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Coupling term makes temperature fields unsymmetric in laminated composite plates
PII: S0045-7949(97)00125-9
Computers & Structures Vol. 66, No. 5, pp. 509±512, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0045-7949/98 $19.00 + 0.00
COUPLING TERM MAKES TEMPERATURE FIELDS UNSYMMETRIC IN LAMINATED COMPOSITE PLATES Antti Pramila{ Department of Mechanical Engineering, University of Oulu, P.O. Box 444, 90571 Oulu, Finland (accepted 1 November 1997) AbstractÐThe importance of the coupling term kxy in heat conduction analysis of laminated, ®bre reinforced plates is considered. A simple proof is given which shows that even with symmetric boundary conditions the temperature ®eld in an o axis ply or in a general laminate cannot be symmetric. Moreover an example problem is presented which indicates that very large errors can occur when the coupling term is neglected. # 1998 Elsevier Science Ltd. All rights reserved Key wordsÐlaminates, heat conduction, ®nite element method, coupling, temperature, composites, unsymmetry
INTRODUCTION
THEORY
In a recent paper Mukherjee and Sinha [1], considered heat conduction analysis of laminated composite plates by using the ®nite element method. The discretization was based on the Galerkin method. They compared the performance of four dierent element types, namely linear triangle, bilinear rectangle, eight and nine noded biquadratic elements. The analytical results for an orthotropic case given in Refs. [2, 3] were used as reference solutions in the steady and transient states, respectively. In their studies on the eect of the stacking sequence they, however, obtain erroneous results. For example, they obtain symmetric temperature ®elds for a laminated plate in cases where it cannot be symmetric. The reason is that they use equations of o axis conductivity coecients which are valid only in cases where the temperature is varying in one direction only. This can be clearly seen from their cited reference [4]. They neglect the coupling term, kxy, which plays an important role when the conductivities in the ®ber direction, k11, and in the transverse direction, k22, dier from each other. The purpose of the present note is to give a simple proof which shows that the temperature ®eld can not be symmetric with symmetric boundary conditions when the coupling term is present and to give an indication of the magnitude of the error made when this coupling term is neglected.
In what follows only the steady state is considered, because it reveals the essential point in question. The standard co-ordinate system of laminate analysis is shown in Fig. 1. The axes of the orthotropic material co-ordinate system are denoted by 1 (®bre direction) and 2 (transverse direction). According to Fourier's law of heat conduction the heat ¯uxes in principal directions of a ply are
{Author to whom all correspondence should be addressed.
Fig. 1. Co-ordinate system used.
q1 ÿk11 T,1 ,
q2 ÿk22 T,2
1
where a comma with subsequent subscript indicates a partial derivative. By using the well known rule for the transformation of vector components one immediately obtains qx ÿkx T,x ÿ kxy T,y ,
509
qy ÿkxy T,x ÿ ky T,y 2
510
A. Pramila
(b) A balanced laminate where the sum of the thicknesses of the yi layers is equal to the sum of the thicknesses of ÿyi layers. By considering the energy balance of an in®nitesimal element of the laminate one obtains the governing equation kx T,xx 2kxy T,xy ky T,yy 0
Fig. 2. The domain and boundary conditions.
where kx k11 cos2 y k22 sin2 y, 2 2 ky k11 sin y k22 cos y,
kxy k11 ÿ k22 sin y cos y,
3
Thus, the correct thermal conductivity matrix of an o axis ply is kx kxy 4 k kxy ky For a laminate having plies in dierent directions the heat ¯uxes in x and y-directions can be summed. Thus, for a laminate consisting of plies with directions yi and thicknesses hi, i = 1, . . ., N and having constant temperature throughout the thickness, the average coecients are N X
kx k11
t1
N X
ky k11
N X
hi cos2 yi k22
h
N X
hi sin2 yi
t1
h
t1
k22
t1
EXAMPLE
5
hi cos2 yi h
6
and N X
kxy k11 ÿ k22
t1
hi sin yi cos yi h
which diers from equation (1) of Ref. [1] by the additional second term. The additional term has clear in¯uence on the nature of the solution. For example, with symmetric boundary conditions with respect to the y-axis (see Fig. 2), a symmetric solution is impossible if kxy is nonzero. The partial derivatives T,xx and T,yy of a symmetric ®eld T are symmetric with respect to the yaxis but the cross derivative T,xy is antimetric with respect to the y-axis. Therefore, if the symmetric ®eld T satis®es the governing dierential equation, Equation (8), at a point indicated by an open circle on the left side, it cannot satisfy it at the point on the right side indicated by a ®lled circle (vide the second term of Equation (8)). The laminates of Ref. [1] in stacking sequence studies were of the type [0/y/0/y] for which kxy is nonzero. Thus, already the symmetry of the solutions obtained reveals that they must be erroneous.
hi sin2 yi h
7
This last coecient has been neglected in the analysis in Ref. [1]. Of course, it is equal to zero for some special laminates, e.g. (a) A cross ply laminate consisting of 08 and 908 plies.
