1
材料导论第四章复合材料
Wood,cellulose fibers in a lignin matrix,
Bone,short and soft collagen fibers embedded in a
mineral matrix called apatite,
Glass fiber reinforced resins have been in
use since about the 1940s,
Naturally Occurring Composites
Comparison between conventional monolithic
materials and composite materials,
Weight Thermal
expansion
Stiffness Strength Fatigue
resistance
CompositesSteel Aluminum
4.1 概述定义1
A mixture of two or more materials that are
distinct in composition and form,each being
present in significant quantities (e.g.,>5%) 。
两种或多种不同组成、不同存在形式材料的混合物，各以显著的量存在
4.1 概述定义2
The union of two or more diverse materials to
attain synergestic or superior qualities to those
exhibited by individual members
两种或多种不同材料的结合体，可获得协同的或优于个别材料的质量复合材料按基体分类陶瓷材料增强材料金属材料高分子材料
MMCCMC
4.1 概述
PMC
2
4.1 概述复合材料按结构分类
4.2混合原理
4.2 混合原理基本假定
?纤维与基体必须紧密结合。
?纤维必须是连续的或在长度方向上搭接的。
?存在一个临界纤维体积分数V
f crit
,高于此值方能发生纤维增强。
?存在一个临界纤维长度，高于此值方能发生增强。
应力符合混合规律：
σ
c
=V
f
σ
f
+V
m
σ
m
V:体积分数，σ:应力，
f与m分别代表纤维与基体。
4.2.1 应力平行于纤维，等应变
σ
f
=E
f
ε
f
，σ
m
= E
m
ε
m
，σ
c
=E
c
ε
c
ε代表应变
E
c
ε
c
=V
f
E
f
ε
f
+V
m
E
m
ε
m
E
c
=E
f
V
f
+ E
m
V
m
由等应变假定
σ
c
=V
f
σ
f
+V
m
σ
m
模量加和规律
σ
f
=E
f
ε
f
，σ
m
= E
m
ε
m
mm
ff
mm
ff
mm
ff
m
f
V
V
lV
lV
A
A
F
F
σ
σ
σ
σ
σ
σ
===
/
/
求受力比由
m
f
m
f
E
E
=
σ
σ
故有
mm
ff
m
f
VE
VE
F
F
=
3
复合材料模量预测（1）
A continuous and aligned glass fiber-reinforced
composite consists of 40 vol% of glass fibers having a
modulus of elasticity of 69 GPa and 60 vol% of a polyester
resin that,when hardened,displays a modulus of 3.4 GPa.
(a) Compute the modulus of elasticity of this composite in
the longitudinal direction.
(b) If the cross-sectional area is 250 mm
2
and a stress of 50
MPa is applied in this longitudinal direction,compute the
magnitude of the load carried by each of the fiber and
matrix phases.
(c) Determine the strain that is sustained by each phase
when the stress in part b is applied.
EXAMPLE PROBLEM 4.1
(a) The modulus of elasticity of the composite is calculated using
Equation E
c
= E
m
V
m
+ E
f
V
f
:
E
c
= (3.4 GPa)(0.6) + (69 GPa)(0.4) = 30 GPa
SOLUTION
A continuous and aligned glass fiber-reinforced composite
consists of 40 vol% of glass fibers having a modulus of elasticity
of 69 GPa and 60 vol% of a polyester resin that,when hardened,
displays a modulus of 3.4 GPa.
(a) Compute the modulus of elasticity of this composite in the
longitudinal direction.
Solution,First find the ratio of fiber load to matrix load,
using Equation,
mm
ff
m
f
VE
VE
F
F
=
mf
m
f
FFor
F
F
5.135.13
GPa)(0.6) (3.4
GPa)(0.4) (69
===
A continuous and aligned glass fiber-reinforced composite
consists of 40 vol% of glass fibers having a modulus of elasticity
of 69 GPa and 60 vol% of a polyester resin that,when hardened,
displays a modulus of 3.4 GPa.
(b) If the cross-sectional area is 250 mm
2
and a stress of 50 MPa
is applied in this longitudinal direction,compute the magnitude of
the load carried by each of the fiber and matrix phases.
The total force sustained by the composite F
c:
F
c
= A
c
σ = (250 mm
2
)(50 MPa) = 12,500 N
This total load is just the sum of the loads carried by fiber and
matrix phases,that is
13.5 F
m
+ F
m
= 12,500 N or F
m
= 860 N
whereas F
f
= F
c
- F
m
= 12,500 N - 860 N = 11,640 N
A continuous and aligned glass fiber-reinforced composite
consists of 40 vol% of glass fibers having a modulus of elasticity
of 69 GPa and 60 vol% of a polyester resin that,when hardened,
displays a modulus of 3.4 GPa.
(b) If the cross-sectional area is 250 mm
2
and a stress of 50 MPa
is applied in this longitudinal direction,compute the magnitude of
the load carried by each of the fiber and matrix phases.
A continuous and aligned glass fiber-reinforced composite
consists of 40 vol% of glass fibers having a modulus of elasticity of
69 GPa and 60 vol% of a polyester resin that,when hardened,
displays a modulus of 3.4 GPa,The cross-sectional area is 250 mm
2
and a stress of 50 MPa is applied.
(c) Determine the strain that is sustained by each phase when the
stress in part b is applied.
For stress calculations,phase cross-sectional areas are
necessary:
A
m
= V
m
A
c
= (0.6)(250 mm
2
) = 150 mm
2
A
f
= V
f
A
c
= (0.4)(250 mm
2
) = 100 mm
2
4
Thus,
MPa
mm
N
A
F
MPa
mm
N
A
F
f
f
f
m
m
m
4.116
100
640,11
73.5
150
860
2
2
===
===
σ
σ
3
3
3
3
1069.1
1069
4.116
1069.1
104.3
73.5
×=
×
==
×=
×
==
MPa
MPa
E
MPa
MPa
E
f
f
f
m
m
m
σ
ε
σ
ε
Finally,strains are computed as
复合材料的应变可表示为：
代入虎克定律：
ε
c
= ε
f
V
f
+ ε
m
V
m
m
m
m
f
f
f
c
c
V
E
V
EE
σ
σ
σ
+=
4.2.2 外力垂直于纤维：等应力由等应力条件：
σ
c
= σ
f
= σ
m
我们得到：
m
m
f
f
c
E
V
E
V
E
+=
1
复合材料模量预测（2）
Compute the elastic modulus of the composite material described in
Example Problem 4.1,but assume that the stress is applied
perpendicular to the direction of fiber alignment.
