SME
Biomaterials
Metals, polymers and ceramics in the body
Use this for “A titanium stem (110 GPa, 100 mm²) is bonded to bone (18 GPa, 300 mm²). Under 2000 N, what share does the implant carry, and what stress is left in the bone?”
Diagram
Bonded,
both
strain
the
same,
so
each
carries
load
in
proportion
to
EA:
share
=
EᵢAᵢ
EᵢAᵢ + EbAb
=
1.1 ×
10
7
N
1.1 ×
10
7
N + 5.4 ×
10
6
N
=
67.1%.
The two shares add to 1: 67.1% + 32.9% = 100%.
Bone
carries
659
N,
so
σ
=
659
N
300
mm²
=
2.2
MPa.
This is an example. Type your own values to start a new problem.
Eᵢ
Implant modulus
example · GPa
Aᵢ
Implant area
example · mm²
Eb
Bone modulus
example · GPa
Ab
Bone area
example · mm²
F
Load
example
share
Implant’s share
calculated · %
Fb
Bone’s force
calculated
σb
Bone stress
calculated
Formulas
share
=
100
×
Eᵢ × Aᵢ
Eᵢ × Aᵢ + Eb × Ab
67.1
=
100
×
110 × 100
110 × 100 + 18 × 300
Fb
=
F
×
100 − share
100
658.5
=
2,000
×
100 − 67.1
100
σb
=
Fb
Ab
2.195
=
658.5
300
Assumptions
•
Implant and bone are bonded, so both strain the same; each carries load in proportion to EA.
•
The load is axial and the stress is even across each member.
•
A stiffer implant takes more of the load, so the bone it shields is stressed less.
Step-by-step
WE KNOW
Eᵢ = 110 GPa, Aᵢ = 100 mm², Eb = 18 GPa, Ab = 300 mm², F = 2,000 N
FIND
implant’s share (share), bone’s force (Fb), bone stress (σb)
Step 1 · Find implant’s share (share)
Use share = 100 × (Eᵢ × Aᵢ) ÷ (Eᵢ × Aᵢ + Eb × Ab)
Both strain the same, so force goes with stiffness EA: the implant’s EA over the total.
share
=
100
×
110 × 100
110 × 100 + 18 × 300
share
=
100
×
11,000
11,000 + 5,400
share
=
100
×
11,000
16,400
share
=
1,100,000
16,400
share = 67.073%
Step 2 · Find bone’s force (Fb)
Use Fb = F × (100 − share) ÷ 100
The bone carries the part of the load the implant does not.
Fb
=
2,000
×
100 − 67.073
100
Fb
=
2,000
×
32.927
100
Fb
=
65,854
100
Fb = 658.5 N
Step 3 · Find bone stress (σb)
Use σb = Fb ÷ Ab
Stress is force over area; one newton on each square millimeter is 1 MPa.
σb
=
658.5
300
σb = 2.195 MPa
Check
Put the numbers back in. Both sides should match.
67.073
=
100
×
110 × 100
110 × 100 + 18 × 300
✓
658.5
=
2,000
×
100 − 67.073
100
✓
2.195
=
658.5
300
✓
Topic check
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