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Composite Structures 71 (2005) 447–452

Thermal stability of composite slab-structures in harsh environment

Mohamed Gadalla *, Hany El Kadi

Mechanical Engineering Department, School of Engineering, American University of Sharjah, Sharjah, United Arab Emirates

Abstract

This paper presents a complete analysis of the effect of harsh environment found in the Arabian Gulf region, especially in UAE, onthermal stability of composite slab-structures with temperature-sensitive internal heat generation. These structures can include buildingslabs, highway bridge slabs and tunnels, cooling towers, silos, and rotundas. Depending on the application, composite slab-structuresmay be subjected to sudden and violent thermal effects on their surface due to different applications and various severe environmentalconditions. The combination of theses conditions with the possible internal heat generation can provoke some defects that may lead tothe degradation of the material properties and the structure function. Therefore, thermal instability of structures will result if the structurefails to dissipate all heat conducted into and generated from/within the structure. It is very important for a designer to select the appro-priate material for an application and predict the structure behavior under different operating conditions. In this regard, it is interesting tonote that the Great Coulee dam in France was poured with pre-chilled concrete just to compensate for the heat release for the exothermicreaction within the concrete long after the pouring process. An analytical model was developed for the generalized problem that includesthe effect of the non-linearity and non-homogeneity of the internal heat generation. Three case studies with combinations of encounteredsevere boundary conditions are presented and analyzed. The results of these analyses indicate that the critical parameters for thermal sta-bility are critically dependent on the environmental curing process, structure size, the material properties, the heat transfer coefficients, andother constants accrued from the structure environment. The critical values for thermal stability of the composite slab structures may besignificant to the service-worthiness of industrial products, to the life span of the structures, as well as to public construction safety.� 2005 Elsevier Ltd. All rights reserved.

Keywords: Composite materials; Slab structures; Thermal stability; Polymeric structures; Harsh environment

1. Introduction

Thermal analysis is increasingly becoming an importanttool in determining the reliability of the selection and appli-cation of composite materials in many engineering fieldssuch as electronic packaging, automotive, aerospace, mili-tary, marine and civil structures. Prediction of thermalbehavior of composite structures due to violent thermaleffects and internal heat generation after the curing processis considered to be an effective and complementary toolduring the design stage. Thermal stability of compositestructures is a key parameter in evaluating the structuresperformance under different operating conditions. Someprevious investigators [1,2] stated that the damage to the

0263-8223/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compstruct.2005.09.019

* Corresponding author. Tel.: +971 50 5144440; fax: +971 6 5152979.E-mail address: mgadalla@aus.edu (M. Gadalla).

structures will ensue in infinite plane slabs and solid cylin-ders if certain parameters exceed specific values. These con-clusions were made without mentioning that thesestructures are uniformly seeded with internal heat sourcesowing to the exothermic chemical reactions in the materialand the values of the prescribed parameters are the limitingvalues below which there will be a steady-state temperaturedistribution within the structures.The final quality of a composite part depends mainly on

the curing process. The temperature in the laminate mustbe kept as uniform as possible to avoid thermal gradi-ents/stresses during and after the cooling process [3–6]. Ifthe maximum temperature of the laminate becomes veryhigh due to the internal heat generation or the effect ofthe sudden variations of surroundings, thermal degrada-tion of the material properties may occur [7–9]; this ismainly due to the thermal instability or thermal viability

448 M. Gadalla, H. El Kadi / Composite Structures 71 (2005) 447–452

of the composite material. This concept asserts from thelogic that a conducting composite-material with tempera-ture-sensitive internal heat generation or under severeboundary conditions will deteriorate under a specific cir-cumstance if it fails to transfer all the heat conducted intoand generated within the material adequately to establish asteady-state temperature distribution in the structure. Thislogic is actually a useful link between inverse analysis forestimating thermo-physical properties [10–14], life span ofcomposite products and even structure reliability.Analysis of thermal instability of composite structures

will predict the threshold below which the structure isunder steady-state temperature behavior. Furthermore,knowing the material response under violent environmen-tal conditions will assist the designer in determining thedesign temperature for thermally loaded composite struc-tures, required material constants, as well as size and struc-ture characteristics. For example, thermal instabilityanalysis of the outer wing of the future composite airlinerA3XX [Megaliner] constitutes an additional thermal loadbeside the mechanical loads [14,15]. The current analysisis capable of predicting the critical parameters under whichthe airliner structure is exposed to solar irradiation, verylow surrounding temperature, and high cooling rate ofthe wing during taxiing and take-off.

