Composa~ Engineering, Vol. 2, No. I, pp. 21-30, 1992. Printed in Great Britain.

0961-9526/92 $5.00+ .@I 0 1992 Pergamon Press plc

A MIXTURE THEORY APPROACH FOR THE CONSTITUTIVE MODELING OF COUPLED

HYGRO-THERMO-ELASTIC PHENOMENA IN FIBER-REINFORCED POLYMERIC COMPOSITE

MATERIALS

MOHAMMAD USMAN, MUKESH V. GANDHI and S. R. KASIVISWANATHAN Composite Materials and Structures Center, Michigan State University,

East Lansing, MI 48824, U.S.A.

(Received 4 June 1991; accepted in final form 13 September 1991)

Abstract-Several theoretical and experimental investigations have highlighted the vulnerability of polymeric fiber-reinforced composite materials to severe hygro-thermal environments. Tradi- tionally, single constituent theories have been employed to model the constitutive response of idealized polymeric composite materials in hygro-thermal environments. Typically, these theories do not incorporate moisture as a distinct constituent in an explicit sense, and the moisture and temperature effects are treated as being uncoupled from the mechanical response of the composite material.

The deficiencies of traditional techniques in modeling the hygro-thermo-elastic response of polymeric composites are addressed in this paper by treating the moisture-composite aggregate within the context of Mixture Theory. This approach not only permits the explicit incorporation of moisture and its effects, but it also allows the possibility of deriving constitutive equations to model the coupled hygro-thermo-elastic response of idealized fiber-reinforced composite materials undergoing small deformations. An illustrative example is presented in order to highlight the significant role of the fibers in modifying the hygro-thermo-elastic characteristics of the overall composite material. The authors believe that this is the first paper which rigorously treats the hygro-thermo-elastic behavior of polymeric composites in the context of Mixture Theory, and it is anticipated that this approach will have significant relevance in solving practical problems in defense, aerospace and manufacturing environments, where significant variations in moisture and temperature conditions are routinely encountered.

1. INTRODUCTION

The tremendous acceleration in the utilization of polymeric composite materials in engineering practice has focused attention, both theoretical and experimental, on under- standing the response of these materials in order to guarantee safe and efficient designs for various engineering environments. The vulnerability of polymeric composite materials to hygro-thermal environments has been well established by numerous theoretical and experimental studies (Sih et al., 1980; Marom and Broutman, 1981; Sih and Ogawa, 1982; Ene, 1986). The presence of moisture and heat produces undesirable dimensional changes, and changes in the mass, stiffness and damping properties of polymeric fiber-reinforced composite materials. Whenever moisture is present in a composite material in amounts which may not be infinitesimal, it is imperative that the moisture be explicitly treated as another constituent in order to realistically address moisture-induced effects in composite materials.

The presence of moisture as another constituent in polymeric fiber-reinforced composite materials may be treated rigorously within the context of Mixture Theory (Atkin and Craine, 1976; Bowen, 1975). The interaction of fluids and nonlinearly elastic polymers for finite deformations has been extensively studied recently to address several phenomena such as swelling, saturation and diffusion (Gandhi et al., 1985, 1987; Gandhi and Usman, 1987, 1988, 1989a; Rajagopal et al., 1986; Shi et al., 1981). Recently, Gandhi and Usman (1989b) have employed Mixture Theory to study constrained polymers in order to demonstrate that the constraint due to the presence of the fibers restricts the ability of the surrounding polymeric material to swell, thereby inducing nonhomogeneous swelling characteristics. The swelling and saturation of polymer-based composite materials

21

22 M. USMAN et al.

have been studied by Gandhi and Usman (1986), and the nonlinear vibrations of laminated composite plates in a hygro-thermal environment have been studied by Gandhi et al. (1988).

