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Poisson's ratio and the interfacial behaviour of composite materials
[~
U T T E R W O R T H E I N E M A N
Composites26 (1995)887-889 ElsevierScienceLimited Printed in Great Britain.All rights reserved 0010-4361/95/$10.00
N
Research Report Poisson's ratio and the interfacial behaviour of composite materials George Laird I1" US Bureau of Mines, Albany Research Center, Albany, OR 97321, USA
and T. C. Kennedy Oregon State University, Department of Mechanical Engineering, Corvallis, OR 97331, USA (Received 16 March 1995)
Composite materials have Poisson's ratios that can range from 0.2 for ceramics to as high as 0.49 for pure elastomers. In this work, Poisson's ratio is shown to influence the magnitude and the location of deleterious stress concentrations that arise along a frictionless interface around a cylindrical reinforcement (i.e. fibre-reinforced materials). Due to the non-linear nature of this problem, finite element models were developed for composites having 5, 25 and 50 vol% of fibres. Both compressive and tensile transverse loading were applied to the simulated composite systems. This particular mode of loading was considered due to the often low fracture strength of many composites in the transverse direction. The US Bureau of Mines is studying such problems to enable the design of new impact and abrasion resistant composite materials. (Keywords: Poisson's ratio; interracial behaviour; impact and abrasion resistant composites)
In two-dimensional isotropic elasticity solutions, elastic properties (e.g. Poisson's ratio) often have no effect upon the calculated stress field. For example, Muskhelishvili's ~ complex variable approach for a frictionless cylindrical inclusion within an infinite elastic plate (plane strain or plane stress) is completely independent of Poisson's ratio effects when calculating stress concentrations :3. Additional limitations of analytical frictionless solutions for both cylindrical 1'4 and spherical reinforcements 5 is the requirement that both radial stress and displacement continuity are enforced across the interface. These requirements impose unrealistic interfacial boundary conditions upon the composite model. That is, if a composite system is modelled as truly frictionless, then tangential shear and tensile radial stresses must be zero along the interface. Work by others 6-9 has investigated the transverse micromechanical response of fibre-reinforced composites; however, the effect of various Poisson's ratios was not considered. In this study, a frictionless interface is assumed to exist between the fibre and the matrix. Furthermore, the modelled composite system is assumed to contain large, elastically stiff fibres embedded within softer matrices. Examples of such composite systems are glass fibres in epoxy matrices l~ or ceramic fibres/ whiskers in glass matrices 12 14. For generic purposes, the elastic moduli ratio between the fibre Er and matrix Em is taken to be five, i.e. Er/Em = 5. (Earlier work 3 has shown that little variation occurs in the micromechanical stress state as the elastic moduli ratio is increased from 5 to infinity.) From this basis, finite element models were
* To whom correspondenceshould be addressed
constructed of composite systems having 5, 25 and 50 vol% fraction of fibres. A simple, repeating periodic structure was used to model the increasing volume fraction of fibres. Such modelling techniques have been shown by others to give good correspondence with real composite systems 6 8j5,16. Full-field moir6 studies 3'Iv on model systems (steel rods embedded in brittle, polymeric resins) have also validated the use of such numerical models to simulate the micromechanical behaviour of composite systems with non-ideal interfaces (e.g. discontinuous or frictional interfaces). Lastly, other modelling work 18 has shown that the frictionless interface yields the highest stress concentrations under compressive and tensile transverse loading. Hence, choosing the frictionless interface as the interfacial boundary condition allows this investigation to explore the extremes of micromechanical behaviour in composite materials with regard to Poisson's ratio mismatch between the fibre and matrix. DISCUSSION OF R E S U L T S Within these three models, the Poisson's ratio of the isotropic matrix Vm was varied (0.20, 0.30, 0.40 and 0.49) under conditions of compressive and tensile transverse loading (O'applied). The Poisson's ratio of the fibre was kept constant at vf = 0.25. Results are presented in the form of normalized hoop stresses (o* = O'0O/Gapplied) along the perimeter of the matrix. Figure 1 shows the quarter-symmetric finite element model (FEM) composed of three- and four-node isoparametric plane strain elements ~9. The FEM was verified against Muskhelishvili's I solution for a frictionless interface and found to be in good agreement (i.e. < 2% error) 2'3. Contact elements 2~ are used to simulate the
