EVALUATION OF THERMAL TRANSPORT PROPERTIES USING A MICRO-CRACKING MODEL FOR WOVEN COMPOSITE LAMINATES

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1 EVALUATION OF THERMAL TRANSPORT PROPERTIES USING A MICRO-CRACKING MODEL FOR WOVEN COMPOSITE LAMINATES C. Luo and P. E. DesJardin* Department of Mechanical and Aerospace Engineering Universit at Buffalo, the State Universit of New York Buffalo, NY * ped3@buffalo.edu SUMMARY This stud concerns the development of a mio-acking model for estimating thermal transport properties of composite laminates in high temperature environments. A modified shear-lag model is developed to account for the effects of thermal epansion and ack gas pressure. Local solutions from the model are presented showing the sensitivit of ack width from thermal epansion and pressure loadings. Kewords: Mio-acking, Thermal mechanical properties, Woven laminates 1. INTRODUCTION Several studies have shown that the highl advantageous strength properties of man composite materials significantl degrade at even relativel low levels of heating [1-5]. In a previous stud b the authors, a thermo-mechanical damage model is developed based on the use of homogenization techniques for which structure response is estimated from heating [6, 7]. This model included basic mechanisms responsible for composite damage at high temperatures including char formation, eation of gas void from resin decomposition, and thermal degradation of elasticit properties. The focus of the present effort is to improve this modelling approach b including the effects of mio-acking that has been observed to be important for understanding earl-time structure response during heating [1, 2]. To estimate the local stress distribution and the stiffness degradation due to mio-acking, Zhang et al. proposed a 2-D shear-lag analsis [8-10]. In their studies, Zhang et al. used a local solution to estimate the degradation of mechanical properties through the calculation of an in-situ damage effective function (IDEF). The effects of mio-acking on thermal transport properties, however, were not considered. In high temperature environments, the effects of gas pressure due to resin decomposition and thermal epansion have not been considered in eisting mio-acking models. Including these effects and use of the mio-acking model to develop estimates of important thermal transport properties, such as gas permeabilit and bulk conductivit, is the focus of this effort. These properties are nearl impossible to determine eperimentall at high temperatures and therefore detailed modeling of them is a necessit. The rest of this stud starts with the estimates of thermal transport properties based on the use of a series of parallel acks that are formed at the interface between two plies. The permeabilit and thermal

2 conductivit estimates are in terms of ack width and number densit. To obtain these parameters, the mio-acking model of Zhang et al. is etended to include both the effects of thermal epansion and pore level gas pressure in Section 2. Results using this analsis are presented in Section 3. Lastl conclusions from this stud are summarized. 2. MATHEMATICAL FORMULATION In the current stud, a composite laminate is considered composed of alternating woven and resin laers with both normal and shear loads, as shown in Fig. 1. The resin acks are assumed to onl eist in the resin laer (region 2) between woven laers (region 1) and are aligned perpendicular to the normal loading ( σ ). The thicknesses of the resin and woven laers are assumed to be 0.2 mm and 0.4 mm, respectivel. The distance between two acks is 2s. The initial composite material is assumed to be composed of fiber, resin and a small amount of gas void. In the resin laer with acks, volume averages are used for estimating conductivit parallel to the acks, and harmonic averages for estimates of conductivit perpendicular to the acks, ( ) ( ) ( ) 1 d k = kz = kr 1 CdW + kgcdw; k = 1 C W / kr + CdW / kg CW d C d where is the ack volume fraction in resin laer is the ack densit, W is the ack width) and kr is resin conductivit. The bulk thermal conductivit of both regions 1 and 2 is modeled b combining the conductivit of acked resin and woven laers as: 2h2 h1 2h2 h1 k = k + k ; k = k + k h1 + 2h2 h1 + 2h2 h1 + 2h2 h1 + 2h2 1 2 h2 /( h1 + 2 h2) h1 /( h1 + 2h2) k z = + k z k z th where kr represents the conductivit of woven laer in i direction. The gas permeabilit induced b formation of mio-acks is estimated as a Poiseuille's flow through the channel with rectangular oss section [ 11] resulting in the following relation, K W W 2h C W 2h d 2 ϕ W C d = h1 + 2h2 12 h1 + h2 ϕ = CW d 2 h2 / h1+ 2h2 = = (3) where ack volume fraction is used ( ). Both the ack width and ack densit are required to use Eqs. and (3). The focus of the following analsis is to determine the ack width using a modified shear-lag analsis with the additional compleit of thermal epansion and pore gas pressure. Estimation of K and k using a shear-lag model The shear lag model of Zhang et al. is modified to desibe the stiffness reduction due to acking [8, 9], with thermal epansion and gas pressure due to decomposition considered. Assuming 1) aial and shear loading on the ends of -direction, 2) changes in the stress along the -direction are negligible ( / 0) and 3) a linear distribution of out-of-plane shear stresses σ z, σ z in the z-direction, the equilibrium equations can be integrated over the z direction resulting in the following set of simplified mechanical equilibrium conditions for region 2,

