- Email: [email protected]
Effective material parameters for composites with randomly oriented short fibres
0045-7949192 $5.00 f 0.00
Compurers & Structures Vol. 44, No. 6, pp. 1179-I 185, 1992 Printed in Great Britain.
I; 1992 Per@mon Press Ltd
EFFECTIVE MATERIAL PARAMETERS FOR COMPOSITES WITH RANDOMLY ORIENTED SHORT FIBRES W. M. G. COURAGE? +TNO,
Building
SDepartment
and Construction
of Mechanical
and P. J. G. SCHREURS~
Research, Department The Netherlands
of Structural
Engineering,
Engineering, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands
Rijswijk,
P.O. Box 513,
(Received 9 July 1991) Abstract-A numerical strategy for the determination of mechanical material parameters for composites is described. Attention is focused on composite materials with randomly oriented short fibres. To determine the mechanical behaviour of the composite material, a representative voiume is modelled in detail and analysed employing the finite element method (FEM). The composite parameters are determined and expressed as a function of microstructural parameters.
INTRODUCTION
A composite is a combination of two or more components. These components may have very different properties. A proper combination results in a composite material with special and very sophisticated characteristics. The composite properties can be tuned to be optimal for a specific application. Composites are used more and more in structural construction parts. In these applications they have to carry loads and thus their stiffness and strength properties are important. Computer simulation of the mechanical behaviour of a structure is essential in the designing process. In the program packages which are being used the material behaviour is described by so-called constitutive equations. The form of these equations is characteristic for a class of materials under certain circumstances. The value of the material parameters in the equations are characteristic for a material in such a class. Determination of the form of the constitutive equation is an experimentai problem. This also holds for the determination of the value of the material parameters. These experiments may be difficult to perform, due to the nonhomogeneity and/or anisotropy of composite materials. Besides that, the experiments have to be repeated when the material is modified. It proves to be advantageous to follow a so-called structural approach. In this case the overall material parameters are determined using info~ation concerning the microstructure of the material. This information is characterized by structural parameters. The overalt parameters which describe the material behaviour in an averaged sense, are called effective parameters. In this paper a numerical technique is described to determine the effective parameters. Courage et al. [l] have described this procedure for particle composites.
Here, attention composites.
will be focused on fibre reinforced
CONFINEMENTS
AND ASSUMPTIONS
The composite material has a two-dimensional morphology. It is a fibre reinforced sheet. Fibres are short and oriented randomly, it is assumed that fibres do not overlap. The fibres are oriented in a square array as is shown in Fig. 1. Spacings between the central points of the fibres in the x- and y-directions are assumed to be equal and the diameter of the fibres is always the same. With these assumptions only two morphological structural parameters remain: the fibre aspect ratio, i.e. the ratio of fibre length to diameter I/d and the volume fraction of the fibres y,. The material ~haviour of fibres and matrix is linear elastic, both components are taken to be isotropic. Perfect bonding is assumed between the components. Under these conditions the composite is expected to show linear elastic material behaviour. Because the fibres are randomly oriented the composite will be statistical isotropic. Thus, the effective Young’s modulus and effective Poisson’s ratio are considered to be the only effective parameters. Because of uncertainties concerning the value of the Poisson’s ratio of most components of a composite material, only the effective Young’s modulus is calculated. It will be determined as a function of the strnctural parameters: the elasticity parameters of fibre and matrix and the morphological structural parameters. PROCEDURE
The procedure consists of four sequential described in Steps l-4 below.