In this work the deficiencies of traditional modeling techniques are addressed by treat- ing the moisture-composite aggregate within the context of Mixture Theory. This approach not only permits the explicit incorporation of moisture and its effects, but also allows the possibility of deriving constitutive equations to model the coupled hygro-thermo-elastic response of idealized fiber-reinforced composite materials undergoing small deformations. An illustrative example is presented in order to highlight the significant role of the fibers in modifying the hygro-thermo-elastic characteristics of the overall composite material.

A brief review of the notations and basic equations relevant to interacting continua is presented in Section 2. The constitutive equations for the mixture of a polymeric solid and an ideal fluid are discussed in Section 3. A model for the infinitesimal hygro-thermo-elastic response of polymeric composite materials is presented in Section 4. In Section 5, an illus- trative example is presented in order to demonstrate the applicability of the proposed model.

2. PRELIMINARIES: NOTATIONS AND BASIC EQUATIONS

The composite-moisture aggregate will be considered as a mixture with S1 repre- senting the composite and S, representing the fluid (moisture). The motion of the composite and the fluid is denoted by

x = x,(X, t) and Y = x,w, 0. (1) These motions are assumed to be one-to-one, continuous and invertible. The various

kinematical quantities associated with the solid S1 and the fluid S, are

Acceleration

Velocity gradient L,au ax ’

M=av ay ’

(3)

(4)

Rate of deformation tensor D = i(L + LT), N = +(M + MT), (5)

where d/dt represents the material time derivative. The deformation gradient associated with the composite is given by F

F=$ (6)

The total density of the mixture and the mean velocity of the mixture are defined, respec- tively, by

P = Pl + Pz, (7)

PW = PlU + P2V, (59

where p1 and p2 are the densities of S1 and S2 at time t, measured per unit volume of the mixture.

The basic equations for Mixture Theory are presented next.

(i) Conservation of mass

Assuming no interconversion of mass between the two interacting continua, the appropriate forms for the conservation of mass for the composite and the moisture are

p,ldetFl = plo, (9) and

where plo is the reference mass density of the composite.

Coupled hygro-thermo-elastic phenomena in composite materials 23

(ii) Conservation of linear momentum

Let ts and R denote the partial stress tensors associated with composite Sr and fluid S,, respectively. Then, assuming that there is no external body force, the balance equations of linear momentum for the composite and moisture are given by

diva - b = prf, (11)

div I[ + b = p2g, (12)

where b denotes the interaction body force.

(iii) Conservation of angular momentum

This condition states that

u + R = UT + nT*

However, the partial stresses o and R need not be symmetric.

(13)

(iv) Thermodynamical considerations

In anticipation of presenting constitutive equations the balance of energy and entropy production inequality will be stated in this subsection. Let U, and U, be the internal energy per unit mass of Sr and S,, respectively. Let rl and r2 denote the heat supply per unit mass of Sr and S, . The heat fluxes from Sr and S, are represented by qi‘) and qk . (‘) The energy balance for the mixture may now be written as

pr - 2 - pg - 4 + bi(Ui - Vi) + UijLji + KijMji = 0. J

where pr = wl + p2r2,

qj = qj”’ + qw J ’

NJ = PI u, + P2&,

u!” = uj - wj, J and UP’ = vj - wj.

J

The entropy production inequality (Green and Naghdi, 1969) will be documented next. Let q, and 1/Z be the entropy per unit mass of Sr and S,, respectively. For a common temperature of St and S, denoted by T, such that T > 0, the entropy production inequality may be written as

pT$!+TW-pr+Tdiv $ 10, 0

(14)

where

P9 = Plrll + P2t721

w = divhtldu - w) + p2t12(v - w)]. (1%

Physically w represents the change in entropy of the mixture due to the interaction between the two constituents S, and S,.

(v) Surface conditions

Let t and p denote the surface traction vectors taken by S, and S, , respectively and let n denote the unit outer normal. Then

t = aTn, p = nTn. (16)

24 M. USMAN et al.

(vi) Volume additivity and incompressibility assumption

Attention is restricted to a mixture of an incompressible solid and an ideal fluid. It is assumed that the volume of the mixture in any deformation state at any given time is the sum of the volumes occupied by the solid and fluid constituents at that time (Mills, 1966). This implies that the motion of the interacting continua is such that it satisfies the following relationship

&+p2=1, PlO PZO

(17)

where pZo is the true mass density of the fluid in the reference state. It may be emphasized that this assumption has a significant bearing on the form of the constitutive equations, and renders the constitutive equations to be more tractable due to the elimination of the density of one of the constituents as an independent variable by virtue of eqn (17).

