ON HEAT CONDUCTION IN PERIODICALLY STRATIFIED COMPOSITES WITH SLANT LAYERING TO BOUNDARIES

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1 THERMAL SCIENCE: Year 05, Vol. 9, No., pp ON HEAT CONDUCTION IN PERIODICALLY STRATIFIED COMPOSITES WITH SLANT LAYERING TO BOUNDARIES by Stanislaw J. MATYSIA a and Dariusz M. PEROWSI b a Institute of Hydrogeology and Engineering Geology, Faculty of Geology, University of Warsaw, Warsaw, Poland b Faculty of Mechanical Engineering, Bialystok University of Technology, Bialystok, Poland Original scientific paper DOI: 0.98/TSCI06909M Introduction The paper deals with the modelling of heat conduction in periodically stratified composites with slant layering to boundaries. The relations of the homogenized model with microlocal parameters for the case of layering parallel to the boundary are transformed to conform to requirements of the composite geometry. The plane problem of periodically stratified half-space with slant layering heated by given boundary temperature is solved analytically. The obtained results are analysed and compared with the solutions for special cases: () homogenized half-space, () half-space with parallel layering to the boundary, and (3) half-space with vertical layering to the boundary. eywords: temperature, heat flu, homogenized model, laminated composite Determination of the temperature, heat flu distributions in non-homogeneous solids plays an important role in many engineering branches. A great deal of progress has been made to the studies of heat conduction problems in structures composed of periodically bounding together two or more materials with different thermal properties. Special attention has been devoted to laminated composites which represent an important type of modern materials. Moreover, the periodically stratified structures can be observed in many rocks and soils. The problems of modeling of heat conduction in laminated bodies were presented in numerous monographs and papers, see for eample [-]. One of the approimated approaches to the description of heat conduction problems in periodically layered composites is the homogenized model with microlocal parameters [7-0]. The main merits of this model are that it permits to evaluate not only mean but also local values of temperature and heat flues in every component of the laminated body, as well as the thermal continuity conditions on interfaces are satisfied. The homogenized model with microlocal parameters has been applied in many boundary value problems for periodically stratified composites with layering parallel or perpendicular to the boundaries, see for instance [-7]. The comparisons of the results obtained within the framework of the homogenized model with the solutions based on the classical heat conduction were presented in [8-0]. It can be observed good consistencies of results for both approaches in considered cases. Corresponding author; s.j.matysiak@uw.edu.pl

2 84 THERMAL SCIENCE: Year 05, Vol. 9, No., pp In the present paper the homogenized model with microlocal parameters will be applied to heat conduction problems for the periodically two-layered composites with the boundaries inclined with an arbitrary angle to the layering. We confirm our considerations to the -D stationary heat conduction problems for the periodically layered half-space heated by given temperature on the boundary plane. The layering is assumed to be inclined to the boundary with an arbitrary angle. The continuity conditions of temperature and heat flu component normal to the layering, and the regularity conditions at infinity are taken into account. The problem will be approimated by ug the homogenized model with microlocal parameters, which satisfies the continuity conditions on interfaces. The governing relations of this model were given in the terms of variables being the co-ordinates connected with the layering [7, 0,, 0]. However, it seems to be more useful to apply Cartesian co-ordinate system with aises normal and parallel to the boundary for solving the considered problems. In this paper the governing relations of the homogenized model with microlocal parameters [7, 0, -0] will be shortly recalled and net main of them will be presented on the co-ordinates connected with the boundary. By ug Fourier transform methods the temperature distribution in the periodically stratified half-plane with inclined layering to the boundary heated by given boundary temperature will be found. The obtained results will be analysed by assumptions of some angles of the layering inclination to the boundary. The determined problem is important in geophysics, engineering geology, because some soils or rocks have the periodically stratified structures (for instance varved clays, sandstone-slates, sandstone-shales). The governing relations of the homogenized model described in the terms of variables connected with the boundary can be developed for the 3-D non-stationary case. Basic equations Consider a non-homogeneous half-space composed of periodically repeated two-layered conductors with slant layering to the boundary plane. The continuity conditions of temperature and heat flu component normal to the layering are taken into considerations. Let (, y, z) comprise the Cartesian co-ordinate system that y-ais is parallel to the layering, see fig.. Let and be the thermal conductivity of the subsequent component of the body, respectively. Moreover, let d and d denote the thicknesses of the layers being the constituents of composite, and d = d + d be the thickness of fundamental unit, fig.. Let the considered body be a half-space (or layer) with boundary plane inclined to the 0 ais with angle a, fig.. Introduce the Cartesian co-ordinates (,y,z) such that the ais 0 is inclined with angle a to the ais 0 and is located on the boundary plane. Let the periodically half-space y 0 will be heated on the boundary y 0 by given temperature J( ), R, and the regularity conditions at infinity will be satisfied. Because of the boundary conditions on interfaces the eact solution of the heat conduc- Figure. The scheme of the periodically layered body tion problem cannot be obtained, the homogenized model with microlocal parameters [7, 0] will be employed in order to seek an approimate solution. The relations of this model determined in the terms of (, y) co-ordinates were presented in many papers, see for instance [7, 0,

