Journal of Composite Materials. On Cyclical Hygrothermal Fields in Laminated Plates. Keywords: cyclical hygrothermal field, laminated plate

Transcription

1 On Cyclical Hygrothermal Fields in Laminated Plates Journal: Journal of Composite Materials Manuscript ID: JCM--00 Manuscript Type: Original Manuscript Date Submitted by the Author: -Sep- Complete List of Authors: Gigliotti, Marco; ENSMA, LMPM Jacquemin, Frederic; Université de Nantes Keywords: cyclical hygrothermal field, laminated plate Abstract: Some results concerning the calculation of cyclical hygrothermal fields in laminated plates are presented. Fick s law is employed for water diffusion, thermal fields are assumed to be uniform: temperature and water concentration are coupled through the coefficient of hygroscopic diffusivity, which follows a classical Ahrrenius law. The problem is characterised by complex boundary conditions, relative humidity and temperature of the environment varying in transient/cyclic way. The solution of the problem is inspired by some ideas put forward by Verchery (Verchery, ) and formally analogous to that presented by Jacquemin and Vautrin (Jacquemin and Vautrin, 0a) for laminated cylinders. Analytical solutions of the cyclical hygrothermal fields in laminated plates can be employed as benchmark for numerical calculations: moreover they can be used for evaluating the induced hygrothermal stress, therefore for structural design and optimisation purposes.

2 Page of 0 Journal of Composite Materials On Cyclical Hygrothermal Fields in Laminated Plates* Marco Gigliotti a and Frédéric Jacquemin b a Institut Pprime CNRS ENSMA Université de Poitiers Département de Physique et Mécanique des Matériaux, Avenue Clément Ader, F, Futuroscope Chasseneuil Cedex, FRANCE (to whom correspondence should be addressed) b Institut de Recherche en Génie Civil et Mécanique (GeM), Université de Nantes, Boulevard de l Université, BP 0, 0, Saint-Nazaire, France Tel.: (Marco Gigliotti) Fax: + 0 (DPMM Secretariat) marco.gigliotti@ensma.fr (Marco Gigliotti) frederic.jacquemin@univ-nantes.fr (Frédéric Jacquemin) * The present paper is dedicated to the memory of our Master, friend and colleague Prof. Alain Vautrin. Abstract Some results concerning the calculation of cyclical hygrothermal fields in laminated plates are presented. Fick s law is employed for water diffusion, thermal fields are assumed to be uniform: temperature and water concentration are coupled through the coefficient of hygroscopic diffusivity, which follows a classical Ahrrenius law. The problem is characterised by complex boundary conditions, relative humidity and temperature of the environment varying in transient/cyclic way. The solution of the problem is inspired by some ideas put forward by Verchery (Verchery, ) and formally analogous to that presented by Jacquemin and Vautrin (Jacquemin and Vautrin, 0a) for laminated cylinders.

3 Page of Analytical solutions of the cyclical hygrothermal fields in laminated plates can be employed as benchmark for numerical calculations: moreover they can be used for evaluating the induced hygrothermal stress, therefore for structural design and optimisation purposes.. Introduction. Hygrothermal conditioning may represent an important source of ageing and degradation, especially in polymer based composite laminate structures. Water diffusion may contribute to material properties degradation and leads to the development of anelastic strains of hygroscopic nature, which contribute to the building of internal stress. High temperature may be also responsible for material properties changes and for the development of thermal stresses, leading possibly to damage onset and propagation. Despite the numerous research studies both theoretical and experimental - dedicated to the effect of an aggressive hygrothermal environment on the behaviour of composite laminated materials (see for instance Weitsman,, for a comprehensive review on the matter), not much attention has been paid to the real hygrothermal conditions encountered by these materials during their service and operating life. Laboratory test conditions put usually samples very far away from the actual environment affecting the structure: the certification authority starts from the principle that often materials subjected to a hygrothermal environment reach a saturated state, in which the residual properties of the material can be comprehensively characterised, and suggests to take into account the properties of the saturated wet state for design. This procedure which is well established and simple openly ignores the transient and cyclical phases of the actual conditioning that precede a saturated state, which sometimes is never attained. A transient/cyclical hygrothermal state may be characterised by strong concentration and materials property gradients, varying with time, which may in turn lead to strain/stress