3 dσ% τ dσ% τ + = 0; + = 0 d h d h 2 2 where the notation indicates an average value aoss the thickness (in z-direction) per unit width (in -direction). The quantities τ and τ are the interface out-of-plane shear stresses at the interface between regions 1 and 2. Following the same approach as Zhang et al., the distribution of shear stress through the laers is assumed to be linear. Global equilibrium relations between the average stress aoss the laer and the applied load are obtained b cutting the laminate verticall to epose the mio-structure stresses on the -plane and -plane, as shown in Fig. 2, resulting in the following, hσ% + h σ% = Hσ ; hσ% + h σ% = 0; hσ% + h σ% = Hσ (5) ( 2) ( 2) The applied load from region 2 is transferred to region 1 (and visa versa) through the interface stresses τ and τ which are related to the deformation in each laer using a shear lag model desiption, ( ); τ ( ) τ = K u% u% = K v% v% (6) where K, K are the shear lag coefficients and can be shown to equal the following, K 0 0 3c55 c55 3c44 c44 = ; K = h c c h c c 0 0 ( + 2χ ) ( + 2χ ) The constants and in Eq. (7) are the material stiffness matrices for regions 1 c ij c ij and 2, respectivel, where the supersipt 0 indicates the undamaged state. The quantit, χ = h 1/ 2h 2 is the height ratio of the constraint to ack laers. Taking d / dderivative of the equilibrium relations of Eq. (4) and the shear-lag desiption of Eq. (6), two independent second order linear ODEs, are derived for the normal and shear stresses. 2 2 d σ% d σ% T T L 2 1σ% + Ω 1σ + ( % ε % ε ) = 0; L 2 2σ% + Ω 2σ = 0 d (8) d ( ) ( ) k T th where % ε = αδt is the thermal strain of the k region and the constants are given ( ) k in [ 8]. The boundar conditions for the solution to Eq. (8) are as follows, % σ =± s = Δ P ; % σ =± s = 0 (9) ( ) ( ) g where ΔPg is the gauge pressure of gas in acks. The effect of thermal epansion and gas pressure in the analsis is new and was not previousl considered in the studies of Zhang et al. [8, 9]. The solutions to Eq. (8) are, T T T T 1σ ( ) 1 ( ) cosh ( ) ε ε σ ε L ε Ω + % % Ω + % % 1 σ% = + p g (10a) L1 L1 cosh ( L1 s ) where in the limit of ΔP g ( L2 ) ( ) cos h Ω2 Ω2 % σ = 1 σ (10b) L 2 L2 cos h L2 s 0 the relations from Zhang et al. are recovered when the thermal strains are zero. The local strains for both laers can be determined using the compliance matri and stresses as, (4) (7)