SOLUTION
According to Equation 17.16,
EXAMPLE PROBLEM 4.2
GPaE
c
5.5
GPa) (0.4)(3.4 GPa) (0.6)(69
GPa) GPa)(69 (3.4
=
+
=
基体发生塑性流动时，复合材料的极限强度可表示为：
σ
cu
= σ
fu
V
f
+ σ’
m
V
m
其中σ
fu
是纤维的极限拉伸强度，
σ’
m
是应变硬化基体的流动应力。
复合材料的极限强度σ
cu
必然高于基体的极限强度：
σ
fu
V
f
+ σ’
m
V
m
≥σ
mu
4.2.3 基体的塑性流动
5
4.2.3 基体的塑性流动可导出发生增强的临界纤维体积分数V
f crit
：
mfu
mmu
fcrit
V
'
'
σσ
σσ
=
4.2.4 应力传递
τ
σ
Stress
Posi
tion
4.2.4 应力传递
24
2
l
d
d
f
τπ
π
σ =
c
cf
c
d
l
τ
σ
2
=
c
fu
c
c
d
l
τ
σ
2
=
临界纤维长度：
临界长径比：
τ
σ
Stress
Posi
tion
临界长度
l/2 l/2
l = l
c
Maximum
applied load
f
σ
f
σ
f
σ
St
r
e
ss
l < l
c
f
σ
St
r
e
ss
f
σ
f
σ
l/2 l/2
f
σ
St
r
e
ss
f
σ
f
σ
l > l
c
非连续纤维整齐排列，复合材料的强度可以按下式修正：
非等长纤维：只要l < l
c
，以上分析仍然适用。
mm
c
ffucu
V
l
l
V '
2
1 σσσ +
=
σ
cu
= σ
fu
V
f
+ σ’
m
V
m
对比
4.3聚合物基体
6
热塑性基体酚酚酚酚环环酚酚不不不不不热固性基体不聚聚聚聚聚酰胺、聚碳酸酯、聚砜、聚苯硫醚、聚醚醚酮、氟塑料
4.3 聚合物基体聚苯硫醚(PPS)
聚砜(PES)
聚醚醚酮(PEEK)
S－－
n
O S
=
O
=
O
－－
n
－
O C
=
O
－－
n
O－
4.3 聚合物基体
4.4增强纤维
1,A small diameter
2,A high aspect ratio
3,A very high degree of flexibility
Important characteristics
Flexibility
4
641
dEMR π
=
The flexibility of a material is determined by shape,
size of the cross section,and its radius of curvature,
We can use the inverse of the product of bending
moment (M) and the radius of curvature (R) as a
measure of flexibility,
Fiber diameter of materials with flexibility equal to that of a
25-μm-diameter nylon fiber.
d (
μ
m)
E (GPa)
100 200 300 400 500
25
20
15
10
5
0
nylon
glass
mullite
Si
3
N
4
C
ZrO
2
Al
2
O
3
BSiC
7
2BX
3
+3H
2
2B+6HX
X,C1,Br,or I
4.4.1 Boron Fibers 硼纤维
Chemical vapor deposition (CVD) 化学气相沉积法钨丝卷取加热加热预热
H
2 BCl
3
/H
2
气相沉积
A boron filament production facility
4.4.1 硼纤维硼纤维的结构比重：2.34 ~ 2.6
拉伸强度：2.1 ~ 5.1GPa
模量：~ 400GPa
Reaction zone
W
2
B
5
+WB
4
W
SiC
B
4.4.2 Carbon Fibers碳纤维碳含量在92~95%之间，模量在344GPa以下的为碳纤维，
碳含量在99%以上，模量在344GPa以上的为石墨纤维。
在1300°C左右热解的称碳纤维在1900°C以上热解的称石墨纤维
4.4.2 碳纤维与石墨纤维
Oxidation
up to 250°C
Carbonization
250~1500°C
Graphitization
1500~2500°C
C
C
H
H
H
C N
C
C
H
H
H
C N
C
C
H
H
H
C N
C
C
H
H
H
C N
C
C
H
H
H
C
N
C
C
H
H
H
C
N
C
C
H
H
H
C
N
C
C
H
H
H
C
N
Ladder molecule
polyacrylonitrile
8
Ex-Pitch Carbon Fibers
Sources of pitch
Polyvinyl chloride (PVC)
Petroleum asphalt
Coal tar
Density Young's Electrical
Precursor (g/m
3
) modulus resistivity
(GPa) (10
-4
cm)
Rayon 1.66 390 10
Polyacrylonitrile 1.74 230 18
Pitch (Kureha)
LT 1.6 41 100
HT 1.6 41 50
Mesophase pitch
LT 2.1 340 9
HT 2.2 690 1.8
Single-crystal graphite 2.25 l000 0.40
Properties of different carbon fibers
Kevlar?
