ANALYTICAL AND FINITE ELEMENT BASED MICROMECHANICS FOR FAILURE THEORY OF COMPOSITES By SAI THARUN KOTIKALAPUDI A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2017
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ANALYTICAL AND FINITE ELEMENT BASED MICROMECHANICS FOR FAILURE THEORY OF COMPOSITES
By
SAI THARUN KOTIKALAPUDI
A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
Introduction to the Three-Phase Model ................................................................... 20 Halpin Tsai Formulation for Composite Properties ................................................. 21 Longitudinal and Hydrostatic Stress Equations ....................................................... 23
Longitudinal Shear Stress in the x-y plane.............................................................. 27
Longitudinal Shear Stress in the x-z plane.............................................................. 32 Biaxial tension/compression in y-z plane ................................................................ 37 Transverse Shear Equations .................................................................................. 43
3 FINITE ELEMENT ANALYSIS AND COMPARISON .............................................. 50
Modelling and analysis of Hexagonal RVE ............................................................. 50 Comparison with analytical model .......................................................................... 61
4 ANALYTICAL MODEL RESULTS AND DISCUSSION ........................................... 65
Results for Kevlar/Epoxy ......................................................................................... 65
2-2 Comparison of macro stresses with average micro stresses for normal Stress and in plane shear stress ................................................................................... 22
3-1 Properties of Kevlar/Epoxy used in the FEA ....................................................... 52
3-2 Coefficients of stiffness matrix obtained from unit strain analysis ....................... 56
3-3 Transverse strengths at various points for Kevlar/Epoxy (plane strain) .............. 60
3-4 Comparison of various transverse strengths for Kevlar/Epoxy (plane strain) ..... 63
3-5 Comparison of maximum principal stress for Kevlar/Epoxy (plane strain) .......... 63
3-6 Comparison of maximum von Mises stress for Kevlar/Epoxy (plane strain) ....... 63
3-7 Comparison of average of top 10% maximum principal stresses for Kevlar/Epoxy (plane strain) ................................................................................ 63
3-8 Comparison of average of top 10% von Mises stresses for Kevlar/Epoxy (plane strain)....................................................................................................... 63
3-9 Comparison of 10th percentile maximum principal stress for Kevlar/Epoxy (plane strain)....................................................................................................... 64
3-10 Comparison of 10th percentile maximum von Mises stress for Kevlar/Epoxy (plane strain)....................................................................................................... 64
4-1 Properties of Kevlar/Epoxy ................................................................................. 66
4-2 Strengths at various points for Kevlar/Epoxy (MPa) ........................................... 69
4-3 Properties of Carbon-T300/Epoxy-5208 ............................................................. 70
4-4 Predicted strengths of T300/5208/Carbon/Epoxy ............................................... 73
4-5 Comparison of strengths for Kevlar/epoxy including interface failure obtained using ADMM ....................................................................................................... 80
4-6 Comparison of strengths for Carbon/Epoxy including interface failure obtained using ADMM ........................................................................................ 81
7
4-7 Comparison of strengths for several composites with analytical model strengths ............................................................................................................. 85
4-8 %Difference of strengths for several composites relative to reference strengths ............................................................................................................. 85
8
LIST OF FIGURES
Figure page 1-1 Depiction of a RVE for the analytical model ....................................................... 16
1-2 Decomposition of macro stresses applied to an RVE of a fiber composite ......... 16
1-3 Macro stresses applied on the unit cell. (similar to π12 , π13 will be acting in the 13 plane and π23 will be acting in the 2-3 plane) .......................................... 17
1-4 Decomposition of applied state of macro stresses into five cases ...................... 18
2-1 Three-phase model ............................................................................................ 20
3-1 Representative volume element of a hexagonal unit cell .................................... 50
3-2 Coordinate system used in ABAQUS and principal coordinate system .............. 51
3-3 Sectional view and dimensions of the RVE ........................................................ 51
3-4 Meshed RVE, red bounded regions represent fiber and green unbounded region represents matrix ..................................................................................... 52
3-5 Element type used for meshing and analysis ..................................................... 53
3-6 Boundary conditions and loading in unit strain analysis (A) Direction 2 (B) direction 3 ........................................................................................................... 54
