Micromechanical analysis of hybrid discontinuous fiber reinforced composite for longitudinal loading.
Srinag, T. ; Murthy, V. Bala Krishna ; Kumar, J. Suresh 等
Introduction
Elastic Properties
Fiber reinforced composites can be tailor made, as their properties
can be controlled by the appropriate selection of the substrata
parameters such as fiber orientation, volume fraction, fiber spacing,
and layer sequence. The required directional properties can be achieved
in the case of fiber reinforced composites by properly selecting various
parameters enlisted above. As a result of this, the designer can have a
tailormade material with the desired properties. Such a material design
reduces the weight and improves the performance of the composite. For
example, the carbon-carbon composites are strong in the direction of the
fiber reinforcement but weak in the other directions. Elastic constants
of continuous fiber reinforced composites with various types of
constituents were determined by Hashin & Rosen [1], Hashin [2],
Whitney [3] and Chen and Chang [4].
It is clear from the above predictions that four of the five
independent composite modulii ([E.sub.1], [E.sub.2], [v.sub.12],
[G.sub.12] and [G.sub.23]) differ only in their expressions for the
fifth elastic constant i.e., transverse shear modulus, which varies
between two bounds that are reasonably close for the cases of practical
interest. In the above terms, the subscript 1- stands for longitudinal
direction and 2- for transverse direction of the fiber respectively. The
values of elastic modulii presented by Hashin and Rosen [1] have very
close bounds. Ishikawa et al. [5] experimentally obtained all the
independent elastic modulii of unidirectional carbon-epoxy composites
with the tensile and torsional tests of co-axis and off-axis specimens.
They confirmed the transverse isotropy nature of the graphite-epoxy
composites. Hashin [6] comprehensively reviewed the analysis of
composite materials with respect to mechanical and materials point of
view. Expressions for [E.sub.1] and [G.sub.12] are derived using the
theory of elasticity approach for continuous fiber-reinforced composites
[7].
Micromechanics
Micromechanics is intended to study the distribution of stresses
and strains within the micro regions of the composite under loading.
This study will be perticularized to simple loading and geometry for
evaluating the average or global stiffnesses and strengths of the
composites [7, 8]. Micromechanics analysis can be carried theoretically
using the principles of continuum mechanics, and experimentally using
mechanical, photo elasticity, ultrasonic tests, etc. The results of
micromechanics will help
* to understand load sharing among the constituents of the
composites, microscopic structure (arrangement of fibers), etc., within
composites,
* to understand the influence of microstructure on the properties
of composite,
* to predict the average properties of the lamina, and
* to design the materials, i.e., constituents volume fractions,
their distribution and orientation, for a given situation.
The properties and behavior of a composite are influenced by the
properties of fiber and matrix, interfacial bond and by its
microstructure. Micro structural parameters that influence the composite
behavior are fiber diameter, length, volume fraction, packing and
orientation of fiber. Sun et al [9] established a vigorous mechanics
foundation for using a Representative Volume Element (RVE) to predict
the mechanical properties of continuous unidirectional fiber composites.
A closed form micromechanical equation for predicting the transverse
modulus, [E.sub.2], of continuous fiber reinforced polymers is presented
[10]. Anifantis [11] predicted the micromechanical stress state
developed within fibrous composites that contain a heterogeneous
interphase region by applying finite element method to square and
hexagonal arrays of fibers. Li [12] has developed two typical idealized
packing systems, which have been employed for unidirectional fiber
reinforced composites, viz. square and hexagonal ones to accommodate
fibers of irregular cross sections and imperfections asymmetrically
distributed around fibers.