3 THERMAL SCIENCE: Year 05, Vol. 9, No., pp ], so we recall only a brief outline of governing equations. The temperature and its gradient will be approimated as: T(, y) q(, y) T(, y) q(, y) h() q(, y) q(, y), h( ) q(, y), () T(, y) q(, y) y y q(, y) is an unknown macro-temperature, q(, y) an unknown thermal micro-parameter, and h() the so-called shape function and it is given in the form: 05. d for 0 d h h ( ). 05 d d for d d, () h h h( d) h( ) h d (3) d Since for every we have that h() < d, then for small d the terms included the d periodic shape function are be small to and are neglected (the derivative h'() is not small and the term included h'() cannot be neglected). Moreover, the shape function () is chosen on such way, that the continuity conditions of heat flu component normal to the layering on interface, are satisfied. The governing equations of the homogenized model for stationary -D case can be written as [8, 9]: q q [ ] q q 0, q [ ] y (4) h h ( h), [ ] h( ), h (5) h The microlocal parameter q can be determined from (4). It leads to: q q 0 (6) y with (7) ( h) h The heat flu vector in a layer of the i-th, i =,, and epressed in the co-ordinate system (, y) is given by: q() i () i () i y q qy q J (, ) (,,) 0, i, 0 y (8) The considered boundaries are assumed to be inclined to the 0 ais with angle a. For this reason it will be more suitable to use co-ordinates (, y), (see fig. ). The relations between (, y) and (, y) can be written as: a y a, y a y a (9) After some calculations, eq.(6) for unknown macro-temperature q(, y) is transformed by ug eq. (9) to the form:

4 86 THERMAL SCIENCE: Year 05, Vol. 9, No., pp q(, y) q a a ( (, ) y ) ( )a y q(, y) ( a a) 0 (0) y The heat flues in the directions and y given by (8) can be rewritten on the basis of eq. (9) in the form: y q ( j ) y (, ) (, ) a q a q, y (, ) (, ( j) y y ) qy i a q a q () y From eqs. (0) and () it can be observed that for the following cases we have: Case Assume that: = = () then from eqs. (5) and (7) it follows that:, [ ] 0, (3) Equations (0) and () reduce to the form: q(, y) q(, y) 0 (4) y and () () (, y) q q a q a q y, () () (, y) qy qy i a q a q (5) y which is agree with the well-known relations for the classical heat conduction problem in a homogeneous body. Case Taking into account the angle: a = 0 (6) from eq. (0) and it follows that: (, ) (, ) q y q y 0 (7) y and () i y q q(, ) () i q(, y) qy i i,,, (8) y The obtained equations are agree with the relations of heat conduction in periodically two-layered composites with the boundary normal to the layering, see for instance [0, ]. The 0 ais will be identical with 0 (and 0y ais will be agree with 0 y). Case 3 Assuming that: a p (9)

