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

QQ

QQ

=

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