Materi ke-13
PERILAKU MEKANIK PADA KOMPOSIT Nurun Nayiroh, M.Si
MATERIAL KOMPOSIT
Tensile Strength pada Komposit yang Diperkuat dengan Fiber yang Unidirectional (Longitudinal) Ec = Ef V f + Em Vm σc = σf V f + σm Vm Isostrain Condition : loading parallel to fiber direction Fiber & Matrix – elastic case Modulus Ec : works reasonably well Strength : does not work well σc Why? : intrinsic property (microstructure insensitive) : extrinsic property (microstructure sensitive) Factors sensitive on strength of composite - Fabrication condition determining microstructure of matrix - Residual stress - Work hardening of matrix - Phase transformation of constituents
Stage III : fiber & matrix – plastic Strength σc = σ′f V f + σ′m Vm Modulus
UTS
dσ dσ Ec = f V f + m Vm d ε f ε dεm ε σcu = σfu V f + σ′m Vm
σfu : ultimate tensile strength of fiber σ′m : flow stress of matrix at the fracture strain of fiber
Effect of Fiber Volume Fraction on Tensile Strength (Kelly and Davies, 1965) Assumption : Ductile matrix ( ε f ,fiber
< ε f ,matrix ) work hardens.
All fibers are identical and uniform. → same UTS Jika serat retak, sebuah matriks menjadi mengeras mengimbangi kehilangan beban-daya dukung. Agar memiliki penguatan komposit dari serat, UTS of composite
UTS of matrix after fiber fracture
σcu = σfu V f + σ′m (1 − Vf ) ≥ σmu (1 − Vf ) fraksi volum fiber minimum: σmu − σ′m Vf ≥ V min = σfu + σmu − σ′m As σ ↓, ↑. Vmin fu As (σmu − σ′m ) ↑, Vmin ↑. degree of work hardening
In order to be the strength of composite higher than that of monolithic matrix, σcu = σfu V f + σ′m (1 − Vf ) ≥ σmu UTS of pure matrix
Critical Fiber Volume Fraction
− ′ Vf ≥ Vcrit = σmu σ m − σfu σ′m As σfu ↓, As
(σ
mu
Vcrit ↑.
− σ′m ) ↑, Vcrit ↑.
degree of work hardening
Note that V crit > V min always! (∵σmu > 0 )
8-2. Kuat Tekan pada Komposit yang diperkuat Fiber yang Searah Kompresi Komposit yang diperkuat Fiber Fiber – merespon elatisitas kolom dalam tekanan. Kegagalan komposit terjadi dengan menekuknya fiber.
Tekuk terjadi ketika kolom menipis di bawah tekanan menjadi tidak stabil terhadap pergerakan lateral bagian tengah. Tegangan kritis sesuai dengan kegagalan tekuk, 2
σc =
π2E d 16 l
where d is diameter, l is length of column.
2 Types of Compressive Deformation 1) In-phase Buckling : melibatkan deformasi geser pada matrix Gm = Em ∝ Gm (or Em ) Vm 2(1 + νm ) Vm Em Q for isostropic matrix, Gm = 2(1 + ν m ) → terutama pada fraksi volum fiber yang besar.