4.4.3 Organic Fibers
N N
C
O
C
O
H H
4.4 增强纤维
4.4.3 有机纤维
Kevlar29：高韧性，
拉伸强度3.4GPa
Kevlar49：模量130GPa
Kevlar149：模量180GPa
比重：1.44
Spectra
CH
2
CH
2
CH
2
CH
2
CH
2
CH
2
CH
2
CH
2
CH
2
CH
2
聚乙烯纤维
4.4.3 有机纤维：聚乙烯纤维烘箱冷却聚乙烯溶液
9
4.4 增强纤维
4.4.3 有机纤维聚乙烯纤维的性质性质Spectra900 Spectra 1000 Kevlar 149
密度(g/cm
3
) 0.97 0.97 1.44
直径（μ）38 27
拉伸强度(GPa) 2.7 3.0
拉伸模量(GPa) 119 175 180
4.4.4 Ceramic Fibers 陶瓷纤维
Three fabrication methods
1,Chemical vapor deposition
2,Polymer pyrolysis
3,Sol-gel techniques
The Sumitomo process of making
a silica- stabilized Al
2
O
3
fiber
Organoalumino
compound
Alkyl aluminum
or alkoxy
aluminum (AlR
3
)
Polymerization
AlR
3
+ H
2
O
－Al－O－
－
R
Organic solvent +
Si-containing compound
(alkyl silicate)
Dry spinning
Precursor fiber
(organoaluminum
polymer and alkyl silicate)
Calcination
Inorganic fiber
Al
2
O
3
,70-100%
SiO
2
,30-0%
Filter Sol reservior
Pump
Draw wheels
Pyrolysis
furnace
Spinneret
High temperature
straightening furnace
Takeup spool
The 3M Co,process of making Al
2
O
3
fiber
Composition Diameter Density Tensile Young’s
No,(wt%) (μm) (g/cm
3
) strength modulus
(MPa) (GPa)
312 Al
2
O
3
62,SiO
2
24,B
2
O
3
14 10-12 2.7 1700 152
440 Al
2
O
3
70,SiO
2
28,B
2
O
3
2 10-12 3.05 2000 186
550 Al
2
O
3
73,SiO
2
27 10-12 3.03 2000 193
610 Al
2
O
3
99+,(SiO
2
,Fe
2
O
3
) 10-12 3.75 1900 370
720 Al
2
O
3
85,SiO
2
15 10-12 3.4 2130 260
Composition and properties of Nextel series fibers
Optical micrograph of Nextel 312(Al
2
O
3
+SiO
2
+B
2
O
3
)
10
CH
3
SiC1
3
(g) SiC(s) + 3
HCl(g)
H
2
CVD Silicon Carbide Fibers
Diameter Density Tensile Young’s
(μm) (g/cm
3
) strength modulus
(MPa) (GPa)
140 3.3 3500 430
Properties
E玻璃：
55%二氧化硅，20%氧化钙，15%氧化铝，10%氧化硼拉伸强度：5GPa（Electrical application）
S玻璃
10%氧化镁，25%氧化铝，65%二氧化硅拉伸强度：7GPa （High Strength）
4.4.5 Glass Fiber
Batch mixing Batch hopper
at furnace
Furnace
melting area
Platinum
bushing
Collet
Filament collecting
and size applicator
Strand traverse
motion
Winding
head unit
Glass fiber manufacture
4.4.5 玻璃纤维
Roving
Fabric
Chopped strand
continuous yarn
单晶接近理论强度（1~2万MPa ）
直径：几μ至几mm
长径比：50~10000
缺点：尺寸不均一性能不均一
4.4.6 Wiskers 晶须
4.4 增强纤维
4.4.6 晶须：谷糠法糠研磨焦化碳管反应器
＞1600 °C
破碎碳化硅晶须拌匀分离多余碳分离残糠烘干脱碳
11
4.4.6 晶须：VLS法液体催化剂含过饱和Si与C
固体SiC
晶须
30μ的固体催化剂
1400°C
H
2
CH
4 SiO
2
3C + SiO
2
SiC + 2CO
The behavior of a composite material is a
result of the combined behavior of the
following three entities:
?Fiber or the reinforcing element
?Matrix
?Fiber/matrix interface
4.5 界面Interface
Solid
Wettability
Liquid
Solid
Complete Wetting θ = 0°
Vapor
Liquid
No Wetting θ = 180°
Vapor
γ
LV
γ
SV
γ
LS
θ
Solid
Liquid
Partial Wetting
Important types of interfacial bonding
? Mechanical bonding
? Physical bonding
? Chemical bonding
Mechanical bonding
(a) Good mechanical bonding
Void
Matrix
Fiber
Matrix
Fiber
(b)Lack of wettability
Any bonding involving weak,secondary or van
der Waals forces,dipolar interactions,and
hydrogen bonding can be classified as physical
bonding.
The bond energy in such physical bonding is
approximately 8-16 kJ/mol.
Physical Bonding
12
Two polymer surfaces may form a bond
owing to the diffusion of matrix molecules to the
molecular network of the fiber,thus forming
tangled molecular bonds at the interface.
Physical Bonding
Atomic or molecular transport,by diffusional
processes,is involved in chemical bonding,Solid
solution and compound formation may occur at
the interface.
This encompasses all types of covalent,ionic,and
metallic bonding,Chemical bonding involves
primary forces and the bond energy in the range
of approximately 40-400 kJ/mol.
Chemical Bonding
Coupling agents (silanes are the most
common ones) are used for glass fibers in resin
matrices
Chemical Bonding
4.6 树脂基复合材料
4.6 树脂基复合材料热塑性树脂基复合材料低性能中温型中温高强型高温型高性能热固性树脂基复合材料树脂基复合材料酚醛不饱和聚酯环氧树脂聚酰亚胺
4.6 树脂基复合材料热塑性树脂基复合材料基体PBT PET尼龙66聚砜组成树脂
40%
玻璃纤维树脂
40%
玻璃纤维树脂
30%
碳纤维
43%
玻璃纤维树脂
30%
玻璃纤维
30%
碳纤维比重1.311.53 - 1.601.141.281.511.241.451.37
工作温度
（°C）
120 140 140 170
拉伸强度
(MPa）
50 120 90 245 170 85 125 160
模量
(GPa）
6.9 13.8 2.5 12.0 3.4 6.7 3.0 7.6
13
4.6 树脂基复合材料酚醛基复合材料（中温型）
组成树脂玻璃短纤维
E玻璃布碳纤维布石墨纤维布比重1.28 1.95 1.90 1.45 1.42
工作温度/°C 150 150 538 3000 3000
拉伸强度/MPa 50 55
280-
420
135 100
模量/GPa 4.0 22 3.3-4.1 17.5 16
4.6 树脂基复合材料环氧树脂基复合材料（中温高强型）
组成树脂玻璃短纤维缠绕玻璃纤维
Kevlar碳纤维石墨纤维比重1.20 1.79 1.86 1.25 1.46 1.58
工作温度/°C 80 150 150 150 150 150
拉伸强度/MPa 70 500 1500 1400 1900 1400
模量/GPa 8.0 23 55 78 145 210
4.6 树脂基复合材料高温型塑料基复合材料基体聚酰亚胺双马来酸酐组成树脂
50%玻璃纤维树脂
68.3%
T300
碳纤维
57.7%
玻璃纤维比重1.43 1.65 1.23 1.60 2.00
工作温度（°C）
260-
370
260 230 230 230
拉伸强度
(MPa）
100
200-
400
70 1600
400-
600
模量(GPa）4.0 10-20 3.4 140 35
4.7复合材料的加工
4.7.1 Lay-up 模面成型
(a)手工涂敷(b)喷涂法
4.7.2 模压成型与树脂转移模压法
14
4.7.3 真空成型
4.7.4 Filament
Winding 缠绕
4.7.5 Pultrusion 拉挤
4.7.6 预制片(Prepreg) 制备
4.7.7 纤维编织
Preform Structures
15
4.8 碳基复合材料
Carbon
fiber
Polymer or
pitch binder
Green fabrication
First carbonization
up to 1000°C
Impregnation
by CVD
Liquid impregnation
with polymer or pitch
Curing of the polymer
Pressure carbonization
of the pitch
Carbonization 1000°C
Intermediate
carbonization
1300-1900°C
Final graphitization
2500-3000°C
CFRC 1000°C
CFRC 2500-3000 °C
The induction furnace susceptor (1) Isothermal (2)
Pressure differential (3) Thermal gradient
Jacket
Carbon
substrate
Hydrocarbon
gas Carrier gas
Induction
coils
Graphite
susceptor
Original
fiber substrate
Deposition
CH
4
H
2
H
2
H
2
The induction furnace susceptor (3) Thermal gradient
Jacket
Carbon
substrate
Hydrocarbon
gas Carrier gas
Induction
coils
Graphite
susceptor
Sleeve
CH
4
H
2
Pitch Resin Processing
Form large sheet-like molecules upon
polymerization,not a cross-linked 3-D
network,as in the case of thermosetting
resins,
Low softening points,low viscosity in the
liquid stage and high graphitic yield.