1179
steps,
W. M, G. COURAGEand P. J. G. SCHREURS
1180
Fig. 1. Arrangement of the fibres in the matrix Fig. 3. RVE with arbitrary boundary load. Step 1. ~jcromechanical volume element
analysis of a representative
A representative volume element (RVE) of the composite is chosen and isolated for detailed observation. For a fully homogeneous material, a continuum, this RVE would be infinitesimally small. Mechanical state variables like strains and stresses and material parameters are defined in every material point. Moreover these quantities meet certain continuity requirements. A composite material is inhomogeneous and thus state and material parameters must be defined in an averaged sense. The averaging is carried out over an RVE. The dimensions of the RVE must be such that the fluctuations of the variables around their average value are sufficiently small. The RVE is modelled in detail employing FEM [2,3]. Because of the two-dimensional character of the RVE, plane stress elements are used. The elements are triangular and the element displacement field is linear. A typical finite element mesh of the RVE is shown in Fig. 2. The RVE is modelled as a
5 x 5 square array of fibres in the matrix. With the assumption concerning non-overlapping fibres, this leads to some restrictions on admissible values for the volume fraction of the fibres. The orientations of the fibres are determined using a random generator, such that the effective mechanical behaviour of the RVE is statistically isotropic. The number of elements needed varied from 2600 to 3200. This corresponds to approximately 1300 and 1600 nodal points. The actual number depends on the actual values for V, and l/d. A boundary load is applied to the RVE as shown in Fig. 3. This load can be arbitrary. The resulting deformation of the RVE is calculated using FEM in a so-called micromechanical analysis. Step 2. Calculation of eflective material parameters In the second stage the composite is regarded to be an homogeneous material. The material behaviour is described by means of a macroscopic model and this behaviour may be non-linear. The above-mentioned RVE is now called HRVE (homogenized RVE). After modelling it using FEM, a boundary load is applied, which is identical to the load applied to the RVE during the micromechanical analysis in step one. The deformation of the HRVE is then calculated. The effective parameters are determined such that the deformation of HRVE and RVE are identical. A
t t t t t t t t t t t”’
Fig. 2. Finite element mesh of the composite with randomly oriented fibres. V,= 0.15 and l/d = 2.
Fig. 4. RVE with specific boundary load.
Composites with randomly oriented short fibres Table 1. Survey of the values for the structural parameters used in calculations. The values for f.?,and E,,, listed in (a) are combined with the 21 combinations of V, and f/d marked in (b) Ef @Pa)
-% @Pa) 10 5
70 70 20 70 120
1 1 I (b) V, = 0.075
0.15
0.225
0.3
0.375
I
l
.
l
l
.
2 3 4 5 6 7 8 9 10
.
l
l
l
. .
l
l
l
l
I/d
*
l l l
. .
Table 2. Additional values for the structural parameters used in calculations E_
(GPaf
EAGPa)
10 5 10 5
(a)
1181
20 20 120 120
lid
0.225 0.225 0.225 0.22s
parameter estimation technique can be used to determine these parameters. This technique originates from system identification theory and has an iterative character. Following the assumptions presented earlier we confine ourselves to linear elastic behaviour. In this case system identification techniques are obsolete if the RVE is loaded in a specific way. A typical boundary load is shown in Fig. 4. The RVE is stretched in one direction by prescribing a uniformly distributed displacement field. The effective Young’s modulus is easily calculated by relating the prescribed displacement to the total force.
(b)
(of E* [GPa] 2.2 v, = 0.375
Et [GPal 20. 1s 16 14. 12
2 I
10
Vf= 0.075
8. 6. 4. 2.
0.8
I 0.
0. 5. 2
3.
4.
J.
6
7. 8. 9. 10.
0. 0.05 0.1 0.u
Vd l-1
0.2 0.X
0.3 0.35 0.4
Vf
i-1
(d)
0. 1. 2. 3. 4 5. 6. 7. 8. 9. 10. Em [GPa]
3 3 3 3
0. 20. 40. 60. Kl. Ku. 17.0. Et IGPal
Fig. 5. Young’s modulus E* versus: (a) aspect ratio I/d (E, = I, E,= 20). (b) volume fraction V, (I/d = 1, E, = i’o), (c) matrix modulus E,,, (I/d = 3, V, = 0.225), (d) fibre modulus E, (I/d = 1, E,, = 1).