3. CONSTITUTIVE EQUATION

The polymeric composite is assumed to be an elastic solid, and the moisture is assumed to be an ideal fluid. Thus we require all the constitutive functions to depend on the following variables

F, VF, pZ, gradp,, T, grad T, u, v, L and M.

The constitutive equations are written in terms of the specific Helmholtz free energy function A defined per unit mass of the mixture. The application of the Clausius-Duhem inequality for the mixture yields

A = A@, ~2, T). (18)

Using the principle of frame-indifference and assuming a nonporous and incompressible polymeric composite, the components of the partial stresses for the composite and moisture, and the interaction body force are given by

(Jki = PgFkj - p&aki, IJ PlO

nki = -pp2aA6ki -pE6,i, ap,

and

bk = -& $j 2 + PI g g - $ 2 + a(& - &). k IJ 10

(21)

(22)

Note that Tki = oki + nki, where Tki are the COIIlpOnerltS Of the total Stress tensor. In eqns (19)-(21)~ is a scalar which arises due to the incompressibility constraint. The inter- action between the composite and the moisture is evident in these equations, where the partial stress of each constituent is affected by the deformation of both the constituents.

4. INFINITESIMAL HYGRO-THERMO-ELASTIC RESPONSE

In the reference configuration the solid and fluid are assumed to have densities p, and p2, both defined with respect to the volume of the reference solid, respectively. Both constituents are assumed to have a constant common temperature T. Subsequent to the initial interaction of the solid and fluid, it is assumed that the displacements and velocities of the solid and fluid, and the temperature changes along with their space and time derivatives, are small. Only first-order terms pertaining to these quantities are systemati- cally retained in the constitutive equations for oij, nij and bi, and hence, only linear and

Coupled hygro-thermo-elastic phenomena in composite materials 25

quadratic terms in the Helmholtz free energy function A are retained. These assumptions yield the following equations:

Xi = Xi + ki, Pz = P2 + /A and T=i:+8, (23)

where iiri is the displacement of a solid or fluid particle which was initially at the reference coordinate Xi, j? is a small perturbation in the reference fluid density p2 and 8 is a small perturbation in the common mixture temperature i? Furthermore, the process of diffusion is assumed to be very slow. The strain-displacement relations and the mass balance for the solid and the fluid, respectively, may be written as follows:

(24)

and (25)

(26)

It is further assumed that the perturbation in the density of the fluid p is small relative to the reference density &, and also (T - F)/F and (q - f)/e are small, and of the same order as eij, where F and ?j are the reference absolute temperature and entropy of the system. The partial stresses and other variables in the constitutive relations (expressed in suitable nondimensional form) are also of the same order. Under these assumptions, the terms of order higher than the first in all the fundamental equations are neglected. The Helmholtz free energy function may be written in the infinitesimal context as

pl = A, + alemm + a2P + 0 + ta4emmenn + a5emnemn

+ +a,B2 + +a,e2 + a,e,,P + a,e,,O + a,,@, (27)

where al-al,, are material constants. The components of the partial stress tensors for the solid and the fluid, the inter-

action body force, and the explicit form of the entropy are given by

ati = (-p + sr e,, + .72/l + s,e + S4)6ij + s5eti, cw

nij = -tP + fiem, + f2P + f3 e + f4Vij 9 (29)

bi = Blemm,i - B2P,i9 (30) and

q = -;[a, + a,8 + a,e,, + 4oP1. (31)