5 THERMAL SCIENCE: Year 05, Vol. 9, No., pp from eqs. (0) and () we obtain: q(, y) q(, y) 0 (0) y and () i y q (, q ) () i q (, y ), qy i y () In this case we have y, y, and the obtained results (0) and () are agree with the relations of the homogenized model for periodically layered composites with the layering parallel to the boundary, see for eample [4-8]. Formulation and solution of the considered problem Consider the -D heat conduction problem for the periodically layered half-space y 0 with the layering inclined with the angle p/ + a to the boundary plane, fig.. The problem is assumed to be independent on the variable z, and stationary, and will be formulated within the framework of the homogenized model presented in the section Basic equations, so the continuity conditions of temperature and heat flu component normal to the layering are satisfied. The boundary y 0 is heated by the given temperature J( ), R, so the following boundary condition will be considered: q(,) 0 J( ), R () satisfies: lim ( J ) 0 (3) Moreover, it will be assumed that: lim (, q (, y ) q y) lim y 0 (4) The considered problem is described by eq. (0) and conditions () and (3). Ug the integral Fourier transform with respect of variable: qg (, ) q(, y y) ig e d, i p from (0) it follows that q should satisfy the following ordinary differential equation: ( ) ( d q i dq a a g a ) y g ( a a) q d dy 0 (6) The general solution of eq. (6) takes the form: g a q(, g ) [ ep( wg ) ep( wg ( ) y D y D y )] ( y a a) g a( ) y i ( a a (7) D and D are constants and: w (8) a a Ug the conditions () and (4) and assuming that: (5)

6 88 THERMAL SCIENCE: Year 05, Vol. 9, No., pp J( ) J( ), R (9) from eq. (7) it follows that the macro-temperature q(, y) can be written in the form: g q(, y y) J( g)ep p a 0 ( ) ( a g y a) dg (30) Jg ( ) J( )( g ) d (3) p 0 The double integral in eq. (30) can be written in the form of convolution. Special case Let the boundary temperature be taken into account as: J( ) J ( 0 H a) (3) J 0 is a given constant and H is the Heaviside`s step function. From eq. (3) we obtain: ( g ) Jg ( ) a J0 (33) p g Substituting (33) into (30) we have: g q(, y y) J0 ep g p 0 a a ( y a g a a ( ag) d g (34) This integral can be calculated analytically. Because [] pa ep b a ( )( ) d tan p a b p s (35) 0 0 for p a b 0 s (36) for p a b0 thus we obtain: q(, ) p J tan y 0 ay a a ( y a ) a y a ( ) 0 ( J s (37) a a y y s 0 for a ( a ( ) a 0 a a) ( ) (38) a a and s = if the opposite inequality to the above ones is valid.

7 THERMAL SCIENCE: Year 05, Vol. 9, No., pp Case It can be observed that for the homogeneous body described by () and (3) from (37) and (38) it follows that q(, ay y) J tan 0 sj y a 0 (39) p s0 for y a, and s for y a (40) Equation (39) together with (40) are agree with the adequate result obtained directly for the homogeneous half-plane. Case If the layering is normal to the boundary, what it leads to a = 0 from eq. (37) we obtain: q(, ay y) J tan 0 p y a sj 0 (4) s0 for y a, and s for y a (4) Case 3 In the case of a = p/ (the layering is parallel to the boundary) from eq. (37) it follows that: q(, ay y) J tan 0 y ( p a) sj 0 (43) s0 for y ( a) 0, and s for y ( a ) 0 (44) The knowledge of temperature distributions in the periodically layered half-space described in the co-ordinates (, y) permits to calculate the macro-temperature q in the co-ordinates (, y) connected with the layering by ug inverse transformation to the relation (9). We have: a y a, y a y a (45) The co-ordinates (, y) seem to be more useful to determine the heat flu vector in every lamina on the basis of equations (8), and (30), or (37) and (45). Numerical results The temperature distributions given in eqs. (37) and (38) can be presented graphically. Introducing the dimensionless co-ordinates (, y ) as:, y d y, and d (46) a a a from eqs.(37) and (38) it follows that: q(, ) p J tan y y 0 ay a a ( a a) y a ( ) ( s (47) a a