Hybrid Composites
To understand the mechanism of the 'hybrid effect' on the
tensile properties of hybrid composites Yiping Qiu & Peter Schwartz
[13] investigated the fiber/matrix interface properties by using single
fiber pull out from a micro composite (SFPOM) test, which showed a
significant difference between the interfacial shear strength of Kevlar
fiber/epoxy in single fiber type and that in the hybrid at a constant
fiber volume fraction, which shortened the ineffective length and
contributed to the failure strain increase of Kevlar 149 fibers in the
hybrid. Mishra & Mohanthy et al [14] investigated the degree of
mechanical reinforcement that could be obtained by the introduction of
glass fibers in bio fiber (pineapple leaf fiber/ sisal fiber) reinforced
polyester composite has been assessed experimentally. Addition of
relatively small amount of glass fiber to the pineapple leaf fiber and
sisal fiber reinforced polyester matrix enhanced the mechanical
properties of the resulting hybrid composites.
Discontinuous Fiber Composites
Pahr and Arnold [15] reviewed the work done in the field of
discontinuous reinforced composites, focusing on the different
parameters which influence the material behaviour of discontinuous
reinforced composites, as well as the various analysis approaches
undertaken and identified the need for the finite element based
micromechanics approach for the analysis of discontinuous reinforced
composites. They conducted an investigation to demonstrate the utility
of utilizing the Generalized Method of Cells (GMC), a semi analytical
micromechanics based approach to simulate the elastic and elastoplastic
material behaviour of aligned short fiber composites.
The works reported in the available literature do not include the
micromechanical analysis of discontinuous hybrid FRP lamina using FEM.
The present work aims to develop a 3-D finite element model for the
micromechanical analysis of discontinuous hybrid fiber reinforced
composite lamina. The analysis includes evaluation of the longitudinal
Young's Modulus [E.sub.1], Poisson's ratios [v.sub.12],
[v.sub.13] and determination of the stresses at the fiber-matrix
interface using 3-D finite element method developed based on theory of
elasticity.
Hexagonal Array of Unit Cells
A schematic diagram of the unidirectional fiber composite is shown
in Figure. 1 where the fibers are arranged in the hexagonal array. It is
assumed that the fiber and matrix materials are linearly elastic. A unit
cell is adopted for the analysis. The volume fraction ([V.sub.f]) is
taken as 55% for the present analysis. In case of continuous fiber
model, which is used for the validation, the gap between fibers becomes
zero and the volume fraction becomes 55.55%.
[FIGURE 1 OMITTED]
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Finite Element Model
The 1-2-3 Coordinate system shown in Figure. 2 is used to study the
behavior of unit cell. The isolated unit cell behaves as a part of large
array of unit cells by satisfying the conditions that the boundaries of
the isolated unit cell remain plane. It is assumed that the geometry,
material and loading of unit cell are symmetric with respect to 1-2-3
coordinate system. Therefore, a one-eighth portion of the unit cell is
modeled for the present work.
Geometry
The dimensions of the finite element model are taken as
* X=25 units,
* Y=43.3 units,
* Z=505 units =half the length of fiber + half the length of
discontinuity = [l.sub.f]/2+e/2
The radius of the fiber is calculated as 19.566 units, so that the
fiber volume fraction becomes 0.55 and the fiber aspect ratio equal to
25.55 (Figure. 3)
[FIGURE 3 OMITTED]
Element type
The element used for the present analysis is SOLID 95 of ANSYS
[16], which is developed, based on three-dimensional elasticity theory
and is defined by 20 nodes having three degrees of freedom at each node:
translations in the nodal x, y and z directions.
Materials
The properties of the constituent materials used for the present
analysis are given in Table 1.
Loading
Uniform tensile load of 1 MPa is applied on the area at Z = 505
units.
Boundary conditions
Due to the symmetry of the problem, the following symmetric
boundary conditions are used
* At the surface x = 0, [U.sub.x] = 0
* At the surface y = 0, [U.sub.y] = 0
* At the surface z = 0, [U.sub.z] = 0
In addition the following multi point constraints are used.
* The [U.sub.x] of all the nodes on the line at x =25 is same
* The [U.sub.y] of all the nodes on the line at y =43.3 is same
* The [U.sub.z] of all the nodes on the line at z = 505 is same
Results
The mechanical properties of the laminae are calculated using the
following expressions.