W. M. G. COURAGEand P. J. G.
1182
Step 3. Variation of the structuraf parameters
The effective Young’s modulus must be determined as a function of the structural parameters. Steps 1 and 2 are repeated for various values of the structural parameters. The structural parameters to be varied are the Young’s modulus of fibres and matrix and the volume fraction and aspect ratio of the fibres. As described before the Poisson coefficients of fibres and matrix are kept at a constant value vI= 0.23 and v, = 0.35, respectively. With regard to the feasibility of the procedure an automatic mesh generator is used. For various V, and l/d the RVE is meshed automatically, guaranteeing statistical isotropy. Procedures from the TRIQUA~ESH meshgenerator [4] are used for the automatic mesh generation. Tables 1 and 2 give the used combinations for Vf and E/d as well as the Young’s modulus of matrix and fibres Ef and E,,, , respectively. Results of the micromechanical analyses are represented in Fig. 5. Step 4. Regression models
SCHREURS
accessible database. Sometimes it is possible to derive closed-form relations by using regression techniques. For linear elastic composites some preliminary requirements can be posed on the format of the relations. We consider the effective Young’s modulus E*. Its value has to be derived as a function of the structural parameters E* = E*(E,,,, E,, V,, l/d).
For linear elastic materials extremum principles lead to bounds for the values of the effective parameters[S]. For the effective Young’s modulus the following bounds can be derived 6%” = ErK/C J’, E, + V&, )
(2)
E&z, = V,,,E, + I/,-$.
(3)
Hence the following format for E* could be assumed E* = E& -t.f(~%, Et, Vr*lldl]E&,
The relation between effective and structural parameters can be represented graphically or stored in an (al
(1)
- E4,], o
(b) 0.225 I/d = 10
0.15 0.075
EffEaf-l
(dl
Cc)
c l-1
0.
1
2
c I-l
3.
a.
5. 6.
7.
8.
9. IO. l/d
1-l
Fig. 6. Function value &’versus: (a) ratio E,/E, (I/d = 3), (b) ratio EJE,, (V, = 0.079, (c) aspect ratio I/d (E,/E, = 20). (d) volume fraction V, (E,/E,,, = 20).
1. (41
Composites with randomly oriented short fihres The requirement concerning the function f leads to rather untransparent restrictions. Instead of the former relation for E* an alternative format for the model is considered. It is based on the generalized Halpin-Tsai fo~ulation [6,7]
where
In this relation parameters
[ is a function
of the structural
i: = CN,r, E,, V/, lid).
(6)
Using eqns (2) and (3), relation (5) can be rearranged into
(7)
1183
yielding
(8) The function 4 has the limiting values 4 = 0 (E* = E&J and i = co (.E* = E&,). The value of E* equals E, and E, for Vf = 0, resp. V, = I, irrespective of 1. The function c is to be determined as a function of the structural parameters, using the calculated values for E*. The function values for [ are obtained from (8). As could be expected from dimensional analysis it was found that the structural parameters Er and E,,, can be combined into one: Ef/Em. Figure 6 shows vatues of 4 as a function of E,jE, and IJd. The function 5 is supposed to be of the form %=a,+a,C,+a*i,f..~+u,l,.
(9)
From the calculated values of [ the regressor variables&(i=l,..., n) must be determined as explicit functions of the structural parameters E,, Em, Vf and I]d.
(b)
2, __
>
I/d = 5
0.
1. 2
3.
1.
I.
d
7.
s
9.
IO.
l/d I-I
Fig. 7. Coefficients b, and b, versus volume fraction Vl and versus aspect ratio l/d. CAS 44/b-B
W. M. G. COURAGE and P. J. G. SCHREURS
II84
88_ 75_ 63_ 50_ 38_ 25 13_ O0 2. d. 6 1. ,B 12.I, 16 111. 20 prediction Fig. 8. Relative residuals
From
E//E,
versus the values predicted model.
by the
Fig. 6 the following relation between J and is deduced
i = b0 + b, exp[-b~(~~~E,~)~.‘l.