COMPOSITES Volume 26 Number 12 1995
887
Poisson's ratio and interracial behaviour: G. Laird II and 7:. C. Kennedy
Figure 1 Q u a r t e r - s y m m e t r i c , p l a n e s t r a i n finite e l e m e n t m e s h s h o w i n g the b o u n d a r y c o n d i t i o n s to s i m u l a t e the effects o f 25 v o l % f r a c t i o n o f fibres. T h e r a t i o R / L varies f r o m 0.252, 0 . 5 6 4 a n d 0.798 f o r 5, 25 a n d 50 v o l % c o n c e n t r a t i o n o f fibres, respectively
frictionless interface in all FEM simulations. These elements are arrayed along the interface and allow interfacial separation or debonding when tensile radial stresses exist across the interface and, logically, transmission of compressive radial forces across the interface. Changing contact element status from contact (transmission of forces) to no contact status requires an iterative solution to reach convergence. Different volume fractions of reinforcements (5, 25 and 50 vol%) were modelled by adjusting the ratio of the radii of the reinforcement R to the unit cell dimension L. Coupling of the nodal Ux or Uy degree-of-freedom along the outer boundary simulated the symmetry boundary conditions. This modelling technique is commonly defined as the 'unit cell' approach for the simulation of materials reinforced by periodic arrangements of fibres or particulates 6-8. Additional modelling details can be found in other references 2'3'18. Figure 2 shows the normalized hoop stresses at 5, 25 and 50 vol% of fibres under uniaxial compressive loading. The loading axis is transverse to the longitudinal fibre axis as depicted in Figure 1. As the Poisson's 1.5
ratio v is increased from 0.20 to 0.49 two interesting phenomena are observed: 1) the tensile hoop stress o-00 at the pole of the fibre (0 ~ decreases as the Poisson's ratio increases; and 2) a stress riser is observed from 35 to 50 ~ which increases in magnitude as the Poisson's ratio increases. The first phenomenon arises from the relationship between elastic stiffness and Poisson's ratio for plane strain conditions, i.e. Eplan e strain -- E l ( 1 - V 2). This effect tends to increase the relative stiffness of the matrix as the Poisson's ratio increases and, hence, tends to lower the stress concentration around the pole of fibre where the matrix and fibre are in direct contact. Furthermore, as the Poisson's ratio is increased, increased lateral expansion occurs along the contact interface (from 0 to ~40~ contributing to the lower tensile o00. The second phenomenon arises from increased matrix dilation as the Poisson's ratio is increased from 0.20 to 0.49. The 'bump' observed in Figure 2 at about 35-50 ~ indicates the approximate location where the matrix has separated from the fibre. As the Poisson's ratio is increased, the point of matrix/fibre separation moves towards the pole of the fibre, e.g. at a volume fraction of 5% fibres and v = 0.20, the stress riser is at 56 ~ while at v -- 0.49 this feature has shifted to 42 ~ The stress riser effect or 'bump' in the hoop stress profile is strictly due to an increase in the lateral movement of the matrix (from the corresponding increase in the Poisson's ratio) as the composite is compressively loaded. Although the magnitude of the stress riser is lower than the tensile o-00 stress near the pole of the fibre (0 to -20~ localized tensile stress concentrations can still lead to fracture in brittle materials containing a range of defect sizes. At the highest Poisson's ratio (0.49) and lowest fibre volume fraction (5%), this effect could lead to crack initiation and fracture away from the pole of the fibre, therefore leading to a lower strength composite material under compressive loading. As the fibre volume fraction increases, this stress riser effect becomes much less dominant and is overshadowed by the much higher tensile o-00 component near the pole of the fibre. This increase in tensile or00 as a function of increasing volume fraction has been covered in prior work 2'~7 for both compressive and tensile loading. Figure 3 shows the normalized hoop stresses under transverse tensile loading. In this case, no Poisson's effect is noted. The only dominant effect shown in Figure 3 is
. . . . .
