltd. IN BRUNO CARBONARO and REMIGIO RUSSO Dipartimento di Matematica e Applicazioni dell’Univetsita di Napoli, via Mezzocannone, 8-80 I34 Napoli, Italy (Communicated by E. SUHUBI) Abstract-By using the method introduced in [ 11,some sufficient conditions on the acoustic tensor and on the kinetic and magnetic fields are given in order that a perturbation initially confined in a proper subset of an unbounded magnetoelastic solid, propagate with finite speed. Moreover, a strong uniqueness theorem is proved for regular solutions to the initial-boundary-value problem of magnetoelastodynamics. 1. INTRODUCTION of the present paper is to give a condition on the acoustic tensor of an unbounded electrically conducting elastic body zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPO B of infinite conductivity, assuring wave propagation, namely, that a perturbation initially confined in a bounded subset of B, have a bounded support in B at each instant. This will be achieved by means of a general domain of influence theorem, which leads, as a simple consequence, to a strong uniqueness result. Let B be an unbounded electrically conducting elastic body, identified with the regular domain D of JR37 it occupies in an assigned reference configuration. We assume that the conductivity of B is infinite, so the system governing the motion of the body is [2] zyxwvutsrqponm THE MAIN PURPOSE pii=V-C[Vu]+p(VXH)XH+b, h = V X (i X H) V-H on Q = D X (0, +a~), = 04. (1.1) In ( 1. l), p is the (positive) mass density, u the displacement, C the (fourth-order) elasticity tensor,@ b the body force per unit volume, p (>O) the magnetic permeability and H the magnetic field. Moreover, the superimposed dot means partial differentiation with respect to time t, V - C[Vu] = zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC dj(C i&&)e i, where & = d/dXk and uh, C$,k are the COIIIpOnentS on R of u and C, respectively, and summation over repeated indexes is implied. The tensor C will be required to be symmetric and positive semidefinite8, i.e. L-C[M] = MaC[L] VL, M E Lin. L-C[L] 2 0 Moreover, denoting by A the acoustic tensor-for the direction m [3], defined by pA(x, m)a = C[a 63 m]m, for any vector a, throughout the paper we shall make the following basic assumption: t As is usual, by R = p; ei} (i = 1, 2, 3) we denote a reference frame on R’; so, Vx E w’, x - o = x’ei = re,, with r = Ix - 01 = (xixi)” . Moreover, S(x, R) stands for the ball centered at x of radius R and ZR stands for the boundary of S, = S(o, R). $ If H is twice differentiable on Q, then (1.Ih, by virtue of ( 1.I)*, is equivalent to the initial condition V - H = 0 on D X (0). QHere C is a linear mapping from Lin (the space of all second-order tensors) into Sym (the space of all symmetric second-order tensors), such that C[L] = 0, for any second-order skew tensor L. ll Recall that these conditions imply the inequality ZL*C[M] 5 [L*C[L] + 5-‘M.C[M], VE > 0, VL, M E Lin. 773 B. CARBONARO and R. RUSSO 774 There exist a smooth, positive, increasing and convex function p(r) on [0, +a) and a positive constant c such that lim p(r) = +co, r-+m Vm : (m( = 1. Vx E D, zyxwvutsrqponmlkjihgfedcb { ~‘(r)}*lA(x, @I I : , Remark 1. It is worth remarking that hypothesis (*) has an interesting physical interpretation. Indeed, let V(r) be the maximum of ]A]“* on Z,. Then, the time employed by a perturbation to reach JZRfrom ZRo is at least equal to i: { V(r)}-‘& L 6 p’(r)dr = P(R) - Q(Ro)* Hence, by letting R - + co, it is readily seen that (*) simply expresses the obvious requirement that a perturbation may not invade the whole space in a finite time. 0 Remark 2. For the sake of completeness, we want to suggest that, as an example of function p(r), we may take any element of the sequence 1log(* . *(log[r + exp(. * - .(expl)]). , * - s)}~~. 