σc =
2) Out-of-phase Buckling : melibatkan kompresi transversal dan tegangan pada matrix dan fiber 1/ 2
V f Em E f σc = 2 V f 3Vm
∝ (Em ⋅E f )1/ 2
→ terutama pada fraksi volum fiber rendah. Faktor-faktor yang mempengaruhi kekuatan kompresi: Gm , Em Ef Vf Interfacial Bond Strength : poor bonding → easy buckling
8-3. Model-model patahan (Fracture) pada Komposit. 1. Single and Multiple Fracture Generally, ε f ,fiber ≠ ε f ,matrix Ketika komponen yang lebih rapuh (brittle) patah, beban yang dibawa oleh komponen rapuh akan dilemparkan ke komponen yang ulet (ductile). Jika komponen yang ulet tidak tahan terhadap beban tambahan ini → Single Fracture Jika komponen yang ulet dapat menanggung beban tambahan ini → Multiple Fracture
1) Single Fracture - terutama pada fraksi volum fiber yang tinggi - semua fiber dan matrix patah pada bidang yang sama - kondisi untuk single fracture σfu V f > σmu Vm − σ′m Vm
stress beared by fiber additional stress which can be supported by matrix
dimana σ′m : tegangan matrix yang sesuai dengan regangan patah fiber 2) Multiple Fracture - predominant at low fiber volume fraction - fibers and matrix are fractured in different planes - condition for multiple fracture
σfu V f < σmu Vm − σ′m Vm
2. Debonding, Fiber Pullout and Delamination Fracture Fracture Process : crack propagation
l
Discontinuous Fiber Reinforced Composite
If distance from crack plane to fiber end < lc 2 → Debond & Pullout
( lc : critical length )
→ Good for toughness If distance from crack plane to fiber end > lc 2 → Fiber Fracture
→ Good for strength
Fracture of Continuous Fiber Reinforced Composite Patahan fiber pada bidang retak atau posisi lain yang tergantung pada posisi cacat. ↓ Pullout of fibers For max. fiber strengthening → fiber fracture is desired. For max. fiber toughening → fiber pullout is desired. Analysis of Fiber Pullout Assumption : Single fiber in matrix rf : fiber radius l : fiber length in matrix σ f : tensile stress on fiber τi : interfacial shear strength
τi σf
Force Equilibrium πrf2σf = 2πrf τil πrf2σfu = 2πrf τilc σfu = lc 2τi r f
( lc : critical length of fiber )
σfu = lc = lc 4 τi 2r f d
1) Condition for fiber fracture, πrf2σ f < 2πrf τil
If l > lc →
σfu l 1 < = 4τi d 2rf
2) Condition for fiber pullout, πrf2σf > 2πrf τil If l < lc →
σfu l 1 ≥ = 4τi d 2rf
Fracture Process of Fiber Reinforced Composites Real fibers - non-uniform properties 3 steps of fracture process 1) Fracture of fibers at weak points near fracture plane : 2) Debonding of fibers : 3) Pullout of fibers : Wp
Wd
Load
W fracture = W d + W p
WP Wd
Displacement
Energy Required for Fracture & Debonding σ 2fu π d2 ⋅ x Wd = ⋅ Ef 24 elastic strain E.
x : debond length
volume
Energy Required for Pullout Let k : distance (lekat) of a broken fiber from crack plane x : pullout distance at a certain moment τi : interfacial shear strength
lc 0 < k < 2
Force to resist the pullout = τi πd(k − x ) fiber contact area
Energy to pullout a distance dx = τiπd(k − x )dx
Total energy(work) to pullout a fiber for distance k k τiπdk 2 W p = τiπd(k − x )dx = 0 2 Average energy to pullout per fiber(considering all fibers with different k, 0 < k < lc ) 2 τiπdl2c τi πdk 2 1 ∴ W p,ave = dk = 0 2 24 lc 2
∫
∫
lc 2
Fracture of Discontinuous Fiber Reinforced Composite l If a fiber is located within a distance, ± c , from crack plane, → pullout 2 Probability for pullout of a fiber with length, l ≈ lc l Average energy to pullout per fiber with length, l 2
l τ πdl Wp,ave = c i c l 24 probability for pullout
energy required for pullout
Energy for Fiber Pullout vs Fiber Length(l)
If l < lc , fiber pullout distance increases with increasing length l. → Wp increases,with increasing length l. Wp ∝ l2 If l > lc , fiber fracture tendency increases with increasing length l. 1 → Wp decreases, with increasing length l. W p ∝ Q lc = constant l W becomes maximum, when l ≈ l .
(
p
)
c
As Wd << Wp
Wfracture = Wd + Wp ≈ Wp Advantage of Composite Material: can obtain strengthening & toughening at the same time Toughening Mechanism in Fiber Reinforced Composite 1) Plastic deformation of matrix - metal matrix composite 2 Vm Energy of fracture ∝ d d : fiber diameter Vf 2) Fiber pullout d Energy of fracture ∝ τi 3) Crack deflection (or Delamination) - ceramic matrix composite Cook and Gordon, Stresses distribution near crack tip
σ
σyy
σxx
σ
If σxx> interfacial tensile strength → delamination → crack deflection
Delamination Fracture in Laminate Composite Fatigue
→ debonding at interface
Fracture
→ repeated crack initiation & propagation
8-4. Statistical Analysis of Fiber Strength Real fiber : nonuniform properties → need statistical approach Brittle fiber (ex. ceramic fibers) - nonuniform strength Ductile fiber (ex. metal fibers) - relatively uniform strength Strength of Brittle Fiber → dependent on the presence of flaws → dependent on the fiber length : "Size Effect“ Weibull Statistical Distribution Function
(
f (σ ) = Lαβσ β−1 exp − Lασβ
)
f (σ ) : probability density function Probability that the fiber strength is between and
α, β : statistical parameters L
.