3-7 Schematic of the procedure followed to obtain Stiffness matrix.......................... 55
3-8 Initial and deformed hexagonal RVE under unit strain in A) 2nd Direction B) 3rd direction C) 2nd and 3rd direction..................................................................... 56
3-9 Schematic of procedure to plot a failure envelope in 2-3 plane .......................... 59
3-10 Failure envelopes of Kevlar/Epoxy in transverse direction obtained through unit strain analysis .............................................................................................. 60
3-11 Comparison of analytical and finite element model failure envelopes using maximum stress theory in the transverse plane (2-3 plane) ............................... 61
3-12 Comparison of analytical and finite element model failure envelopes using quadratic theory in the transverse plane (2-3 plane) .......................................... 62
4-1 Comparison of MMN and QQN failure envelopes of Kevlar/Epoxy on π1 β π2 plane ................................................................................................................... 66
9
4-2 Comparison of MMN and QQN failure envelopes of Kevlar/Epoxy in the π2 βπ3 plane .............................................................................................................. 67
4-3 Comparison of MMN and QQN failure envelopes of Kevlar/Epoxy for longitudinal shear in the 1-2 or 1-3 plane ........................................................... 67
4-4 Comparison of MMN and QQN failure envelopes of Kevlar/Epoxy subjected to both longitudinal and transverse shear stresses............................................. 68
4-5 Comparison of MMN and QQN failure envelopes of Kevlar/Epoxy for shear in longitudinal direction and stress in fiber direction ............................................... 68
4-6 Comparison of MMN and QQN failure envelopes of Carbon/Epoxy in 1-2 plane ................................................................................................................... 70
4-7 Comparison of MMN and QQN failure envelopes of Carbon/Epoxy in 2-3 plane ................................................................................................................... 71
4-8 Comparison of MMN and QQN failure envelopes of Carbon/Epoxy for shear in longitudinal directions ..................................................................................... 71
4-9 Comparison of MMN and QQN failure envelopes of Carbon/Epoxy subjected to both longitudinal and transverse shear stresses............................................. 72
4-10 Comparison of MMN and QQN failure envelopes of Carbon/Epoxy for longitudinal shear and normal stress in fiber direction ........................................ 72
4-11 Interface effects on failure envelopes for Kevlar/Epoxy in 1-2 plane using maximum stress theory ...................................................................................... 74
4-12 Interface effects on failure envelopes for Kevlar/Epoxy in 1-2 plane using quadratic theory .................................................................................................. 75
4-13 Interface effects on failure envelopes for Kevlar/Epoxy in 2-3 plane using maximum stress theory ...................................................................................... 75
4-14 Interface effects on failure envelopes for Kevlar/Epoxy in 2-3 plane using quadratic theory .................................................................................................. 76
4-15 Interface effects on failure envelopes subjected to both longitudinal and transverse shear stresses using maximum stress theory ................................... 76
4-16 Interface effects on failure envelopes subjected to both longitudinal and transverse shear stresses using quadratic theory .............................................. 77
4-17 Interface effects on failure envelopes for Carbon/Epoxy in 1-2 plane using maximum stress theory ...................................................................................... 77
10
4-18 Interface effects on failure envelopes for Carbon/Epoxy in 1-2 plane using quadratic theory .................................................................................................. 78
4-19 Interface effects on failure envelopes for Carbon/Epoxy in 2-3 plane using maximum stress theory ...................................................................................... 78
4-20 Interface effects on failure envelopes for Carbon/Epoxy in 2-3 plane using quadratic theory .................................................................................................. 79
4-21 Interface effects on failure envelopes for Carbon/Epoxy for envelopes subjected to both longitudinal and transverse shear stresses ............................ 79
4-22 Interface effects on failure envelopes for Carbon/Epoxy longitudinal shear and normal stress in fiber direction using maximum stress theory ..................... 80
11
LIST OF ABBREVIATIONS
ADMM Analytical Direct Micromechanics Method
DMM
FEA
PBC
RVE
Direct Micromechanics Method
Finite Element Analysis
Periodic boundary Conditions
Representative Volume Element
12
Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science
ANALYTICAL AND FINITE ELEMENT BASED MICROMECHANICS FOR FAILURE
THEORY OF COMPOSITES
By
Sai Tharun Kotikalapudi
December 2017
Chair: Bhavani V. Sankar Major: Mechanical Engineering
An analytical method using elasticity equations to predict the failure of a
unidirectional fiber-reinforced composite under multiaxial stress is presented. This
technique of calculating micro-stresses using elasticity equations and estimating the
strengths of a composite is based on the Direct Micromechanics Method (DMM).
