drying-induced stresses in porous bodies

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 11,1175-1 184 (1977)

DRYING-INDUCED STRESSES IN POROUS BODIES

R. W. LEWIS

University College of Swansea, Wales, U.K.

M. STRADA AND G. COMINI

Istituto di Fisica Tecnica e Laboratorio fer la Tecnica del Freddo del C.N.R. Padova, Italy

INTRODUCTION

The analysis presented in this paper is an application of the finite element method to the determination of stresses induced by drying processes in porous capillary bodies. 1-3,6 The problem involves two distinct stages where first the distribution of the thermal and of moisture potential fields are determined' and then an elastic analysis is carried out to find the stress field set up by these drying p r o c e s s e ~ . ~ * ~ The first stage involves the solution of the non-linear heat and mass transfer problem which has been solved successfully by the authors in situations involving complex geometry and/or boundary conditions. Subsequently, these results are then used to determine the elastic stresses induced during the drying processes for the cooling of bricks, ceramic electric insulators and the drying of timber planks.' The zones of potential cracking are then predicted and hence geometry or surface drying conditions can be modified to alleviate the position.

THEORY OF THE DETERMINATION OF THERMAL STRESSES

The stresses induced by drying processes in building and ceramic materials are determined by first solving the heat and mass transfer problem' viz:

at pCq-= k q V 2 t + d p C m -

a 6

aL4

a 6 pC,-= k,SV2t + k,V2u

with boundary conditions

t = t w on rl

kqVtn+ jq +aq(t - f a ) + (1 - ~ ) A a , ( u - u,) = 0 on r2 U = U , on r3

kmVun+jm + k,SVtn+a,(u -Ma) = 0 on r4

0 1977 by John Wiley & Sons, Ltd

Received 7September 1976

1175

1176 R. W. LEWIS. M. STRADA AND G. COMINI

The problem defined by equations (1)-(5) can be rewritten in a generalized two-dimensional form;

with boundary conditions: T = T , on rl

and

In the above, dimensionless variables: T = t / to , U = u/uo, 6 = 8/90, X = x / l , Y = y / l are utilized, with to(=t , ) , uo(=u,), 1 9 ~ and 1 taken as reference values.

Generalized capacities C’s, generalized transfer coefficients K’s and generalized ‘equivalent’ fluxes.J*’s are also referred to. Boundary conditions (8), (10) are formulated in such a manner as to retain the symmetry of the problem. Also, with suitable definition of the generalized

6) coefficients, K, can always be made equal to K8 thus making the system of equations symmetric. If, in particular, the following condition is imposed

( K, = K8 = d k m 6 / k q

from equations (1)-(5) it follows that:

Kq = (kq +;:krna to6 Ehk,Ug )(&); Krn=-; kqto

and = A, ( T - T,) + A, (U - U, ) +J, ;

= A8 (T - T,) + A, (U - U,) + J, ;

DRYING-INDUCED STRESSES IN POROUS BODIES 1177

The solution of Luikov's system of equations yields the temperature and moisture potential fields for most low intensity drying processes, where total pressure can be taken as constant throughout the body.223

The values of temperature and moisture potential:

can then he used to evaluate the deformations induced by the drying processes.

are given as follows: These stresses and deformations are determined according to Reference 4 where the strains

1 E,, = s y y =a,t+a,u(=E,,)

Exy = O( = E y z = EZ, )

where a,["Kp'] and a, ["M-'1 are linear expansion coefficients for the temperature and moisture variations respectively.

In most practical situations it has been found that

aft + a,u = a,u

since e.g. the coefficient a, for building and ceramic materials is of the order of lops to lop6 OK-' with a, being in the range lop3 to lop4 O M - ' . The variations of t and u in drying processes are of the order of lo2 OK and lo2 O M respectively. This assumption is not necessary however as the program has already calculated both aft and a,u.