To indicate the magnitude of the error made by neglecting the coupling term, kxy, a square anisotropic plate made of one carbon/epoxy ply (k11=18 W/mK and k22=1 W/mK) with dierent ply angles is considered. The boundary conditions are as follows: the temperature of the upper edge is 5008C whereas the temperature of the three other edges is 08C, see Fig. 2. The calculations have been done by using 10 10 mesh of bilinear elements. A summary of the results obtained for the temperature of the center of the plate is given in Table 1. This shows clearly how important it is to take into account the eect of the coupling term. Indeed, the error is not at its maximum at the center of the plate as can be seen from Fig. 3, which has been computed for the convenience of plotting by using over 1000 triangular elements.
Table 1. Angle (8) 10 30 45 50 70 90
kx 17.487 13.75 9.5 8.024 2.989 1
ky 1.513 5.25 9.5 10.976 16.011 18.0
8
kxy
T(a/2, a/2) correct
T(a/2, a/2) simpli®ed kxy=0
2.907 7.361 8.500 8.371 5.464 0
1.146 30.481 139.09 179.36 247.59 249.59
2.774 50.452 126.87 154.06 234.96 249.59
Error (%) 142 66 ÿ9 ÿ14 ÿ5 0
Coupling term makes temperature ®elds unsymmetric
511
Fig. 3. Solution (a) without and (b) with kxy. CONCLUSIONS
In heat conduction analysis of composite plates it is important to take into account properly the coupling between the heat ¯uxes an temperature gradients in the x- and y-directions, i.e. to use also
kxy. By neglecting it one obtains results which are already incorrect qualitatively, e.g. one can obtain a symmetric solution in a case where it is impossible. Moreover, the error in temperature can be over 100%.
512
A. Pramila REFERENCES
1. Mukherjee, N. and Sinha, P. K. A comparative ®nite element heat conduction analysis of laminated composite plates. Comput. Struct., 1994, 52, 505±510. 2. Carslaw, J. S. and Jaeger, J. C., Conduction of Heat in Solids. Clarendon Press, Oxford, 1959.
3. Bruch, J. C. and Zyvoloski, G. Transient two dimensional heat conduction problems solved by the ®nite element method. Int. J. Numer. Methods Eng., 1974, 8, 481±494. 4. Chen, C.-H. and Springer, G. S. Moisture absorption and desorption of composite materials. J. Comp. Mater., 1976, 10, 2±20.
Computers & Structures Vol. 66, No. 5, pp. 509±512, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0045-7949/98 $19.00 + 0.00
COUPLING TERM MAKES TEMPERATURE FIELDS UNSYMMETRIC IN LAMINATED COMPOSITE PLATES Antti Pramila{ Department of Mechanical Engineering, University of Oulu, P.O. Box 444, 90571 Oulu, Finland (accepted 1 November 1997) AbstractÐThe importance of the coupling term kxy in heat conduction analysis of laminated, ®bre reinforced plates is considered. A simple proof is given which shows that even with symmetric boundary conditions the temperature ®eld in an o axis ply or in a general laminate cannot be symmetric. Moreover an example problem is presented which indicates that very large errors can occur when the coupling term is neglected. # 1998 Elsevier Science Ltd. All rights reserved Key wordsÐlaminates, heat conduction, ®nite element method, coupling, temperature, composites, unsymmetry
INTRODUCTION
THEORY
In a recent paper Mukherjee and Sinha [1], considered heat conduction analysis of laminated composite plates by using the ®nite element method. The discretization was based on the Galerkin method. They compared the performance of four dierent element types, namely linear triangle, bilinear rectangle, eight and nine noded biquadratic elements. The analytical results for an orthotropic case given in Refs. [2, 3] were used as reference solutions in the steady and transient states, respectively. In their studies on the eect of the stacking sequence they, however, obtain erroneous results. For example, they obtain symmetric temperature ®elds for a laminated plate in cases where it cannot be symmetric. The reason is that they use equations of o axis conductivity coecients which are valid only in cases where the temperature is varying in one direction only. This can be clearly seen from their cited reference [4]. They neglect the coupling term, kxy, which plays an important role when the conductivities in the ®ber direction, k11, and in the transverse direction, k22, dier from each other. The purpose of the present note is to give a simple proof which shows that the temperature ®eld can not be symmetric with symmetric boundary conditions when the coupling term is present and to give an indication of the magnitude of the error made when this coupling term is neglected.
In what follows only the steady state is considered, because it reveals the essential point in question. The standard co-ordinate system of laminate analysis is shown in Fig. 1. The axes of the orthotropic material co-ordinate system are denoted by 1 (®bre direction) and 2 (transverse direction). According to Fourier's law of heat conduction the heat ¯uxes in principal directions of a ply are
{Author to whom all correspondence should be addressed.