(10)
The values of bO,6, and b2 are now fitted on the values of < using a non-linear regression procedure, based on the Marquard algorithm. The coefficient b2 was found to scatter around the value 0.5. The exponential term in (IO) was therefore simplified by assuming that the coefficient b, has always the value 0.5. The coefficients b, and h, are now once more fitted on the data. A linear least-squares fit could now be performed. The resulting values are plotted in Fig. 7. From these results it is concluded that 6, is linear in both V, and l/d, b, is linear in Viand quadratic in I/d. This results in the following regression model for 5 i =bo+b,ex~[-~,o(EflE,)051
(11)
with b, = a, + a, l/d + a2 V&Id + a3 V, b, = a, + a, V, + a,l/d + a, V,ljd f a&d)’
+ a, V,(l/d)‘.
The regression coefficients aj are calculated according to the following procedure: (I) the model (11) is fitted on the data for [, using a non-linear fit-procedure. (2) If there are coefficients likely to be zero their corresponding terms in the model are removed and Step 1 is repeated. The following model for [ is obtained c = b. + b, exp[-0.51(E,/E,)‘,‘] with b, = 0.68 + 1.22V, + 0.23(//d) b, = -0.3O~Z/d) - 0.01(~~d)2.
6.
-0.755
(12)
Fig. 9. Histogram
of residuals
0.755
in the data points calculated.
It appears that a good agreement between calculated and fitted values for E* is obtained. In Fig. 8 a scatter diagram of the relative residuals is presented. It is seen that these residuals are all within 1% of the predicted values. In Fig. 9 these relative residuals are presented in a histogram.
CONCLUSIONS
The numerical micromechanical analyses appeared to be rather time-consuming. Results depend on type and dimensions of the finite elements. After some preliminary research, accurate results could be achieved. Using regression models closed-form relations between effective and structural parameters can be derived. The use of the regression techniques demand much experience from the user. The application is hardly formalized. It appeared difficult to choose a proper regression model. This is caused by the fact that there is no physical model available for the relation between effective and structural parameters. This is also the reason that the regression model can not be used for extrapolations.
.4cknowledg~ments_Mechanical analyses of RVEs and HRVEs were carried out using the finite element package DIANA (DIANA {SJ) running on Apollo Domain system. The SAS program package (SAS 191)was employed to carry out the parameter fits.
REFERENCES 1. W. M. G. Courage, P. J. G. Schreurs and J. D. Janssen, Estimation of mechanical parameter values of composites with the use of finite element and system identification techniaues. Comnur. Snuct. 14, 231-237 ( 1990). 2. 0. C. Zienkiewicz, The Finite Etemenr Method, 3rd Edn. McGraw-Hill (1977). 3. K. J. Bathe, Finite Element Procedures in Engineering Analvsis. Prentice-Hall (1982). 4. A. J: G. Schoofs, L. H. Th. M. van Beukering and M. L. C. Sluiter, A general purpose two-dimensional
Composites with randomly oriented short fibres mesh generator. In Engineering Sofware (Edited by R. A. Adey). Pentech Press, London (1979). 5. 13. Paul, Prediction of elastic constants of multiphase materials. Truns Mefall, Sot. AIME 218, Xi-41 (1960). 6. J. C. Halpin and J. L. Kardos, The Halpin-Tsai equations: a review. J. Polymer Engng ScE. 16, 344-352 (1976).
II85
7. J. C. Halpin, Primev on Composite Materials: Analysis. Technomic Publishing (1984). 8. DIANA, DIANA finite element analysis user manuals. TN0 Institute far Building Materials and Building Structures, Rijswijk (1989). 9. SAS, SAS User’s Guide. SAS institute Inc., Cary, NC (1985).
