Formula Sheet: Mechanical Properties of Materials | Solid Mechanics - Mechanical Engineering PDF Download

 Page 1


Mec hanical Prop erties of M aterials F orm ula Sheet
for Mec hanical GA TE
Stress and Strain
• Normal Stress : F orce p er unit area p erp endicular to the surface.
s =
F
A
(Units: Pa or Nm
-2
)
where F is the force, A is the cross-sectional area.
• Shear Stress : F orce p er unit area parallel to the surface.
t =
F
A
• Normal Strain : Deformation p er unit length.
? =
?L
L
where ?L is the c hange in length, L is the original length.
• Shear Strain : Angular deformation.
? = tan?˜? (for small angles)
• P oisson’s Ratio : Ratio of lateral strain to axial strain.
? =-
?
lateral
?
axial
Elastic Constan ts
• Ho ok e’s La w : Stress is prop ortional to strain within elastic limit.
s =E?
where E is Y oung’s mo dulus (Pa ).
• Shear Mo dulus (Mo dulus of Rigidit y) : Ratio of shear stress to shear strain.
G =
t
?
• Bulk Mo dulus : Ratio of v olumetric stress to v olumetric strain.
K =-
?P
?V/V
where ?P is the c hange in pressure, ?V/V is the v olumetric str ain.
• Relation b et w een Elastic Constan ts :
E = 2G(1+?), E = 3K(1-2?), E =
9KG
3K +G
1
Page 2


Mec hanical Prop erties of M aterials F orm ula Sheet
for Mec hanical GA TE
Stress and Strain
• Normal Stress : F orce p er unit area p erp endicular to the surface.
s =
F
A
(Units: Pa or Nm
-2
)
where F is the force, A is the cross-sectional area.
• Shear Stress : F orce p er unit area parallel to the surface.
t =
F
A
• Normal Strain : Deformation p er unit length.
? =
?L
L
where ?L is the c hange in length, L is the original length.
• Shear Strain : Angular deformation.
? = tan?˜? (for small angles)
• P oisson’s Ratio : Ratio of lateral strain to axial strain.
? =-
?
lateral
?
axial
Elastic Constan ts
• Ho ok e’s La w : Stress is prop ortional to strain within elastic limit.
s =E?
where E is Y oung’s mo dulus (Pa ).
• Shear Mo dulus (Mo dulus of Rigidit y) : Ratio of shear stress to shear strain.
G =
t
?
• Bulk Mo dulus : Ratio of v olumetric stress to v olumetric strain.
K =-
?P
?V/V
where ?P is the c hange in pressure, ?V/V is the v olumetric str ain.
• Relation b et w een Elastic Constan ts :
E = 2G(1+?), E = 3K(1-2?), E =
9KG
3K +G
1
Stress-Strain Relationships
• Generalized Ho ok e’s La w (3D) :
?
x
=
1
E
[s
x
-?(s
y
+s
z
)]
Similar for ?
y
and ?
z
.
• Shear Stress-Strain :
?
xy
=
t
xy
G
• Thermal Strain : Strain due to temp erature c hange.
?
T
=a?T
where a is the co e?icien t of the rmal expansion, ?T is the temp erature c hange.
Shear F orce and Bending Momen t
• Shear F orce : In ternal force resisting shear stress.
dV
dx
=w(x)
where V is shear force, w(x) is the distributed load.
• Bending Momen t : In ternal momen t resisting b ending.
dM
dx
=V
where M is the b ending momen t.
• Flexural F orm ula : Stress due to b ending.
s =
My
I
where y is the distance from the neutral a xis, I is the momen t of inertia.
• Shear Stress in Beams :
t =
VQ
Ib
where Q is the first momen t of area, b is the width of the b e am.
T o rsion
• T orsion F orm ula : Shear stress in a circular shaft.
t =
Tr
J
where T is the torque, r is the radius, J is the p olar momen t of inertia.
• Angle of T wist :
? =
TL
GJ
where L is the length of the shaft.
2
Page 3


Mec hanical Prop erties of M aterials F orm ula Sheet
for Mec hanical GA TE
Stress and Strain
• Normal Stress : F orce p er unit area p erp endicular to the surface.
s =
F
A
(Units: Pa or Nm
-2
)
where F is the force, A is the cross-sectional area.
• Shear Stress : F orce p er unit area parallel to the surface.
t =
F
A
• Normal Strain : Deformation p er unit length.
? =
?L
L
where ?L is the c hange in length, L is the original length.
• Shear Strain : Angular deformation.
? = tan?˜? (for small angles)
• P oisson’s Ratio : Ratio of lateral strain to axial strain.
? =-
?
lateral
?
axial
Elastic Constan ts
• Ho ok e’s La w : Stress is prop ortional to strain within elastic limit.
s =E?
where E is Y oung’s mo dulus (Pa ).
• Shear Mo dulus (Mo dulus of Rigidit y) : Ratio of shear stress to shear strain.
G =
t
?
• Bulk Mo dulus : Ratio of v olumetric stress to v olumetric strain.
K =-
?P
?V/V
where ?P is the c hange in pressure, ?V/V is the v olumetric str ain.
• Relation b et w een Elastic Constan ts :
E = 2G(1+?), E = 3K(1-2?), E =
9KG
3K +G
1
Stress-Strain Relationships
• Generalized Ho ok e’s La w (3D) :
?
x
=
1
E
[s
x
-?(s
y
+s
z
)]
Similar for ?
y
and ?
z
.
• Shear Stress-Strain :
?
xy
=
t
xy
G
• Thermal Strain : Strain due to temp erature c hange.
?
T
=a?T
where a is the co e?icien t of the rmal expansion, ?T is the temp erature c hange.
Shear F orce and Bending Momen t
• Shear F orce : In ternal force resisting shear stress.
dV
dx
=w(x)
where V is shear force, w(x) is the distributed load.
• Bending Momen t : In ternal momen t resisting b ending.
dM
dx
=V
where M is the b ending momen t.
• Flexural F orm ula : Stress due to b ending.
s =
My
I
where y is the distance from the neutral a xis, I is the momen t of inertia.
• Shear Stress in Beams :
t =
VQ
Ib
where Q is the first momen t of area, b is the width of the b e am.
T o rsion
• T orsion F orm ula : Shear stress in a circular shaft.
t =
Tr
J
where T is the torque, r is the radius, J is the p olar momen t of inertia.
• Angle of T wist :
? =
TL
GJ
where L is the length of the shaft.
2
Material Prop erties
• Yield Strength : Stress at whic h plastic deformation b egins.
• Ultimate T ensile Strength : Maxim um stress a material can withstand.
• T oughness : Energy absorb ed b efore fracture (area under stress-strain curv e).
• Ductilit y : Abilit y to deform plastically without fracture.
% Elongation =
L
f
-L
0
L
0
×100
where L
f
is the final length, L
0
is the original length.
Key N otes
• Standard v alues: E
steel
˜ 200GPa , ?
steel
˜ 0.3 , G
steel
˜ 77GPa .
• Use SI units for GA TE calculations.
• Chec k for plane stress or plane strain conditions in problems.
• F or torsion, assume circular sections unless sp ecified otherwise.
3