SME
Advanced Solid Mechanics (Solid Mechanics II) · Problem type
Elastic constants from E and ν
Use this for “Steel has E = 200 GPa and ν = 0.3. Find its shear modulus, bulk modulus and Lamé constant λ.”
Diagram
Two constants fix the rest.
G
=
E
2(1 + ν)
=
76.92
GPa.
K
=
E
3(1 − 2ν)
=
166.7
GPa.
λ
=
Eν
(1 + ν)(1 − 2ν)
=
115.4
GPa.
Each cube shows what its modulus resists: a pull (E, with ν the sideways shrink), a shear (G) and an even squeeze (K); the changes are drawn larger.
This is an example. Type your own values to start a new problem.
G (GPa)
=
E (GPa)
2(1
+
)
K (GPa)
=
E (GPa)
3(1
−
2
×
)
λ (GPa)
=
E (GPa)
×
(1
+
)(1
−
2
×
)
Formulas
G
=
E
2 × (1 +
ν
)
76.9231
=
200
2 × (1 + 0.3)
K
=
E
3 × (1 − 2 ×
ν
)
166.6667
=
200
3 × (1 − 2 × 0.3)
λ
=
E
×
ν
(1 +
ν
) × (1 − 2 ×
ν
)
115.3846
=
200
×
0.3
(1 + 0.3) × (1 − 2 × 0.3)
Assumptions
•
Linear elastic and isotropic: any two elastic constants fix all the others.
•
ν stays below 0.5; at 0.5 the material is incompressible and K has no finite value.
•
G is also Lamé’s μ; λ is the other Lamé constant of the 3D stress–strain law.
•
Moduli are worked in MPa (1 GPa = 1000 MPa); steel: E = 200 GPa, ν = 0.3.
Step-by-step
WE KNOW
E = 200 GPa, ν = 0.3
FIND
shear modulus (G), bulk modulus (K), first Lamé constant (λ)
Step 1 · Find shear modulus (G)
Use G = E ÷ (2 × (1 + ν))
Shear
and
tension
describe
the
same
isotropic
solid,
so
G
follows
from
E
and
ν
.
G
=
200
2 × (1 + 0.3)
G
=
200
2 × 1.3
G
=
200
2.6
G
=
76.9231
GPa
Step 2 · Find bulk modulus (K)
Use K = E ÷ (3 × (1 − 2 × ν))
Equal
pressure
on
every
face
shrinks
each
edge
by
(1
−
2
ν
)
p
E
,
so
the
volume
by
three
times
that.
K
=
200
3 × (1 − 2 × 0.3)
K
=
200
3 × (1 − 0.6)
K
=
200
3 × 0.4
K
=
200
1.2
K
=
166.6667
GPa
Step 3 · Find first Lamé constant (λ)
Use λ = E × ν ÷ ((1 + ν) × (1 − 2 × ν))
Lamé’s
λ
is
the
part
of
the
stress
that
follows
the
volume
change;
it
grows
without
limit
as
ν
nears
0.5.
λ
=
200
×
0.3
(1 + 0.3) × (1 − 2 × 0.3)
λ
=
200
×
0.3
1.3 × (1 − 0.6)
λ
=
200
×
0.3
1.3 × 0.4
λ
=
200
×
0.3
0.52
λ
=
60
0.52
λ
=
115.3846
GPa
Check
Put the numbers back in. Both sides should match.
76.9231
=
200
2 × (1 + 0.3)
✓
166.6667
=
200
3 × (1 − 2 × 0.3)
✓
115.3846
=
200
×
0.3
(1 + 0.3) × (1 − 2 × 0.3)
✓
Topic check
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