Compurers & Structures Vol. 44, No. 6, pp. 1179-I 185, 1992 Printed in Great Britain.
I; 1992 Per@mon Press Ltd
EFFECTIVE MATERIAL PARAMETERS FOR COMPOSITES WITH RANDOMLY ORIENTED SHORT FIBRES W. M. G. COURAGE? +TNO,
Building
SDepartment
and Construction
of Mechanical
and P. J. G. SCHREURS~
Research, Department The Netherlands
of Structural
Engineering,
Engineering, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands
Rijswijk,
P.O. Box 513,
(Received 9 July 1991) Abstract-A numerical strategy for the determination of mechanical material parameters for composites is described. Attention is focused on composite materials with randomly oriented short fibres. To determine the mechanical behaviour of the composite material, a representative voiume is modelled in detail and analysed employing the finite element method (FEM). The composite parameters are determined and expressed as a function of microstructural parameters.
INTRODUCTION
A composite is a combination of two or more components. These components may have very different properties. A proper combination results in a composite material with special and very sophisticated characteristics. The composite properties can be tuned to be optimal for a specific application. Composites are used more and more in structural construction parts. In these applications they have to carry loads and thus their stiffness and strength properties are important. Computer simulation of the mechanical behaviour of a structure is essential in the designing process. In the program packages which are being used the material behaviour is described by so-called constitutive equations. The form of these equations is characteristic for a class of materials under certain circumstances. The value of the material parameters in the equations are characteristic for a material in such a class. Determination of the form of the constitutive equation is an experimentai problem. This also holds for the determination of the value of the material parameters. These experiments may be difficult to perform, due to the nonhomogeneity and/or anisotropy of composite materials. Besides that, the experiments have to be repeated when the material is modified. It proves to be advantageous to follow a so-called structural approach. In this case the overall material parameters are determined using info~ation concerning the microstructure of the material. This information is characterized by structural parameters. The overalt parameters which describe the material behaviour in an averaged sense, are called effective parameters. In this paper a numerical technique is described to determine the effective parameters. Courage et al. [l] have described this procedure for particle composites.
Here, attention composites.
will be focused on fibre reinforced
CONFINEMENTS
AND ASSUMPTIONS
The composite material has a two-dimensional morphology. It is a fibre reinforced sheet. Fibres are short and oriented randomly, it is assumed that fibres do not overlap. The fibres are oriented in a square array as is shown in Fig. 1. Spacings between the central points of the fibres in the x- and y-directions are assumed to be equal and the diameter of the fibres is always the same. With these assumptions only two morphological structural parameters remain: the fibre aspect ratio, i.e. the ratio of fibre length to diameter I/d and the volume fraction of the fibres y,. The material ~haviour of fibres and matrix is linear elastic, both components are taken to be isotropic. Perfect bonding is assumed between the components. Under these conditions the composite is expected to show linear elastic material behaviour. Because the fibres are randomly oriented the composite will be statistical isotropic. Thus, the effective Young’s modulus and effective Poisson’s ratio are considered to be the only effective parameters. Because of uncertainties concerning the value of the Poisson’s ratio of most components of a composite material, only the effective Young’s modulus is calculated. It will be determined as a function of the strnctural parameters: the elasticity parameters of fibre and matrix and the morphological structural parameters. PROCEDURE
The procedure consists of four sequential described in Steps l-4 below.
