Abstract. 1 Introduction

Transcription

1 Dynamic thermoelasticity problem for a plate with a moving semi-infinite cut A.I. Zhornik,* V.A. Zhornik,' E.M. KartashoV Department of Theoretical Physics, Taganrog Pedagogical Institute, 48 Initsiativnaya Street, Taganrog , Russia *Department ofhigher and Applied Mathematics, Lomonosov Moscow Institute of Fine Chemical Technology, 1 Malaya Pirogovskaya Street, , Russia Abstract The heat-shock, i.e. action of stresses created by a sharp change in temperature of a thin plate with a semi-infinite cut, whose tip is moving at a constant velocity since the initial time is considered. The lateral surfaces of the plate are subjected to a linear heat transfer by radiation to the surroundings. 1 Introduction In a thin isotropic infinite plate of 2S thickness along the ray of J/ = 0, X < 0 there is a semi-infinite cut that starts at the initial time to move to the region of y'= 0,.%'> Oat a constant velocity. The plate heats symmetrically with respect to the middle plane by heat exchange with the medium for temperature 7^ which surrounds its lateral surfaces. The initial temperature 7^ of the plate is not equal to the surrounding temperature. In no time the temperature 1^ arises on the edges of the cut. It is considered that the origin of the fixed (jc', j>')-system is at the point where the end of the cut is at the initial time. The origin of the moving x,y -coordinate system is chosen to remain at the cut tip, i.e., x = x' - Vj,t, y= /, t = t' This problem represents a case of plane-stress state, a case of plane strain which is realized by zero heat exchange with the medium and by replacing of elastic constants combinations. In consequence of the symmetry with respect to the X -axis, the problem is considered for a half-plane y >0 with the boundary and initial conditions given in the moving system of coordinates. 2 Mathematical statement of the problem.

2 550 Advanced Computational Methods in Heat Transfer Mathematical problem has the form, T(x,y,t)=T,, x<0,y=0 (1) dy., (x,y,t)=t,, / = 0 (3)»,(x,y,t) = Q, x<0,y = Q,t>0 (4) *,.V,0 = 0> -w<^<oo,y = 0, f>0 (5>,;/, 0=0, *>0,j=0,/>0 (6),y,t) = 0, t<q (7) dt =o,,,o 3 Solution of the thermoelasticity problem The problem formulated in eqns (l)-(3) for TQ = 7^. was solved by Zhornik & Kartashov [1] considering an analogous thcrmoclastic problem in the quasistatic statement. For the same reason the solution for temperaturefieldwill be the same and has the form, T(x,y,t)-T.=0(x,y,t).e-"+(T,-T.).e-"*' (9) 0(x,y,t) is a function which has the following form according to the Laplace-Fourier transformations, where y = V^ / 2k, k = A, / pc - thermal diftusivity of the plate; X - its heat conduction; /?- density; C- specific heat; a- the coefficient of linear heat transfer to the medium embracing the lateral surfaces of the plate. The case of the stationary problem of heat conduction for / -xx> (5 ->0) was investigated by Salganik & Chertkov [2] and the problem of heat conduction for a fixed cut was considered by Poberezhny & Gaivas [3], Kozlov, Mazya & Parton [4]. The solution of the dynamic problem of thermoelasticity will have the form

3 Advanced Computational Methods in Heat Transfer 551 <J, j and U. (11) (12) - the problem of thermoelasticity for a plate without cut satis-fies the boundary conditions ofeqns (5), (7), (8) and its solution is discover-ed by means of the thermoelastic potential of displacement F(x, V, f)in the Jt, y - coordinates from the equation below (13) -a where a = l/c^ = ^p(l - v)/2g - longitudinal wave slowness; d = 1/Fy, -lowness of the cut end; G - the shear modulus; v - Poisson's ratio; OC T - the coefficient of linear thermal expansion. The normal stress which is necessary for the subsequent solution formulated according to Laplace- Fourier has the form = X l/2. \l/2 where a = -a + a-+sa, -<- sa a, =fl/(l + a/d), a, = a/(l - a/d),

4 552 Advanced Computational Methods in Heat Transfer h = \ I c± = ^] p I G - shear wave slowness. The boundary conditions for <J y, U i; - solutions of isothermic theory of elasticity have the form of eqns (5)-(8), and also x < 0, v = 0 (15) P T Tp To find solutions (or(7. and U. it is considered a fundamental solution * T T * <J ^ U, of elasticity theory about a stressed state of the plate with a semiinfinite cut moving at a constant velocity Vj, when normal concentrated forces J, moving subsequently at a constant distance/ from the cut end, instantly apply on the edges of the cut at the distance / from its end. The problem as in the preceding item is considered in the moving x,y- coordinates of eqns (5)-(8) and also <r^(x,y,f) = -f8(x + l)h(t\ x<q,y = Q (16) where S (*)- delta Dirac function; H(t)- Heaviside function. For solution of this problem it is used an original method proposed by Freund [5] for a fixed cut. The stress intensity factor Kj(t)tor further consideration, has the form which is necessary for 02 xlc, - (17) where J ar tan[4 if a (+%) x /T a 2,a i (18) a = (1 ~ C I ' (1 C I d\ C = 1 / ^ - Rayleigh wave velocity.