Young's modulus in fiber direction [E.sub.1] =
[[sigma].sub.1]/[[epsilon].sub.1]
Poisson's ratio [v.sub.12] =
-[[epsilon].sub.2]/[[epsilon].sub.1]
[v.sub.13] = -[[epsilon].sub.3]/[[epsilon].sub.1]
Where [[sigma].sub.1] = Stress in 1-direction (Z).
[[epsilon].sub.1] = Strain in 1-direction (Z)
[[epsilon].sub.2] = Strain in 2-direction (X)
[[epsilon].sub.3] = Strain in 3-direction (Y)
Sufficient numbers of convergence tests are made and the present
finite element model is validated by comparing the Young's modulus
for continuous fiber with the value obtained from exact elasticity
theory [7] and with Rule of mixtures for continuous hybrid fiber and
found close agreement (Table 2). Later the finite element models are
used to evaluate the properties [E.sub.1], [v.sub.12], [v.sub.13] and
the stresses at the fiber matrix interface of a discontinuous hybrid
fiber composite with T300 and S-Glass fibers.
Analysis of Results
Table 3 presents the mechanical properties predicted from the
present analysis. It is observed that the Young's modulus of the
composite with T300 fibers is more when compared to the Young's
modulus of the composite with S-glass fibers. This is due to the reason
that the longitudinal Young's modulus of T300 fiber is more than
the Young's modulus of S-glass fiber. The composite with both the
fibers shows the resultant value of Young's modulus in the
longitudinal direction. The similar trend can be observed in
Poisson's ratios also.
Figures. 4-19 show the variation of the fiber-matrix interfaces
near the end of the fiber at discontinuity. The normal stress in fiber
at bottom and top interfaces ([[sigma].sup.f.sub.n]) is shown in
Figures. 4 and 5 respectively. This stress is maximum near the ends and
at [theta]=[90.sup.0] at bottom interface and at [theta]=[67.5.sup.0] at
top interface. Fibermatrix debond may occur at these locations. It is
observed that the normal stress at top interface is greater than that of
at bottom interface. This may be due to the higher transverse stiffness
of the top fiber (S-glass) than the bottom fiber (T300).
The normal stress in matrix at bottom and top interfaces
([s.sup.m.sub.n]) is shown in Figure s. 6 and 7 respectively. This
stress is maximum near the ends and at [theta]=[45.sup.0] at bottom
interface and at [theta]=[67.5.sup.0] at top interface. Fiber-matrix
debond may occur at these locations. It is observed that the normal
stress at top interface is greater than that of at bottom interface.
This may be due to the higher transverse stiffness of the top fiber
(S-glass) than the bottom fiber (T300).
The shear stress in fiber at bottom and top interfaces
([[tau].sup.f.sub.nc]) is shown in Figures. 8 and 9 respectively. This
stress is maximum at the ends and at [theta]=[22.5.sup.0] at bottom
interface and approximately at [theta]=[35.sup.0] at top interface.
Fiber damage may occur at these locations. It is observed that the shear
stress at top interface is greater than that of at bottom interface.
This may be due to the higher transverse stiffness of the top fiber
(S-glass) than the bottom fiber (T300).
The shear stress in matrix at bottom and top interfaces
([[tau].sup.m.sub.nc]) is shown in Figures. 10 and 11 respectively. This
stress is maximum at the ends and at [theta]=[22.5.sup.0] at bottom
interface and approximately at [theta]=[35.sup.0] at top interface.
Matrix damage may occur at these locations.
The circumferential normal stress in fiber at bottom and top
interfaces ([[sigma].sup.f.sub.c]) is shown in Figures. 12 and 13
respectively. This stress is maximum near the ends and at
[theta]=[90.sup.0] at both the interfaces. This is due to the mismatch
of transverse poisson's ratios of fiber and matrix materials. Fiber
damage may occur at these locations due to this stress.