1179
steps,
W. M, G. COURAGEand P. J. G. SCHREURS
1180
Fig. 1. Arrangement of the fibres in the matrix Fig. 3. RVE with arbitrary boundary load. Step 1. ~jcromechanical volume element
analysis of a representative
A representative volume element (RVE) of the composite is chosen and isolated for detailed observation. For a fully homogeneous material, a continuum, this RVE would be infinitesimally small. Mechanical state variables like strains and stresses and material parameters are defined in every material point. Moreover these quantities meet certain continuity requirements. A composite material is inhomogeneous and thus state and material parameters must be defined in an averaged sense. The averaging is carried out over an RVE. The dimensions of the RVE must be such that the fluctuations of the variables around their average value are sufficiently small. The RVE is modelled in detail employing FEM [2,3]. Because of the two-dimensional character of the RVE, plane stress elements are used. The elements are triangular and the element displacement field is linear. A typical finite element mesh of the RVE is shown in Fig. 2. The RVE is modelled as a
5 x 5 square array of fibres in the matrix. With the assumption concerning non-overlapping fibres, this leads to some restrictions on admissible values for the volume fraction of the fibres. The orientations of the fibres are determined using a random generator, such that the effective mechanical behaviour of the RVE is statistically isotropic. The number of elements needed varied from 2600 to 3200. This corresponds to approximately 1300 and 1600 nodal points. The actual number depends on the actual values for V, and l/d. A boundary load is applied to the RVE as shown in Fig. 3. This load can be arbitrary. The resulting deformation of the RVE is calculated using FEM in a so-called micromechanical analysis. Step 2. Calculation of eflective material parameters In the second stage the composite is regarded to be an homogeneous material. The material behaviour is described by means of a macroscopic model and this behaviour may be non-linear. The above-mentioned RVE is now called HRVE (homogenized RVE). After modelling it using FEM, a boundary load is applied, which is identical to the load applied to the RVE during the micromechanical analysis in step one. The deformation of the HRVE is then calculated. The effective parameters are determined such that the deformation of HRVE and RVE are identical. A
t t t t t t t t t t t”’
Fig. 2. Finite element mesh of the composite with randomly oriented fibres. V,= 0.15 and l/d = 2.
Fig. 4. RVE with specific boundary load.
Composites with randomly oriented short fibres Table 1. Survey of the values for the structural parameters used in calculations. The values for f.?,and E,,, listed in (a) are combined with the 21 combinations of V, and f/d marked in (b) Ef @Pa)
-% @Pa) 10 5
70 70 20 70 120
1 1 I (b) V, = 0.075
0.15
0.225
0.3
0.375
I
l
.
l
l
.
2 3 4 5 6 7 8 9 10
.
l
l
l
. .
l
l
l
l
I/d
*
l l l
. .
Table 2. Additional values for the structural parameters used in calculations E_
(GPaf
EAGPa)
10 5 10 5
(a)
1181
20 20 120 120
lid
0.225 0.225 0.225 0.22s
parameter estimation technique can be used to determine these parameters. This technique originates from system identification theory and has an iterative character. Following the assumptions presented earlier we confine ourselves to linear elastic behaviour. In this case system identification techniques are obsolete if the RVE is loaded in a specific way. A typical boundary load is shown in Fig. 4. The RVE is stretched in one direction by prescribing a uniformly distributed displacement field. The effective Young’s modulus is easily calculated by relating the prescribed displacement to the total force.
(b)
(of E* [GPa] 2.2 v, = 0.375
Et [GPal 20. 1s 16 14. 12
2 I
10
Vf= 0.075
8. 6. 4. 2.
0.8
I 0.
0. 5. 2
3.
4.
J.
6
7. 8. 9. 10.
0. 0.05 0.1 0.u
Vd l-1
0.2 0.X
0.3 0.35 0.4
Vf
i-1
(d)
0. 1. 2. 3. 4 5. 6. 7. 8. 9. 10. Em [GPa]
3 3 3 3
0. 20. 40. 60. Kl. Ku. 17.0. Et IGPal
Fig. 5. Young’s modulus E* versus: (a) aspect ratio I/d (E, = I, E,= 20). (b) volume fraction V, (I/d = 1, E, = i’o), (c) matrix modulus E,,, (I/d = 3, V, = 0.225), (d) fibre modulus E, (I/d = 1, E,, = 1).
