MECH4301 2022 Fall 
Intermediate Mechanics of Materials 
• Ch.1 Introduction: Mechanical Design and Tensor analysis 
• Ch.2 Review on Mechanics of Materials : Basic Notations 
• Ch.3 Mechanical Behaviors of Materials 
• Ch.4 Elastic Behaviors of Materials 
• Ch.5 2D Elasticity 
• Ch.6 Yield Criteria 
MECH4301 2022 Fall 
Ch.1 Mechanical Design and Tensor analysis 
Tensor transformation 
~ useful for the coordinate transformation for general tensors(vector, tensors) 
~ vector transformation 
ji j iQ e e′ ⋅ 
i i j j 
j ji i 
a e a e 
a Q a 
′ ′= = 
′ = 
Α 
1 11 12 13 1 
2 21 22 23 2 
3 31 32 33 3 
a Q Q Q a 
a Q Q Q a 
a Q Q Q a 
′      
     ′ ′= ⋅ ⇔ =      
′           
Α Q Α 
ij i j k k 
k ki j ij 
a e e a e e 
a Q Q a 
′ ′ ′= ⊗ = ⊗ 
′ = 
Α   
  
    
~ tensor transformation 
11 12 13 11 12 13 11 12 13 11 21 31 
21 22 23 21 22 23 21 22 23 12 22 32 
31 32 33 31 32 33 31 32 33 13 23 33 
T 
a a a Q Q Q a a a Q Q Q 
a a a Q Q Q a a a Q Q Q 
a a a Q Q Q a a a Q Q Q 
′ ′ ′        
       ′ ′ ′ ′= ⋅ ⋅ ⇔ =        
′ ′ ′               
Α Q Α Q 
MECH4301 2022 Fall 
Fundamental variables(unknowns) 
Ch.2 Elementary Mechanics of Materials 
~ Displacement = −u x X 
X 
2x 
Initial configuration 
Current configuration 
x 
u 
~ Strain 
1 
2 
ji 
ij 
j i 
uu 
x x 
ε 
 ∂∂ 
= +  ∂ ∂  
~ Stress i ij jt nσ= 
: internal traction force per unit areat 
: normal vector of the surfacen 
MECH4301 2022 Fall 
Fundamental equations(governing equations) 
Ch.2 Elementary Mechanics of Materials 
~ Equilibrium :  
1 
2 
ji 
ij 
j i 
uu 
x x 
ε 
 ∂∂ 
= +  ∂ ∂  
, 0ij jσ = 
~ Kinematics :  
~ Constitutive relation :  
ij ijk kCσ ε=   
0, 0, 0xy yx yy yz zyxx xz zx zz 
x y z x y z x y z 
σ σ σ σ σσ σ σ σ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ 
+ + = + + = + + = 
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ 
1111 1122 1133 1123 1113 111211 
2211 2222 2233 2223 2213 221222 
3311 3322 3333 3323 3313 331233 
2311 2322 2333 2323 2313 231223 
1311 1322 1333 1323 1313 131213 
1211 122212 
C C C C C C 
C C C C C C 
C C C C C C 
C C C C C C 
C C C C C C 
C C C 
σ 
σ 
σ 
σ 
σ 
σ 
  
  
  
  
=  
  
  
   
  
11 
22 
33 
23 
13 
1233 1223 1213 1212 12C C C 
ε 
ε 
ε 
ε 
ε 
ε 
   
   
   
   
   
   
   
     
   
31 2 
11 22 33 
1 2 3 
3 31 2 2 1 
12 23 13 
2 1 3 2 3 1 
, , 
1 1 1, , 
2 2 2 
uu u 
x x x 
u uu u u u 
x x x x x x 
ε ε ε 
ε ε ε 
∂∂ ∂ 
= = = 
∂ ∂ ∂ 
     ∂ ∂∂ ∂ ∂ ∂ 
= + = + = +    ∂ ∂ ∂ ∂ ∂ ∂      
, , , , 0ij k k ij ik j j ikε ε ε ε+ − − =   ~ Compatibility:  
MECH4301 2022 Fall 
Elastic 
Ch.3 Mechanical Behaviors of Materials 
~ deformation which is  
“instantaneous” and “recoverable” 
Anelastic 
~ deformation which is  
“recoverable but “time dependent” 
Plastic 
~ deformation which is “permanent” 
MECH4301 2022 Fall 
Ch.4 Elastic Behaviors of Materials 
Material constants in isotropic elasticity 
~ Young’s modulus, Shear modulus, Bulk modulus, Compressibility,  
Poisson’s ratio, Lame’s constants~ 2 independent variables 
( )2 1 
Eµ 
ν 
= 
+ ( )( ) ( ) 
2 
1 1 2 1 2 
Eν νµλ 
ν ν ν 
= = 
+ − − ( ) ( ) 
2 3 2 
2 2 2 3 
E B 
B 
λ µ µν 
µ λ µ µ 
− − 
= = = 
+ + 
( ) 
( ) ( ) 
3 2 92 1 
9 
BE 
B 
µ λ µ λ µµ ν 
µ λ µ 
+ 
= = + = 
+ + 
3 2 1 
3 
B 
K 
λ µ+ 
= = 
Generalized Hooke’s Law 
~ linear relation between strain components and stress components 
~ for an isotropic elastic solids 
2ij ij kk ijσ µε λε δ+= 
1 
ij ij kk ijE E 
ν νε σ σ δ+= − 
MECH4301 2022 Fall 
Ch.4 Elastic Behaviors of Materials 
Anisotropic Linear Elasticity 
~ Independent components in stiffness tensor: (depending on crystal symmetry) 
21(triclinic) 13(monoclinic) 9(Orthotropic) 
6(tetragonal) 3(Cubic) 2(Isotropic) 
MECH4301 2022 Fall 
Ch.5 2D Elasticity 
~ Equilibrium :  
~ Constitutive 
relation:  
0 
0 
0 
xyxx xz 
yx yy yz 
zyzx zz 
x y z 
x y z 
x y z 
σσ σ 
σ σ σ 
σσ σ 
∂∂ ∂ 
+ + = ∂ ∂ ∂ 
∂ ∂ ∂ + + = 
∂ ∂ ∂ 
 ∂∂ ∂ 
+ + = 
∂ ∂ ∂ 
1 
2 
,  
yx 
xy 
yx 
xx yy 
uu 
y x 
uu 
x y 
ε 
ε ε 
∂  ∂ 
= +  ∂ ∂   
 