0 h times h times As far as the smoothness assumptions on the data are concerned, we suppose that? P E C(D), b E C,(d zyxwvutsrqponmlkjihgfedcbaZYXW X (0, +a)), C E C:(D) n Cd@), and we shall consider only solutions (u, H) to system ( 1.1) such that uE c:(Q) n Cl@ x (0, +co)) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQP n C;(D x [0, +co)), H E C;(Q) fl C,@ X (0, +co)) fl C,(D X [0, +a)). Let now (u, H) and (u + v, H + h) be two solutions to system (1.1) corresponding to body forces b and b + f, respectively. Then the perturbation (v, h) satisfies the system pt = V.C[Vu] li = V X (ti X H) + ~(0 X h) - V X 2. A DOMAIN X h+ [(H + h) p[(V X h) X (H + h)] + f, X i]. (1.2) OF INFLUENCE THEOREM This section deals with the definition of domain of influence of data and with the statement and proof of the related theorem. Let us assign three vector fields v. E C;(D) and Vo, he E C,(D). To system (1.2) we append the initial conditions v = vo, iJ = Go, h=b on D x (0). Consider the set D*(t) of the points x E D such that (i) x E D * 3s E (0, t]: qo(x) > 0 or f(x, s) Z 0; (ii) x E aD * 3s E (0, t]: {(i - C[Vv] + ph X [(H + h) where 77(x, t) = i(pti’ + Vv.C[Vv] + X i] - +ph*ti). n}(x, s) # O$ ph2), q(x, 0) = vo(x), aD is the boundary of D t We denote by Cp(G), G c @ X [0, +to), m, k E No, the set of all kth order tensor-valued functions on G, whose components are continuously differentiable up to the order m inclusive. $ Throughout the paper, n will denote the outward unit normal to any boundary we shall consider. 775 Wave propagation and uniqueness in magneto-elastodynamics and d = zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA D zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA U aD. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA We shall call domain of influence of the initial and boundary data and body.forces at instant t the set D&) = {xg E D: D*(t) f7 S(Q, p-‘[p(r,) + ct] - ro> Z 0}. The following theorem holds. Domain ofinfluence theorem. Let (u, H), (u + v, H + h) be two solutions to system (1.1) corresponding to body forces b and b + f, respectively. If Ii11+ {p-'p}"21H + hi 5 G {p’(r)}-‘, for the same p and c as in (*), and vo, hp-‘f 2, (i - C[Vv] + ph X [(H + h) X 61 - $ph21;)- n are locally summable on D, D X (0, t), dD X (0, t), respectively, Vt > 0, then v=h=O on {fi - Dp} X [0, t]. •! Remark 3. The above condition is quite similar to the assumption on the acoustic tensor. The second term in the left-hand side is the Alfven velocity in the motion whose displacement field is u + v. The presence of hii simply means that the unperturbed motion influences the propagation speed of the wave. 0 The proof of the above theorem will be carried out as a simple consequence of the following: Lemma (domain of zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFED dependence inequality). If the hypotheses of the foregoing theorem are met, then, VT > 0, s 7(x T)dV S eXfT) &(x)dV + to lT D(wwR) ’ IJ Woo,R) e-“(“ds[ s,,,,,, ip-tf2(x, s)dV + dDnS(MRs,{(ir.C[Vv] + ph X [(H + h) X i] - ~~h2U).n}(x, s)dL7 s , (2.1) where DQq, R) = D n S(% , R), R, = p-‘[p(R + ro) + c(T - s)] - ro, r. = Ix,, - 01, X(T) = so [ti’ + m(s)]ds, with m(s) = SUPS(~,R,){ (~-‘/.L)“~IV X HI + 4&71iI} and to a positive constant with time dimension. 0 Proof Let R, T> 0, $6, 4 = AR + ro) + c(T - s) - p(lx - zyxwvutsrqponmlkjihgfedcbaZYXWVUTSR xo l+ ro), and let g6 E Coo(R) (6 > 0) be a nondecreasing function, vanishing on (-co, 0] and equal to 1 on [6, +co). Then the support of the function g(x, s) = ga(c-‘$(x, s)), defined by the condition J/(x, S) 2 0, is certainly compact in Q3 X [0, +co)j-. Multiply both sides of ( 1.2), by S; and both sides of ( 1.2), by ~gh. By using the identities gi - V * C[Vv] = V * {gi - C[Vv]} - gVl - C[Vv] + gbp’i - C[Vv]e,, g[(V X h) X (H + h)] -i =g[(H+h)X~].(VXh)=gV.{hX[(H+h)Xv]}+gh.VX[(H+h)Xir] =V.