4 Page of 0 Journal of Composite Materials gradients and interact with other sources of degradation affecting the durability of the structure in an unpredictable way. Even from the modelling side, the available tools to handle the problem of laminated plates under hygrothermal conditioning usually reduce to the classical Fick s law with simple boundary conditions, mainly uniform concentration at the structure edges for which analytical solutions are available (Crank, ). The calculation of the hygrothermal fields for laminated cylinders exposed to complex transient/cyclical environment has been carried out by Jacquemin and Vautrin, 0a, but an analogous formulation for laminated plates has never been presented. The present article is concerned with the case of a laminated plate subjected to cyclical hygrothermal conditions. Temperature and relative humidity vary in a periodical (transient/cyclical) manner on the external surfaces of the plate, while the values of hygroscopic concentration within the solid are governed by the Fick s law of diffusion, which is formally identical to the Fourier s law for thermal conduction. Analytical solutions of the cyclical hygrothermal fields in laminated plates can be employed as benchmark for numerical calculations: moreover they can be used for evaluating the induced hygrothermal stress (see, for instance Jacquemin and Vautrin, 0b, and Gigliotti et al., 0, 0), therefore for structural, design and optimisation purposes. The present formulation generalises an approach for homogeneous diffusion already presented by the authors (Gigliotti et al., 0) to the case of laminated plates, in which different plies have different material properties. In order to better understand the phenomenon, we will shortly recall an analogous problem already treated by Kelvin (Kelvin, ) and concerned with the effects of solar radiation temperature fluctuations on the hearth surface temperature: that is, the approximately cyclical fluctuations occurring due to day to night excursions or during a solar year.

5 Page of Kelvin noted that fluctuations of superficial temperature occurring due to day to night excursions induce temperature fluctuations within the soil up to a distance of almost meter, while fluctuations of superficial temperature during a solar year induce temperature fluctuations within the soil up to a distance of almost meters. A crude schematisation of the phenomenon consists in considering a semi-infinite space x > 0 submitted at x = 0 to cyclical sinusoidal surface temperature fluctuations (fig. ) of the type: () where t is the time. The temperature within the space, T int, as a function of time, is governed by the equation: () where k is the thermal conductivity. The searched solution has the form (Carslaw and Jaeger, ): () with period π/ω. By substitution u(x) must satisfy the following equation: () The solution of which is finite for, from which it follows:

6 Page of 0 Journal of Composite Materials () Finally, the solution with period π/ω is given by: () with Equation represents a temperature wave with wave number K and wave length λ, given by: () where n is the frequency ω/π. Three important properties of this solution are the followings: ) the amplitude of temperature oscillations diminishes as: () and diminishes rapidly for high values of ω. At a distance equal to a wave length the waves are much damped and diminish by a factor e -π = 0.00, ) there is a progressive lag Kx in the phase of the temperature wave, this lag increases with ω, ) temperature fluctuations propagate with speed:

7 Page of () For instance, the external temperature varies with time as indicated in figure, with A = C, ω = π/ hrs -, ε = 0; then, k = 0.0 m hrs -. Figure shows the temperature distribution within the semi-infinite space at several times and, within a period, between and hours. Fluctuations propagate up to a distance λ, which can be rewritten as: () where τ is the period of the external fluctuations. In this example λ =.m. It can be seen that λ is one of the fundamental parameters of the problems, since it gives the distance at which fluctuations persist within a body whose surface is exposed to temperature conditions which vary in a periodical manner. It should be noted that λ depends only on the material properties of the body (in this case the conductivity k) and on the period of the external fluctuations. A formula analogous to that for λ will be established in the following sections pertaining to laminated plates subjected to cyclical hygrothermal conditions. The analogous problem of hygrothermal fluctuations in laminated plates will be solved analytically, following the theoretical developments of Section. Section will provide a simplified version of the model, valid for homogeneous plates and finally Section will provide for the scope of illustration some results issued from simulations in case of industrial/practical interest.

