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Effect of cyclic hygrothermal conditions on the stresses
near the surface of a thick composite pipe
Frédéric Jacquemin, Alain Vautrin
To cite this version:
Frédéric Jacquemin, Alain Vautrin. Effect of cyclic hygrothermal conditions on the stresses near the
surface of a thick composite pipe. Composites Science and Technology, Elsevier, 2002, 62, pp.567-572.
�10.1016/S0266-3538(01)00150-6�. �hal-01006775�
The effect of cyclic hygrothermal conditions on the stresses near the
surface of a thick composite pipe
Fre´de´ric Jacquemin, Alain Vautrin*
Mechanical and Materials Engineering Department, Ecole des Mines de Saint-Etienne, 158, cours Fauriel, F-42023 Saint-Etienne cedex 02, France
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 hygrothermal stresses induced are computed by using the classical equations of solid mechanics and assuming a hygrothermoelastic orthotropic behaviour for every ply. The aim of this paper is to model the effects of the periodic boundary conditions on the hygrothermal stresses.
Keywords:A. Polymer-matrix composites; B. Hygrothermal effect; Durability
1. Introduction
We consider a thick laminated pipe, whose outer and inner radii are a and b, respectively, submitted to tem-perature and relative humidity cycles of the same period, . Assuming that the thermal equilibrium is reached instantaneously, the temperature is considered to be uniform 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 and initial conditions Eq. (2):
@c @t¼DðtÞ @2c @r2þ 1 r @c @r ; a < r < b ð1Þ cða; tÞ ¼ caðtÞand cðb; tÞ ¼ cbðtÞ cðr;0Þ ¼ 0 ð2Þ The diffusion coefficient D(t) is only a time depen-dent function which depends on the temperature through an Arrhe´nius’ law. ca(t)and cb(t)are the
cyc-lic boundary concentrations related to the relative humidity. 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 converges towards a periodic solution of period in the vicinity of the external surfaces.
cðrÞ ¼ Ð 0DðtÞcÐ bðtÞdt 0DðtÞdt þ Ð 0DðtÞðcaðtÞ cbðtÞÞdt lna b ð 0 DðtÞdt lnðr bÞ ð3Þ
Therefore, the permanent solution (3) is only valid up to a distance from the edge where the cyclic boundary conditions are applied. The extent e0 of the periodic
solution, which will be determined by using a finite dif-ference scheme, has been estimated by Verchery [2] for a semi-infinite plate, depends on the diffusion coefficient, the temperature and the period of the cycles:
e0¼2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð 0 DðtÞdt s ð4Þ 2. Mechanical problem
In this part, the hygrothermal stresses are computed by using the classical equations of solid mechanics for every ply at any time: constitutive laws of hygro-thermoelastic orthotropic materials (5),
strain/displace-* Corresponding author.
ment relationship, compatibility and equilibrium equa-tions. ¼ L : " T Tð ð 0Þ m mð 0ÞÞ with m ¼mwater ms ¼ c s ð5Þ
c and s are, respectively, the moisture concentration
and the mass density of the dry material.
Introducing L0, we consider the following reduced
variables:
¼ =L0; L ¼ L=L0; ðw; u; vÞ ¼ ðw; u; vÞ=b; r ¼ r=b
Displacement with respect to x and , respectively uðx; rÞand vðx; rÞ are then expressed:
uðx; rÞ ¼ Sryxr R3 r þSxrxrR4lnr þ R1x þ R5; vðx; rÞ ¼ R2xr Sryry 2 R3 r SryxrR4þR6r; R1; R2; R3; R4; R5; R6 are constants and S ¼ L1:
8 > > > < > > > : ð6Þ
The radial component of the displacement field, w, satisfies the following expression:
r2@ 2w @r2 þr @w @r Lyy Lrr w ¼ r ½ðLxyLxrÞR1þ ðLsy2LrsÞR2r Lrr þ r½ðI1I2ÞðT T0Þ þ ðK1K2Þ ðm m0Þ þK1r @m @r Lrr ð7Þ with, I1¼LxrxxþLryyyþLrrrrþLrsxy, I2¼Lxyxx þLyyyyþLryrrþLsyx , K1¼Lxr xxþLry yyþ Lrr rrþLrs xy, K2 ¼Lxy xxþLyy yyþLry rrþLsy xy.