i__: _:__" _:__: L__25% 0.5 0 O'*
-1.5 : -2
~--~-
,~-4
"~...,i....I.._l,...I 10 20 30 40 50 60 70 g0 90 Degrees
io 20 30 40 50 60 70 80 90 Degrees
0 10 20 30 40 50 60 70 80 90 Degrees . 9 e _ _
Poisson's Poisson's Poisson's Poisson's
ratio = 0.20 ratio = 0.30 ratio = 0.40 ratio = 0.49
Figure 2 N o r m a l i z e d h o o p stresses o'00 f r o m the q u a r t e r - s y m m e t r i c , p l a n e s t r a i n F E M as the v o l u m e f r a c t i o n is i n c r e a s e d f r o m 5 to 50~ u n d e r c o m p r e s s i v e l o a d i n g . T h e inset figures s h o w details o f the stress riser effect f r o m 35 to 50 ~
888
COMPOSITES Volume 26 Number 12 1995
fibres
Poisson's ratio and interfacial behaviour: G. Laird II and T. C. Kennedy 6 5
:s%
!~
i-i!
4
3
%._
2
_
. ,
•
1 0 :....' . . . . . . . . . . . . 0
10
20
30
40
50
60
70
80
90
10 20
30
Degrees
40
50
60
70
80
90
0
Degrees
10
20
30
i........ 40
50
,...' ....... 60
70
80
90
Degrees
3 Normalized hoop stresses o'e0from the quarter-symmetric, plane strain FEM as the volume fraction is increased from 5 to 50% fibres under tensile loading Figure
the increase in tensile o00 as the v o l u m e fraction o f fibres is increased f r o m 5 to 50%. This o b s e r v a t i o n is in c o n t r a s t with t h a t shown in Figure 2 for c o m p r e s s i v e loading. As the m a t r i x is l o a d e d u n d e r tension, the m a t r i x separates f r o m the fibre f r o m 0 to - 7 0 ~ This effect leaves only a small p o r t i o n o f the m a t r i x in c o n t a c t with the fibre. As such, it is r e a s o n a b l e t h a t little effect is n o t e d for an increasing P o i s s o n ' s ratio. CONCLUDING
REMARKS
In this investigation, the v a r i a t i o n o f P o i s s o n ' s r a t i o from 0.20 to 0.49 u n d e r c o m p r e s s i v e l o a d i n g was shown to create small stress risers a n d to lower the p e a k tensile h o o p stress n e a r the p o l e o f the fibres as the P o i s s o n ' s r a t i o was increased. T h e m a g n i t u d e o f these stress risers was shown to decrease as the v o l u m e fraction o f fibre was increased from 5 to 50%. N o P o i s s o n ' s effect was n o t e d for c o m p o s i t e s u n d e r tensile loading. Overall, this w o r k has shown t h a t the effect o f P o i s s o n ' s r a t i o is negligible for c o m p o s i t e s h a v i n g frictionless interfaces that are transversely l o a d e d u n d e r c o m p r e s s i o n or tension. These results s h o u l d also lay to rest s p e c u l a t i o n a b o u t a n y adverse affects o f P o i s s o n ' s r a t i o m i s m a t c h between the reinforcing p h a s e a n d the matrix. T h a t is, P o i s s o n ' s r a t i o m i s m a t c h s h o u l d have little o r no effect u p o n the tensile or c o m p r e s s i v e transverse strengths m e a s u r e d in c o m p o s i t e materials.