The circumferential normal stress in matrix at bottom and top
interfaces ([[sigma].sup.m.sub.c]) is shown in Figures. 14 and 15
respectively. This stress is maximum at the ends and at
[theta]=[90.sup.0] at both the interfaces. This is due to the mismatch
of transverse poisson's ratios of fiber and matrix materials.
Matrix damage may occur at these locations due to this stress.
The fiber directional normal stress in fiber at bottom and top
interfaces ([[sigma].sup.f.sub.1]) is shown in Figures. 16 and 17
respectively. This stress is maximum near the ends and at
[theta]=[22.5.sup.0] at bottom interface and approximately at
[theta]=[35.sup.0] at top interface. This is due to the mismatch of
longitudinal Poisson's ratios of fiber and matrix materials. Fiber
damage may occur at these locations due to this stress.
The fiber directional normal stress in matrix at bottom and top
interfaces ([[sigma].sup.m.sub.1]) is shown in Figures. 18 and 19
respectively. This stress is maximum at the ends and at
[theta]=[90.sup.0] at both the interfaces. This is due to the mismatch
of longitudinal Poisson's ratios of fiber and matrix materials.
Matrix damage may occur at these locations due to this stress.
The longitudinal stresses at the plane of discontinuity on both
sides are shown in Figures. 20 and 21. It can be observed that the
stress is maximum at lower fiber (T300) on each side. This might be due
to the higher longitudinal stiffness T300 fiber than S-glass fiber. This
may cause separation of fiber and matrix and/ or the damage of matrix
material.
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Conclusions
The micromechanical behaviour of discontinuous hybrid FRP lamina
has been studied using finite element method. The Young's modulus
and Poison's ratios are predicted for 55% fiber volume fraction.
The stresses at the fiber-matrix interface are also computed. The
critical locations where the stresses are maximum are identified. The
reasons for the stresses and the mode of failure due to each stress are
stated.
This analysis can be further extended to find the effect of
variation of the individual fiber volume fractions in the composite and
to find the suitability of various combinations for a particular
application for better strength with minimum cost.
Nomenclature
[E.sub.1] = Young's modulus in fiber direction (z-direction)
[E.sub.2] = Young's modulus in in-plane transverse direction
(x-direction)
[E.sub.3] = Young's modulus in out-of-plane transverse direction
(y-direction)
[v.sub.12] = Poisson ratio in 1-2 plane
[v.sub.13] = Poisson ratio in 1-3 plane
[v.sub.23] = Poisson ratio in 2-3 plane
[G.sub.12] = Shear modulus in 1-2 Plane
[G.sub.13] = Shear modulus in 1-3 Plane
[G.sub.23] = Shear modulus in 2-3 Plane
[[sigma].sup.f.sub.n] = Normal stress in the fiber at the
interface
[[sigma].sup.m.sub.n] = Normal stress in the matrix at the
interface
[[tau].sup.f.sub.nc] = Shear stress in the fiber at the
interface.
[[tau].sup.m.sub.nc] = Shear stress in the matrix at the
interface.
[[sigma].sup.f.sub.c] = Circumferential stress in the fiber at
the interface
[[sigma].sup.m.sub.c] = Circumferential stress in the matrix
at the interface
[[sigma].sup.f.sub.1] = Fiber directional stress in the fiber at
the interface
[[sigma].sup.m.sub.1] = Fiber directional stress in the matrix at
the interface
[theta] = angle measured in CCW direction from bottom face of
unit cell for bottom interface and from top face of unit cell for
top interface
References
[1] Hashin, Z. and Rosen, B.W., 1964, "The elastic moduli of
fiber reinforced materials", Trans. ASME Journal of Applied
Mechanics, 31, pp. 223-232.
[2] Hashin, Z. 1965, "On elastic behavior of fiber-reinforced
materials of arbitrary transverse phase geometry", Journal of the
mechanics and physics of solids, 13, pp. 119-134.
[3] Whitney, J.M., 1967, "Elastic moduli of unidirectional
composites with anisotropic filaments", Journal of Composite
Materials, 1, pp.188-193.