If the coefficients a,t and a,u are of the same order of magnitude, equation (16) can still be used by defining an 'equivalent' moisture potential ueq as:

(17)

where aeq = 1 for example. From this point the determination of the stresses in the porous body can be made by assuming

some constitutive relationship for the stress-strain behaviour. It is realized that the stress-strain behaviour is probably complex but as a first approximation to the problem the authors have utilized a linear elastic model as proposed in Reference 5.

(16)

ueq = (art + auu)/aeq

EXAMPLES

In all the examples presented it is assumed that

aft c a,u (18)

Thus, only moisture potential distributions have a significant influence on drying-induced stresses in the processes considered. However temperature distributions in porous bodies influence moisture distributions and, therefore, the problem under investigation is of a coupled nature and a heat and mass transfer program must be used in the determination of moisture potential fields.

Drying of a brick

This problem is almost identical to that solved by Comini and Lewis' and the thermophysical property values are as given in that reference. However, the boundary and initial conditions are

1178 R. W. LEWIS, M. STRADA AND G. COMINI

somewhat different and are given as follows:-

?, = 60°C U, = 10"M

ti = 30°C ui = 180"M

In the stress analysis the following values for the mechanical properties were assumed:

Modulus of elasticity E = 0.25 X lo1' N/m2

Poisson's ratio p = 0.25 a, = 2.4 x lop4 OM-'

and a,? << a,u

The section of the brick under analysis is shown in Figure 1. As the problem is symmetrical a quarter section only was analysed and it was found that the moisture and thermal gradients

' ,-.

- Section under investigation

Figure 1. Cross-section of brick

DRYING-INDUCED STRESSES IN POROUS BODIES 1179

became insignificant away from the corner of the brick.' Hence, the finite element spatial discretization was carried out on the shaded area only and this zone is shown in detail in Figure 2.

The boundaries AF and DC are assumed to be non-conducting with convective boundary conditions acting on faces FED and ABC respectively. The assumption was also made that the coefficient on the outside of the brick was twice that on the inside due to air currents being able to move more freely.

0 5 10 mm

Figure 2. Finite element discretization

The distribution of moisture potential at a time of 600 sec was calculated and the resulting state of stress within the brick at this time is shown in Figure 3. As bricks are composed of a brittle material the contours of principal stress uI are shown as possible damage would depend on the magnitude of these stresses. These results clearly indicate the zones of imminent failure in such a brittle material.

Drying of building timber

The cross-section of the timber investigated is shown in Figure 4(a). It is necessary to investigate only a quarter of the section due to the symmetry of the problem. Non-conducting boundary conditions are assumed along the sides AB and BC with convective boundary


.

Y A

I I I

- + - - 7 I

\

Figure 3 . Distribution of principal stresses cI (MN/m2) for the moisture potential distribution shown in Figure 3

conditions along AD and DC. The cross-hatched section is then subdivided into the finite element mesh shown in Figure 4(b).

The thermophysical properties for this problem are as follows7x8:

p = 600 kg/m3;

6 = 6.7 OM.K-';

C, = 2100 J/kg.K;

C, = 3 x

k, = 0.2 W/m.K;

kgmoisture/kgdry body OM;

A = 2.5 X lo6 J/kg; E = 0.3; k , = 2.5 X kgm0istJm.s. OM.

The convective coefficients at the external surfaces AC and CD are:

a, = 40 W/m2.K; a, = 1 X kgmoisture/m2.s.K.

The equilibrium potentials are:

f, = 60°C; u, = 50"M


- - -_

1181

- - - - - - - -

I I I

- o x

Figure 4(b). Finite element mesh of timber board

Constant initial conditions are assumed as follows:

ti = 25°C; ~i = 1OO"M

The mechanical properties of the timber are obtained by averaging Ex and EY with similar reasoning applied to the values of a,. This results in the following properties.