Pressure
melted pitch 650°C,108.4 MPa 24 h
induction furnace 2300°C
specific gravity > 1.6
16
Preparation of Rocket nozzles
3-D woven fabrics
impregnated
with
a phenolic resin
trimming and
pyrolyzing
impregnated
with
furfuryl alcohol
and pyrolyzed
silica and
alumina
powder
1650°C,silicon
carbide formed
treatment with tetraethylorthosilicate
and curing form silicon dioxide
Oxidation Behavior and Protection
A 2.3mm
B 2.7mm
C 5.5mm
D 7.9mm
A
0 200 400 600 800 1000
Temperature °C
We
ight
loss
(
%
)
0
1
2
3
4
5
6
7
D
C
B
Air 200ml/min
T/?t= 10°C/min
C/C Sheet
Silicon carbide,titanium carbide and boron nitride
(高温效果不佳)
Boron oxide or borates
(1000°C 以上或水蒸气存在下效果不佳)
Borates also were found to he more effective when
they were mixed with particulate refractory phases
such as zirconium and hafnium borides,
C/C composites with successive layers of boron
oxide,zirconium boride and silicon carbide can
prevent oxidation at temperature as high as 1500°C in
dry air,
Coating through CVD
Conversion of the carbon surface of the
composite to an oxidation resistant carbide,
Chemical reaction between carbon and melted
silicon will form silicon carbide layer on the
surface of C/C composite,
Surface Carbide Conversion
碳碳复合材料的典型性能（括号中为1900°C下数据）
单维增强三维增强高强处理高模处理
Z方向
X-Y
方向拉伸强度
(MPa）
600 575
310
（400）
103
（124）
模量
(GPa）
125 220
152
（159）
62
（83）
?Aerospace industry (jet-aircraft brake linings)
? Medical implantation,as they offer
biocompatibility,Carbon fibers have successfully
been used as ligament replacements
?Chemically abrasive environments,since
elemental carbon is one of the most corrosion-
resistant materials
?Metallurgy,molds,outer rings in gas
turbines
Applications
17
4.9 STRUCTURAL COMPOSITES
4.9.1 LAMINAR
COMPOSITES
4.9.2 Sandwich Composites
例4-3：设计一棒材，长度为3m。以环氧树脂为基体。受到2.2kN的外力时，伸长不能超过2.5mm，不能发生脆性断裂。已知环氧树脂弹性模量为3.5GPa，屈服强度为
80MPa，价格为$1.75/kg。
解：假定棒材全部用环氧树脂制造（无纤维）：
mmdmm
F
A
E
mm
mm
314.753
92.2
2200
MPa92.2)1083.0)(105.3(
1083.0
3000
5.2
2
33
max
3
max
====
=××==
×==
直径即棒
σ
εσ
ε
环氧树脂的密度ρ＝1.25Mg/m
3
，故棒的重量为(1.25) (753.4×10
－6
)
(3)=2.83kg，其价格为(2.83kg)($1.75/kg)=$4.95
例题再来看复合材料的情况。最低要求是80MPa应力下的应变为0.83×10
－3
，则弹性模量为
GPa4.96
1083.0
80
3
max
=
×
=>
ε
σ
c
E
玻璃纤维的模量低于96.4GPa，应考虑更强的纤维。高模碳纤维的模量为531GPa，密度为1.9Mg/m
3
，价格为$66/kg。为使复合材料具有96.4GPa的模量，最低碳纤维体积分数为：
176.04.96)5.3)(1()531( =>?+=
fffc
VVVE
环氧树脂体积分数为0.824。
例4-3：设计一棒材，长度为3m。以环氧树脂为基体。受到2.2kN的外力时，伸长不能超过2.5mm，不能发生脆性断裂。已知环氧树脂弹性模量为3.5GPa，屈服强度为
80MPa，价格为$1.75/kg。
如果纤维全部断裂，截面积82.4%上的环氧树脂必须承受
2.2kN的外力且使应力不大于80MPa。由此决定棒材的截面积：
mmdmmA
mm
F
AA
c
cm
5.64.33
824.0
5.27
5.27
80
2200
824.0
2
2
===
====
即
σ
例4-3：设计一棒材，长度为3m。以环氧树脂为基体。受到2.2kN的外力时，伸长不能超过2.5mm，不能发生脆性断裂。已知环氧树脂弹性模量为3.5GPa，屈服强度为
80MPa，价格为$1.75/kg。
18
42.2$)/75.1)($103.0()/66)($034.0(
103.0)137.0)((755.0(
034.0)137.0)(245.0(
245.0
)25.1)(824.0()19.0)(176.0(
)19.0)(176.0(
137.010137.0)101.0)](824.0)(25.1()176.0)(9.1[(
101.0)3)(104.33(
33
3326
=+=
==
==
=
+
=
=×=×+=
×=×=
kgkgkgkg
kg
kg
kgMg
mmm
棒材价格环氧树脂的重量碳纤维的重量碳纤维的重量分数棒材重量棒材体积对比2.83kg，$4.95
采用碳纤维复合后的材料直径不足只用环氧树脂的四分之一，重量只有5％，价格还降低了一半。
例4-3：设计一棒材，长度为3m。以环氧树脂为基体。受到2.2kN的外力时，伸长不能超过2.5mm，不能发生脆性断裂。已知环氧树脂弹性模量为3.5GPa，屈服强度为
80MPa，价格为$1.75/kg。
4.6 树脂基复合材料热变形温度
材料导论第四章复合材料
Wood,cellulose fibers in a lignin matrix,
Bone,short and soft collagen fibers embedded in a
mineral matrix called apatite,
Glass fiber reinforced resins have been in
use since about the 1940s,
Naturally Occurring Composites
Comparison between conventional monolithic
materials and composite materials,
Weight Thermal
expansion
Stiffness Strength Fatigue
resistance
CompositesSteel Aluminum
4.1 概述定义1
A mixture of two or more materials that are
distinct in composition and form,each being
present in significant quantities (e.g.,>5%) 。
两种或多种不同组成、不同存在形式材料的混合物，各以显著的量存在
4.1 概述定义2
The union of two or more diverse materials to
attain synergestic or superior qualities to those
exhibited by individual members
两种或多种不同材料的结合体，可获得协同的或优于个别材料的质量复合材料按基体分类陶瓷材料增强材料金属材料高分子材料
MMCCMC
4.1 概述
PMC
2
