Finite element methods for coupled thermoelasticity and coupled consolidation of clay

Three hnear two-dimensional coupled problems are considered dynamical ihermoelasticity, quasistatical thermoelasticity and consolidation ofclay In thejïrst two cases an équation parabolic with respect to the température T is coupled with a System of équations either hyperbolic or elhptic with respect to the displacement vector u, in the third case an équation elliptic with respect to the pressure T is coupled with a System elhptic with respect to the displacement vector u The problems are solved approximately using both tnangular and curved tnangular fimte éléments in the space discretization and v-step Astable différence methods (v = 1 or 2) in the time discretization. The effect ofnumencal intégration is also considered The resulting schemes are unconditionally stable Resumé — Nous traitons trois problèmes linéaires en deux dimensions la thermoélasticité dynamique, la thermoélasticité quasi statique et la consolidation de Vargile Dans les deux premiers cas une équation parabolique par rapport à la température T est couplée avec un système adéquations hyperboliques ou bien elliptiques par rapport au vecteur u des déplacements Dans le troisième cas, une équation elliptique par rapport à la pression T est couplée avec un système elliptique par rapport au vecteur u des déplacements Ces problèmes sont approchés en utilisant la méthode des éléments finis avec les triangles rectihgnes ou curvilignes pour la discrétisation spatiale et les méthodes des différences finis Astables a v pas (v = 1 ou 2) pour la discrétisation en temps Les schémes qui en résultent sont inconditionnellement stables 1. FORMULATION OF THE PROBLEM According to [2] the dynamical two-dimensional problem of coupled linear thermoelasticity can be formulated in the following way : Let Q be a bounded domain in the xv x2-plane with a sufficiently smooth boundary F. Find a displacement vector n(xv x2, t) and a température T(xl9 x2, t) which satisfy the following équations and boundary and initial conditions (for a greater sim(*) Received in Apnl 1982, revised in May 1983 O Computing Center of the Technical Umversity, Obrâncû miru 21, 602 00 Brno, Czechoslovakia R A I R O Analyse numénque/Numencal Analysis, 0399-0516/84/02/183/23/$ 4 30 © AFCET Bordas-Dunod 184 A. ZENISEK plicity we restrict ourselves to the case of Dirichlet boundary conditions) : T.n + Q = c x t + c2 ùltl in Q x (0, /*] (1) alJtJ + Xl = c4 ux (i = 1, 2) in Q x (0, t*] (2) r(x l s x2, t) | r = T(xv x2), t > 0 (3) «,(*!, *2> 0 Ir = «>(*i» *2) 0' = 1» 2) , f > 0 (4) cx T(xl9 x29 0) = cx T0(x19 x2), (xls x2) e Q (5) u£xu x2, 0) = ul0(xv x2), (x15 x2) e O (1 = 1, 2) (6) c 4 w t(^ l5 x2 ï 0) = c4 i?l0(JCi, ^ 2 ) . (*i> ^2) G Q 0" = !» 2 ) () where " v = DvJfoJu) a(T Tr) 6 J (8) öu f c m = DJlkm = D t m i J (9) £ yW = (»« + "J /2 (10) O . j t m ^ ^ ^ ^ o ^ ^ V^ = ^ £ J R (11) where (i0 = const > 0. A summation convention over a repeated subscript is adopted. A comma is employed to dénote partial differentiation with respect to spatial coordinates and a dot dénotes the derivative with respect to time t Thus équation (1) is the coupled heat équation and équations (2) are Cauchy's équations of equilibnum. The symbol Q dénotes a prescribed sufficiently smooth rate of internai heat génération per unit volume, the symbols Xv X2 dénote prescribed sufficiently smooth components of body forces per unit volume. The symbols cl5 c2, c4 are positive constants; c1 and c4 depend only on the material of a considered body, c2 = c2 Tr where ~c2 is a positive constant depending only on the material and Tr is a positive constant which has the meaning of the température for which the material is stress-free. The fonctions on the right-hand sides of relations (3)-(7) are prescribed sufficiently smooth fonctions. In relation (8) a is the coefficient of linear thermal expansion, hXJ is the Kronecker delta and Dljkm are constants depending on the material only. We shall consider isotropic materais only ; in this case m §km = c3 6y , c3 = const > 0 . (12) If we set c1 = c4 = 0, replace (6) by «M(*I> *I> °) = 0 . (^i. x2) e Q (6*) R A.I R O Analyse numénque/Numencal Analysis COUPLED THERMOELASTICITY AND CONSOLIDATION OF CLAY 185 and define atJ by ^ = Z W * > ) ~ TbtJ (13) then problem (l)-(4), (6*), (9)-(13) represents the two-dimensional problem of coupled consolidation of clay in the case of incompressible pore water [1, 3, 9], The symbol T has now the meaning of pore water pressure, the constant c2 dépends on the material only and 6 = 0. Numerical tests [9] show that the hnear model (l)-(4), (6*), (9)-(ll), (13) gives satisfactory results. (Let us note that a nonlmear elastoplastic model is studied in [8].) Now we present a vanational formulation of our three problems. Before doing it let us introducé some notation. By H(Q) we dénote the Sobolev space of real functions which together with their generalized derivatives up to order m inclusive are square integrable over Q. The inner product and the norm are denoted by (.,-)m,n and || . ||mn, respectively. i/ H(ü) which have continuous derivatives up to order m on [0, t*].L(H\£ï)) is the space of strongly measurable functions/: (0, t*) -> H(Q) such that f Jo Multiplying équation (1) by w e HQ(Ü) and using Green's theorem we easily find D(T9 w) + ct(f, w)Qta + c2(ülV w)oa = (Q, w)0 a VweffoHQXteCO,/*] (14) where D(v9w)= vtlwtldx9 (v,w)on= vwdx. (15) J J 9 )= tldx9 (v,w)on= Jn a Multiplying équation (8) or (13) by &tJ;(y), where ve [^(Q)] 2 , integrating over Q, using relations (2), (9), (10) and Green's theorem we find «(«, v) + c4(ll, v)Ojft c3(T rr5 vtXa = = (X,y)OtQ V V G ^ O H Q ) ] 2 , ïe(0,/*] (16)