8 Page of 0 Journal of Composite Materials Analytical model. The equation of unidirectional Fick s diffusion (along the direction z, for instance the thickness), for a laminated plate with thickness e and made by n laminae/layers with thickness e i and for 0 < z i < e i reads: () where c i is the water concentration within the laminated plate and with the following boundary and interface conditions: () where c a (t), c b (t) are the surface water concentrations, both depending on time. D i (t) is the diffusion coefficient within the i th layer. In equations and boundary conditions and the diffusion coefficient depend on time: in particular, the diffusion coefficient depends on temperature through the Arrhenius law: ()

9 Page of A i and B i are material constants. Temperature is supposed to be uniform along the laminated plate, that is, thermal transport phenomena are supposed to be much faster than water concentration diffusion phenomena, which is physically reasonable in most practical situations. α i,i+ is a constant which expresses the ratio between the saturated concentration of the i th layer and that of the adjacent (i+) th layer. Usually, at layer interfaces, continuity of the water concentration is imposed. A more correct condition consists in imposing continuity of the chemical potential: the use of a proper constant coefficient α i,i+ (not necessarily equal to ) allows for this condition to be treated. Let us now consider the case in which hygrothermal conditions are cyclical, that is, temperature and water concentration boundary conditions wary with time periodically with period τ. In order to solve analytically the problem a change of variables is needed: the initial problem equations - can be then substituted by an analogous problem in which the boundary conditions and the diffusion coefficient are constant. Let us consider the following change of variables: It follows then: () ()

10 Page of 0 Journal of Composite Materials When the activation energies B i are the same for all layers, it follows: () and the problem to solve becomes then: () () where c a (u), c b (u) are cyclical with period. Boundary conditions depend on u and have to be converted into analogous conditions independent of time. In order to do so, the so called average mobile concentration can be introduced. () from which:

11 Page of () The problem to solve becomes: () () Introducing adimensional variables: () Then:

12 Page of 0 Journal of Composite Materials () () Boundary conditions do not depend anymore on time (t or u), therefore the problem can be solved in a relatively simple way. In the Laplace domain the solution of - takes the form: () where is the Laplace transform of the solution: () ()

13 Page of or: () where: (0) Boundary and interface conditions in the Laplace domain now read: () By applying the theorem of residues the solution of the problem in the time domain is: () Where u and s are the determinants of the nxn (a) and (b) matrices (equations -).

14 Page of 0 Journal of Composite Materials A * i (ω m ) and B * i (ω m ) are the determinants calculated in ω m (ω m in the m th positive root of the equation u (ω) = 0) of the matrices with components a ij in which the columns (i-) and i are substituted by the constant vector {g} (equation ). A i and B i are the determinants calculated in ω m of the matrices with components a ij in which the columns (i-) and i are substituted by the constant vector {g}. Finally, u (ω m ) is the derivative of u with respect to ω and evaluated at ω m. Non-zero elements of matrices (a) and (b) and of vector {g} are given by: () for i = n-: () for i = n-: ()

15 Page of For a homogeneous plate (with respect to water diffusion) matrices (a) and (b) become: () () It is worth noting that the solution of the hygroscopic problem in its analytic form (equation ) is formally analogous to that of a laminated cylinder under the same conditions. In that case however the linear term of the stationary solution is replaced by an equivalent logarithmic term, while functions sin / cos of the transient solution are replaced by Bessel functions of zero th order. In the sequel, equation will be expressed as a function of the number of cycles by adopting a recursive technique. Since: () it follows: () If N is the number of cycles at saturation, then:

16 Page of 0 Journal of Composite Materials but it is also: and: (0) () is defined by: () () () ()

17 Page of () Finally: () After (N-k) cycles: In equation boundary conditions and diffusion coefficients are constant: therefore solution is valid only at a certain distance from the surface boundaries. In other words since the analytical solution employs average boundary conditions and diffusion coefficients, fluctuations close to the external surfaces cannot be simulated by this solution. ()

18 Page of 0 Journal of Composite Materials Homogeneous plate. In the case of a homogeneous plate it can be shown that the distance at which the analytical solution becomes applicable is given by: () which is then the characteristic length within which external fluctuations are effective. It is interesting to note that this characteristic length is analogous to that given by Kelvin (equation ) for pure thermal problems and depends on the material, through the diffusion coefficient D(t), and on the period of a cycle τ. For a homogeneous plate with diffusion coefficient D(t) is then: (0) For a homogeneous plate with identical boundary conditions on the two lateral surfaces, after (N-k) cycles: () where