2.1. Radial component of the displacement field within the pipe
The general solution of the differential Eq. (7) is the sum of a solution of the homogeneous equation and of a particular solution. Considering the per-manent moisture concentration (3), we obtain the radial component of the displacement field within the pipe: w ¼R7r ffiffiffiffiffi L yy Lrr q þR8r ffiffiffiffiffi Lyy Lrr q þ ðLxyLxrÞR1r Lrrð1 Lyy Lrr Þ þðLsy2LrsÞR2r 2 Lrrð4 Lyy Lrr Þ þ ðI1I2ÞðT T0 Þr Lrrð1 Lyy Lrr Þ ðK1K2Þm0r Lrrð1 Lyy Lrr Þ þ ðK1K2Þr sLrrð1 Lyy Lrr Þ Ð 0DðtÞcÐ bðtÞdt 0DðtÞdt ½ðK1K2Þrlnr þ K1r sLrrð1 Lyy Lrr Þ 2ðK1K2Þr sLrrð1 Lyy Lrr Þ2 Ð 0DðtÞðcaðtÞ cbðtÞÞdt lna b ð 0 DðtÞdt ð8Þ for Lyy Lrr 6¼1; Lyy Lrr 6¼4:
2.2. Radial component of the displacement field in the vicinity 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 concentration and assume on each subdivision a parabolic moisture concentration (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(K1) 2, 3(K1) 1 2, 3
ci¼A0ir2þB 0 ir þ C
0
i ð9Þ
Thus, we obtain the radial component of the dis-placement field, solution of Eq. (7), for each subdivi-sion: w ¼ R7r ffiffiffiffiffi Lyy Lrr q þR8r ffiffiffiffiffi Lyy Lrr q þðLxyLxrÞR1r Lrrð1 Lyy Lrr Þ þðLsy2LrsÞR2r 2 Lrrð4 Lyy Lrr Þ þðI1I2ÞðT T0 Þr Lrrð1 Lyy Lrr Þ ðK1K2Þm0r Lrrð1 Lyy Lrr Þ þ K1 sLrr B0 ir2 ð4 Lyy Lrr Þ þ 2A 0 ir3 ð9 Lyy Lrr Þ 2 6 6 6 4 3 7 7 7 5 þðK1K2Þ sLrr C0 ir ð1 Lyy Lrr Þ þ B 0 ir2 ð4 Lyy Lrr Þ þ A 0 ir3 ð9 Lyy Lrr Þ 2 6 6 6 4 3 7 7 7 5 ð10Þ
Finally, the displacement through every ply depends on eight constants to be determined : Rifor 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 each interply;
. continuity of the transverse shear stress at each interply;
. continuity of the normal stress at each interply and its 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 outer
and inner radii are, respectively, a=10 mm and b=30 mm. The hygroscopic properties [3] and the mechanical properties [4] are presented in Tables 1and 2.The pipe is homogeneous from the hygroscopic point of view, every ply has the same hygroscopic properties (Table 1), but it is heterogeneous from the mechanical point of view because of the different orientations of the plies. The pipe is submitted to relative humidity (Fig. 1and 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 shows that the oscillations of the periodic con-centration disappear at a distance e0 from the edge.
Therefore, at a distance e0from the edge the permanent
concentration holds with a constant value because of the symmetrical hygrothermal loading. We observe that fluctuating concentration gradients are important for the points C and E where the relative humidity changes roughly. Figs. 5 and 6 depict the radial stress and the normal ply stress in the transverse direction to the fibres, respectively. We observe that the periodic con-centration gradients have not any influence on the radial stress but induce strong gradients of the normal ply stress. In this periodic concentration regions, the normal
ply stress gradients are so important that theirs values firstly negatives become positives. The radial stress and the normal ply stress within the pipe are dependent on the temperature changing: for (A,C) and (D,E), corre-sponding to identical temperature but to different rela-tive humidity, the stresses are identical. The temperature decreases between A and B induce tensile stresses and the temperature increase between C and D induce com-pressive stresses.
4. Conclusion
We propose an approach which allows to measure the influence of the periodic field, close to the surfaces, of the moisture concentration on the internal stresses for thick laminated pipes. For the internal stresses induced by cyclic hygrothermal conditions, we dissociate the thermal effects and the hygroscopic effects. This solu-tion provides a helpful tool for the design of thick composite pipes under hygrothermal fatigue, since it leads to the knowledge of the stress evolution which can be strong within a narrow region near to the surfaces where 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.
