The effect of cyclic hygrothermal conditions on the stresses near thesurface of a thick composite pipe

Frederic Jacquemin, Alain Vautrin*

Mechanical and Materials Engineering Department, Ecole des Mines de Saint-Etienne, 158, cours Fauriel, F-42023 Saint-Etienne cedex 02, France

Received 17 November 2000; accepted 26 July 2001

Abstract

It is necessary to estimate the moisture concentration and the hygrothermal internal stress fields to evaluate the durability of thick

composite pipes submitted to cyclic environmental conditions. After some time, the moisture concentration, induced by tempera-ture and relative-humidity cycles, is permanent within the pipe and periodic close to the inner and outer surfaces. The hygrothermalstresses induced are computed by using the classical equations of solid mechanics and assuming a hygrothermoelastic orthotropicbehaviour for every ply. The aim of this paper is to model the effects of the periodic boundary conditions on the hygrothermal

stresses. # 2002 Elsevier Science Ltd. All rights reserved.

Keywords: A. Polymer-matrix composites; B. Hygrothermal effect; Durability

1. Introduction

We consider a thick laminated pipe, whose outer andinner radii are a and b, respectively, submitted to tem-perature and relative humidity cycles of the same period,�. Assuming that the thermal equilibrium is reachedinstantaneously, the temperature is considered to beuniform over the thickness of the pipe at any time.The moisture concentration, c(r,t), is solution of the

following system with Fick’s Eq. (1) and boundary andinitial conditions Eq. (2):

@c

@t¼ DðtÞ

@2c

@r2þ1

r

@c

@r

� �; a < r < b ð1Þ

cða; tÞ ¼ caðtÞ and cðb; tÞ ¼ cbðtÞcðr; 0Þ ¼ 0

�ð2Þ

The diffusion coefficient D(t) is only a time depen-dent function which depends on the temperaturethrough an Arrhenius’ law. ca(t) and cb(t) are the cyc-lic boundary concentrations related to the relativehumidity. D(t), ca(t) and cb(t) are periodic time func-tions of the same period �.The general solution of this problem, studied by Jac-

quemin and Vautrin [1], comprises a transient part,

which converges towards a permanent solution (3)within the pipe and a fluctuating part which convergestowards a periodic solution of period � in the vicinity ofthe external surfaces.

cðrÞ ¼

Ð �0DðtÞcbðtÞdtÐ �

0DðtÞdtþ

Ð �0DðtÞðcaðtÞ � cbðtÞÞdt

lna

b

ð�0

DðtÞdt

lnðr

bÞ ð3Þ

Therefore, the permanent solution (3) is only valid upto a distance from the edge where the cyclic boundaryconditions are applied. The extent e0 of the periodicsolution, which will be determined by using a finite dif-ference scheme, has been estimated by Verchery [2] for asemi-infinite plate, depends on the diffusion coefficient,the temperature and the period of the cycles:

e0 ¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�

ð�0

DðtÞ dt

sð4Þ

2. Mechanical problem

In this part, the hygrothermal stresses are computedby using the classical equations of solid mechanics forevery ply at any time: constitutive laws of hygro-thermoelastic orthotropic materials (5), strain/displace-

0266-3538/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved.

PI I : S0266-3538(01 )00150-6

Composites Science and Technology 62 (2002) 567–570

www.elsevier.com/locate/compscitech

* Corresponding author.

E-mail address: vautrin@emse.fr (A. Vautrin).

ment relationship, compatibility and equilibrium equa-tions.

� ¼ L : "� � T� T0ð Þ � m�m0ð Þð Þ

with m ¼mwater

ms¼c

sð5Þ

c and s are, respectively, the moisture concentrationand the mass density of the dry material.Introducing L0, we consider the following reduced

variables:

�� ¼ �=L0; L ¼ L=L0; ðw� ; u�; v�Þ ¼ ðw; u; vÞ=b; r� ¼ r=b

Displacement with respect to x and �, respectivelyu�ðx�; r�Þ and v�ðx�; r�Þ are then expressed:

u�ðx�; r�Þ ¼ �S�ryxrR3

r�þ S� xrxrR4lnr�þ R1x� þ R5;

v�ðx�; r�Þ ¼ R2x�r��S� ryry2

R3

r�� S� ryxrR4 þ R6r�;