4.1 概述复合材料按结构分类
4.2混合原理
4.2 混合原理基本假定
?纤维与基体必须紧密结合。
?纤维必须是连续的或在长度方向上搭接的。
?存在一个临界纤维体积分数V
f crit
,高于此值方能发生纤维增强。
?存在一个临界纤维长度，高于此值方能发生增强。
应力符合混合规律：
σ
c
=V
f
σ
f
+V
m
σ
m
V:体积分数，σ:应力，
f与m分别代表纤维与基体。
4.2.1 应力平行于纤维，等应变
σ
f
=E
f
ε
f
，σ
m
= E
m
ε
m
，σ
c
=E
c
ε
c
ε代表应变
E
c
ε
c
=V
f
E
f
ε
f
+V
m
E
m
ε
m
E
c
=E
f
V
f
+ E
m
V
m
由等应变假定
σ
c
=V
f
σ
f
+V
m
σ
m
模量加和规律
σ
f
=E
f
ε
f
，σ
m
= E
m
ε
m
mm
ff
mm
ff
mm
ff
m
f
V
V
lV
lV
A
A
F
F
σ
σ
σ
σ
σ
σ
===
/
/
求受力比由
m
f
m
f
E
E
=
σ
σ
故有
mm
ff
m
f
VE
VE
F
F
=
3
复合材料模量预测（1）
A continuous and aligned glass fiber-reinforced
composite consists of 40 vol% of glass fibers having a
modulus of elasticity of 69 GPa and 60 vol% of a polyester
resin that,when hardened,displays a modulus of 3.4 GPa.
(a) Compute the modulus of elasticity of this composite in
the longitudinal direction.
(b) If the cross-sectional area is 250 mm
2
and a stress of 50
MPa is applied in this longitudinal direction,compute the
magnitude of the load carried by each of the fiber and
matrix phases.
(c) Determine the strain that is sustained by each phase
when the stress in part b is applied.
EXAMPLE PROBLEM 4.1
(a) The modulus of elasticity of the composite is calculated using
Equation E
c
= E
m
V
m
+ E
f
V
f
:
E
c
= (3.4 GPa)(0.6) + (69 GPa)(0.4) = 30 GPa
SOLUTION
A continuous and aligned glass fiber-reinforced composite
consists of 40 vol% of glass fibers having a modulus of elasticity
of 69 GPa and 60 vol% of a polyester resin that,when hardened,
displays a modulus of 3.4 GPa.
(a) Compute the modulus of elasticity of this composite in the
longitudinal direction.
Solution,First find the ratio of fiber load to matrix load,
using Equation,
mm
ff
m
f
VE
VE
F
F
=
mf
m
f
FFor
F
F
5.135.13
GPa)(0.6) (3.4
GPa)(0.4) (69
===
A continuous and aligned glass fiber-reinforced composite
consists of 40 vol% of glass fibers having a modulus of elasticity
of 69 GPa and 60 vol% of a polyester resin that,when hardened,
displays a modulus of 3.4 GPa.
(b) If the cross-sectional area is 250 mm
2
and a stress of 50 MPa
is applied in this longitudinal direction,compute the magnitude of
the load carried by each of the fiber and matrix phases.
The total force sustained by the composite F
c:
F
c
= A
c
σ = (250 mm
2
)(50 MPa) = 12,500 N
This total load is just the sum of the loads carried by fiber and
matrix phases,that is
13.5 F
m
+ F
m
= 12,500 N or F
m
= 860 N
whereas F
f
= F
c
- F
m
= 12,500 N - 860 N = 11,640 N
A continuous and aligned glass fiber-reinforced composite
consists of 40 vol% of glass fibers having a modulus of elasticity
of 69 GPa and 60 vol% of a polyester resin that,when hardened,
displays a modulus of 3.4 GPa.
(b) If the cross-sectional area is 250 mm
2
and a stress of 50 MPa
is applied in this longitudinal direction,compute the magnitude of
the load carried by each of the fiber and matrix phases.
A continuous and aligned glass fiber-reinforced composite
consists of 40 vol% of glass fibers having a modulus of elasticity of
69 GPa and 60 vol% of a polyester resin that,when hardened,
displays a modulus of 3.4 GPa,The cross-sectional area is 250 mm
2
and a stress of 50 MPa is applied.
(c) Determine the strain that is sustained by each phase when the
stress in part b is applied.
For stress calculations,phase cross-sectional areas are
necessary:
A
m
= V
m
A
c
= (0.6)(250 mm
2
) = 150 mm
2
A
f
= V
f
A
c
= (0.4)(250 mm
2
) = 100 mm
2
4
Thus,
MPa
mm
N
A
F
MPa
mm
N
A
F
f
f
f
m
m
m
4.116
100
640,11
73.5
150
860
2
2
===
===
σ
σ
3
3
3
3
1069.1
1069
4.116
1069.1
104.3
73.5
×=
×
==
×=
×
==
MPa
MPa
E
MPa
MPa
E
f
f
f
m
m
m
σ
ε
σ
ε
Finally,strains are computed as
复合材料的应变可表示为：
代入虎克定律：
ε
c
= ε
f
V
f
+ ε
m
V
m
m
m
m
f
f
f
c
c
V
E
V
EE
σ
σ
σ
+=
4.2.2 外力垂直于纤维：等应力由等应力条件：
σ
c
= σ
f
= σ
m
我们得到：
m
m
f
f
c
E
V
E
V
E
+=
1
复合材料模量预测（2）
Compute the elastic modulus of the composite material described in
Example Problem 4.1,but assume that the stress is applied
perpendicular to the direction of fiber alignment.