8 90 THERMAL SCIENCE: Year 05, Vol. 9, No., pp y s 0 for ( g a ( ) 0 a a) ( (48) a a) and s = for the opposite inequality to the above ones. The isothermal lines for the homogeneous half-plane ( / = ) and the periodically layered half-plane with angles a =0,p/4, p/, and / = 4, are shown in fig.. Figure. The isothermal lines for the considered half-plane for: (a) / =,(b)a =0,(c)a = p/4, (d) a = p/, and / =4,h = 0.5 The dimensionless temperature q/j 0 as a function of angle a at the point 0, y = 0.05 (under the centre of heated boundary region) for / =, 4, and 8, and h = 0.5 is presented in fig. 3(a). It is seen that the highest values of temperature at this point are achieved for a = 0 (the layering is perpendicular to the boundary) and the smallest values for a = p/ (the layering is parallel to the boundary). Figure 3(b) shows the dimensionless temperature q/j 0 as a function of parameter h for a =0,p, and p/, and the / = 4 and 8, at the same point as in fig. 3(a). The maimal or minimal values of the dimensionless temperature are achieved for h = 0.5. The dimensionless components of heat flu vector are presented in figs The component q a/(j 0 ) as a function of (see fig. ) on the depths y = 0.05, 0.5, 050 for / = (the homogeneous half-plane) as well as / =4,h = 0.5 and three cases of angle a, a =0, p/4, and p/ is shown in fig. 4. The components q () i i, i =, are continuous on the interfaces, so we denote q q ().It can be observed that the etreme values of q are achieved at points = (under the ends of heated range). For the angle of layering inclination a = 0 (the layering is normal to the boundary) q is positive for 0, but when a = p/4 or a = p/ the component q changes sign.

9 THERMAL SCIENCE: Year 05, Vol. 9, No., pp Figure 3. The dimensionless temperature q/j 0 at the point 0 y =0.05: (a) as a function of a, (b) as a function of h, black line is for / = 4 and grey line is for / =8 Figure 4. The dimensionless component of heat flu q () i a /(J 0 ), i =,, as a function of : (a) / =, (b) a =0,(c)a = p/4, (d) a = p/, and / =4,h = 0.5, y = 0.05, y = 0.5, 3 y = 0.50

10 9 THERMAL SCIENCE: Year 05, Vol. 9, No., pp Figure 5. The dimensionless component of heat flu ( qy i a) /(J 0 ), i =,, as a function of : (a) / =, (b) a = 0, (c) a = p/4, (d) a = p/, and / =4,h = 0.5, y = 0.05, y = 0.5, 3 y = 0.50, and i = black line; i = grey line () Figure 6. The dimensionless component of heat () flu ( qy i a) /(J 0 ), i =,, as a function of y for = 0.0 The components q () i y, i =, are discontinuous on the interfaces. Figure 5 shows the dimensionless component of flu vector () i ( qy a) /(J 0 ) as a function of on the depths y = 0.05, 0.5, and 0.50 for homogeneous body / = (in this case q () y q () y as well as for / =4,h = 0.5, and a =0,p/4, and p/. () The black curves represent the heat flu q y (in the layers of first kind), the grey curves correspond with q y, and both kinds of curves are e- () tended for a better visibility. To show the jumps of the heat flu q () i y fig. 6 is added. Figure 6 presents the dimensionless () i ( qy a) /(J 0 ) as a function of y (the depth) for = 0.0, / =4,h = 0.5, a = p/4, d =, then

11 THERMAL SCIENCE: Year 05, Vol. 9, No., pp d = / d/a. It can be observed that q greatest near the boundary plane. () () y y q for y nd (on the interfaces), and the jumps are Final remarks The presented results for temperature and heat distributions in the periodically layered half-plane with the layering inclined to the boundary possess the same characteristics as adequate solutions within the framework of the classical description. The temperature and heat flu are continuous and component q () i y is discontinuous on the interfaces, By ug co-ordinates (, y) connected with boundary, then both components of heat flues in the, and y -directions eperience jumps on the interfaces. The derived relations given by (0) and () permit to solve by analytical methods some boundary value problems for heat conduction in laminated composites with slant layering to boundaries. Nomenclature a half of length of heated range, [m] h() shape function of the homogenized model with microlocal parameters, [m] i comple unit (= / ), [ ], coefficients of thermal conductivity of the subsequent component of the body, [Wm ], effective thermal modulus on the homogenized model with microlocal parameters, [Wm ] q unknown thermal micro-parameter of the homogenized model with microlocal parameters, [m ] q (i) heat flu vector in a layer of the i-th kind, i =,,[W], y Cartesian co-ordinates connected with, y the layering, [m] Cartesian co-ordinates connected with the boundary, [m] (, y ) Cartesian dimensionless co-ordinates, [ ] T temperature, [] Greek symbols a angle of inclination of layering to ais [rad], d, d thickness of the layers being the constituents of composite, [m] d thickness of fundamental unit, (= d + d ), [m] h saturation coefficient of fundamental unit by the first kind of material, (= d /d), [ ] q macro-temperature in homogenized model with microlocal parameters, [] J( ) boundary temperature, [] Indees i kind of sublayer; i = the first kind or i = the second kind of the subsequent layers References [] Aurialt, J. L., Effective Macroscopic Description for Heat Conduction in Periodic Composites, International Journal of Heat and Mass Transfer, 6 (983), 6, pp [] Backhalov,N.S., Panasenko,G.P., Averaged Processes in Periodic Media (in Russian), Nauka, Moscow, 984 [3] Bedford, A., Stern, M., Toward Diffug Continuum Theory of Composite Materials, Journal of Applied Mechanics, 38 (97),, pp. 8-4 [4] Bufler, H., Stationary Heat Conduction in a Macro- and Microperiodically Layered Solid, Archives of Applied Mechanics, 70 (000), -3, pp [5] Furmasnki, P., Heat Conduction in Composites: Homogenization and Macroscopic Behavior, Applied Mechanics Reviews, 50 (997), 6, pp [6] Ignaczak, J., Baczynski, Z. F., On a Refined Heat-Conduction Theory of Microperiodic Layered Solids, Journal of Thermal Stresses, 0 (997), 7, pp [7] Wozniak, Cz., A Nonstandard Method of Modeling of Thermoelastic Composites, International Journal of Engineering Science, 5 (987), 5, pp

