More cross section does not automatically mean more force

Track the same material fibres as they shorten. Then compare cuts in series and fibre groups in parallel.

Scope: authored bookkeeping examples. No anatomical arm, equilibrium solver, biological calibration, or fluid transport. Six Real algebra contracts do not validate anatomy or browser arithmetic. These controls prescribe states rather than predict contraction.

1. Follow one material bundle

Reference geometric area is 100 mm². Nominal contractile stress scale is an authored 300 kPa. Packing is applied once to the reference area. The apparent area is the cut projected normal to the current fibre, not an arbitrary anatomical slice.

Reference contractile area: 80.00 mm²
Current projected area: 100.00 mm²
Nominal stress P: 300.00 kPa
Cauchy fibre stress σ: 240.00 kPa
Axial force: 24.00 N
Tendon-directed component: 20.78 N

Force–length is held at 1 to isolate the area/stress transformation.

A⊥ = A₀ J / λ
σ = P λ / J
T = σ A⊥ = P A₀
T along tendon = T cos α

At fixed nominal stress P, shortening increases projected area and lowers Cauchy stress. Fibre count and axial force remain unchanged. If P changes through a force–length law, force changes through that law. Neither result predicts a real arm’s equilibrium. Changing φ compares different reference bundles; it is not acute fibre creation.

Default material-cut case
Reference contractile area	80.00 mm²
Current projected area	100.00 mm²
Nominal stress P	300.00 kPa
Cauchy fibre stress σ	240.00 kPa
Axial force	24.00 N
Tendon-directed component	20.78 N

Authored default: geometric reference area 100 mm², contractile fraction 0.80, fibre stretch 0.80, volume ratio 1.00 and pennation 30°. The force–length multiplier is held at 1. The tendon-directed component is rounded to two decimals.

2. A taper is a series load path

Three reference cuts have areas 100, 200, and 100 mm². With 30 N axial force and no lateral load, all three transmit 30 N; nominal stress is 300, 150, and 300 kPa. These are section bookkeeping values, not a solved tapered three-dimensional body.

100 mm²
30 N300 kPa

200 mm²
30 N150 kPa

100 mm²
30 N300 kPa

If instead each cut is assigned 300 kPa, force becomes 30 → 60 → 30 N. Static balance requires lateral forces −30 N and +30 N between these cuts. Those forces must have an explicit extracellular-matrix or external load path. The wider cut alone cannot create net tendon force.

Tᵢ₊₁ − Tᵢ + Qᵢ = 0

Qᵢ is the net external/lateral force on a cell, positive in the chosen axial direction. Internal exchanges cancel in total force but can store or dissipate energy when adjacent materials move differently.

3. Fibre counts are not area weights

Two synthetic groups each contain 50,000 fibres. Group A has 1,000 µm² per fibre and 100 kPa nominal stress; group B has 2,000 µm² and 300 kPa. Their count fractions are both one half, but their area fractions are one third and two thirds.

Total reference area 150.00 mm² · summed force 35.00 N · area-weighted nominal stress 233.33 kPa

Summing individual forces gives 35 N. Multiplying total area by the count-averaged stress would give 30 N. These invented group sizes and stresses demonstrate the arithmetic; no human fibre distribution is inferred.

What this leaves open

The anatomical candidate’s fitted 3.60–20.38 MPa stress scales are model-reference coefficients with unresolved stationarity and compression. They are not measured physiological specific tensions. This lesson does not replace them.

PCSA conventions also differ across papers. Murray and colleagues use V/Lf, then apply the pennation projection separately. Ward and Lieber include cos α in their PCSA formula. Projecting a projected area again applies cos α twice.

ECM bonding, pressure, perfusion, geometry, and joint attachments need their own data and constitutive validation. A near-constant total volume does not establish local incompressibility. Shared external skin remains a separate unresolved anatomical layer.

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