SOLUTION
According to Equation 17.16,
EXAMPLE PROBLEM 4.2
GPaE
c
5.5
GPa) (0.4)(3.4 GPa) (0.6)(69
GPa) GPa)(69 (3.4
=
+
=
基体发生塑性流动时，复合材料的极限强度可表示为：
σ
cu
= σ
fu
V
f
+ σ’
m
V
m
其中σ
fu
是纤维的极限拉伸强度，
σ’
m
是应变硬化基体的流动应力。
复合材料的极限强度σ
cu
必然高于基体的极限强度：
σ
fu
V
f
+ σ’
m
V
m
≥σ
mu
4.2.3 基体的塑性流动
5
4.2.3 基体的塑性流动可导出发生增强的临界纤维体积分数V
f crit
：
mfu
mmu
fcrit
V
'
'
σσ
σσ
=
4.2.4 应力传递
τ
σ
Stress
Posi
tion
4.2.4 应力传递
24
2
l
d
d
f
τπ
π
σ =
c
cf
c
d
l
τ
σ
2
=
c
fu
c
c
d
l
τ
σ
2
=
临界纤维长度：
临界长径比：
τ
σ
Stress
Posi
tion
临界长度
l/2 l/2
l = l
c
Maximum
applied load
f
σ
f
σ
f
σ
St
r
e
ss
l < l
c
f
σ
St
r
e
ss
f
σ
f
σ
l/2 l/2
f
σ
St
r
e
ss
f
σ
f
σ
l > l
c
非连续纤维整齐排列，复合材料的强度可以按下式修正：
非等长纤维：只要l < l
c
，以上分析仍然适用。
mm
c
ffucu
V
l
l
V '
2
1 σσσ +
=
σ
cu
= σ
fu
V
f
+ σ’
m
V
m
对比
4.3聚合物基体
6
热塑性基体酚酚酚酚环环酚酚不不不不不热固性基体不聚聚聚聚聚酰胺、聚碳酸酯、聚砜、聚苯硫醚、聚醚醚酮、氟塑料
4.3 聚合物基体聚苯硫醚(PPS)
聚砜(PES)
聚醚醚酮(PEEK)
S－－
n
O S
=
O
=
O
－－
n
－
O C
=
O
－－
n
O－
4.3 聚合物基体
4.4增强纤维
1,A small diameter
2,A high aspect ratio
3,A very high degree of flexibility
Important characteristics
Flexibility
4
641
dEMR π
=
The flexibility of a material is determined by shape,
size of the cross section,and its radius of curvature,
We can use the inverse of the product of bending
moment (M) and the radius of curvature (R) as a
measure of flexibility,
Fiber diameter of materials with flexibility equal to that of a
25-μm-diameter nylon fiber.
d (
μ
m)
E (GPa)
100 200 300 400 500
25
20
15
10
5
0
nylon
glass
mullite
Si
3
N
4
C
ZrO
2
Al
2
O
3
BSiC
7
2BX
3
+3H
2
2B+6HX
X,C1,Br,or I
4.4.1 Boron Fibers 硼纤维
Chemical vapor deposition (CVD) 化学气相沉积法钨丝卷取加热加热预热
H
2 BCl
3
/H
2
气相沉积
A boron filament production facility
4.4.1 硼纤维硼纤维的结构比重：2.34 ~ 2.6
拉伸强度：2.1 ~ 5.1GPa
模量：~ 400GPa
Reaction zone
W
2
B
5
+WB
4
W
SiC
B
4.4.2 Carbon Fibers碳纤维碳含量在92~95%之间，模量在344GPa以下的为碳纤维，
碳含量在99%以上，模量在344GPa以上的为石墨纤维。
在1300°C左右热解的称碳纤维在1900°C以上热解的称石墨纤维
4.4.2 碳纤维与石墨纤维
Oxidation
up to 250°C
Carbonization
250~1500°C
Graphitization
1500~2500°C
C
C
H
H
H
C N
C
C
H
H
H
C N
C
C
H
H
H
C N
C
C
H
H
H
C N
C
C
H
H
H
C
N
C
C
H
H
H
C
N
C
C
H
H
H
C
N
C
C
H
H
H
C
N
Ladder molecule
polyacrylonitrile
8
Ex-Pitch Carbon Fibers
Sources of pitch
Polyvinyl chloride (PVC)
Petroleum asphalt
Coal tar
Density Young's Electrical
Precursor (g/m
3
) modulus resistivity
(GPa) (10
-4
cm)
Rayon 1.66 390 10
Polyacrylonitrile 1.74 230 18
Pitch (Kureha)
LT 1.6 41 100
HT 1.6 41 50
Mesophase pitch
LT 2.1 340 9
HT 2.2 690 1.8
Single-crystal graphite 2.25 l000 0.40
Properties of different carbon fibers
Kevlar?