Prediction of failure using phenomenological failure criteria such as Maximum Stress,
Maximum Strain and Tsai-Hill theories have been prevalent in the industry. However,
DMM has not been used in practice due to its prohibitive computational effort such as
the finite element analysis (FEA). The present method replaces the FEA in DMM by
analytical methods, thus drastically reducing the computational effort.
A micromechanical analysis of unidirectional fiber-reinforced composites is
performed using the three-phase model. A given state of macro-stress is applied to the
composite, and the micro-stresses in the fiber and matrix phases and along the fiber-
matrix interface are calculated. The micro-stresses in conjunction with failure theories
for the constituent phases are used to determine the integrity of the composite. The
analytical model is first verified by comparing with results from finite element based
micro-mechanics. Then, it is used to study the failure envelopes of various composites.
13
The effects of fiber-matrix interface on the strength of the composite is studied. The
results are compared with those available in the literature. It is found that the present
analytical Direct Micro-Mechanics (ADMM) predicts the strength of composites
reasonably well.
14
CHAPTER 1 INTRODUCTION
Literature Review
With the growing application of fiber composites, and with the tremendous
progress in low-cost manufacturing of composite structures, e.g., wind turbine blades,
there is a need to develop efficient predictive methodologies for the behavior of
composites. This should include probabilistic methods to aid in nondeterministic
optimization tools used in design. While methods to predict stiffness properties are well
established, methods to predict failure and fracture properties are still evolving.
Computational material science is the new field of study which attempts to use modern
computational analysis tools to perform multiscale analysis beginning from atomic scale
all the way up to structural scale.
While large scale computational methods are being advanced, there is always a
need for simple and efficient analytical methodologies. This is especially true for
strength prediction and failure behavior of composites. Currently available methods for
strength prediction either use numerical simulations such as finite element analysis [1]
or very simple methods such as mechanics of materials models (MoM) [2]. The former
can be expensive and time consuming, and the latter is only an approximate estimate to
be useful in practical design applications. The current study is aimed at developing an
analytical micromechanics method that is better than MoM models, but still not as
complicated as FEA based micromechanics. To this end we use the principles of Direct
Micromechanics Method [3] developed in 1990s in conjunction with the classical three-
phase elasticity model for unidirectional fiber composites [4].
15
Several failure theories for composite materials are available in the literature.
Majority of them use experimental results along with an empirical (phenomenological)
approach to plot the failure envelopes.
Contemporary failure theories, developed for unidirectional composites such as
Maximum Stress Theory, Maximum Strain Theory and Tsai-Hill Theory have been
thoroughly studied and implemented by various researchers and design engineers.
Direct micromechanics method (DMM) has a propitious approach to predict failure
strengths for an orthotropic composite material. First proposed by Sankar, it has been
widely used to analyze various phenomenological failure criteria, e.g., Marrey and
Sankar [5], Zhu et al. [6], Stamblewski et al. [7], karkakainen and Sankar. DMM
encompasses analytical techniques which is an alternative approach for physical testing
and experimental procedures. A micromechanical model is subjected to multiple macro
stresses which produce micro stresses in each element of the finite element model. The
micro stresses are used to devise a failure envelope considering various failure criteria
for fiber and matrix such as maximum principal stress theory and von Mises criterion.