12 94 THERMAL SCIENCE: Year 05, Vol. 9, No., pp [8] Maewal, A., Homogenization for Transient Heat Conduction, ASME Journal of Applied Mechanics, 46 (979), 4, pp [9] Manevith, L. I., et al., Mechanics of Periodically Layered Heterogenous Structures, Springer, Berlin, 00 [0] Matysiak, S. J., Wozniak, Cz., On the Modelling of Heat Conduction Problem in Laminated Bodies, Acta Mechanica, 65 (986), -4, pp [] Wozniak, Cz., Wierzbicki, E., Averaging Techniques in Thermomechanics of Composite Solids. Tolerance Averaging versus Homogenization, Wydawnictwo Politechniki Czestochowskiej, Czestochowa, Poland, 000 [] Matysiak, S. J., et al., Temperature Field in a Microperiodic Two -layered Composite Caused by a Circular Laser Heat Source, Heat and Mass Transfer, 34 (998), -3, pp [3] Matysiak, S. J., et al., Distribution of Friction Heat during Cold Rolling of Metals by Ug Composite Rolls, Numerical Heat Transfer, Part A, 34 (998), 3, pp [4] Matysiak, S. J., Pauk, V., Plane Contact Problem for Periodic Laminated Composite Involving Frictional Heating, Archive of Applied Mechanics, 66 (995), -, pp [5] aczynski, A., Matysiak,,S. J., On the Three Dimensional Problem of an Interface Crack under Uniform Heat Flow in a Bimaterial Periodically-Layered Space, International Journal of Fracture, 3 (003), 3-4, pp [6] aczynski, A., Matysiak, S. J., Thermal Stresses in a Laminate Composite with a Row of Interface Cracks, International Journal of Engineering Science, 7 (989),, pp [7] Matysiak, S. J., et al., Temperature Field in the Process of Braking of a Massive Body with Composite Coating, Materials Sciences, 43 (007),, pp [8] ulchytsky-zhyhailo, R., Matysiak, S. J., On Heat Conduction Problem in a Semi-Infinite Periodically Laminated Layer, International Communications in Heat and Mass Transfer, 3 (005), -, pp. 3-3 [9] ulchytsky-zhyhailo, R., Matysiak, S. J., On Some Heat Conduction Problem in a Periodically Two-Layered Body: Comparative Results, International Communications in Heat and Mass Transfer, 3 (005), 3-4, pp [0] Matysiak, S. J., Perkowski, D. M., On Heat Conduction in a Semi-Infinite Laminated Layer. Comparative Results for Two Approaches, International Communications in Heat and Mass Transfer, 37 (00), 4, pp [] Gradshteyn, I. S., Ryzhik, I. M., Tables of Integrals, Series and Products (in Russian), Science, Moscow, 97 Paper submitted: June 9, 0 Paper revised: January 4, 03 Paper accepted: February 7, 03