In eqns (28)-(30) the material constants are redefined as

Pl 1 s1 = a, - -a,,

P s2 = -al + ag,

P

s3 = a9, s4 = 6, s5 = 2(a, + ad, - -

fi = asp2 - ya2, f2 = ?a2 + a6p2,

f3 = alo, f4 = a2p2, B1 = $al, and B =!?ia 2

P 2*

In the following section, attention is focused on the equilibrium formulation only, hence the part of the interaction body force in eqn (30) which arises due to relative motion of the constituents is dropped. It is also useful to record the representation for the components of the total stress tensor

Tj = Im2P + Csl - fi)emm + 62 - f2lP + 63 - f3v + 64 - f4)lsij + wij. (32)

26 M. USMAN et al.

5. ILLUSTRATIVE EXAMPLE

The problem of swelling of a cylindrical polymeric material with a rigid core is considered within the context of infinitesimal Mixture Theory. In this section, the swelling characteristics of an idealized polymeric composite representative element are investigated by assuming that:

(i) The fiber is located at the cylindrical axis and perfectly bonded to the surrounding polymeric matrix material.

(ii) The polymeric matrix material surrounding the fiber is isotropic. (iii) The representative element is exposed to an ideal fluid bath with isothermal

conditions. (iv) The cylindrical representative element is free from initial stresses.

Consider a hollow cylinder of a nonlinearly elastic material described by internal and external radii Ri and R,, respectively, and a length LO in the reference configuration as shown in Fig. 1. This hollow cylinder is assumed to be perfectly bonded to a rigid cylindrical core of radius Ri . The coordinates of a typical material particle in the reference configuration will be denoted by cylindrical coordinates (R, 0,Z). In the deformed swollen state the coordinates of the same particle are assumed to be described by

r = r(R), e = 0, and 2 = AZ, (33)

where (r, 8, z) denote the coordinates of the particle at (R, 0, Z) in the deformed swollen configuration, I, being a constant axial stretch ratio assumed to be unity.

Reference Configuration Deformed Configuration

Fig. 1. Swelling of elastic cylinder with rigid core.

For the assumed deformation field (33) the infinitesimal strain tensor may be written as 0.R 0 0 e= [ 1 0 o/R 0 (34) 0 0 0

where o is the radial displacement defined as

co = r(R) - R. (35)

It is useful to record that the volume additivity assumption and the incompressibility constraint (17) along with the mass balance of the solid constituent yield

B = P20emm - ~72. (36)

It may be pointed out that due to the volume additivity assumption and incompressibility constraint eqn (36), /I may be eliminated from governing equations in favor of e,,.

Coupled hygro-thermo-elastic phenomena in composite materials 21

The equations of equilibrium which are appropriate for the infinitesimal deformation being considered are documented next. Since the assumed form of the deformation implies that the stresses depend only on the radial coordinate R, the equations of equilibrium for the solid constituent, namely (ll), reduce to

da, + =rr - gee dR R

- b, = 0, (37)

where crrr and ~~~ denote the appropriate components of o, and b, denotes the component of the interaction body force b in the radial direction. The equilibrium equations for the fluid constituent, namely (12), reduce to

dn, I 71, - nee dR R

+ b, = 0,

where 7c, and ?rgO denote the components of n. The definition of the total stress along with eqns (28) and (29) yield

dTr dR+

Trr - Tee R =

0, (3%

which is the equation of equilibrium for the mixture, and T, and Too denote the total radial and total tangential stresses, respectively.

It is sufficient to satisfy any two of the three equilibrium equations (37)-(39). By employing eqn (34), eqns (28)-(30) are substituted in the equilibrium eqns (37) and (39), to yield

-2P.R + 61 - f, + S&,RR + 61 - f, + $1 y - 61 - fi + s5) 5 + @2 - fi)fi,R = OS

(40) and

-P,R + 61 + %b,RR + 61 + %) y - 61 + %,$ + S2b.R *

The indeterminate scalar p may be eliminated to yield the governing equation

(41)

O,RR + y - w = 0. R2 (42)

Equation (42) has a solution given by

o(R) = c,R + c,/R (43)

where ci and c, are integration constants and are to be evaluated from the boundary conditions.