[4] Chen, C.H. and Cheng, S., 1970, "Mechanical properties of
anisotropic Fiberreinforced Composites", Trans. ASME Journal of
Applied Mechanics, 37, pp. 186-189.
[5] Takashi Ishiwaka, Koyama, K. and Kobayashi, S., 1977,
"Elastic moduli of carbon-epoxy composites and carbon fibers",
Journal of Composite Materials, 11, pp. 332-344.
[6] Hashin, Z., 1983, "Analysis of composite materials--A
survey. Trans. ASME Journal of Applied Mechanics, 50, pp. 481-505.
[7] Hyer, M.W., 1998, "Stress Analysis of Fiber-Reinforced
Composite Materials", Mc. GRAW- HILL International edition.
[8] Mohana Rao, K., 1986, "Work Shop on Introduction to
Fiber-Reinforced Composites", NSTL.
[9] Sun, C.T. and Vaidya, R.S., 1996, "Prediction of composite
properties from a representative volume element", Composites
Science and Technology, 56, pp. 171-179.
[10] Morais, A.B., 2000, "Transverse moduli of
continuous-fiber-reinforced polymers", Composites Science and
Technology, 60, pp. 997-1002.
[11] Anifantis, N. K., 2000, "Micromechanical stress analysis
of closely packed fibrous composites", Composites Science and
Technology, 60, pp. 1241-1248.
[12] Li, S., 2000, "General unit cells for micromechanical
analyses of unidirectional composites", Composites: part A, 32, pp.
815-826.
[13] Yiping Qiu and Peter Schwartz, 1993, "Micromechanical
behaviour of Kevlar 149/S-Glass hybrid seven fiber microcomposites: I.
Tensile strength of the hybrid composite", Composites Science and
Technology, 47, pp. 289-301.
[14] Mishra and Mohanty, 2003, "Studies of mechanical
performance of biofiber/glass reinforced polyester hybrid
composites", Composites Science and Technology, 63, pp.1377-1385.
[15] Pahr, D.H. and Arnold, S. M., 2001, "The applicability of
the Generalized Method of Cells for analyzing discontinuously reinforced
composites", Composites Part B, 33, pp. 153-170.
[16] ANSYS Reference Manuals 2006.
T. Srinag (1), V. Bala Krishna Murthy (2), J. Suresh Kumar (3) and
G. Sambasiva Rao (2)
(1) Lecturer, Mech. Engg. Dept, K. L. College of Engineering,
Vaddeswaram, A.P, India, E-mail:
[email protected]
(2) Professor, Mech. Engg. Dept., P. V. P. Siddhartha Institute of
Technology, Vijayawada, A.P, India, E-mail:
[email protected]
(3) Associate Professor & Additional Controller of Exams,
J.N.T.U, Hyderabad, A.P, India.
Table 1: Properties of Constituents
S. No. Material E (GPa) v G (GPa)
1 T300 220.632-axial 0.2 (long. 8.963
Fiber 13.789-radial Plane) (long.
(Orthotropic) 0.25 (Tran. Plane)
Plane) 4.826
(Tran.
Plane)
2 S-Glass 85.495 0.2 --
Fiber
(Isotropic)
3 Epoxy 5.171 0.35 --
Matrix
(Isotropic)
Table 2: Validation of present FE model
T 300-epoxy S glass-epoxy
Ref [7] FEM Ref [7] FEM
Elasticity Elasticity
Theory Theory
[E.sub.1] (GPa) 124.897 124.879 49.835 49.827
Hybrid-epoxy
Ref [7] FEM
Rule of
Mixtures
[E.sub.1] (GPa) 87.073 87.355
Table 3: Predicted properties for discontinuous fiber
T300-epoxy S glass-epoxy Hybrid-epoxy
[E.sub.1] (GPa) 102.653 46.335 75.328
[v.sub.2] 0.28 0.25 0.26
[v.sub.3] 0.28 0.25 0.26