2. Problem formulation

Consider an infinite composite slab-structure(0 6 x 6 L) having a constant thermal conductivity (K)and seeded with a non-uniform distribution of heat source.As a result, the rate of internal heat generation is assumedto obey a simple exponential function of the spatial coordi-nate x; in other words1:

q ¼ q0e�axebðT�T 0Þ ð1Þ

where q0 and b are material constants, a is the power of thenon-uniformity factor, and T0 is the ambient temperature.The spatial exponential function representing the inter-

nal heat generation source of a composite material is afunction of the location within the material.A customary heat balance yields the following differen-

tial equation for the temperature distribution in the com-posite structure-slab:

Kd2Tdx2

þ q0e�axebðT�T 0Þ ¼ 0 ð2Þ

Introducing the new variables

h ¼ T � T 0; k ¼ aL and f ¼ x=L ð3ÞEq. (2) can thus be transformed into the following form:

d2bh

df2þ Cebh�kf ¼ 0 ð4Þ

1 The present analysis become applicable to polymers if a = 0.

where

C ¼ bq0L2

Kð5Þ

Defining

W ðfÞ ¼ bhðfÞ � kf

Eq. (4) can be written as

d2W

df2þ CeW ðfÞ ¼ 0 ð6Þ

The solution of Eq. (6) is

W ðfÞ ¼ C1f þ C3 � 2 lnð1þ C2eC1fÞ ð7ÞUsing the original variables, Eq. (7) becomes:

bhðfÞ ¼ ðC1 þ kÞf þ C3 � 2 lnð1þ C2eC1fÞ ð8Þwhich in its derivative form, becomes:

dbhdf

¼ C1 þ k � 2C1C2eC1f

1þ C2eC1fð9Þ

The three constants of integration in Eq. (8) require onemore condition than the slab-structure boundaries can pro-vide. This extra condition can be secured by substitutingEq. (8) back into Eq. (4). This results in the extra condition:

CeC3 ¼ 2C21C2 ð10Þ

3. Case studies

The developed formulation will now be used to predictthe critical thermal-stability parameters for compositeand polymeric structures subjected to three differentboundary conditions. These case studies show that thesolution can be used in predicting the structure responseand selecting the required material constants and the struc-ture size for various engineering applications. Fig. 1 showsthe Landau slab [1] and the boundary conditions of theinvestigated case studies.

3.1. Case A: Top and bottom boundaries at constant

temperature

The boundary conditions of this case are

bhð0Þ ¼ 0 and bhð1Þ ¼ 0 ð11ÞUsing Eqs. (8) and (9), the first boundary condition yields:

eC3 ¼ ð1þ C2Þ2 ð12Þwhile the second boundary condition gives:

C1 þ k þ C3 ¼ lnð1þ C2eC1Þ2 ð13ÞThe critical solution of Eqs. (12), (13) and (10) will be firstcarried out by eliminating C3 between Eqs. (12) and (13),resulting in

C1 þ k ¼ ln 1þ C2eC1

1þ C2

� �2ð14Þ

To

To

To

To

Landau slab

To

To

X Case A

To

X

insulated

Case B

To

X

X

)( oTTxo eeqq

T

Case

1

C

βα – –=

)( oTTo eqq β –

=

)( oTTxo eeqq βα – –

=)( oTTxo eeqq βα – –

=

Fig. 1. Definition of a Landau slab and the investigated cases.