REFERENCES 1 2 3 4 5 6 7
8 9 10 11 12 13 14 15 16 17 18 19 20
Muskhelishvili, N.I. in 'Some Basic Problems of the Mathematical Theory of Elasticity, 3rd edn, P. Noordhoff, Groningen, The Netherlands, 1953, p. 215 Laird II, G., Micromechanics of heterogeneous materials under compressive loading PhD Thesis Oregon State University, 1993 Laird II, G. and Kennedy, T.C. Composites 1994, 25, 593 Mura, T. and Furuhashi, R. J. Appl. Mech. 1984, 51, 308 Ghahremani, F. Int. J. Solid Structures 1980, 16, 825 Broutman, L.J. and Agarwal, B.D. Polym. Eng. Sci. 1974, 14, 581 Sohi, M., Adams, J. and Mahapatra, R. in 'Proc. 3rd Int. Conf. on Constitutive Laws for Engineered Materials' (Eds C.S. Desai, E. Krempl, G. Frantziskonis and H. Saadatmanesh), ASME, 1991, p. 617 Yeh, J.R. Int. J. Solid Structures 1992, 29, 2493 Achenbach, J.D. and Zhu, H. J. Mech. Phys. Solids 1989, 37, 381 Peters, P.WM. and Chou, T.W. Composites 1987, 18, 40 Cinquin, J., Chabert, B., Chaucbard, J., Morel, E. and Trotignon, J.P. Composites 1990, 21, 141 Brennan, J.J. and Prewo, K.M.J. Mater. Sci. 1982, 17, 2371 Vekinis, G., Ashby, M.F., Shercliff, H. and Beaumont, P.W.R. Compos. Sci. Technol. 1993, 48, 325 Mall, S. and Kim, R.Y. Composites 1992, 23, 215 Agarwal, B.D. and Broutman, L.J. Fibre Sci. Technol. 1974, 7, 63 Furno, J.S. and Nauman, E.B.J. Mater. Sci. 1992, 27, 1428 Laird II, G., Epstein, J.S. and Kennedy, T.C. Exp. Mech. in press Laird II, G. and Kennedy, T.C. Eng. Fract. Mech. in press Cook, R.D. in 'Concepts and Applications of Finite Element Analysis', 2nd edn, Wiley, New York, 1981, p. 113 'ANSYS Engineering Analysis System - Theoretical Manual', Swanson Analysis Systems, Inc., PA 1993
COMPOSITES Volume 26 Number 12 1995
889
U T T E R W O R T H E I N E M A N
Composites26 (1995)887-889 ElsevierScienceLimited Printed in Great Britain.All rights reserved 0010-4361/95/$10.00
N
Research Report Poisson's ratio and the interfacial behaviour of composite materials George Laird I1" US Bureau of Mines, Albany Research Center, Albany, OR 97321, USA
and T. C. Kennedy Oregon State University, Department of Mechanical Engineering, Corvallis, OR 97331, USA (Received 16 March 1995)
Composite materials have Poisson's ratios that can range from 0.2 for ceramics to as high as 0.49 for pure elastomers. In this work, Poisson's ratio is shown to influence the magnitude and the location of deleterious stress concentrations that arise along a frictionless interface around a cylindrical reinforcement (i.e. fibre-reinforced materials). Due to the non-linear nature of this problem, finite element models were developed for composites having 5, 25 and 50 vol% of fibres. Both compressive and tensile transverse loading were applied to the simulated composite systems. This particular mode of loading was considered due to the often low fracture strength of many composites in the transverse direction. The US Bureau of Mines is studying such problems to enable the design of new impact and abrasion resistant composite materials. (Keywords: Poisson's ratio; interracial behaviour; impact and abrasion resistant composites)
In two-dimensional isotropic elasticity solutions, elastic properties (e.g. Poisson's ratio) often have no effect upon the calculated stress field. For example, Muskhelishvili's ~ complex variable approach for a frictionless cylindrical inclusion within an infinite elastic plate (plane strain or plane stress) is completely independent of Poisson's ratio effects when calculating stress concentrations :3. Additional limitations of analytical frictionless solutions for both cylindrical 1'4 and spherical reinforcements 5 is the requirement that both radial stress and displacement continuity are enforced across the interface. These requirements impose unrealistic interfacial boundary conditions upon the composite model. That is, if a composite