The interface of fiber/matrix played a pivotal role in the failure of envelopes which was
also considered in DMM. Interfacial tensile stress and interfacial shear stress in the
composite are very sensitive properties that depend on various factors.
Research Scope
In this section, the research procedure followed will be discussed in detail. The
RVE for the analytical approach has been modeled as circular fiber surrounded by an
annular region of matrix. The description of analytical model is presented in Chapter 2.
16
Figure 1-1. Depiction of a RVE for the analytical model.
The applied macro stresses on the composite are denoted by the Cauchy stress
tensor [ππ₯ ππ¦ ππ§ ππ¦π§ ππ₯π¦ ππ₯π§], Note that throughout this study the principal material
coordinate system of the fiber composite will be denoted either by the standard 1-2-3
coordinates or x-y-z coordinates used in the commercial finite element software
ABAQUS. The two normal stresses ππ¦ and ππ§ can be decomposed into two cases
hydrostatic stress state such that ππ¦ = ππ§ = ππ» and a biaxial tension/compression
such that ππ¦ = βππ§. The chart in Figure1-2 depicts various analysis of the stress cases
needed to complete the DMM.
Figure 1-2. Decomposition of macro stresses applied to an RVE of a fiber composite
Applied stress
Normal stress
Axail and hydrostaic
Biaxial tension and compresion
Shear stress
Transverse
Shear YZ
Longitudinal Shear XZ
Longitudinal Shear XY
Fiber
Matrix
Composite
17
Application of the stress field on the Representative Volume Element (RVE) of
the composite through individual cases generates macro strains. Every case has
distinct analytical equations for calculating the micro stresses in the fiber and matrix
phases. The load factors are calculated based on the type of failure criterion used either
maximum stress or some form of quadratic failure criterion, e.g. von Mises for isotropic
materials.
Figure 1-3. Macro stresses applied on the unit cell. (similar to π12 , π13 will be acting in the 13 plane and π23 will be acting in the 2-3 plane)
π12
π12
π12
π12
π1
π3
π3
π2 π2
π1
18
Figure 1-4. Decomposition of applied state of macro stresses into five cases A) Hydrostatic and longitudinal stress; B) Biaxial tension and compression; C) Shear in 2-3 plane; D) Shear in 1-2 plane; E) Shear in 1-3 plane
π23
π23 π23
π1
π1
ππ» ππ»
ππ»
ππ» =(π2 + π3)
2
(A) Case i
πππΆ
βπππΆ
βπππΆ
πππΆ =(π2 β π3)
2
(B) Case ii
(E) Case v
π13
(C)Case iii
π23
(D)Case iv
π12
19
Figure 1-5. Schematic depiction of DMM followed to obtain failure envelopes
Figure 1-3 shows the six stresses acting on the unit cell of the composite which
are divided into five cases as shown in figure 1-4. For each case the micro stresses are
calculated at several locations using stress equations which is explained in detail in
chapter 2. Figure 1-5 portrays a schematic of the process followed in Direct
Micromechanics Method (DMM) to obtain the failure envelopes and strengths. Chapter
2 elaborates the stress equations used for micromechanical analysis and validation of
using energy methods. The analytical equations employed are further validated in
Chapter 3 through unit strain analysis in finite element analysis software ABAQUS.
Chapter 4 consists of the results obtained from the analytical model wherein a thorough
study is performed by considering two different materials i.e. isotropic and transversely
isotropic. A meticulous comparison on different strengths of composites with present
data is included in Chapter 4. A study is performed in Chapter 4 to understand the effect
of fiber volume fraction on the strength properties for few materials.
Macro stressesIsolating
stresses/formulating stress equations
Micro stresses
Eigen values/principal
stressesLoad Factors
Failure envelopes and strengths
20
CHAPTER 2 ANALYTICAL EQUATIONS
Introduction to the Three-Phase Model
In this section, the three-phase concentric cylinder composite assemblage model
is described. The three phase model proposed by Christensen [8] had been
successfully used in the past for predicting the elastic constants of fiber composites,
e.g. Flexible-resin/glass-fiber composite lamina [9]. In the present study we investigate
the use of three phase model to predict the strengths of unidirectional fiber composites.