R1;R2;R3;R4;R5; R6 are constants and S ¼ L�1:

8>>><>>>:

ð6Þ

The radial component of the displacement field, w� ,satisfies the following expression:

r�2@2w�

@r�2þ r�

@w�

@r��L� yy

L� rrw� ¼

r� ½ðL� xy � L� xrÞR1 þ ðL� sy � 2L� rsÞR2r��

L� rrþ

r�½ðI1 � I2ÞðT� T0Þ þ ðK1 � K2Þ ðm�m0Þ þ K1r�@m

@r��

L� rrð7Þ

with, I1 ¼ L� xr�xx þ L� ry�yy þ L� rr�rr þ L� rs�xy, I2 ¼ L� xy�xx

þL� yy�yy þ L� ry�rr þ L� sy�x�, K1 ¼ L� xrxx þ L� ryyy þ

L� rrrr þL� rsxy, K2 ¼ L� xyxx þ L� yyyy þ L� ryrr þ L� syxy.

2.1. Radial component of the displacement field withinthe pipe

The general solution of the differential Eq. (7) isthe sum of a solution of the homogeneous equationand of a particular solution. Considering the per-manent moisture concentration (3), we obtain theradial component of the displacement field within thepipe:

w� ¼R7r�

ffiffiffiffiffiL�yyL�rr

qþ R8r�

�

ffiffiffiffiffiL�yyL�rr

qþ

ðL� xy � L� xrÞR1r�

L� rrð1�L� yy

L� rrÞ

þðL� sy � 2L� rsÞR2r�

2

L� rrð4�L� yy

L� rrÞ

þ

ðI1 � I2ÞðT� T0 Þr�

L� rrð1�L� yy

L� rrÞ

�ðK1 � K2Þm0r�

L� rrð1�L� yy

L� rrÞ

þ

ðK1 � K2Þr�

sL� rrð1�L� yy

L� rrÞ

Ð �0DðtÞcbðtÞdtÐ �

0DðtÞdt

½ðK1 � K2Þr�lnr�þ K1r�

sL� rrð1�L� yy

L� rrÞ

�2ðK1 � K2Þr�

sL� rrð1�L� yy

L� rrÞ2

�

Ð �0DðtÞðcaðtÞ � cbðtÞÞdt

lna

b

ð�0

DðtÞdt

ð8Þ

forL� yy

L� rr6¼ 1;

L� yy

L� rr6¼ 4:

2.2. Radial component of the displacement field in thevicinity of the surfaces

In the vicinity of the surfaces, the periodic concentra-tion is determined by using a finite difference scheme.To propose a close form solution of the displacement,we subdivide the extent of the periodic concentrationand assume on each subdivision a parabolic moistureconcentration (9):

Table 1

Hygroscopic properties

Diffusion coefficient (mm2/s) D(t)=0.57exp(�4993/T(t))

Ambient moisture

concentration (kg/m3)

c=0.2385 H

Table 2

Mechanical properties in the orthotropic reference frame (1, 2, 3)

Material E1 (Gpa) E2, E3 (Gpa) �12, �13 �23 G12 (Gpa) �1 (K�1) �2, �3 (K

�1) 1 2, 3

T300/5208 181 10.3 0.28 0.43 7.17 0.02 10�6 22.5 10�6 0 0.6

568 F. Jacquemin, A. Vautrin / Composites Science and Technology 62 (2002) 567–570

ci ¼ A 0ir

2 þ B 0ir�þ C

0i ð9Þ

Thus, we obtain the radial component of the dis-placement field, solution of Eq. (7), for each subdivi-sion:

w� ¼ R7r�

ffiffiffiffiffiL�yyL�rr

qþ R8r�

�

ffiffiffiffiffiL�yyL�rr

qþðL� xy � L� xrÞR1r�

L� rrð1�L� yy

L� rrÞ

þðL� sy � 2L� rsÞR2r�

2

L� rrð4�L� yy

L� rrÞ

þðI1 � I2ÞðT� T0 Þr�

L� rrð1�L� yy

L� rrÞ

�ðK1 � K2Þm0r�

L� rrð1�L� yy

L� rrÞ

þK1

sL� rr

B0ir�2

ð4�L� yy

L� rrÞ

þ2A0

ir�3

ð9�L� yy

L� rrÞ

26664

37775

þðK1 � K2Þ

sL� rr

C0ir�

ð1�L� yy

L� rrÞ

þB0ir�2

ð4�L� yy

L� rrÞ

þA0

ir�3

ð9�L� yy

L� rrÞ

26664

37775 ð10Þ

Finally, the displacement through every ply dependson eight constants to be determined : Ri for i=1..8.