system is modelled as truly frictionless, then tangential shear and tensile radial stresses must be zero along the interface. Work by others 6-9 has investigated the transverse micromechanical response of fibre-reinforced composites; however, the effect of various Poisson's ratios was not considered. In this study, a frictionless interface is assumed to exist between the fibre and the matrix. Furthermore, the modelled composite system is assumed to contain large, elastically stiff fibres embedded within softer matrices. Examples of such composite systems are glass fibres in epoxy matrices l~ or ceramic fibres/ whiskers in glass matrices 12 14. For generic purposes, the elastic moduli ratio between the fibre Er and matrix Em is taken to be five, i.e. Er/Em = 5. (Earlier work 3 has shown that little variation occurs in the micromechanical stress state as the elastic moduli ratio is increased from 5 to infinity.) From this basis, finite element models were
* To whom correspondenceshould be addressed
constructed of composite systems having 5, 25 and 50 vol% fraction of fibres. A simple, repeating periodic structure was used to model the increasing volume fraction of fibres. Such modelling techniques have been shown by others to give good correspondence with real composite systems 6 8j5,16. Full-field moir6 studies 3'Iv on model systems (steel rods embedded in brittle, polymeric resins) have also validated the use of such numerical models to simulate the micromechanical behaviour of composite systems with non-ideal interfaces (e.g. discontinuous or frictional interfaces). Lastly, other modelling work 18 has shown that the frictionless interface yields the highest stress concentrations under compressive and tensile transverse loading. Hence, choosing the frictionless interface as the interfacial boundary condition allows this investigation to explore the extremes of micromechanical behaviour in composite materials with regard to Poisson's ratio mismatch between the fibre and matrix. DISCUSSION OF R E S U L T S Within these three models, the Poisson's ratio of the isotropic matrix Vm was varied (0.20, 0.30, 0.40 and 0.49) under conditions of compressive and tensile transverse loading (O'applied). The Poisson's ratio of the fibre was kept constant at vf = 0.25. Results are presented in the form of normalized hoop stresses (o* = O'0O/Gapplied) along the perimeter of the matrix. Figure 1 shows the quarter-symmetric finite element model (FEM) composed of three- and four-node isoparametric plane strain elements ~9. The FEM was verified against Muskhelishvili's I solution for a frictionless interface and found to be in good agreement (i.e. < 2% error) 2'3. Contact elements 2~ are used to simulate the
COMPOSITES Volume 26 Number 12 1995
887
Poisson's ratio and interracial behaviour: G. Laird II and 7:. C. Kennedy
Figure 1 Q u a r t e r - s y m m e t r i c , p l a n e s t r a i n finite e l e m e n t m e s h s h o w i n g the b o u n d a r y c o n d i t i o n s to s i m u l a t e the effects o f 25 v o l % f r a c t i o n o f fibres. T h e r a t i o R / L varies f r o m 0.252, 0 . 5 6 4 a n d 0.798 f o r 5, 25 a n d 50 v o l % c o n c e n t r a t i o n o f fibres, respectively
frictionless interface in all FEM simulations. These elements are arrayed along the interface and allow interfacial separation or debonding when tensile radial stresses exist across the interface and, logically, transmission of compressive radial forces across the interface. Changing contact element status from contact (transmission of forces) to no contact status requires an iterative solution to reach convergence. Different volume fractions of reinforcements (5, 25 and 50 vol%) were modelled by adjusting the ratio of the radii of the reinforcement R to the unit cell dimension L. Coupling of the nodal Ux or Uy degree-of-freedom along the outer boundary simulated the symmetry boundary conditions. This modelling technique is commonly defined as the 'unit cell' approach for the simulation of materials reinforced by periodic arrangements of fibres or particulates 6-8. Additional modelling details can be found in other references 2'3'18. Figure 2 shows the normalized hoop stresses at 5, 25 and 50 vol% of fibres under uniaxial compressive loading. The loading axis is transverse to the longitudinal fibre axis as depicted in Figure 1. As the Poisson's 1.5