4.4.3 Organic Fibers
N N
C
O
C
O
H H
4.4 增强纤维
4.4.3 有机纤维
Kevlar29：高韧性，
拉伸强度3.4GPa
Kevlar49：模量130GPa
Kevlar149：模量180GPa
比重：1.44
Spectra
CH
2
CH
2
CH
2
CH
2
CH
2
CH
2
CH
2
CH
2
CH
2
CH
2
聚乙烯纤维
4.4.3 有机纤维：聚乙烯纤维烘箱冷却聚乙烯溶液
9
4.4 增强纤维
4.4.3 有机纤维聚乙烯纤维的性质性质Spectra900 Spectra 1000 Kevlar 149
密度(g/cm
3
) 0.97 0.97 1.44
直径（μ）38 27
拉伸强度(GPa) 2.7 3.0
拉伸模量(GPa) 119 175 180
4.4.4 Ceramic Fibers 陶瓷纤维
Three fabrication methods
1,Chemical vapor deposition
2,Polymer pyrolysis
3,Sol-gel techniques
The Sumitomo process of making
a silica- stabilized Al
2
O
3
fiber
Organoalumino
compound
Alkyl aluminum
or alkoxy
aluminum (AlR
3
)
Polymerization
AlR
3
+ H
2
O
－Al－O－
－
R
Organic solvent +
Si-containing compound
(alkyl silicate)
Dry spinning
Precursor fiber
(organoaluminum
polymer and alkyl silicate)
Calcination
Inorganic fiber
Al
2
O
3
,70-100%
SiO
2
,30-0%
Filter Sol reservior
Pump
Draw wheels
Pyrolysis
furnace
Spinneret
High temperature
straightening furnace
Takeup spool
The 3M Co,process of making Al
2
O
3
fiber
Composition Diameter Density Tensile Young’s
No,(wt%) (μm) (g/cm
3
) strength modulus
(MPa) (GPa)
312 Al
2
O
3
62,SiO
2
24,B
2
O
3
14 10-12 2.7 1700 152
440 Al
2
O
3
70,SiO
2
28,B
2
O
3
2 10-12 3.05 2000 186
550 Al
2
O
3
73,SiO
2
27 10-12 3.03 2000 193
610 Al
2
O
3
99+,(SiO
2
,Fe
2
O
3
) 10-12 3.75 1900 370
720 Al
2
O
3
85,SiO
2
15 10-12 3.4 2130 260
Composition and properties of Nextel series fibers
Optical micrograph of Nextel 312(Al
2
O
3
+SiO
2
+B
2
O
3
)
10
CH
3
SiC1
3
(g) SiC(s) + 3
HCl(g)
H
2
CVD Silicon Carbide Fibers
Diameter Density Tensile Young’s
(μm) (g/cm
3
) strength modulus
(MPa) (GPa)
140 3.3 3500 430
Properties
E玻璃：
55%二氧化硅，20%氧化钙，15%氧化铝，10%氧化硼拉伸强度：5GPa（Electrical application）
S玻璃
10%氧化镁，25%氧化铝，65%二氧化硅拉伸强度：7GPa （High Strength）
4.4.5 Glass Fiber
Batch mixing Batch hopper
at furnace
Furnace
melting area
Platinum
bushing
Collet
Filament collecting
and size applicator
Strand traverse
motion
Winding
head unit
Glass fiber manufacture
4.4.5 玻璃纤维
Roving
Fabric
Chopped strand
continuous yarn
单晶接近理论强度（1~2万MPa ）
直径：几μ至几mm
长径比：50~10000
缺点：尺寸不均一性能不均一
4.4.6 Wiskers 晶须
4.4 增强纤维
4.4.6 晶须：谷糠法糠研磨焦化碳管反应器
＞1600 °C
破碎碳化硅晶须拌匀分离多余碳分离残糠烘干脱碳
11
4.4.6 晶须：VLS法液体催化剂含过饱和Si与C
固体SiC
晶须
30μ的固体催化剂
1400°C
H
2
CH
4 SiO
2
3C + SiO
2
SiC + 2CO
The behavior of a composite material is a
result of the combined behavior of the
following three entities:
?Fiber or the reinforcing element
?Matrix
?Fiber/matrix interface
4.5 界面Interface
Solid
Wettability
Liquid
Solid
Complete Wetting θ = 0°
Vapor
Liquid
No Wetting θ = 180°
Vapor
γ
LV
γ
SV
γ
LS
θ
Solid
Liquid
Partial Wetting
Important types of interfacial bonding
? Mechanical bonding
? Physical bonding
? Chemical bonding
Mechanical bonding
(a) Good mechanical bonding
Void
Matrix
Fiber
Matrix
Fiber
(b)Lack of wettability
Any bonding involving weak,secondary or van
der Waals forces,dipolar interactions,and
hydrogen bonding can be classified as physical
bonding.
The bond energy in such physical bonding is
approximately 8-16 kJ/mol.
Physical Bonding
12
Two polymer surfaces may form a bond
owing to the diffusion of matrix molecules to the
molecular network of the fiber,thus forming
tangled molecular bonds at the interface.
Physical Bonding
Atomic or molecular transport,by diffusional
processes,is involved in chemical bonding,Solid
solution and compound formation may occur at
the interface.
This encompasses all types of covalent,ionic,and
metallic bonding,Chemical bonding involves
primary forces and the bond energy in the range
of approximately 40-400 kJ/mol.
Chemical Bonding
Coupling agents (silanes are the most
common ones) are used for glass fibers in resin
matrices
Chemical Bonding
4.6 树脂基复合材料
4.6 树脂基复合材料热塑性树脂基复合材料低性能中温型中温高强型高温型高性能热固性树脂基复合材料树脂基复合材料酚醛不饱和聚酯环氧树脂聚酰亚胺
4.6 树脂基复合材料热塑性树脂基复合材料基体PBT PET尼龙66聚砜组成树脂
40%
玻璃纤维树脂
40%
玻璃纤维树脂
30%
碳纤维
43%
玻璃纤维树脂
30%
玻璃纤维
30%
碳纤维比重1.311.53 - 1.601.141.281.511.241.451.37
工作温度
（°C）
120 140 140 170
拉伸强度
(MPa）
50 120 90 245 170 85 125 160
模量
(GPa）
6.9 13.8 2.5 12.0 3.4 6.7 3.0 7.6
13
4.6 树脂基复合材料酚醛基复合材料（中温型）
组成树脂玻璃短纤维
E玻璃布碳纤维布石墨纤维布比重1.28 1.95 1.90 1.45 1.42
工作温度/°C 150 150 538 3000 3000
拉伸强度/MPa 50 55
280-
420
135 100
模量/GPa 4.0 22 3.3-4.1 17.5 16
4.6 树脂基复合材料环氧树脂基复合材料（中温高强型）
组成树脂玻璃短纤维缠绕玻璃纤维
Kevlar碳纤维石墨纤维比重1.20 1.79 1.86 1.25 1.46 1.58
工作温度/°C 80 150 150 150 150 150
拉伸强度/MPa 70 500 1500 1400 1900 1400
模量/GPa 8.0 23 55 78 145 210
4.6 树脂基复合材料高温型塑料基复合材料基体聚酰亚胺双马来酸酐组成树脂
50%玻璃纤维树脂
68.3%
T300
碳纤维
57.7%
玻璃纤维比重1.43 1.65 1.23 1.60 2.00
工作温度（°C）
260-
370
260 230 230 230
拉伸强度
(MPa）
100
200-
400
70 1600
400-
600
模量(GPa）4.0 10-20 3.4 140 35
4.7复合材料的加工
4.7.1 Lay-up 模面成型
(a)手工涂敷(b)喷涂法
4.7.2 模压成型与树脂转移模压法
14
4.7.3 真空成型
4.7.4 Filament
Winding 缠绕
4.7.5 Pultrusion 拉挤
4.7.6 预制片(Prepreg) 制备
4.7.7 纤维编织
Preform Structures
15
4.8 碳基复合材料
Carbon
fiber
Polymer or
pitch binder
Green fabrication
First carbonization
up to 1000°C
Impregnation
by CVD
Liquid impregnation
with polymer or pitch
Curing of the polymer
Pressure carbonization
of the pitch
Carbonization 1000°C
Intermediate
carbonization
1300-1900°C
Final graphitization
2500-3000°C
CFRC 1000°C
CFRC 2500-3000 °C
The induction furnace susceptor (1) Isothermal (2)
Pressure differential (3) Thermal gradient
Jacket
Carbon
substrate
Hydrocarbon
gas Carrier gas
Induction
coils
Graphite
susceptor
Original
fiber substrate
Deposition
CH
4
H
2
H
2
H
2
The induction furnace susceptor (3) Thermal gradient
Jacket
Carbon
substrate
Hydrocarbon
gas Carrier gas
Induction
coils
Graphite
susceptor
Sleeve
CH
4
H
2
Pitch Resin Processing
Form large sheet-like molecules upon
polymerization,not a cross-linked 3-D
network,as in the case of thermosetting
resins,
Low softening points,low viscosity in the
liquid stage and high graphitic yield.