Figure 2-1. Three-phase model
The model shown in Figure 2-1. consists of a single cylindrical inclusion (fiber)
embedded in a cylindrical annular region of matrix material. The composite cylinder is in
turn embedded in infinite medium properties of which are equal to that of the composite
material studied. The fiber and the composite are assumed transversely isotropic and
the matrix is isotropic. This enables us to use a polar coordinate system for the analysis.
Furthermore, the entire assemblage is in a state of generalized plane strain as the strain
2
1
3
r b
β
a Fiber
Matrix
Composite
21
ν1 must be uniform and the same in all three phases. Thus, the problem becomes a
plane problem. Since we are using the model to calculate the micro-stresses in the fiber
and matrix for a give macro-stress state, the elastic constants of all three phases must
be available for the stress analysis. As a first step, the Rule of Mixtures and Halpin-Tsai
equations are used to estimate the elastic constants of the composite. In each analysis
energy equivalence verifies the validity of the input composite elastic constants as
explained in subsequent sections.
Halpin Tsai Formulation for Composite Properties
Halpin-Tsai equation is a widely used semi-empirical formulation for transverse
moduli πΈ2, πΊ12 of unidirectional fiber composites. The general form of Halpin-Tsai
equations for a property, say π, is as follows:
ππ = ππ (1 + ππππ
1 β πππ) (2-1)
Where,
π = ((ππ/ππ) β 1
(ππ/ππ) + π) (2-2)
ππ: Property of the composite
ππ: Property of the fiber
ππ: Property of the matrix
π: Curve fitting parameter
ππ: Fiber volume fraction
The above formula was obtained using curve fitting the result for square array of
circular fibers. It is found that for π = 2, an excellent fit is obtained for transverse
modulus πΈ2. Whereas for shear modulus πΊ12, the value π = 1 was in excellent
22
agreement with the Adams and Doner [10] solution. In both cases a fiber volume
fraction of ππ = 0.55 is used.
Since the analytical model in this paper has a circular fiber in an annular region
of matrix, the curve fitting parameter has been adjusted to formulate more accurate
predictions of moduli - πΈ2, πΊ12. The curve fitting parameter π was estimated for the
present case by comparing the applied macro stresses in each case to the volume
average of the corresponding micro stresses. The modified values of π are: π = 1.16 for
transverse Youngβs modulus πΈ2, π = 1 for transverse shear modulus πΊ23 and π = 1 for
longitudinal or axial shear modulus πΊ12.
Table 2-1. Comparison of macro stresses with average micro stresses for longitudinal shear stress
Figure 4-6. Comparison of MMN and QQN failure envelopes of Carbon/Epoxy in 1-2
plane
71
Figure 4-7. Comparison of MMN and QQN failure envelopes of Carbon/Epoxy in 2-3
plane
Figure 4-8. Comparison of MMN and QQN failure envelopes of Carbon/Epoxy for shear
in longitudinal directions
72
Figure 4-9. Comparison of MMN and QQN failure envelopes of Carbon/Epoxy subjected
to both longitudinal and transverse shear stresses
Figure 4-10. Comparison of MMN and QQN failure envelopes of Carbon/Epoxy for
longitudinal shear and normal stress in fiber direction
Figures 4-6 to 4-10 shows the failure envelopes of Carbon/Epoxy for various
combinations of normal and shear stresses. The strength properties of Carbon/epoxy
73
have been extracted from the above plots and are tabulated in table 4-4. The strengths
obtained showed excellent comparison with the properties from principles of composite
material mechanics by Gibson [18].
