SME
Biomechanics
Tissue mechanics
Use this for “A tendon 50 mm long with a 50 mm² cross-section carries 2000 N. With E = 1.2 GPa, how far does it stretch?”
Diagram
Tissue: a toe region while the fibres straighten, then a straight part of slope E.
Here ε counts from where the straight part meets the axis.
The
point:
σ
=
Eε
=
1,200
MPa
×
0.0333
=
40
MPa.
σ
=
F
A
=
2,000
N
50
mm²
=
40
MPa.
ε
=
ΔL
L
=
1.67
mm
50
mm
=
0.0333.
ΔL
is
drawn
5
times
larger
to
be
seen.
This is an example. Type your own values to start a new problem.
σ (MPa)
=
F (N)
A (mm²)
ε
=
σ (MPa)
E (GPa)
ΔL (mm)
=
ε
×
L (mm)
Formulas
σ
=
F
A
40
=
2,000
50
ε
=
σ
E
0.03333
=
40
1,200
ΔL
=
ε
×
L
1.67 = 0.03333 × 50
Formulas are worked in MPa. The step-by-step shows the conversions.
Assumptions
•
Past the toe region the tissue is linear elastic; ε counts from where the straight part starts.
•
σ is engineering stress: the force over the starting cross-section area.
•
The load runs along the fibers, and the stress is even across the section.
Step-by-step
WE KNOW
F = 2,000 N, A = 50 mm², E = 1.2 GPa, L = 50 mm
FIND
stress (σ), strain (ε), stretch (ΔL)
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 = 1.2 GPa = 1,200 MPa (1 GPa = 1000 MPa)
Step 2 · Find stress (σ)
Use σ = F ÷ A
Stress is force over area; one newton on each square millimeter is 1 MPa.
σ
=
2,000
50
σ
=
40
MPa
Step 3 · Find strain (ε)
Use ε = σ ÷ E
On the straight part of the curve, strain is stress over the modulus.
ε
=
40
1,200
ε
=
0.03333
Step 4 · Find stretch (ΔL)
Use ΔL = ε × L
Strain is the stretch per millimeter of resting length, so multiply by the length.
ΔL = 0.03333 × 50
ΔL = 1.67 mm
Check
Put the numbers back in. Both sides should match.
40
=
2,000
50
✓
0.03333
=
40
1,200
✓
1.67 = 0.03333 × 50 ✓
Topic check
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