4 T 0 0 T % ε = S % 11 σ + S % 12 σ + % ε ; % ε = S % 11 σ + S % 12 σ + % ε (11) Using the overall equilibrium relations in Eq. (5), substituting in the local stress solutions of Eq. (6), and averaging the local strain solutions of Eq. (11) between two acks, the average strain for each laer can be determined and are given as, s T ε = ε d ( a1s12 S22) σ 1 S22 a1s12 2s % = χ 2χ 2χ s σ ε (12a) s T = d ( a1s12 S22 ) a2s12 2s % = + + σ + ε (12b) s ε ε σ where difference between the average strains of two laers can be used to define a normalized ack width. * W W = ε ε = ( S12 a1 S22 ) ( a1s12 S22 ) σ s χ T ( 2) T ( ) S a2 S12 a2s12 σ + ε ε 2χ 2χ With Eq. (13) defining the ack width, estimates of the bulk thermal conductivit given in Eq. and gas permeabilit given in Eq. (3) can now be determined for a specified ack densit. 3. RESULTS AND DISCUSSION Figure 3 show the shear loading τ and τ on the acked laer (region 2) for two cases of α / α = 2 and α / α = 0.5. For the first case, the constrained laer (region 1) epands more than the acked laer resulting in a tension loading in region 2 for which τ is negative for / s< 0 and positive for /s>0. For the second case, the opposite is true, where the acked laer now epands more than the constrained laer. For this case, the resulting shear loading on region 2 is compressive. The resulting local stress and strain distributions for the shear loadings are given in Figs. 4 and 5. For all cases, the tensile load applied to the constraining laer deeases from the ends to the mid-point because of the transfer of the load through the shear to the acked laer. It is interesting to note that for both cases, laer 2 much more readil epands or contracts than laer 1 because of the presence of the mio-acking. In effect, the entire load is being applied to laer 1 at the ends thereb limiting the etent for which that laer can epand or contract, however, for laer 2 the ends are free to move due to the presence of ( the acks. For the case widthα / α 2) = 2, the tension % σ / σ stress of the acked laer ineases and would promote additional acking, although the process of ack multiplication is not accounted for in the present analsis. For the case α / α = 0.5, which is more tpical of laminated composites where the pure resin epands more than the weave matri, the effects of thermal epansion suppresses ack formation since the acked laer is placed in compression. Net, the effects of gas pressure are considered. Figures 6 & 7 present the stress and strain response of the constrained laer and the acked laer for applied internal pore pressures of Δ P g = 1,2,4atm. The internal pore pressure places a compressive loading to the acked laer which in turn is transferred to the constrained laer through the τ shear loading that is shown in Fig. 8. As the ack pressure ineases, the (13)

5 compressive τ shear loading ineases, thereb reducing the tensile loads in both the constrained and acked laers. If the pressure is high enough, the ends of the acked laer are in compression while the center remains in tension. Also, given in Figs. 3 and 8 is the in-plane shear stress τ which is shown to be unaffected b differential changes in thermal epansion or pore pressure resulting in no change in % σ / σ from the baseline case, as shown in Fig. 9. The reason for this is due to the limitations of the present 1D analsis. An etension of the current analsis to multi-dimensions would be required to account for changes in τ from thermal epansion and pressure effects but this also adds the compleit of desibing multidimensional mio-acking. From the local strains, the average strains over the constrained ( ε ) and acked ( ε ) laers are computed as defined b Eq. (11). The difference between the strains is the * W normalized ack width, W = ε ε, which is used to determine bulk thermal s transport properties. Figure 10 show the average strains from effects of thermal epansion and ack pressure. Figures 10 and show the strains and their difference for α / α = 2and α / α = 0.5, respectivel. For the first case, the overall effect of thermal epansion serves to open the ack width while for the latter case the effect of thermal epansion serves to reduce the ack width. Figure 10(c) shows the effects of pressure onl result in an inease in the ack width. The pressure shown in Fig. 10(c) is normalized b the maimum pressure considered of Δ Pg = 4atm. At this high pressure the average strain in the acked laer is nearl zero corresponding to a case where the entire acked laer is in compression. Figure 11 shows the effects of both thermal epansion and gas pressure on ack width for α / α = 2and α / α = 0. 5, respectivel. For the first case, an inease in temperature and pressure both serve to inease the ack width linearl. For the second case, the effects of gas pressure and thermal epansion have an opposing effect on the ack width. The second case is more tpical of woven sstem where the epansion coefficient of resin is often higher than that of the weave. It is interesting to see that for some combinations of pressure and temperature the model predicts a negative ack width which is nonphsical. In this case the effects of thermal epansion are sufficientl large to close the ack even at high pressures. With the ack width determined, the ack permeabilit, K, and effective conductivities, k & k = k z for region 2 are computed using Eqs. (3) and, respectivel, and are summarized in Figs. 12 through 14. The thermal conductivit of resin and woven matri are both assumed to be kr = ki = 0.43W / m K and the conductivit of decomposed gas is assumed to be k g = W / m K [12]. Properties are plotted over a range of temperature and pressures and for α / α = 2and α / α = 0.5. A non-linear dependence is observed for K and due to their cubic and inverse dependence on W, respectivel. k