Table 4-4. Predicted strengths of T300/5208/Carbon/Epoxy
Failure Criteria ππΏ+ ππ
+ ππΏπ πππ
MMN 1456 49 57 43
QQN 1456 62 33 31
Effects of Interface
In this section, effects of interface on different strength properties and envelopes
will be discussed. Fiber/ matrix interface plays a pivotal role in determining the strength
of the composites. Two failure criteria are used to study the effects of interface on
strengths. It is assumed that compressive stresses will not affect the interface.
Maximum stress criteria used is discussed as follows:
The normal stress acting at the interface is ππ i.e. radial tensile stress acting at
the interface.
Interfacial tensile (ππ > 0)
ππ < ππΌπ (4-4)
Interfacial shear
β(πππ2 + πππ₯
2 ) < ππΌπ (4-5)
Where ππΌπ is interfacial tensile strength and ππΌπ is interfacial shear strength
74
The quadratic interface theory is explained below:
ππ2
ππΌπ2 +
(πππ2 + πππ₯
2 )
ππΌπ2 = 1 (4-6)
Interface strength determination through experiments is inaccurate and difficult
as very limited data is available. Significant amount of work was done by Huges [19] on
carbon/Epoxy properties. Many factors were considered and there was no simple
conclusion found about the nature of Carbon/Epoxy interface. It was found out that the
interface has a very high stress concertation. The following plots demonstrates the
effect of interface on failure envelopes.
Figure 4-11. Interface effects on failure envelopes for Kevlar/Epoxy in 1-2 plane using
maximum stress theory
75
Figure 4-12. Interface effects on failure envelopes for Kevlar/Epoxy in 1-2 plane using
quadratic theory
Figure 4-13. Interface effects on failure envelopes for Kevlar/Epoxy in 2-3 plane using
maximum stress theory
76
Figure 4-14. Interface effects on failure envelopes for Kevlar/Epoxy in 2-3 plane using
quadratic theory
Figure 4-15. Interface effects on failure envelopes subjected to both longitudinal and
transverse shear stresses using maximum stress theory
77
Figure 4-16. Interface effects on failure envelopes subjected to both longitudinal and
transverse shear stresses using quadratic theory
Figure 4-17. Interface effects on failure envelopes for Carbon/Epoxy in 1-2 plane using
maximum stress theory
78
Figure 4-18. Interface effects on failure envelopes for Carbon/Epoxy in 1-2 plane using
quadratic theory
Figure 4-19. Interface effects on failure envelopes for Carbon/Epoxy in 2-3 plane using
maximum stress theory
79
Figure 4-20. Interface effects on failure envelopes for Carbon/Epoxy in 2-3 plane using
quadratic theory
Figure 4-21. Interface effects on failure envelopes for Carbon/Epoxy for envelopes
subjected to both longitudinal and transverse shear stresses using maximum stress theory
80
Figure 4-22. Interface effects on failure envelopes for Carbon/Epoxy longitudinal shear
and normal stress in fiber direction using maximum stress theory
Figures 4-11 to 4-22 shows the failure envelopes of Carbon/Epoxy for various
combinations of normal and shear stresses. The strength properties of Kevlar/Epoxy
and Carbon/epoxy have been extracted from the above plots and are tabulated in tables
4-5 and 4-6. The strengths obtained showed acceptable comparison with the properties
obtained through finite element micromechanical analysis by Zhu, Marrey and Sankar
[20].
Table 4-5. Comparison of strengths for Kevlar/epoxy including interface failure obtained using ADMM
Failure Criteria ππΏ+ ππ
+ ππΏπ πππ
MMN 1573 53 56 44
QQN 1592 75 32 39
MMM 1573 23.4 22.55 20.38
QQQ 1592 23.4 18.15 20.38
81
Table 4-6. Comparison of strengths for Carbon/Epoxy including interface failure obtained using ADMM
Failure Criteria ππΏ+ ππ
+ ππΏπ πππ
MMN 1456 49 57 43
QQN 1456 62 33 31
MMM 1456 38.6 36 36.5
QQQ 1456 39 37 37
Volume fraction analysis
In this section, the variation of strengths with volume fraction will be discussed for
composites Kevlar/Epoxy and Carbon /Epoxy. Figures 4-23 to 4-28 depicts the variation
of ππΏ+, ππ
+, ππΏπ , πππ with ππ.