19 Page of () When the initial concentration c i (z,0) is different than zero, than: () It is interesting to note that the analytical solutions given by equations, and can be subjected to truncation errors of the series, in particular for a low number of cycles, that is, far from saturation. In order to illustrate schematically and qualitatively the problem and its solutions let us consider a homogeneous plate with thickness e subjected to cyclical hygrothermal conditions on the two external lateral surfaces, see figure. Assuming unidirectional diffusion from a physical point of view means that the plate has a length and a width much bigger than its thickness, in other words the length and the width of the plate are assumed to be infinite. Environmental conditions (temperature and relative humidity) vary, for instance, as illustrated in figure : temperature varies between T 0 and T (figure a), the relative humidity of the external environment (in %) varies between HR 0 and HR (figure b). Fluctuations are periodic of period τ. Fluctuations of relative humidity induce water concentration fluctuations on the external surfaces of the plate following, for instance, the law:

20 Page of 0 Journal of Composite Materials () (C and b are material constants) while the diffusion coefficient varies as a function of temperature as stated, for instance, by equation. As mentioned, it is assumed that at each time the temperature is uniform within the homogeneous plate: this approximation is admissible when considering that thermal and water diffusion transfers differ by many orders of magnitude. The distribution of water concentration c(t) within the homogeneous plate is illustrated schematically in figure. In figure, the schematic results of finite difference simulations are also reported. Water concentration fluctuates at the external lateral boundaries between c 0 and c, the extent of the fluctuating zone within the plate is equal to e 0. The fluctuating part of the solution can be simulated by finite difference methods. The analytical solution on the other hand employs constant average boundary conditions, namely: () In the transient state (after n cycles) zones close to the external surfaces are influenced by external fluctuations with period τ, the extent of the fluctuating zone is e 0, while internal zones are in a transient regime: in the steady state (N cycles), water concentration is uniform along almost all the thickness of the plate with value ; fluctuations persists along e 0, that is, in zones close to the external surfaces, and are still of period τ.

21 Page of Application to aircraft plate structures subjected to supersonic flight environments. An important practical application of the proposed method concerns aircraft structures: these are typically subjected to environmental conditions of almost constant period (the time of a flight). Figure presents schematically a hygrothermal cycle which is related to supersonic flight conditions and to which supersonic aircraft structures are subjected. Point A represents the start of a cycle/flight (ground conditions: C, HR 0%), point B represents the start of a subsonic phase, characterised by low temperatures and relative humidity (typically - C HR 0%). Point C represents the start of a supersonic phase: the exposed structures are subjected to high temperatures due to aerodynamic friction (typically 0 C) and to low HR. Finally, point D represents the end of a cycle/flight. Figures, and present the hygrothermal fields after, respectively,, and 00 flight/cycles. Both finite difference and analytical (average along a cycle) solutions are reported. Hygrothermal cycling is preceded by a conditioning phase on ground ( C, HR 0% for months) simulating the time of a long maintenance at an airport. The start of cycle (point A) is thus characterised by the effects of this on-ground maintenance, by a non homogeneous initial water concentration (labelled by initial humid state). Plate hygroscopic properties used for simulations are reported in table. From figures the following conclusions can be drawn: - the extent of the fluctuating zone e 0 calculated by an explicit expression, equation, is in agreement with finite difference simulations and is constant with varying the number of cycles; in this specific application e 0 equals 0, mm, thus fluctuations extend within a consistent zone of the structure (/ of the total thickness) and engender consistent concentration gradients,

