SME
Biomechanics
Viscoelasticity
Use this for “A cartilage model with E = 10 MPa and η = 500 MPa·s is held at 5% strain. What stress is left after 50 s?”
Diagram
τ
=
η
E
=
500
10
=
50
s.
σ₀
=
Eε₀
=
10
×
0.05
=
0.500
MPa.
σ
=
σ₀
e
−t/τ
=
0.500
×
e
−50 ÷ 50
=
0.184
MPa:
36.8%
of
σ₀
left.
After one τ, 37% of the stress is left.
This is an example. Type your own values to start a new problem.
τ (s)
=
η (MPa·s)
E (MPa)
σ₀ (MPa)
=
E (MPa)
×
ε₀
σ (MPa)
=
σ₀ (MPa)
e
−
s
/
s
Formulas
τ
=
η
E
50
=
500
10
σ₀
=
E
×
ε₀
0.5 = 10 × 0.05
σ
=
σ₀
×
e
−
t
÷
τ
0.1839
=
0.5
×
e
−50 ÷ 50
Assumptions
•
A Maxwell solid: a spring and a dashpot in series, stretched to ε₀ and held there.
•
At first the spring takes all the stretch; the dashpot slowly gives way and the stress relaxes.
•
After one time constant τ, 37% of the stress is left (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
E = 10 MPa, η = 500 MPa·s, ε₀ = 0.05, t = 50 s
FIND
time constant (τ), starting stress (σ₀), stress at t (σ)
Step 1 · Find time constant (τ)
Use τ = η ÷ E
The dashpot’s viscosity over the spring’s modulus: how long the dashpot takes to undo the spring.
τ
=
500
10
τ
=
50
s
Step 2 · Find starting stress (σ₀)
Use σ₀ = E × ε₀
At first only the spring stretches: Hooke’s law.
σ₀
=
10
×
0.05
σ₀
=
0.5
MPa
Step 3 · Find stress at t (σ)
Use σ = σ₀ × e−t ÷ τ
The
stress
falls
by
the
factor
e
−
t
τ
as
the
dashpot
gives
way.
σ
=
0.5
×
e
−50 ÷ 50
σ
=
0.5
×
0.36788
σ
=
0.1839
MPa
Check
Put the numbers back in. Both sides should match.
50
=
500
10
✓
0.5 = 10 × 0.05 ✓
0.1839
=
0.5
×
e
−50 ÷ 50
✓
Topic check
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