E = 0.12 x 10'l N/m2; p = 0.4; a, = 3 X OM-'

which represent typical conditions during the primary stages of the timber drying processes. The results for the distribution of the moisture potential at time 1200 sec are shown in Figure

5(a). As would be expected the drying process is still superficial at this time and the bulk of the timber is reasonably unaffected.

The stress field induced by the drying process as this time is shown in Figures 5(b) and 5(c). As timber is not an isotropic material the damage incurred by the drying process will depend on the magnitude of both the tangential and radial distribution of stresses. The results are realistic and indicate that the corner region is the zone most likely to crack under the drying process.

Drying of a ceramic electric insulator

The last example investigated is that of a typical ceramic insulator as used in the electrical industry. This was first solved by Comini and Lewis' for the heat and mass transfer problem and further details may be obtained from that reference regarding the thermophysical properties

1182 R. W. LEWIS. M. STRADA AND G. COMINI

Y I

Ua= 50' M Ui = 100' M rx Figure 5(a). Distribution of moisture potential at 4 = 1200 sec - -

y !

r40 ---c

-3 X

Figure 5(b). Distribution of tangential stresses (a,) at 4 = 1200 sec

Figure 5(c). Distribution of radial stresses (u,,) at 4 = 1200 sec

used. Also the same values of convection coefficients, boundary and initial potentials were used.

The mechanical properties of the insulator were assumed to be

E = 0.69 x 10l1 N/m2; p = 0.25;

The symmetry of the insulator was used to advantage and only one of the sections was analysed. The finite element mesh for this section is indicated in Figure 6.

The distribution of moisture potential is as reported previously by Comini and Lewis and is not reproduced in this paper. The resulting stresses uI are shown in Figure 7 and indicate clearly the zones of potential failure in the ceramic insulator which again is treated as a brittle material.

a, = ( - 1 x lop4 O M - '

CONCLUSIONS

A demonstration has been made that the finite element method can be successfully applied to the problem of stress determination during the drying process in porous capillary bodies. A linear stress-strain relationship has been assumed and this is realized to be a first approximation


- - 0 2 4 6 8 Drnm

I i z

Figure 6. Finite element mesh for insulator

1183

I t z

Figure 7. Distribution of principal stress u, (MN/m2) at 8 = 3600 sec


to the problem. More complex constitutive relationships can be applied if the experimental data is available. The results clearly indicate the potential zones of failure in the three examples considered and give an insight into the modifications necessary if failure is to be avoided. This could be achieved by either changing the geometry of the sections analysed-which would obviously be impractical in most cases-or modifying the rate of the drying processes. The advantages of a numerical model are obvious in this instance as these changes could be made by altering a few computer cards.

REFERENCES

1. G. Comini and R. W. Lewis, ‘A numerical solution of two-dimensional problems involving heat and mass transfer’,

2. A. V. Luikov, Heat and Mass Transfer in Capillary Porous Bodies, Pergamon, Oxford, 1966. 3. A. V. Luikov, ‘Systems of differential equations of heat and mass transfer in capillary-porous bodies’, (Review), Int.

4. N. I. Nikitenko, I. M. Piyevskiy and A. S. Khomenko, ‘Determination of stresses arising in drying of ceramic blocks’,

5. 0. C. Zienkiewicz, The Finite Element Method in Engineering Science, McGraw-Hill, London, 1971. 6. 0. Krischer, Die Wissenschoftlichen Grundlagen der Trocknungstechnik, Springer, Berlin, 1963. 7. F. Kneule, Le Sichoge, Eyrolles, Paris, 1964. 8. A. Dascalescu, Le Skchoge et ses Applications Industrielles, Dunod, Paris, 1969. 9. R. B. Keey, Drying Principles and Practice, Pergamon, Oxford, 1975.

Int. J . HeatMass Transfer, 19, 1387-1392 (1976).

J. Heat Mass Transfer, 18, 1-14 (1975).

Heat Transfer-Soviet Research 5, 176-179 (1973).

drying-induced stresses in porous bodies

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