Two of the appropriate boundary conditions for evaluating the integration constants are

o(R,) = 0, (44) and

TAR,) = 0. (45)

For the equilibrium problem considered herein, the indeterminate scalarp (Gandhi, 1984) is given by

P = -~2~ f2 + 2

( >

em, + P2f2* (46)

The total radial stress may be written by using eqn (46) in (32) as

T, = -al e,,4j + %eij - P&2 + f2Vij9 (47) where

a1 = -KS1 +fA + P2oCf2 + s2)ls and cY2 = s5.

28 M. USMAN et al.

The integration constants c1 and c1 may be determined using eqns (43) and (47) and the boundary conditions (44)-(45), and these constants are given by

(48)

c, = -c,R;. (49)

For the computational work the following material properties were employed (Sih et al., 1986):

Density of the polymeric matrix in the reference state p1 = 1590 kg mm3

Density of the fluid in the reference state p2 = 27.49 kg me3

True density of the fluid in the reference state p20 = 997 kg mm3

Poisson’s ratio of the matrix v = 0.493 Shear modulus of the matrix G = 24.166 GPa Swelling coefficient of the matrix y = 3.56 x 10m3 cm cm-’ (wt%H,O)-‘.

The following relationships are employed for the calculation of the material constants:

a _ 3GU + v) 2 6- I-2v yy

a, = -2s~.

The computational results are presented in Figs 2 and 3. Figure 2 shows the variation of the nondimensional radial displacement ci, = o/R, with respect to the nondimensional reference radial coordinate R/R, for various values of Ri/R, . It may be pointed out that Ri/R, represents the ratio of the radus of the rigid core to the outer radius of the polymeric composite cylinder, and is numerically bounded by 0 and 1. When the ratio Ri/R, is in the neighborhood of 0.0, then it represents a “thick” polymeric cylinder bonded to a relatively thin fiber-like rigid core. Similarly, when the ratio RJR, is in the neighborhood of 1.0, then it represents a polymeric “membrane” cylinder bonded to a relatively thick rigid core. In the absence of the rigid core the swelling of the unconstrained polymeric cylinder would exhibit a linear relationship between the radial displacement w and the radial coordinate R. It is clearly evident from Fig. 2 that the presence of the rigid core induces nonhomogeneous swelling characteristics in the immediate neighborhood of the core.

Non-dimensional Radial Coordinate R/R,

Fig. 2. Variation of the radial displacement with the radial coordinate.

Coupled hygro-thermo-elastic phenomena in composite materials

k 6.0OE-02

(t-

I S.OOE-02-

5 cn 4.00E-02 0 .- -0 I?

J.OOE-02

i i -I.clOE-020 Z

0.0 0.2 0.4 0.6 0.0

Non-dimensional Radial Coordinate R/R,

Fig. 3. Variation of the radial stress with the radial coordinate.

Figure 3 presents the variation of the nondimensional radial stress er = T,/G with the nondimensional radial coordinate R/R, for various values of Ri/R,. The radial stresses are tensile and considerably higher in the immediate neighborhood of the rigid core where the polymeric cylinder is bonded, and zero at the outer radial surface which is assumed to be traction-free.

6. CONCLUDING REMARKS

A methodology has been presented for the constitutive modeling of coupled hygro- thermo-elastic phenomenon in fiber-reinforced polymeric composite material in the context of Mixture Theory. The methodology is based on a linearization assumption which is imposed subsequent to the initial interaction of the composite material and moisture. Furthermore, phenomena involving stress-dependent diffusivity and non- homogeneous swelling characteristics can be adequately modeled in the context of this methodology. It is anticipated that this approach will have significant relevance to solving practical problems where significant variations in moisture and temperature conditions are routinely encountered.