2.3. Determination of eight constants per ply

The eight constants are determined from the follow-ing conditions:

. rigid body motions restrained;

. continuity of the displacement components at eachinterply;

. continuity of the transverse shear stress at eachinterply;

. continuity of the normal stress at each interply andits nullity on the boundaries surfaces;

. global force balance of the pipe.

3. Case study

We consider a thick pipe made up of five carbon/epoxy plies of equal thickness alternatively oriented at+55 or �55 versus the longitudinal axis. The outerand inner radii are, respectively, a=10 mm and b=30mm. The hygroscopic properties [3] and the mechanicalproperties [4] are presented in Tables 1 and 2.The pipe ishomogeneous from the hygroscopic point of view, everyply has the same hygroscopic properties (Table 1), but itis heterogeneous from the mechanical point of viewbecause of the different orientations of the plies. Thepipe is submitted to relative humidity (Fig. 1 and Fig. 2)and temperature (Fig. 3) cycles of 4 week period.

Fig. 1. Hollow laminated cylinder.

Fig. 2. Cyclic concentration on the surfaces.

Fig. 3. Temperature cycle.

Fig. 4. Moisture concentration for different points of the cycles.

F. Jacquemin, A. Vautrin / Composites Science and Technology 62 (2002) 567–570 569

Fig. 4 shows that the oscillations of the periodic con-centration disappear at a distance e0 from the edge.Therefore, at a distance e0 from the edge the permanentconcentration holds with a constant value because ofthe symmetrical hygrothermal loading. We observe thatfluctuating concentration gradients are important forthe points C and E where the relative humidity changesroughly. Figs. 5 and 6 depict the radial stress and thenormal ply stress in the transverse direction to thefibres, respectively. We observe that the periodic con-centration gradients have not any influence on the radialstress but induce strong gradients of the normal plystress. In this periodic concentration regions, the normal

ply stress gradients are so important that theirs valuesfirstly negatives become positives. The radial stress andthe normal ply stress within the pipe are dependent onthe temperature changing: for (A,C) and (D,E), corre-sponding to identical temperature but to different rela-tive humidity, the stresses are identical. The temperaturedecreases between A and B induce tensile stresses andthe temperature increase between C and D induce com-pressive stresses.

4. Conclusion

We propose an approach which allows to measure theinfluence of the periodic field, close to the surfaces, ofthe moisture concentration on the internal stresses forthick laminated pipes. For the internal stresses inducedby cyclic hygrothermal conditions, we dissociate thethermal effects and the hygroscopic effects. This solu-tion provides a helpful tool for the design of thickcomposite pipes under hygrothermal fatigue, since itleads to the knowledge of the stress evolution which canbe strong within a narrow region near to the surfaceswhere holds the periodic moisture concentration.

References

[1] Jacquemin F, Vautrin A. Thick laminated pipes submitted to

cyclic environmental conditions. In: 9th European Conference on

Composite Materials, Brighton 2000.

[2] Verchery G. Moisture diffusion in polymer matrix composites

with cyclic environmental conditions. In: 5th European Con-

ference on Composite Materials, Bordeaux 1992.

[3] Loos AC, Springer GS. Moisture absorption of graphite-epoxy

composition immersed in liquids and in humid air. In: Springer

GS, editor. Environmental effects on composite materials,

Springer G. S., Technomic 1981, p. 51–62.

[4] Tsai SW. Composite design., Think composites, 1987.

Fig. 5. Radial stress for different points of the cycles.

Fig. 6. Normal in-plane stress in the transverse direction to the fibres.

570 F. Jacquemin, A. Vautrin / Composites Science and Technology 62 (2002) 567–570