∂∂ = = ∂ ∂ 
1 
1 
1 
, ,  
2 2 2 
xx xx yy zz 
yy xx yy zz 
zz xx yy zz 
xy yz zx 
xy yz zx 
E E E 
E E E 
E E E 
ν νε σ σ σ 
ν νε σ σ σ 
ν νε σ σ σ 
σ σ σε ε ε 
µ µ µ 
 = − − 
 
 = − + − 
 
 
 = − − + 
 
 
 = = = 
 
0 
0 
xyxx 
yx yy 
x y 
x y 
σσ 
σ σ 
∂∂ 
+ = ∂ ∂ 
∂ ∂ + = ∂ ∂ 
1 
1 
2 
xx xx yy 
yy xx yy 
xy 
xy 
E E 
E E 
νε σ σ 
νε σ σ 
σ 
ε 
µ 
 
= − 
 
 = − + 
 
 
= 
 
~ Kinematics:  
33 
1 1, 
2 2 
1 
2 
, ,  
y yx z 
xy yz 
x z 
xz 
yx z 
xx yy 
u uu u 
y x z y 
u u 
z x 
uu u 
x y z 
ε ε 
ε 
ε ε ε 
∂ ∂    ∂ ∂ 
= + = +    ∂ ∂ ∂ ∂    
 ∂ ∂  = +  ∂ ∂  
 ∂∂ ∂ 
= = = 
∂ ∂ ∂ 
MECH4301 2022 Fall 
Ch.5 2D Elasticity 
~ Airy Stress Function( ): φ ,xx yyσ φ= ,yy xxσ φ= ,xy xyσ φ= − 
~ Governing equation : (bi-harmonic equation)  
, , ,2 0xxxx yyyy xyxyφ φ φ+ + = or 4 0φ∇ = 
~ General solution procedure 
Step I : Determine the highest order of polynomial 
( )~q O n- Distributed normal loading: ( )~ 5O nφ +→ Airy Fn: 
Step II : Write down a polynomial function up to order n+5 
Step III : Check the governing equation 
Step IV : Check the B.C.’s 
Step V : Solve for Cis 
MECH4301 2022 Fall 
Ch.5 2D Elasticity 
~ cylindrical vessel: θθσ (hoop stress), zzσ (axial stress), rrσ (radial stress) 
Inner 
surface 
Outer  
surface 
θθσ 
zzσ 
rrσ 
2 
pR 
t 2 
pR 
t 
pR 
t 
pR 
t 
p− 0 
Inner 
surface 
Outer  
surface 
θθσ 
rrσ − 

2 

20 
~ spherical vessel: θθσ (hoop stress), rrσ (radial stress) 
MECH4301 2022 Fall 
Ch.6 Yield Criteria 
( ) 0I II IIIf f σ σ σ= > > = 
~ function of principal stresses in isotropic solids:  
I ff σ σ= − 
at yielding(onset of plastic deformation)  
~ Max. normal stress(Rankine): Stress at failure by yielding 
~ Max. shear stress(Tresca) : max 2 2 
I III Y 
Cf 
σ σ σ 
τ τ 
− 
= − = − 
yield stress in uniaxial loading 
~ Max. distortional energy (Mises) : 
( ) ( ) ( )2 2 2 
2 
I II II III III I 
Yf 
σ σ σ σ σ σ 
σ 
− + − + − 
= − 
~ Mohr-Coulomb : ( ) ( )sin 2 cosI III I IIIf Cσ σ σ σ φ φ= − + + − 
~ Drucker-Prager : 
( ) ( ) ( ) ( ) 
2 2 2 
1 2 6 
I II II III III I 
I II IIIf I J K K 
σ σ σ σ σ σ 
α α σ σ σ 
− + − + − 
= + − = + + + − 
~ Hill: ( ) ( ) ( )2 2 2 2 2 222 33 33 11 11 22 23 31 122 2 2 1f F G H L M Nσ σ σ σ σ σ σ σ σ= − + − + − + + + − 
Hydrostatic stress 
Anisotropy