Interface failure strengths are also considered and as explained in Chapter 3,
MMM refers to maximum stress theory for fiber, matrix and interface.
Figure 4-23. Variation of longitudinal shear strength with volume fraction in
Kevlar/Epoxy
82
Figure 4-24. Variation of Transverse tensile strength with volume fraction in
Kevlar/Epoxy
Figure 4-25. Variation of Transverse shear strength with volume fraction in Kevlar/Epoxy
83
Figure 4-26. Variation of longitudinal shear strength with volume fraction in
Carbon/Epoxy
Figure 4-27. Variation of transverse tensile strength with volume fraction in
Carbon/Epoxy
84
Figure 4-28. Variation of transverse shear strength with volume fraction in
Carbon/Epoxy
Summary
In this section, a detailed comparison of strengths obtained from the analytical
model with reference strengths extracted from Gibson [21] will be discussed.
Table 4-7 shows the comparison of properties obtained from maximum principal stress
theory and quadratic failure theory with the properties available in literature. von Mises
theory was used for an isotropic fiber and Hashinβs [22] criteria was used for
transversely isotropic fibers.
Table 4-8 shows the percentage difference with reference strengths. It can be
noticed that the model showed an excellent comparison with ππΏ+ in both MMN and QQN
except for boron/aluminum and E-glass/epoxy. In the case of boron/aluminum, yielding
plays a crucial role in determining the strengths which the present model will not
consider.
85
Table 4-7. Comparison of strengths for several composites with analytical model strengths
Reference MMN QQN
Composite ππ ππΏ+ ππ
+ ππΏπ ππΏ+ ππ
+ ππΏπ ππΏ+ ππ
+ ππΏπ
Boron/5505 Boron/epoxy
0.5 1586 62.7 82.7 1593 46 65 1593 63 38
AS/3501 Carbon/epoxy
0.6 1448 48.3 62.1 1456 49 57 1456 62 33
T300/5208 Carbon/Epoxy
0.6 1488 44.8 62.1 1456 48 57 1456 62 33
IM7/8551-7 Carbon/Epoxy
0.6 2578 75.8 --- 2492 67 69 1492 86 42
AS4/APC2 carbon/PEEK
0.58 2060 78 157 2169 63 68 2169 81 41
B4/6061 Boron/Aluminum
0.5 1373 118 128 400 96 95 400 115 57
Kevalr49/epoxy aramid/epoxy
0.6 1379 27.6 60 1332 38 50 1332 45 31
Scocthply1002 E-glass/epoxy
0.45 1100 27.6 82.7 440 35 36 440 44 23
E-glass/470-36 E-glass/Vinyl ester
0.3 584 43 64 478.5 43 40 478.5 50 24
Table 4-8. %Difference of strengths for several composites relative to reference
strengths MMN QQN
Composite ππ ππΏ+ ππ
+ ππΏπ ππΏ+ ππ
+ ππΏπ
Boron/5505 Boron/epoxy
0.5 0 1 21 0 -31 54
AS/3501 Carbon/epoxy
0.6 0 -1 8 0 -28 47
T300/5208 Carbon/Epoxy
0.6 2 -7 8 2 -38 47
IM7/8551-7 Carbon/Epoxy
0.6 3 10 --- 3 -13 ---
AS4/APC2 carbon/PEEK
0.58 -5 18 57 -5 -5 74
B4/6061 Boron/Aluminum
0.5 67 9 26 67 -7 55
Kevalr49/epoxy aramid/epoxy
0.6 3 -38 17 3 -63 48
Scothply1002 E-glass/epoxy
0.45 60 -37 56 60 -59 72
E-glass/470-36 E-glass/Vinyl ester
0.3 18 0 38 18 -16 63
86
ππ+ comparison has been acceptable for all the composites and maximum
principal stress theory is found out to be more suitable to calculate transverse tensile
strength. On the other hand, there is a lot of discrepancy for ππΏπ but maximum stress
theory proved to be best considering the cases of carbon fibers. Predicted strengths for
carbon-IM7/epoxy have also been presented in table 4-7. Overall maximum principal
stress theory overlooked quadratic theories for predicting strengths through Direct
Micromechanics Method (DMM).