W. M. G. COURAGEand P. J. G.
1182
Step 3. Variation of the structuraf parameters
The effective Young’s modulus must be determined as a function of the structural parameters. Steps 1 and 2 are repeated for various values of the structural parameters. The structural parameters to be varied are the Young’s modulus of fibres and matrix and the volume fraction and aspect ratio of the fibres. As described before the Poisson coefficients of fibres and matrix are kept at a constant value vI= 0.23 and v, = 0.35, respectively. With regard to the feasibility of the procedure an automatic mesh generator is used. For various V, and l/d the RVE is meshed automatically, guaranteeing statistical isotropy. Procedures from the TRIQUA~ESH meshgenerator [4] are used for the automatic mesh generation. Tables 1 and 2 give the used combinations for Vf and E/d as well as the Young’s modulus of matrix and fibres Ef and E,,, , respectively. Results of the micromechanical analyses are represented in Fig. 5. Step 4. Regression models
SCHREURS
accessible database. Sometimes it is possible to derive closed-form relations by using regression techniques. For linear elastic composites some preliminary requirements can be posed on the format of the relations. We consider the effective Young’s modulus E*. Its value has to be derived as a function of the structural parameters E* = E*(E,,,, E,, V,, l/d).
For linear elastic materials extremum principles lead to bounds for the values of the effective parameters[S]. For the effective Young’s modulus the following bounds can be derived 6%” = ErK/C J’, E, + V&, )
(2)
E&z, = V,,,E, + I/,-$.
(3)
Hence the following format for E* could be assumed E* = E& -t.f(~%, Et, Vr*lldl]E&,
The relation between effective and structural parameters can be represented graphically or stored in an (al
(1)
- E4,], o
(b) 0.225 I/d = 10
0.15 0.075
EffEaf-l
(dl
Cc)
c l-1
0.
1
2
c I-l
3.
a.
5. 6.
7.
8.
9. IO. l/d
1-l
Fig. 6. Function value &’versus: (a) ratio E,/E, (I/d = 3), (b) ratio EJE,, (V, = 0.079, (c) aspect ratio I/d (E,/E, = 20). (d) volume fraction V, (E,/E,,, = 20).
1. (41
Composites with randomly oriented short fihres The requirement concerning the function f leads to rather untransparent restrictions. Instead of the former relation for E* an alternative format for the model is considered. It is based on the generalized Halpin-Tsai fo~ulation [6,7]
where
In this relation parameters
[ is a function
of the structural
i: = CN,r, E,, V/, lid).
(6)
Using eqns (2) and (3), relation (5) can be rearranged into
(7)
1183
yielding
(8) The function 4 has the limiting values 4 = 0 (E* = E&J and i = co (.E* = E&,). The value of E* equals E, and E, for Vf = 0, resp. V, = I, irrespective of 1. The function c is to be determined as a function of the structural parameters, using the calculated values for E*. The function values for [ are obtained from (8). As could be expected from dimensional analysis it was found that the structural parameters Er and E,,, can be combined into one: Ef/Em. Figure 6 shows vatues of 4 as a function of E,jE, and IJd. The function 5 is supposed to be of the form %=a,+a,C,+a*i,f..~+u,l,.
(9)
From the calculated values of [ the regressor variables&(i=l,..., n) must be determined as explicit functions of the structural parameters E,, Em, Vf and I]d.
(b)
2, __
>
I/d = 5
0.
1. 2
3.
1.
I.
d
7.
s
9.
IO.
l/d I-I
Fig. 7. Coefficients b, and b, versus volume fraction Vl and versus aspect ratio l/d. CAS 44/b-B
W. M. G. COURAGE and P. J. G. SCHREURS
II84
88_ 75_ 63_ 50_ 38_ 25 13_ O0 2. d. 6 1. ,B 12.I, 16 111. 20 prediction Fig. 8. Relative residuals
From
E//E,
versus the values predicted model.
by the
Fig. 6 the following relation between J and is deduced
i = b0 + b, exp[-b~(~~~E,~)~.‘l.