M. Gadalla, H. El Kadi / Composite Structures 71 (2005) 447–452 449

Solving for C2:

C2 ¼1� ek � ek=2 e

C12 � e�

C12

� �ek � eC1 ð15Þ

Consequently, from Eq. (12),

eC3 ¼1� eC1 � ek=2 e

C12 � e�

C12

� �ek � eC1

24

352

ð16Þ

Substitution of Eqs. (15) and (16) into Eq. (10) yields:

C ¼ 2C21ðek � eC1Þ 1� ek � ek=2 e

C12 � e�

C12

� �h i1� eC1 � ek=2 e

C12 � e�

C12

� �h i2 ð17Þ

Maximizing C with respect to C1, the critical parameters ofthermal stability for different material and size structurescan be obtained by taking the logarithms of both sides ofEq. (17), i.e.

lnC2¼ 2 lnC1 þ lnðek � eC1Þ

þ ln 1� ek � ek=2 eC12 � e�

C12

� �h i� 2 ln 1� eC1 � ek=2 e

C12 � e�

C12

� �h ið18Þ

Differentiating of Eq. (18) with respect to C1 and subse-quently equating the derivative to zero yields:

2

C1c� eC1c

ek � eC1c ¼ek=2 e

C1c2 þ e�

C1c2

� �2 1� ek � ek=2 e

C1c2 � e�

C1c2

� �h i

�2eC1c þ ek=2 e

C1c2 þ e�

C1c2

� �1� eC1c � ek=2 e

C1c2 � e�

C1c2

� � ð19Þ

For the special case of polymers (k = 0), Eq. (19) becomes:

2

C1c� 12¼ � e�

C1c2

1þ e�C1c2

ð20Þ

and Eq. (17) becomes:

Cc ¼ 2C21ce�C1c2 1þ e�

C1c2

� ��2ð21Þ

Whence C1c = 4.79872 (see Appendix A), the critical valueof C1 is subsequently substituted into Eq. (21) to yieldCc = 3.5138, which is Landau�s finding for a Landau slab[1] (see Appendix A).For composite slab-structures, with non-zero values of a

or k, we obtain different results that mainly depend on theselected material and the thickness of the slab structure.Fig. 2 shows the results obtained for the critical thermal-stability parameters for different material constants and/or structure thickness.

3.2. Case B: Top boundary insulated, bottom boundary

at constant temperature

In this case, the upper boundary of the slab (x = 0) isinsulated while the lower boundary (x = L) is kept at theambient temperature T0. The boundary conditions for thiscase are

bh0ð0Þ ¼ 0 and bhð1Þ ¼ 0 ð22Þ

Using Eqs. (8) and (9), these conditions can be transformedinto the following forms:

C2 ¼C1 þ kC1 � k

ð23Þ

C3 þ C1 þ k ¼ 2 lnð1þ C2eC1Þ ð24Þ

0

0.5

1

1.5

2

2.5

3

LA/L

B

Polymeric Slab

-1.5 -1-2 -0.5 0 0.5 1 1.5 2 2.5

λ

Fig. 3. Slab size ratio vs. k for a material subjected to boundaryconditions of Cases A and B.

0.1

1

10

-1.5 -1-2-2.5 -0.5 0 0.5 1 1.5 2 2.5

Polymeric Slab

3.5138

0.8785

CASE A

CASE B

Γc

λ

Fig. 2. Critical parameter Cc for polymeric and composite slab structuresof Cases A and B.

2 When k = 0, Eq. (31) reduces to C2 ¼ e�C12 which is identical to Eq.

(A.10).

