Post on 17-Jan-2016
EML 4230 Introduction to Composite Materials
Chapter 2 Macromechanical Analysis of a Lamina
2D Stiffness and Compliance Matrix for Unidirectional Lamina
Dr. Autar KawDepartment of Mechanical Engineering
University of South Florida, Tampa, FL 33620
Courtesy of the TextbookMechanics of Composite Materials by Kaw
Compliance and Stiffness Matrix Elements in Terms of Elastic Constants
(a) Tensile load in direction 1
(b) Tensile load in direction in 2
(c) Pure shear stress in plane of 1-2FIGURE 2.18Application of stresses tofind engineering constantsof a unidirectional lamina
Pure Axial Load in Direction 1
Apply a pure axial load in direction 1
000 1221 = τ, = σ, σ
τ
σ
σ
S
SS
SS
=
γ
ε
ε
12
2
1
66
2212
1211
12
2
1
00
0
0
S=
ε
σE
111
11
1
.S
S =ε
ε ν11
12
1
212
E=S
111
1
E
ν=S1
1212
012
1122
1111
=γ
σS=ε
σS=ε
Pure Axial Load in Direction 2
Apply a pure axial load in direction 2
000 1221 =τ, σ, σ
τ
σ
σ
S
SS
SS
=
γ
ε
ε
12
2
1
66
2212
1211
12
2
1
00
0
0
S=
ε
σE
222
22
1
.S
S =ε
ε ν22
12
2
121
E=S
222
1
E
ν=S2
2112
012
2222
2121
=γ
σS=ε
σS=ε
Reciprocal Relationship
E
ν=S2
2112
E
ν=S1
1212
E
ν=E
ν
2
21
1
12
Pure Shear Load in Plane 12
Apply a pure shear load in Plane 12
000 1221 τ, =σ,σ
τ
σ
σ
S
SS
SS
=
γ
ε
ε
12
2
1
66
2212
1211
12
2
1
00
0
0
126612
2
1
0
0
=Sγ
=ε
=ε
S =G
6612
1212
1
12
66
1
G=S
Compliance Matrix
τ
σ
σ
S
SS
SS
=
γ
ε
ε
12
2
1
66
2212
1211
12
2
1
00
0
0
τ
σ
σ
G
EE
νE
ν
E
=
γ
ε
ε
12
2
1
12
21
12
1
12
1
12
2
1
100
01
01
Coefficients of Stiffness Matrix
γ
ε
ε
Q
=
τ
σ
σ
12
2
1
66
2212
1211
12
2
1
00
0
0
S = Q
S SS
S = Q
S SSS = Q
S SS
S = Q
6666
2122211
1122
2122211
1212
2122211
2211
1
γ
ε
ε
G
νν E
νν Eν
νν E ν
νν - E
=
τ
σ
σ
12
2
1
12
1221
2
1221
212
1221
212
1221
1
12
2
1
00
011
011
END