22 Page of 0 Journal of Composite Materials the analytical solution is capable to well represent the average water concentration along a cycle; the analytical solutions converges rapidly starting from n = cycles, - the analytical solution is able to well represent the behaviour of the structure along all its cycling: the structure passes from an almost humid state (cycle/flight ) to an almost dry state after 00 hundred cycles. The final permanent concentration is well reproduced by the explicit analytical expression, equation. Concluding remarks. An analytical solution for laminated plates subjected to cyclical hygrothermal conditions is provided. Thermal transfers are discarded; it is assumed that temperature diffusion phenomena are much faster than water concentration diffusion. The solution can be employed as benchmark for simulations based on numerical procedures and can be employed for evaluating the induced hygrothermal stress, therefore for structural, design and optimisation purposes. The solution is unidirectional, therefore is applicable for slender plates. References. Carslaw, H.S., Jaeger, J.C.,. Conduction of Heat in Solids. Oxford University Press, Second Edition. Crank, J.,. The Mathematics of Diffusion. Oxford University Press, London, Second edition. Gigliotti, M., Jacquemin, F., Vautrin, A., 0. Internal Stresses in Composite Laminates due to Cyclical Hygrothermal Loading. AIAA Journal, 0. Gigliotti, M., Jacquemin, F., Vautrin, A., 0. Assessment of Approximate Models to Evaluate Transient and Cyclical Hygrothermoelastic Stress in Composite Plates. International Journal of Solids and Structures, -.

23 Page of Jacquemin, F., Vautrin, A., 0a. Modelling of the Moisture Concentration Field due to Cyclical Hygrothermal Conditions in Thick Laminated Pipes. Eur. J. Mech. A/Solids, -. Jacquemin, F., Vautrin, A., 0b. A Closed-Form Solution for the Internal Stresses in Thick Composite Cylinders Induced by Cyclical Environmental Conditions. Composite Structures, -. Kelvin, L.,. The Reduction of Observation of Underground Temperature. Trans. Royal Soc. Edin., 0. Verchery, G.,. Moisture Diffusion in Polymer Matrix Composites with Cyclic Environmental Conditions, in: Proc. th European Conference on Composites Materials ECCM-, - Avril, Bordeaux, France, pp. 0-. Weitsman, Y.J.,. Effects of Fluids on Polymeric Composites. A Review. In: Talreja, R. (Ed.), Polymer Matrix Composite. Pergamon Press, pp. 0.

24 Page of 0 Journal of Composite Materials List of tables. Table : Plate hygroscopic properties. List of figures. Figure : Semi-infinite spatial region subjected to thermal cyclical (sinusoidal) fluctuations at its boundary (external solid surface). Figure : Time evolution of the external temperature T ext (t). Figure : Temperature distribution in a semi-infinite solid subjected to external temperature fluctuations (figure ). Figure : Schematic representation of a homogeneous plate subjected to environmental temperature and relative humidity cyclical fluctuations at its external lateral boundaries. Figure : Schematic example of cyclical temperature (a) and relative humidity (b) fluctuations at the external lateral boundaries of the plate. Figure : Schematic representation of the water concentration solution c(t): (a) transient regime (n cycles), (b) permanent regime (N cycles). Figure : Hygrothermal cycle for a structure exposed to a supersonic flight environment. Figure : Hygrothermal cycles within the plate (cycle ). Figure : Hygrothermal cycles within the plate (cycle 0). Figure : Hygrothermal cycles within the plate (cycle 00).

25 Page of Tables Table : Plate hygroscopic properties.

26 Page of 0 Journal of Composite Materials Figures. Figure : Semi-infinite spatial region subjected to thermal cyclical (sinusoidal) fluctuations at its boundary (external solid surface). Figure : Time evolution of the external temperature T ext (t).

27 Page of Figure : Temperature distribution in a semi-infinite solid subjected to external temperature fluctuations (figure ). Figure : Schematic representation of a homogeneous plate subjected to environmental temperature and relative humidity cyclical fluctuations at its external lateral boundaries.

28 Figure : Schematic example of cyclical temperature (a) and relative humidity (b) fluctuations at the external lateral boundaries of the plate. Figure : Schematic representation of the water concentration solution c(t): (a) transient state (n cycles), (b) steady state (N cycles).

29 Page of Figure : Hygrothermal cycle for a structure exposed to a supersonic flight environment.

30 Page of 0 Journal of Composite Materials Figure. Hygrothermal cycles within the plate (cycle ). Figure. Hygrothermal cycles within the plate (cycle 0).

31 Page 0 of Figure. Hygrothermal cycles within the plate (cycle 00). 0