Temperature-dependent shear modulus in torsion

Part VI · VerificationVI–B8

Part VI

Verification

B. Elements and loads · Temperature-dependent shear modulus in torsion

A hollow straight pipe four metres long, 0.12 m outside and 0.10 m inside diameter, is fixed at one end and twisted by a torque of 12,000 N·m at the other. Its shear modulus is given at named points and depends on temperature. What tip rotation follows from the modulus at an exact point, and from one interpolated between two temperatures?

A base modulus is also present. It is chosen so that a program which quietly fell back to it would miss both targets by a wide margin.

Fig. VI–B8.—Temperature-dependent shear modulus in torsion

1.Inputs.

Illustrative values, taken from no standard and chosen so the arithmetic can be followed by hand. Most do not describe a real pipe; read them in any consistent set of units.

Quantity	Symbol	Value
Pipe length	L	4.0 m
Outside diameter	D_o	0.12 m
Inside diameter	D_i	0.10 m
Tip torque	T	12,000 N·m
Base shear modulus	G_base	80.0 × 10⁹ Pa
Shear modulus at the exact point	G_exact	50.0 × 10⁹ Pa
Lower bracket temperature	T₁	300.0 K
Shear modulus at lower bracket	G₁	60.0 × 10⁹ Pa
Upper bracket temperature	T₂	500.0 K
Shear modulus at upper bracket	G₂	40.0 × 10⁹ Pa
Selected temperature	T_s	425.0 K

2.Method.

The torsion constant of the hollow circular section comes first. The exact-point case uses that point’s modulus directly. The temperature case interpolates linearly between the two adjacent points, and the base value plays no part in either.

For a circular shaft in pure torsion the tip rotation is TL/(GJ). The same formula with the base modulus gives the rotation a wrong fallback would produce. It differs from the two targets by 37.5% and 40.625%.

J = (π/32)(D_o⁴ − D_i⁴) = 1.0540043352793751 × 10⁻⁵ m⁴(1)

f = (T_s − T₁) / (T₂ − T₁) = (425 − 300) / (500 − 300) = 0.625(2)

G_interp = G₁ + f(G₂ − G₁) = 47.5 × 10⁹ Pa(3)

θ = TL / (GJ)(4)

3.Results.

Section, selected moduli and tip rotations
Quantity	Expected
Torsion constant J	1.054004 × 10⁻⁵ m⁴
Exact-point shear modulus	5 × 10¹⁰ Pa
Interpolated shear modulus	4.75 × 10¹⁰ Pa
Tip rotation, exact-point modulus	9.108122 × 10⁻² rad
Tip rotation, interpolated modulus	9.587497 × 10⁻² rad
Tip rotation, base modulus (must not be selected)	5.692576 × 10⁻² rad
Base-modulus miss against exact-point target	0.37500 (37.5%)
Base-modulus miss against interpolated target	0.40625 (40.625%)

The tests check that:

The solver’s tip rotation with the exact-point modulus matches the hand value.
The solver’s tip rotation with the interpolated modulus matches the hand value.
The tip rotation with the base modulus misses the two targets by 37.5% and 40.625%, far outside the tolerance.
Separate tests of the material-selection code check that an exact point uses its own modulus, and that interpolation uses the two adjacent points and records both.
Those tests also check that selection stops with an error at or beyond the ends of the stored range, and never extrapolates.
They check that a missing, non-positive, non-finite or wrongly dimensioned modulus stops selection, as do duplicate temperatures and conflicting selectors.
They check that selection never falls back to the base modulus, and that the base modulus is used when no temperature basis is named.
The torsion comparison is independent of the selection code and does not by itself show how selection behaves.

What it shows. Compared with an independent hand calculation.

Path exercised. The solver’s components directly, and, for the properties at temperature, the program’s own model-to-solve path.

Agreement. Each tip rotation must match the reference within a relative difference of 1.0 × 10⁻⁹. Long values are shown here to seven significant figures; the tests compare the full values in the record.

For the student

A good check is built so that the wrong answer cannot pass by accident. Here the base modulus sits well away from both selected values, so a program that used it by mistake would miss the target by more than a third.

4.Run it yourself.

cd projects/chirality-piping
cargo test --offline --manifest-path validation/benchmarks/mechanics/Cargo.toml temperature_indexed_shear_modulus_torsion_matches_independent_oracle
cargo test --offline --manifest-path core/product_physics/Cargo.toml dec092_fixture_consumes_exact_and_interpolated_g_with_provenance_and_combination_carry_through
cargo test --offline --manifest-path core/product_physics/Cargo.toml selected_point_g_is_base_independent_and_selected_g_sensitive
cargo test --offline --manifest-path core/product_physics/Cargo.toml selected_basis_blocks_missing_invalid_or_dimensionally_wrong_point_g
cargo test --offline --manifest-path core/product_physics/Cargo.toml temperature_g_interpolation_uses_adjacent_points_and_duplicate_temperatures_still_block
cargo test --offline --manifest-path core/product_physics/Cargo.toml interpolation_blocks_at_and_beyond_stored_range_edges
cargo test --offline --manifest-path core/product_physics/Cargo.toml exact_and_interpolated_basis_fields_are_mutually_exclusive
cargo test --offline --manifest-path core/product_physics/Cargo.toml base_material_values_are_used_when_no_modulus_basis_is_named

Hand calculation: validation/hand_calcs/mechanics/tp_dec092_temperature_indexed_shear_modulus_torsion.md. Test record, with the recorded run of 2026-08-03: mech-tp-dec092-temperature-indexed-shear-modulus-torsion.md.