SME
Advanced Solid Mechanics (Solid Mechanics II)
Generalized Hooke's law
Use this for “A steel block carries σₓ = 100 MPa, σᵧ = 50 MPa and σz = −20 MPa. Find the three strains and the volumetric strain.”
Diagram
Each
strain
is
its
own
stress
over
E
less
ν
times
the
other
two:
εₓ
=
σₓ − ν(σᵧ + σz)
E
=
4.55 ×
10
-4
,
εᵧ
=
1.3 ×
10
-4
,
εz
=
−3.25 ×
10
-4
.
e
=
εₓ
+
εᵧ
+
εz
=
2.6 ×
10
-4
:
the
volume’s
share
of
change.
The dashed shape draws the strains 310 times larger.
This is an example. Type your own values to start a new problem.
εₓ
=
σₓ (MPa)
E (GPa)
−
ν
×
σᵧ (MPa)
+
(
σz (MPa)
)
E (GPa)
εᵧ
=
σᵧ (MPa)
E (GPa)
−
ν
×
σz (MPa)
+
σₓ (MPa)
E (GPa)
εz
=
σz (MPa)
E (GPa)
−
ν
×
σₓ (MPa)
+
σᵧ (MPa)
E (GPa)
e
Volumetric strain
calculated
Formulas
εₓ
=
σₓ −
ν
× (σᵧ + σz)
E
+4.55
×
10
-4
=
(100
−
0.3
×
(50
+
(−20)))
÷
200,000
εᵧ
=
σᵧ −
ν
× (σz + σₓ)
E
+1.3
×
10
-4
=
50 − 0.3 × (−20 + 100)
200,000
εz
=
σz −
ν
× (σₓ + σᵧ)
E
−3.25 ×
10
-4
=
−20 − 0.3 × (100 + 50)
200,000
e
=
εₓ
+
εᵧ
+
εz
+2.6
×
10
-4
=
4.55 ×
10
-4
+
(1.3
×
10
-4
)
+
(−3.25
×
10
-4
)
Formulas are worked in MPa. The step-by-step shows the conversions.
Assumptions
•
Linear elastic and isotropic: one E and one ν in every direction, below the proportional limit.
•
Tension is +; shear stresses change no normal strain.
•
For small strains the volumetric strain e = ΔV ÷ V is the sum of the three normal strains.
•
Steps work in N, mm and MPa (1 MPa = 1 N/mm²); steel: E = 200 GPa, ν = 0.3.
Step-by-step
WE KNOW
σₓ = +100 MPa, σᵧ = +50 MPa, σz = −20 MPa, E = 200 GPa, ν = 0.3
FIND
normal strain in x (εₓ), normal strain in y (εᵧ), normal strain in z (εz), volumetric strain (e)
Step 1 · Convert to the formula’s units
The formulas don’t work directly in the units you chose, so convert first (to MPa).
E = 200 GPa = 200,000 MPa (1 GPa = 1000 MPa)
Step 2 · Find normal strain in x (εₓ)
Use εₓ = (σₓ − ν × (σᵧ + σz)) ÷ E
Its
own
stress
stretches
the
x
edge
by
σₓ
E
;
Poisson’s
effect
of
the
other
two
stresses
takes
off
ν
E
times
their
sum.
εₓ = (100 − 0.3 × (50 + (−20))) ÷ 200,000
εₓ
=
100 − 0.3 × 30
200,000
εₓ
=
100 − 9
200,000
εₓ
=
91
200,000
εₓ
=
+4.55
×
10
-4
Step 3 · Find normal strain in y (εᵧ)
Use εᵧ = (σᵧ − ν × (σz + σₓ)) ÷ E
Its
own
stress
stretches
the
y
edge
by
σᵧ
E
;
Poisson’s
effect
of
the
other
two
stresses
takes
off
ν
E
times
their
sum.
εᵧ
=
50 − 0.3 × (−20 + 100)
200,000
εᵧ
=
50 − 0.3 × 80
200,000
εᵧ
=
50 − 24
200,000
εᵧ
=
26
200,000
εᵧ
=
+1.3
×
10
-4
Step 4 · Find normal strain in z (εz)
Use εz = (σz − ν × (σₓ + σᵧ)) ÷ E
Its
own
stress
stretches
the
z
edge
by
σz
E
;
Poisson’s
effect
of
the
other
two
stresses
takes
off
ν
E
times
their
sum.
εz
=
−20 − 0.3 × (100 + 50)
200,000
εz
=
−20 − 0.3 × 150
200,000
εz
=
−20 − 45
200,000
εz
=
−
65
200,000
εz
=
−3.25 ×
10
-4
Step 5 · Find volumetric strain (e)
Use e = εₓ + εᵧ + εz
For
small
strains
the
volume
grows
by
the
sum
of
the
three
normal
strains:
e
=
ΔV
V
.
e
=
4.55 ×
10
-4
+
(1.3
×
10
-4
)
+
(−3.25
×
10
-4
)
e
=
0.000585
+
(−3.25
×
10
-4
)
e
=
+2.6
×
10
-4
Check
Put the numbers back in. Both sides should match.
+4.55
×
10
-4
=
(100
−
0.3
×
(50
+
(−20)))
÷
200,000
✓
+1.3
×
10
-4
=
50 − 0.3 × (−20 + 100)
200,000
✓
−3.25 ×
10
-4
=
−20 − 0.3 × (100 + 50)
200,000
✓
+2.6
×
10
-4
=
4.55 ×
10
-4
+
(1.3
×
10
-4
)
+
(−3.25
×
10
-4
)
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.