Acknowledgements-Mukesh V. Gandhi gratefully acknowledges the support of the National Science Foundation through Grant No. MSM8514087, the Defense Advanced Research Projects Agency through Contract No. DAAL03-87-K-0018, the U.S. Army Research Office through Contract Nos. DAAL03-88-K-0022, DAAL03-89-G-0091, DAAL02-90-G-0052 and the Michigan Research Excellence and Economic Development Fund.

REFERENCES

Atkin, R. J. and Craine, R. E. (1976). Continuum theories of mixtures: basic theory and historical development. Q.J. Mech. Appl. Math. 29(2), 209-244.

Bowen, R. M. (1975). Continuum Physics (Edited by A. C. Eringen), Vol. 3. Academic Press, New York. Ene, H. T. (1986). On the hygrothermomechanical behavior of a composite material. Int. J. Engng Sci. 24(5),

841-847. Gandhi, M. V. (1984). Large deformations of a non-linear elastic solid and the diffusion of an ideal fluid treated

as interacting continua. Ph.D. Dissertation, University of Michigan, Ann Arbor, MI. Gandhi, M. V., Rajagopal, K. R. and Wineman, A. S. (1985). A universal relation in torsion for a mixture of

solid and fluid. J. Elasticity 15(2), 155-165. Gandhi, M. V., Rajagopal, K. R. and Wineman, A. S. (1987). Some non-linear diffusion problems within the

context of the theory of interacting continua. Int. J. Engng Sci. 25(11/12), 1441-1457. Gandhi, M. V. and Usman, M. (1986). Swelling and saturation of polymer based composite materials. Presented

at the Third Japan-U.S. Conf. on Composite Materials, Tokyo, Japan, June 1986. Gandhi, M. V. and Usman, M. (1987). The flexure problem in the context of the theory of interacting continua.

SES Paper No. ESP 24.87016, presented at the 24th Annual Technical Meeting of the Society of Engng Sci., Salt Lake City, Utah, September 1987. SES Inc.

Gandhi, M. V. and Usman, M. (1988). Exact solutions for the uniaxial extension of a mixture slab. Arch. Mech. 40(2-3), 275-286.

29

0

30 M. USMAN et al.

Gandhi, M. V. and Usman, M. (1989a). Combined extension and torsion of a swollen cylinder within the context of mixture theory. Acta Mechanica 79, 81-95.

Gandhi, M. V. and Usman, M. (1989b). On the non-homogeneous finite swelling of a non-linearly elastic cylinder with a rigid core. Znt. J. No&n. Mech. 24(4), 251-261.

Gandhi, M. V., Usman, M. and Chao, L. (1988). Nonlinear vibration of laminated composite plates in hygrothermal environments. ASME J. Engng Mater. Technol. 110(2), 140-145.

Green, A. E. and Naghdi, P. M. (1969). On basic equations for mixtures. Q.J. Mech. Appl. Math. 22(4), 427-438.

Marom, G. and Broutman, L. J. (1981). Moisture penetration into composites under external stress. Polymer Composites 3(3), 132-136.

Mills, N. (1966). Incompressible mixtures of Newtonian fluids. Inf. J. Engng Sci. 4, 97-l 12. Rajagopal, K. R., Wineman, A. S. and Gandhi, M. V. (1986). On boundary conditions for a certain class of

problems in mixture theory. Znt. J. Engng Sci. 24(8), 1453-1463. Shi, J. J., Rajagopal, K. R. and Wineman, A. S. (1981). Applications of the theory of interacting continua to

the diffusion of a fluid through a non-linear elastic medium. ht. J. Engng Sci. 9, 871-889. Sih, G. C., Michopoulos, J. G. and Chou, S. C. (1986). Hygrothermoelasticity. Martinus Nijhoff, Dordrecht. Sih, G. C. and Ogawa, A. (1982). Transient thermal change on a solid surface: coupled diffusion of heat and

moisture. J. Thermal Stresses 5, 265-282. Sih, G. C., Shih, M. T. and Chou, S. C. (1980). Transient hygrothermal stresses in composites: coupling of

moisture and heat with temperature varying diffusivity. Znt. J. Engng Sci. 18, 19-42.