87
CHAPTER 5 CONCLUSIONS AND FUTURE WORK
The three-phase composite model for unidirectional composites which is
originally proposed for homogenization is extended for a method called Analytical Direct
Micro-Mechanics (ADMM) to predict the strength properties of composites. In the three-
phase model the fiber is surrounded by an annular region of matrix and the fiber-matrix
assemblage is embedded in an infinite medium of composite. The elastic constants of
the composite are evaluated using a modified Halpin-Tsai type equations. In the ADMM
the micro-stresses are calculated in the fiber and matrix phases for a given macro-
stress applied to the composite. The micro-stresses in conjunction with failure theories
for fiber, matrix, and fiber/matrix interface are used to determine the failure of the
composites.
The ADMM is evaluated by comparing the results with that of finite element
based micromechanics. The ADMM results compare reasonably well with FEA based
DMM results for failure envelopes. Then the ADMM is extended to various composite
systems and compared with available results for strength values. The ADMM is able to
predict the strength reasonably well in majority of the cases. The strength in the fiber
direction compares well. The transverse strength properties are different in the 2 and 3
directions in the FEA model because of the hexagonal unit cell. However, the three-
phase model being axisymmetric predicts the same strength in the 2 and 3 directions,
which is closer to practical fiber composites. Since it is based on elasticity solution, it is
much faster than FEA based micro-mechanics. Due to its speed, ADMM can be used in
understanding the effects of variability of constituent properties on the composite
88
strength. The model will be very much suited for non-deterministic design of composite
structures.
Although present failure theories have been partially successful, many theories
have to be modified to keep up with the growing composite technology. The need for
unprecedented methods for predicting failure strengths is everlasting.
The normal strengths obtained from Direct Micromechanics Method (DMM) were
reasonable and are in good agreement with the properties of the composites currently
used in the industry. The modified Halpin-Tsai formulations played a productive role in
calculating the effective composite properties for the three-phase model. Although there
were slight discrepancies while comparing ADMM and FEA in principal direction 2 and
3, it still validates the analytical model since symmetry is not observed in a hexagonal
RVE. The unit strain finite element analysis of a hexagonal RVE validated the analytical
model. Interface strength has no effect on longitudinal tensile strength.
Future work. The newly developed Analytical Direct Micromechanics Method
(ADMM) has a lot of potential for future research. Since a preliminary model was
constructed, modifications can be done to consider buckling of fibers under
compression. The ADMM model with correction response surfaces can be a useful
analytical tool wherein a polynomial function of design variables is used to rectify the
difference from the experimental strength. It is also amenable to probabilistic
micromechanics, in which the effect of variability in fiber and matrix properties on the
stiffness and strength properties can be studied. The model can be extended to fracture
analysis (crack propagation in matrix and interface). Since there was a relatively
noticeable difference in longitudinal shear with properties available in literature, more
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modifications to the model should be made. Thus, the ADMM paves the way for detailed
analysis of the interface.
90
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[15] Zhu, H., B. V. Sankar, and R. V. Marrey. "Evaluation of failure criteria for fiber composites using finite element micromechanics." Journal of composite materials 32.8 (1998): 766-782
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BIOGRAPHICAL SKETCH
Sai Tharun Kotikalapudi was born in Hyderabad, India, in 1991. He grew up in
various southern states of India. He received a Bachelor of Technology in the field of
mechanical engineering from SASTRA University, India. From there he proceeded to
work for a year at Tata Consultancy Services Engineering and Infrastructure Services,
in Bangalore, India. He has worked under the guidance of personnel from various
internationally recognized companies such as Johnson & Johnson and Beckman
Coulter. His work involved developing and redesigning medical equipment in
SolidWorks. He defended his M.S. thesis in October 2017.