ratio v is increased from 0.20 to 0.49 two interesting phenomena are observed: 1) the tensile hoop stress o-00 at the pole of the fibre (0 ~ decreases as the Poisson's ratio increases; and 2) a stress riser is observed from 35 to 50 ~ which increases in magnitude as the Poisson's ratio increases. The first phenomenon arises from the relationship between elastic stiffness and Poisson's ratio for plane strain conditions, i.e. Eplan e strain -- E l ( 1 - V 2). This effect tends to increase the relative stiffness of the matrix as the Poisson's ratio increases and, hence, tends to lower the stress concentration around the pole of fibre where the matrix and fibre are in direct contact. Furthermore, as the Poisson's ratio is increased, increased lateral expansion occurs along the contact interface (from 0 to ~40~ contributing to the lower tensile o00. The second phenomenon arises from increased matrix dilation as the Poisson's ratio is increased from 0.20 to 0.49. The 'bump' observed in Figure 2 at about 35-50 ~ indicates the approximate location where the matrix has separated from the fibre. As the Poisson's ratio is increased, the point of matrix/fibre separation moves towards the pole of the fibre, e.g. at a volume fraction of 5% fibres and v = 0.20, the stress riser is at 56 ~ while at v -- 0.49 this feature has shifted to 42 ~ The stress riser effect or 'bump' in the hoop stress profile is strictly due to an increase in the lateral movement of the matrix (from the corresponding increase in the Poisson's ratio) as the composite is compressively loaded. Although the magnitude of the stress riser is lower than the tensile o-00 stress near the pole of the fibre (0 to -20~ localized tensile stress concentrations can still lead to fracture in brittle materials containing a range of defect sizes. At the highest Poisson's ratio (0.49) and lowest fibre volume fraction (5%), this effect could lead to crack initiation and fracture away from the pole of the fibre, therefore leading to a lower strength composite material under compressive loading. As the fibre volume fraction increases, this stress riser effect becomes much less dominant and is overshadowed by the much higher tensile o-00 component near the pole of the fibre. This increase in tensile or00 as a function of increasing volume fraction has been covered in prior work 2'~7 for both compressive and tensile loading. Figure 3 shows the normalized hoop stresses under transverse tensile loading. In this case, no Poisson's effect is noted. The only dominant effect shown in Figure 3 is
. . . . .
i__: _:__" _:__: L__25% 0.5 0 O'*
-1.5 : -2
~--~-
,~-4
"~...,i....I.._l,...I 10 20 30 40 50 60 70 g0 90 Degrees
io 20 30 40 50 60 70 80 90 Degrees
0 10 20 30 40 50 60 70 80 90 Degrees . 9 e _ _
Poisson's Poisson's Poisson's Poisson's
ratio = 0.20 ratio = 0.30 ratio = 0.40 ratio = 0.49
Figure 2 N o r m a l i z e d h o o p stresses o'00 f r o m the q u a r t e r - s y m m e t r i c , p l a n e s t r a i n F E M as the v o l u m e f r a c t i o n is i n c r e a s e d f r o m 5 to 50~ u n d e r c o m p r e s s i v e l o a d i n g . T h e inset figures s h o w details o f the stress riser effect f r o m 35 to 50 ~
888
COMPOSITES Volume 26 Number 12 1995
fibres
Poisson's ratio and interfacial behaviour: G. Laird II and T. C. Kennedy 6 5
:s%
!~
i-i!
