SME
Biomechanics · Problem type
Creep in a Kelvin–Voigt solid
Use this for “A ligament model with E = 10 MPa and τ = 50 s carries 0.5 MPa. What is its strain after 100 s?”
Diagram
τ
=
η
E
=
500
10
=
50
s.
It
creeps
toward
σ
E
=
0.5
10
=
0.0500
and
never
flows.
ε
=
(
σ
E
)(1
−
e
−t/τ
)
=
0.0500
×
(1
−
e
−100 ÷ 50
)
=
0.0432:
86.5%
of
the
way.
After one τ, the strain is 63% of the way there.
This is an example. Type your own values to start a new problem.
τ (s)
=
η (MPa·s)
E (MPa)
ε∞
=
σ (MPa)
E (MPa)
ε
=
ε∞
(1
−
e
−
s
/
s
)
Formulas
τ
=
η
E
50
=
500
10
ε∞
=
σ
E
0.05
=
0.5
10
ε
=
ε∞
×
(1
−
e
−
t
÷
τ
)
0.04323
=
0.05
×
(1
−
e
−100 ÷ 50
)
Assumptions
•
A Kelvin–Voigt solid: a spring and a dashpot side by side, under a stress held from t = 0.
•
The dashpot slows the stretch; the strain creeps toward σ ÷ E, where the spring takes all the load, and never flows past it.
•
After one time constant τ the strain is 63% of the way there (1 − e⁻¹).
•
Viscosity η is in MPa·s and E in MPa, so τ = η ÷ E comes out in seconds.
Step-by-step
WE KNOW
σ = 0.5 MPa, E = 10 MPa, τ = 50 s, t = 100 s
FIND
viscosity (η), final strain (ε∞), strain at t (ε)
Step 1 · Find viscosity (η)
Use τ = η ÷ E
The viscosity is the time constant times the modulus.
η
=
τ
×
E
η
=
50
×
10
η
=
500
MPa·s
Step 2 · Find final strain (ε∞)
Use ε∞ = σ ÷ E
In the end the spring carries the whole stress.
ε∞
=
0.5
10
ε∞ = 0.05
Step 3 · Find strain at t (ε)
Use ε = ε∞ × (1 − e−t ÷ τ)
The
share
of
the
final
strain
reached
by
t
is
1
−
e
−
t
τ
.
ε
=
0.05
×
(1
−
e
−100 ÷ 50
)
ε
=
0.05
×
(1
−
0.1353)
ε
=
0.05
×
0.8647
ε
=
0.04323
Check
Put the numbers back in. Both sides should match.
50
=
500
10
✓
0.05
=
0.5
10
✓
0.04323
=
0.05
×
(1
−
e
−100 ÷ 50
)
✓
Topic check
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