450 M. Gadalla, H. El Kadi / Composite Structures 71 (2005) 447–452

Substituting from Eqs. (23) and (24) into Eq. (10):

C ¼ 2C21C1þkC1�k

e�ðC1þkÞ 1þ C1þkC1�k e

C1

� �2 ð25Þ

Maximizing C with respect to C1, we obtain the criticalequation for C1c,

eC1c ¼C1c þ 2� 2kC1c

C21c�k2

ðC1c � 2Þ C1cþkC1c�k �

2kC1cðC1c�kÞ2

ð26Þ

The root C1c and the corresponding critical value of ther-mal stability (Cc) are evaluated for several values of k.The critical parameter of thermal stability Cc is graphicallydepicted as a function of the material constant and/orstructure size (k) in Fig. 2.For structures made of polymers, (k = 0), Eq. (25)

reduces to the following form:

C ¼ 2C21eC1ð1þ eC1Þ�2 ð27Þ

Differentiating of Eq. (27) with respect to C1 gives

dC=2dC1

¼ C1eC1

ð1þ eC1Þ32þ C1 þ eC1ð2� C1Þ� �

A polymeric slab-structure (k = 0) under these boundaryconditions (Case B) will have C1c = 2.39936 andCc = 0.87846 which is 25% of that obtained for the samepolymer in Case A. As evident from the relative magnitudesof the critical thermal stability parameters, the polymericslab-structure of Case B is equivalent to a polymeric slab-structure of Case A having a thickness of 2L and of identicalmaterial. This relation is graphically demonstrated in Fig. 3.

3.3. Case C: Top boundary at a temperature higher

than bottom boundary

In this case, the upper and lower boundaries are at tem-peratures T1 and T0 (where T1 > T0); respectively. Thus theboundary conditions are

bhð0Þ ¼ bðT 1 � T 0Þ ¼ B and bhð1Þ ¼ 0 ð28ÞUsing Eq. (8), these conditions become:

C3 ¼ Bþ lnð1þ C2Þ2 ð29Þand

C3 ¼ �C1 � k þ ln 1þ C2eC1 �2 ð30Þ

Eliminating C3 between Eqs. (29) and (30), we have2

C2 ¼1� eBþk

2 eC12

eBþk2 � e

C12

� �eC12

ð31Þ

substituting in Eq. (30),

eC3 ¼ e�C1e�k eBþk2 ð1� eC1ÞeBþk2 � e

C12

" #2ð32Þ

Substituting from Eqs. (31) and (32) in Eq. (10) yields:

C2¼ C21e

�B

ð1� eC1Þ eC1=2 1� eBþk

2 eC1=2� �

eBþk2 � eC1=2

� �ð33Þ

To maximize C/2 with respect to C1, we differentiate Eq.(33) and subsequently equating the derivative to zero. Thisleads to

� 4

C1cþ 1

� �¼ 4eC1c

1� eC1c � eC1c=2

eBþk2

1� eBþk2 eC1c=2

þ 1

eBþk2 � eC1c=2

!

ð34ÞEq. (34) is now solved for C1c corresponding to given val-ues of B and k. When this critical root C1c is substitutedinto Eq. (33), we obtain the corresponding critical para-meter Cc. Fig. 4 shows the effect of the material constantand/or structure size k on the critical thermal stabilityparameter Cc for different values B.

B=0

B=0.5

B=1

Polymeric Slab

CASE A

3.5138

0.1

1

10

Γc

-1.5 -1-2-2.5 -0.5 0 0.5 1 1.5 2 2.5

λ

Fig. 4. Critical parameter Cc for polymeric and composite slab structuresof Case C.

M. Gadalla, H. El Kadi / Composite Structures 71 (2005) 447–452 451

4. Discussion and conclusion

According to the definition of a critical parameter, Cc,there is a critical or limiting relationship among the mate-rial constants and the thickness of the slab structure, withinwhich a steady-state temperature distribution can and willexist in the structure. Such a critical parameter has beendetermined for the polymeric and composite slab structuresunder a variety of temperature boundary conditions.Fig. 2 shows that for both Cases A and B, the increase

in the composite slab thickness (manifested by the increasein the value of k) is associated with an increase in the valueof the critical parameter of thermal stability. For each ofthe two cases, the critical parameter, Cc, for polymericslabs (k = 0) is also shown in the figure. As evidenced fromthe relative magnitudes of their critical parameters, thepolymeric slab-structure of Case B is equivalent to a poly-meric slab-structure of Case A of identical material havingits thickness doubled as shown in Fig. 3. It can be con-cluded that the critical parameter of thermal stability of acomposite material increases as the material constant andthe slab thickness increase.Unlike the first two cases, the critical parameter of ther-