5 Advanced Computational Methods in Heat Transfer 553 The dependence S+( -- ) on Vb for various cut tip velocities bid was obtained by Zhornik & Kartashov [6]. Here is V, velocity of edge dislocation which is moving in the positive direction of the X axis and starts the motion from the cut tip. The dislocation velocity is measered within 0 < V < Cjj - Vy. The curves for afixedcut bl d = 0 are given by Parton & Boriskovsky [7]. In case when the load (J (,X,0, f)is applied to the cut edges, the stress intensity factor is represented as 0 where AT/(.%,f) results from eqn (17) by replacing / with X > 0, and we also take f = 1. Then to eqn (19) the Laplace transform is applied - GO (20) And then using the Parceval ratio by Sneddon [8] Kj (s) has the form 00 = = where C7 (^,0,^) is given in eqn (14) for T^ = 7],, A^(,5) is obtained from eqn (17) using the above-mentioned replacements and applying to eqn (17) the Laplace-Fourier transform with parameters of S and and it will have the form ^ «/" "\T x» 5 * v'l i ^ 1 (22) The substitution of eqn (14) and (22) into eqn (21) comes K, (s) to the form (23)

6 554 Advanced Computational Methods in Heat Transfer ca* Jl + ;/, = VI - a^ / d* ^a^s + i^a^s - ig, For afixedcut d > QO eqn (23) is obtained by general asymptotic method by Kozlov, Mazya & Parton [9]. In expression of eqn (23) integration is conducted along the real axis embracing the origin of the coordinates from below. Calculation of Kj (»$) is connected with calculating of the contour integral in the lower half-plane of, which embraces the cut along the imaginary axis from -ia^s till -/(/?- /) Having made the integration and proceeding to the original off, large and little intervals comes to the form j (t ) for (24) x (25) Jt) /

7 Advanced Computational Methods in Heat Transfer 555 Figure 1: Relation between K' and r for various y* Figure 2: Relation between Kj and T for various y *

8 556 Advanced Computational Methods in Heat Transfer r» at (l + r = kt I ^-thefourier criterion; 7* = y 8, % = %S, ^ is the generalized hypergeometrical function; F(#) is the gamma function. is* The dependence of A ^ on T for great intervals under different intensities of * * heat exchange % and moving cut velocities y is given on Fig For y* = 0 case eqns (24),(25) were obtained by Kozlov, Mazya & Parton [9], eqn (25) for d 0 was obtained by Zhornik & Kartashov [1]. References 1. Zhornik, A.I. & Kartashov, E.M. Temperaturefieldsand temperature stresses arising in a plate with a moving semi-infinite crack, Melody i algoritmy parametricheskogo analiza lineinyh i nelineinyh modeley perenosa, 1988, 6, Salganik, R.L. & Chertkov, V.Y. On strength reduction under the action of shrinkage strain, Mekhanika l verdogo tela, 1969, 3, Poberezhny, O.V. & Gaivas', I.V. Influence of non-stationary temperature field and of heat output of a plate upon stress intensity factors, Prikiadnaya Mekhanika, 1982, 18, 6, Kozlov, V.A., Mazya, V.G. & Parton, V.Z. Asymptotics of stress intensity factors in quasi- static temperature problems for the cut region, Prikiadnaya Matematika I Mekhanika, 1985, 49, 4, Freund, L.B. The stress intensity factor due to normal impact loading of the faces of a crack, Int. Journal of Engineering and ScL, 1974, Vol. 1 2, pp Zhornik, A.I. & Kartashov, E.M. Dynamic problem of elasticity theory for a space with a semi-infinite crack, International Applied Mechanics, 1992, Vol.2, 12, Parton, V.Z. & Boriskovsky, V.G. Dynamic Mechanics of Fracture, Mashinostroyeniye, Moscow, Sneddon, I.N. Fourier Transforms, McGraw-Hiill Book Company, Inc., New-York, Kozlov, V.A., Mazya, V.G. & Parton, V.Z. On heat shock in the region of a crack, Prikiadnaya Matematika I Mekhanika, 1988,52, 2,