6 4. CONCLUSIONS A modified shear-lag model is developed to account for the material damage caused b mio-acking, and the effect of both thermal epansion and decomposed gas pressure on material damage is studied. Crack width is dependent on not onl the mechanical load, but also thermal epansion and gas pressure. Using porous media relations and stead-state heat transfer solutions, relations for the ack permeabilit and effective conductivit are developed and eplored as a function of gas pressure and temperature. Depending on the relative ratio of thermal epansion coefficients of the constrained to acked laers, the ack width ma either inease or deease. The gas pressure, however, alwas results in an inease in the ack width. At high enough temperatures, the ack can close when the thermal epansion coefficient of the acked laer is larger than the constrained laer which is epected to be the case for man laminated composites where the resin epands more than the weave matri. For this case, gas permeabilit is reduced and bulk thermal conductivit ineases dramaticall. Figure 1: Sketch of representative volume for woven composite sstem that includes interpl resin mio-acks. Figure 2: Mechanical equilibrium of the representative volume for - and - cutting plane.

7 ( 2 ) Figure 3: Shear loading stresses, τ / σ and τ / σ, for α / α = 2 and α / α = 0.5. ( k Figure 4: Local normalized stresses, ) ( 2 ) σ% / σ, α / α = 2 and α / α = 0.5. ( k ) Figure 5: Local strains, ε%, α ( 2 / α ) = 2 and α / α = 0.5.

8 ( k Figure 6: Local normal stresses, σ% ) / σ, for laer 1 (constraint laer) and laer 2 (acked laer) for ack gas gauge pressures of Pg = 1, 2 and 4 atm. ( k ) Figure 7: Local normal strains, ε%, for laer 1 (constraint laer) and laer 2 (acked laer) for ack gas gauge pressures of _Pg = 1, 2 and 4 atm. Figure 8: Shear loading stresses, τ / σ and τ / σ, for ack gas gauge pressures of _Pg = 1, 2 and 4 atm. Figure 9: Local in-plane shear stresses, ( k σ% ) / σ, for different thermal and pressure conditions.

9 (c) ( k ) Figure 10: Average strain, ε, with ineasing temperature for α ( 2 ) / α = 2 and α / α = 0.5 and (c) with ineasing gas pressure ( 2 ) Figure 11: Crack width vs. gas pressure and temperature for α / α = 2 α / α = 0.5. and

10 Figure 12: Gas permeabilit, K, vs. gas pressure and temperature for α ( 2 ) / α = 2 and α / α = 0.5. Figure 13: Thermal conductivit, α / α ( 2 ) k vs. gas pressure and temperature for = 2 and α / α = 0.5 Figure 14: Thermal conductivities, k or, vs. gas pressure and temperature for k z ( 2 ) α / α = 2 and α / α = 0.5. ACKNOWLEDGEMENTS This work is supported b the Office of Naval Research (ONR) under Grant No. N with Dr. Luise Couchman serving as technical monitor. References

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