mal stability of the composite slab of Case C is a functionof the temperature difference (manifested by the parameterB). Fig. 4 shows the variation of the critical parameter ofthermal stability, Cc, with the thickness of the slab (mani-fested by the parameter k) for different values of tempera-ture differences. For the same material, with a constantvalue of k, a higher value of B (implying that the materialis subjected to a higher temperature difference) results in alower value of Cc. When B = 0, the solution is the same asthat of a composite slab-structure of Case A, as it should.For B > 0, the critical parameter of thermal stability de-creases with increasing B, as expected. This means that amaterial subjected to low temperature difference, has theability to withstand more heat and subsequently has ahigher critical parameter of thermal stability. We note theupper seal for thermal stability of Case B is the composite

slab-structure of Case A. It can be concluded that the crit-ical parameter of thermal stability can be utilized as a sup-plementary and a complimentary tool to predict thestructure behavior and select the material properties as wellas the structure size during the design stage for any engi-neering application.

Appendix A. Landau slab structure and solution

Landau [1] analyzed the thermal stability of a slab struc-ture having a uniformly populated internal heat sourcewithin the material. This source generates heat at steadystate rate of

q ¼ q0ebðT�T 0Þ ðA:1Þ

and which is subjected to the boundary conditions:

T ð0Þ ¼ T 0 and T ðLÞ ¼ T 0 ðA:2Þ

A steady-state heat balance establishes the governing differ-ential equation for the temperature distribution in the slabstructure, i.e.

Kd2Tdx2

þ q0eBðT�T 0Þ ¼ 0 ðA:3Þ

Introducing the new variables:

h ¼ T � T 0 and f ¼ x=L; ðA:4Þ

we rewrite Eq. (A.3) as

d2bh

df2þ Cebh ¼ 0 ðA:5Þ

where

C ¼ bq0L2

KðA:6Þ

Using Eq. (8), the boundary conditions

bhð0Þ ¼ 0 and bhð1Þ ¼ 0 ðA:7Þ

become

eC3 ¼ ð1þ C2Þ2 ðA:8Þ

and

C1 þ C3 ¼ 2 lnð1þ C2eC1Þ ðA:9ÞEliminating C3 between Eqs. (A.8) and (A.9), we have

C2 ¼ e�C1=2 ðA:10ÞSubstituting in (A.9) gives:

eC3 ¼ ð1þ e�C1=2Þ2 ðA:11ÞThus, Eq. (10) yields:

C ¼ 2C21e�C1=2

ð1þ e�C1=2Þ2ðA:12Þ

452 M. Gadalla, H. El Kadi / Composite Structures 71 (2005) 447–452

With a view to extremize C, we differentiate both sides ofEq. (A.12) with respect to C1,

dC=2dC1

¼ C1e�C1=2

ð1þ e�C1=2Þ22� C1

1

2� e�C1=2

1þ e�C1=2

� �� �ðA:13Þ

Equating the left hand side of Eq. (A.13) to zero, weobtain:

1

2� 2

C1c¼ eC1c=2

1þ e�C1c=2

Simplification results in

e�C1c=2 ¼ C1c � 4C1c þ 4

ðA:14Þ

where the subscript c indicates criticality.Eq. (A.14) is now solved for the root

C1c ¼ 4:79872 ðA:15ÞConsequently, Eq. (A.10) gives

C2c ¼ 0:0907763 ðA:16Þand Eq. (A.11) yields

eC3 ¼ 1:18979 ðA:17ÞSubstituting Eqs. (A.15)–(A.17) into Eq. (A.12), the criticalvalue of thermal stability for which a limiting steady-statetemperature distribution exists in the polymeric slab, is cal-culated as

Cc ¼ 3:5138 ðA:18Þ

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