Fig. 1. Co-ordinate system used.
q1 ÿk11 T,1 ,
q2 ÿk22 T,2
1
where a comma with subsequent subscript indicates a partial derivative. By using the well known rule for the transformation of vector components one immediately obtains qx ÿkx T,x ÿ kxy T,y ,
509
qy ÿkxy T,x ÿ ky T,y 2
510
A. Pramila
(b) A balanced laminate where the sum of the thicknesses of the yi layers is equal to the sum of the thicknesses of ÿyi layers. By considering the energy balance of an in®nitesimal element of the laminate one obtains the governing equation kx T,xx 2kxy T,xy ky T,yy 0
Fig. 2. The domain and boundary conditions.
where kx k11 cos2 y k22 sin2 y, 2 2 ky k11 sin y k22 cos y,
kxy k11 ÿ k22 sin y cos y,
3
Thus, the correct thermal conductivity matrix of an o axis ply is kx kxy 4 k kxy ky For a laminate having plies in dierent directions the heat ¯uxes in x and y-directions can be summed. Thus, for a laminate consisting of plies with directions yi and thicknesses hi, i = 1, . . ., N and having constant temperature throughout the thickness, the average coecients are N X
kx k11
t1
N X
ky k11
N X
hi cos2 yi k22
h
N X
hi sin2 yi
t1
h
t1
k22
t1
EXAMPLE
5
hi cos2 yi h
6
and N X
kxy k11 ÿ k22
t1
hi sin yi cos yi h
which diers from equation (1) of Ref. [1] by the additional second term. The additional term has clear in¯uence on the nature of the solution. For example, with symmetric boundary conditions with respect to the y-axis (see Fig. 2), a symmetric solution is impossible if kxy is nonzero. The partial derivatives T,xx and T,yy of a symmetric ®eld T are symmetric with respect to the yaxis but the cross derivative T,xy is antimetric with respect to the y-axis. Therefore, if the symmetric ®eld T satis®es the governing dierential equation, Equation (8), at a point indicated by an open circle on the left side, it cannot satisfy it at the point on the right side indicated by a ®lled circle (vide the second term of Equation (8)). The laminates of Ref. [1] in stacking sequence studies were of the type [0/y/0/y] for which kxy is nonzero. Thus, already the symmetry of the solutions obtained reveals that they must be erroneous.
hi sin2 yi h
7
This last coecient has been neglected in the analysis in Ref. [1]. Of course, it is equal to zero for some special laminates, e.g. (a) A cross ply laminate consisting of 08 and 908 plies.
To indicate the magnitude of the error made by neglecting the coupling term, kxy, a square anisotropic plate made of one carbon/epoxy ply (k11=18 W/mK and k22=1 W/mK) with dierent ply angles is considered. The boundary conditions are as follows: the temperature of the upper edge is 5008C whereas the temperature of the three other edges is 08C, see Fig. 2. The calculations have been done by using 10 10 mesh of bilinear elements. A summary of the results obtained for the temperature of the center of the plate is given in Table 1. This shows clearly how important it is to take into account the eect of the coupling term. Indeed, the error is not at its maximum at the center of the plate as can be seen from Fig. 3, which has been computed for the convenience of plotting by using over 1000 triangular elements.
Table 1. Angle (8) 10 30 45 50 70 90
kx 17.487 13.75 9.5 8.024 2.989 1
ky 1.513 5.25 9.5 10.976 16.011 18.0
8
kxy
T(a/2, a/2) correct
T(a/2, a/2) simpli®ed kxy=0
2.907 7.361 8.500 8.371 5.464 0
1.146 30.481 139.09 179.36 247.59 249.59
2.774 50.452 126.87 154.06 234.96 249.59
Error (%) 142 66 ÿ9 ÿ14 ÿ5 0
Coupling term makes temperature ®elds unsymmetric
511
Fig. 3. Solution (a) without and (b) with kxy. CONCLUSIONS
In heat conduction analysis of composite plates it is important to take into account properly the coupling between the heat ¯uxes an temperature gradients in the x- and y-directions, i.e. to use also
kxy. By neglecting it one obtains results which are already incorrect qualitatively, e.g. one can obtain a symmetric solution in a case where it is impossible. Moreover, the error in temperature can be over 100%.
512
A. Pramila REFERENCES
1. Mukherjee, N. and Sinha, P. K. A comparative ®nite element heat conduction analysis of laminated composite plates. Comput. Struct., 1994, 52, 505±510. 2. Carslaw, J. S. and Jaeger, J. C., Conduction of Heat in Solids. Clarendon Press, Oxford, 1959.
3. Bruch, J. C. and Zyvoloski, G. Transient two dimensional heat conduction problems solved by the ®nite element method. Int. J. Numer. Methods Eng., 1974, 8, 481±494. 4. Chen, C.-H. and Springer, G. S. Moisture absorption and desorption of composite materials. J. Comp. Mater., 1976, 10, 2±20.