Pressure
melted pitch 650°C,108.4 MPa 24 h
induction furnace 2300°C
specific gravity > 1.6
16
Preparation of Rocket nozzles
3-D woven fabrics
impregnated
with
a phenolic resin
trimming and
pyrolyzing
impregnated
with
furfuryl alcohol
and pyrolyzed
silica and
alumina
powder
1650°C,silicon
carbide formed
treatment with tetraethylorthosilicate
and curing form silicon dioxide
Oxidation Behavior and Protection
A 2.3mm
B 2.7mm
C 5.5mm
D 7.9mm
A
0 200 400 600 800 1000
Temperature °C
We
ight
loss
(
%
)
0
1
2
3
4
5
6
7
D
C
B
Air 200ml/min
T/?t= 10°C/min
C/C Sheet
Silicon carbide,titanium carbide and boron nitride
(高温效果不佳)
Boron oxide or borates
(1000°C 以上或水蒸气存在下效果不佳)
Borates also were found to he more effective when
they were mixed with particulate refractory phases
such as zirconium and hafnium borides,
C/C composites with successive layers of boron
oxide,zirconium boride and silicon carbide can
prevent oxidation at temperature as high as 1500°C in
dry air,
Coating through CVD
Conversion of the carbon surface of the
composite to an oxidation resistant carbide,
Chemical reaction between carbon and melted
silicon will form silicon carbide layer on the
surface of C/C composite,
Surface Carbide Conversion
碳碳复合材料的典型性能（括号中为1900°C下数据）
单维增强三维增强高强处理高模处理
Z方向
X-Y
方向拉伸强度
(MPa）
600 575
310
（400）
103
（124）
模量
(GPa）
125 220
152
（159）
62
（83）
?Aerospace industry (jet-aircraft brake linings)
? Medical implantation,as they offer
biocompatibility,Carbon fibers have successfully
been used as ligament replacements
?Chemically abrasive environments,since
elemental carbon is one of the most corrosion-
resistant materials
?Metallurgy,molds,outer rings in gas
turbines
Applications
17
4.9 STRUCTURAL COMPOSITES
4.9.1 LAMINAR
COMPOSITES
4.9.2 Sandwich Composites
例4-3：设计一棒材，长度为3m。以环氧树脂为基体。受到2.2kN的外力时，伸长不能超过2.5mm，不能发生脆性断裂。已知环氧树脂弹性模量为3.5GPa，屈服强度为
80MPa，价格为$1.75/kg。
解：假定棒材全部用环氧树脂制造（无纤维）：
mmdmm
F
A
E
mm
mm
314.753
92.2
2200
MPa92.2)1083.0)(105.3(
1083.0
3000
5.2
2
33
max
3
max
====
=××==
×==
直径即棒
σ
εσ
ε
环氧树脂的密度ρ＝1.25Mg/m
3
，故棒的重量为(1.25) (753.4×10
－6
)
(3)=2.83kg，其价格为(2.83kg)($1.75/kg)=$4.95
例题再来看复合材料的情况。最低要求是80MPa应力下的应变为0.83×10
－3
，则弹性模量为
GPa4.96
1083.0
80
3
max
=
×
=>
ε
σ
c
E
玻璃纤维的模量低于96.4GPa，应考虑更强的纤维。高模碳纤维的模量为531GPa，密度为1.9Mg/m
3
，价格为$66/kg。为使复合材料具有96.4GPa的模量，最低碳纤维体积分数为：
176.04.96)5.3)(1()531( =>?+=
fffc
VVVE
环氧树脂体积分数为0.824。
例4-3：设计一棒材，长度为3m。以环氧树脂为基体。受到2.2kN的外力时，伸长不能超过2.5mm，不能发生脆性断裂。已知环氧树脂弹性模量为3.5GPa，屈服强度为
80MPa，价格为$1.75/kg。
如果纤维全部断裂，截面积82.4%上的环氧树脂必须承受
2.2kN的外力且使应力不大于80MPa。由此决定棒材的截面积：
mmdmmA
mm
F
AA
c
cm
5.64.33
824.0
5.27
5.27
80
2200
824.0
2
2
===
====
即
σ
例4-3：设计一棒材，长度为3m。以环氧树脂为基体。受到2.2kN的外力时，伸长不能超过2.5mm，不能发生脆性断裂。已知环氧树脂弹性模量为3.5GPa，屈服强度为
80MPa，价格为$1.75/kg。
18
42.2$)/75.1)($103.0()/66)($034.0(
103.0)137.0)((755.0(
034.0)137.0)(245.0(
245.0
)25.1)(824.0()19.0)(176.0(
)19.0)(176.0(
137.010137.0)101.0)](824.0)(25.1()176.0)(9.1[(
101.0)3)(104.33(
33
3326
=+=
==
==
=
+
=
=×=×+=
×=×=
kgkgkgkg
kg
kg
kgMg
mmm
棒材价格环氧树脂的重量碳纤维的重量碳纤维的重量分数棒材重量棒材体积对比2.83kg，$4.95
采用碳纤维复合后的材料直径不足只用环氧树脂的四分之一，重量只有5％，价格还降低了一半。
例4-3：设计一棒材，长度为3m。以环氧树脂为基体。受到2.2kN的外力时，伸长不能超过2.5mm，不能发生脆性断裂。已知环氧树脂弹性模量为3.5GPa，屈服强度为
80MPa，价格为$1.75/kg。
4.6 树脂基复合材料热变形温度