4
3
%._
2
_
. ,
•
1 0 :....' . . . . . . . . . . . . 0
10
20
30
40
50
60
70
80
90
10 20
30
Degrees
40
50
60
70
80
90
0
Degrees
10
20
30
i........ 40
50
,...' ....... 60
70
80
90
Degrees
3 Normalized hoop stresses o'e0from the quarter-symmetric, plane strain FEM as the volume fraction is increased from 5 to 50% fibres under tensile loading Figure
the increase in tensile o00 as the v o l u m e fraction o f fibres is increased f r o m 5 to 50%. This o b s e r v a t i o n is in c o n t r a s t with t h a t shown in Figure 2 for c o m p r e s s i v e loading. As the m a t r i x is l o a d e d u n d e r tension, the m a t r i x separates f r o m the fibre f r o m 0 to - 7 0 ~ This effect leaves only a small p o r t i o n o f the m a t r i x in c o n t a c t with the fibre. As such, it is r e a s o n a b l e t h a t little effect is n o t e d for an increasing P o i s s o n ' s ratio. CONCLUDING
REMARKS
In this investigation, the v a r i a t i o n o f P o i s s o n ' s r a t i o from 0.20 to 0.49 u n d e r c o m p r e s s i v e l o a d i n g was shown to create small stress risers a n d to lower the p e a k tensile h o o p stress n e a r the p o l e o f the fibres as the P o i s s o n ' s r a t i o was increased. T h e m a g n i t u d e o f these stress risers was shown to decrease as the v o l u m e fraction o f fibre was increased from 5 to 50%. N o P o i s s o n ' s effect was n o t e d for c o m p o s i t e s u n d e r tensile loading. Overall, this w o r k has shown t h a t the effect o f P o i s s o n ' s r a t i o is negligible for c o m p o s i t e s h a v i n g frictionless interfaces that are transversely l o a d e d u n d e r c o m p r e s s i o n or tension. These results s h o u l d also lay to rest s p e c u l a t i o n a b o u t a n y adverse affects o f P o i s s o n ' s r a t i o m i s m a t c h between the reinforcing p h a s e a n d the matrix. T h a t is, P o i s s o n ' s r a t i o m i s m a t c h s h o u l d have little o r no effect u p o n the tensile or c o m p r e s s i v e transverse strengths m e a s u r e d in c o m p o s i t e materials.
REFERENCES 1 2 3 4 5 6 7
8 9 10 11 12 13 14 15 16 17 18 19 20
Muskhelishvili, N.I. in 'Some Basic Problems of the Mathematical Theory of Elasticity, 3rd edn, P. Noordhoff, Groningen, The Netherlands, 1953, p. 215 Laird II, G., Micromechanics of heterogeneous materials under compressive loading PhD Thesis Oregon State University, 1993 Laird II, G. and Kennedy, T.C. Composites 1994, 25, 593 Mura, T. and Furuhashi, R. J. Appl. Mech. 1984, 51, 308 Ghahremani, F. Int. J. Solid Structures 1980, 16, 825 Broutman, L.J. and Agarwal, B.D. Polym. Eng. Sci. 1974, 14, 581 Sohi, M., Adams, J. and Mahapatra, R. in 'Proc. 3rd Int. Conf. on Constitutive Laws for Engineered Materials' (Eds C.S. Desai, E. Krempl, G. Frantziskonis and H. Saadatmanesh), ASME, 1991, p. 617 Yeh, J.R. Int. J. Solid Structures 1992, 29, 2493 Achenbach, J.D. and Zhu, H. J. Mech. Phys. Solids 1989, 37, 381 Peters, P.WM. and Chou, T.W. Composites 1987, 18, 40 Cinquin, J., Chabert, B., Chaucbard, J., Morel, E. and Trotignon, J.P. Composites 1990, 21, 141 Brennan, J.J. and Prewo, K.M.J. Mater. Sci. 1982, 17, 2371 Vekinis, G., Ashby, M.F., Shercliff, H. and Beaumont, P.W.R. Compos. Sci. Technol. 1993, 48, 325 Mall, S. and Kim, R.Y. Composites 1992, 23, 215 Agarwal, B.D. and Broutman, L.J. Fibre Sci. Technol. 1974, 7, 63 Furno, J.S. and Nauman, E.B.J. Mater. Sci. 1992, 27, 1428 Laird II, G., Epstein, J.S. and Kennedy, T.C. Exp. Mech. in press Laird II, G. and Kennedy, T.C. Eng. Fract. Mech. in press Cook, R.D. in 'Concepts and Applications of Finite Element Analysis', 2nd edn, Wiley, New York, 1981, p. 113 'ANSYS Engineering Analysis System - Theoretical Manual', Swanson Analysis Systems, Inc., PA 1993
COMPOSITES Volume 26 Number 12 1995
889