{ghX[(H+h)X~]}+gbp’(hX[(H+h)Xv]}.e,+gh.VX[(H+h)Xv], g[V X (ti X h)] - h = g(h * Vu) - h - $0. {gh2U} - $gh2V * u - fgbp’h2b. e,, t Observe that g E C”(W’ X [0,+m)) even if + is not smooth along the straight line x = Q. Indeed, in the set {(x, s)E w3 X (0,+m): Ix- ~1 5 p-‘[p(R + ro) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA + c(T - s - a)] - r,,, s S T}, which is not empty for sufficiently small 8, one has g(x, S) = I. 776 B. CARBONARO and R. RUSSO and integrating over D, = (x E D: Ix - y[ 2 w > 0, zyxwvutsrqponmlkjihgfedcbaZYXWVUT Vy f dD}; the simple application of the divergence theorem leads to s s s DwmdV= - D, g;ivdV + c-’ + gip’{ i *C[Vv] + ph X [(H + h) X +] - $&%I} *e,d V s DW gb4G’ X H) x hl - i DU - $gh2VG + p(h*Vti).h + f.ii}dV s + J& g{i. C[Vv] + ph X [(H + h) X V] - $ph*ti) . ndZ, (2.2) for any s S T. We may majorize the terms in the third and the fourth integral of (2.2) as follows: C[; @ e,]e, C’p’V - C[Vv]e, 5 $Vv - C[Vv] + c-“p”i. = $Vv - C[Vv] + c-*p”(lx - ~1 -f r,JA)pti* 5 iv, pc-‘p’{h X [(H + h) X ;] - ih*ti} .e, S @‘P’{(hl(itllH + hl + $h21rll} 5 f~~‘p’{(p-‘~)“~JH + hj(pti2 + ph2) + ph2jri(} 5 fq, p{[(V X H) X h]*ir - $h2V.ti + (haVti)*h) 5 ;(p-‘fi)1’2)V x Hl(pti* + ph*) + 4p)VUlh2 i m(s)rj, f. i 5 ;t&NY + ft(jp-‘j-2. Thus the sum of the second and the third integral in (2.2) is not positive. Hence it follows that . (to’ + m(s))gvd I’ + ; to D, Pf2dV s . +s g{i*C[Vv] -t- ph x [(H -t h) X i] - $h2ti} .ndZ. (2.3) aD, Next, by integrating (2.3) over (0, T) and using a well-known differential inequality [namely,f’@) 5 @)f(s) + g(s) -f(T) 5 exp(S,*a(s)ds){ f(0) + Jtexp(-Ji @)ds)g(t)dt)], we have s gqdV 5 eAtT) (g&dV+ D&S {S D‘s + %emAis)[i to lw gp-‘f2dV s a& (2.4) g{ir . C[Vv] + ph X [(H + h) X +] - zyxwvutsrqponmlkjihgfedcbaZYXWVUTS ;ph%} . ndZ Observe now that, as 6 - 0, g tends boundedly to the characteristic function of the set {(x, S) E R3 X [0, +a~): Ix - ~01S R,, s 5 T}. So, letting first w - O,t then 6 - 0 in (2.4) and using Lebesgue’s dominated convergence theorem yield (2.1). 0 Proof of the domain of influence theorem. Let (xg, E) E (6 - D&t)} X [0, t]. and let R = p-‘[p(r,-,) + c(t - Q] - ro. Then, (2.1) implies t Note that the introduction of the set ou in the proof of the domain of dependence inequality allows us to avoid the smoothness assumptions ii E C,(D X (0. +co)), Vi E CT’*@ X zyxwvutsrqponmlkjihgfedcbaZYXWV (0, +co)), p E C(D) [3]. 777 Wave propagation and uniqueness in magneto-elastodynamics s qodV + lr q(x t)d V 5 eX(o IS D(xo,Ro) D(xo,R) ’ + e&(‘)[, to s,,,,, p_tf’dP’ {+ - C[Vv] + Fh zyxwvutsrqponmlkjihgfedcbaZYXWVUT X [(H + h) X i] - fph2ti} . ndZ ds . s ~Dn.SWo,Rs) 11 Hence the desired result follows, by noting that all the integral at the right-hand side vanish. •i 3. A UNIQUENESS THEOREM Let {a, D,a2D, d3D} be a partition of aD and let LYbe a continuous and positive function on aD.To system ( 1.2) we append the following boundary conditions: zyxwvutsrqponmlkjihgfe on &D x (0, +a)), C[Vv]n = 0 on a2D x C[Vv]n = 0 on d3D x (0, +oo), on dD x (0,+oo). v=O av + h=O (0, +001, (3.1) We may claim the following Uniqueness theorem. Let (v, h) be a solution to system (1.2)-(3.1) such that v E C,(Q). Under the assumption of the domain of influence theorem, if v. = V. = ho = 0 on D, then v=hono. Cl Proof To prove this theorem it suffices to notice that (2.1) may be now written as which in turn implies s vdV+; D(xo,R) s a3D-wxo,Rs) cuv2dZ:5 0, au*dZ = - i eAcT)ST &)e-‘(‘)ds s 0 a3Dmb,Rs) whence uniqueness immediately follows. REFERENCES [_I1 _ B. CARBONARO and R. RUSSO. Enerav ineaualities and the domain of influence theorem in classical _ elastodynamics, J. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Elasticity 14, 163 (1984). PARIA, Magneto-elasticity and Magneto-thermo-elasticity, M. E. GURTIN, The linear thedry of elasticity, in Hundbuch l-275. Springer-Verlag, New York ( I97 I). (21 G. [3] A& App. M ech. 10, 73 (1967). der Physik (Ed. C. (Received 30 M arch 1985) Truesdell), Vol. VIal2, PP.