(10)
The values of bO,6, and b2 are now fitted on the values of < using a non-linear regression procedure, based on the Marquard algorithm. The coefficient b2 was found to scatter around the value 0.5. The exponential term in (IO) was therefore simplified by assuming that the coefficient b, has always the value 0.5. The coefficients b, and h, are now once more fitted on the data. A linear least-squares fit could now be performed. The resulting values are plotted in Fig. 7. From these results it is concluded that 6, is linear in both V, and l/d, b, is linear in Viand quadratic in I/d. This results in the following regression model for 5 i =bo+b,ex~[-~,o(EflE,)051
(11)
with b, = a, + a, l/d + a2 V&Id + a3 V, b, = a, + a, V, + a,l/d + a, V,ljd f a&d)’
+ a, V,(l/d)‘.
The regression coefficients aj are calculated according to the following procedure: (I) the model (11) is fitted on the data for [, using a non-linear fit-procedure. (2) If there are coefficients likely to be zero their corresponding terms in the model are removed and Step 1 is repeated. The following model for [ is obtained c = b. + b, exp[-0.51(E,/E,)‘,‘] with b, = 0.68 + 1.22V, + 0.23(//d) b, = -0.3O~Z/d) - 0.01(~~d)2.
6.
-0.755
(12)
Fig. 9. Histogram
of residuals
0.755
in the data points calculated.
It appears that a good agreement between calculated and fitted values for E* is obtained. In Fig. 8 a scatter diagram of the relative residuals is presented. It is seen that these residuals are all within 1% of the predicted values. In Fig. 9 these relative residuals are presented in a histogram.
CONCLUSIONS
The numerical micromechanical analyses appeared to be rather time-consuming. Results depend on type and dimensions of the finite elements. After some preliminary research, accurate results could be achieved. Using regression models closed-form relations between effective and structural parameters can be derived. The use of the regression techniques demand much experience from the user. The application is hardly formalized. It appeared difficult to choose a proper regression model. This is caused by the fact that there is no physical model available for the relation between effective and structural parameters. This is also the reason that the regression model can not be used for extrapolations.
.4cknowledg~ments_Mechanical analyses of RVEs and HRVEs were carried out using the finite element package DIANA (DIANA {SJ) running on Apollo Domain system. The SAS program package (SAS 191)was employed to carry out the parameter fits.
REFERENCES 1. W. M. G. Courage, P. J. G. Schreurs and J. D. Janssen, Estimation of mechanical parameter values of composites with the use of finite element and system identification techniaues. Comnur. Snuct. 14, 231-237 ( 1990). 2. 0. C. Zienkiewicz, The Finite Etemenr Method, 3rd Edn. McGraw-Hill (1977). 3. K. J. Bathe, Finite Element Procedures in Engineering Analvsis. Prentice-Hall (1982). 4. A. J: G. Schoofs, L. H. Th. M. van Beukering and M. L. C. Sluiter, A general purpose two-dimensional
Composites with randomly oriented short fibres mesh generator. In Engineering Sofware (Edited by R. A. Adey). Pentech Press, London (1979). 5. 13. Paul, Prediction of elastic constants of multiphase materials. Truns Mefall, Sot. AIME 218, Xi-41 (1960). 6. J. C. Halpin and J. L. Kardos, The Halpin-Tsai equations: a review. J. Polymer Engng ScE. 16, 344-352 (1976).
II85
7. J. C. Halpin, Primev on Composite Materials: Analysis. Technomic Publishing (1984). 8. DIANA, DIANA finite element analysis user manuals. TN0 Institute far Building Materials and Building Structures, Rijswijk (1989). 9. SAS, SAS User’s Guide. SAS institute Inc., Cary, NC (1985).