: fiber length
σ
σ + dσ
8-6. Fatigue of Composite Materials Fatigue Failure in Homogeneous Monolithic Materials → Initiation and growth of a single crack perpendicular to loading axis. Fatigue Failure in Fiber Reinforced Laminate Composites Pile-up of damages - matrix cracking, fiber fracture, fiber/matrix debonding, ply cracking, delamination → Crack deflection (or Blunting) → Reduction of stress concentration A variety of subcritical damage mechanisms lead to a highly diffuse damage zone.
Constant-stress-amplitude Fatigue Test Damage Accumulation vs Cycles
Crack length in homogeneous material - accelerate (∵ increase of stress concentration) Damage (crack density) in composites - accelerate and decelerate (∵ reduction of stress concentration)
S-N Curves of Unreinforced Plolysulfone vs Glassf/Polysulfone, Carbonf/Polysulfone
Carbon Fibers : higher stiffness & thermal conductivity → higher fatigue resistance S-N Curves of Unidirectional Fiber Reinforced Composites (B/Al, Al2O3/Al, Al2O3/Mg)
Fatigue of Particle and Whisker Reinforced Composites For stress-controlled cyclic fatigue or high cycle fatigue, particle or whisker reinforced Al matrix composites show improved fatigue resistance compared to Al alloy, which is attributed to the higher stiffness of the composites. For strain-controlled cyclic fatigue or low cycle fatigue, the composites show lower fatigue resistance compared to Al alloy, which is attributed to the lower ductility of the composites. Particle or short fibers can provide easy crack initiation sites. The detailed behavior can vary depending on the volume fraction, shape, size of reinforcement and mostly on the reinforcement/matrix bond strength.
Fatigue of Laminated Composites Crack Density, Delamination, Modulus vs Cycles i) Ply cracking ii) Delamination iii) Fiber fatigue
Modulus Reduction during Fatigue Ogin et al. Modulus Reduction Rate n 2 1 dE σmax − =A 2 E0 dN E0 (1 − E / E0 ) where E : current modulus E0 : initial modulus σmax: peak fatigue stress 1 dE log − E0 dN
vs
σ σmax σm
time
σmin
N : number of cycles A, n : constants
σ2 log 2 max plot E0 (1 − E / E0 )
→ linear fitting
Integrate the equation to obtain a diagram relating modulus reduction to number of cycles for different stress levels. → used for material design
8-7. Thermal Fatigue of Composite Materials Thermal Stress Tegangan Thermal terjadi di dalam material komposit karena secara umum adanya perbedaan koefisien ekspansi termal (α) yang luas pada reinforcement dan matrix. Perlu ditekankan bahwa tegangan termal dalam komposit akan timbul jika suhu berubah secra seragam di seluruh volume komposit. σ ∝ ∆α ⋅ ∆T
Thermal Fatigue (Kelelahan Termal) Ketika suhu berulang kali berubah, tegangan termal akan menghasilkan kelelahan termal, karena tegangan yang berputar adalah asal dari adanya termal. Kelelahan termal dapat menyebabkan retak pada matriks yang brittle rapuh atau deformasi plastik dari matriks yang ductile-ulet. Kavitasi dalam matriks dan fiber/matriks debonding adalah bentuk lain dari kerusakan yang teramati karena kelelahan termal komposit. Thermal fatigue dalam matriks dapat dikurangi dengan memilih matriks yang memiliki kuat tarik yang tinggi dan regangan gagal yang besar. Fiber/matriks debonding hanya dapat dihindari dengan memilih bahan dasar sehingga perbedaan koefisien ekspansi termal pada fiber dan matriks rendah.
