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 is four meters long, with outside diameter 0.12 m and inside diameter 0.10 m. One end is fixed. A torque of 12,000 N·m acts at the other end. Shear modulus depends on temperature and is given at named points. What tip rotations follow from an exact point and from interpolation between two temperatures?

A base modulus is also supplied. Its value makes an incorrect fallback miss both targets by a wide margin.

This page preserves a recorded benchmark and its expected values. The dated run below is not a new run of the current product.

This benchmark isolates selected shear modulus and its recorded torsion response. The current exact straight-pressure route selects a common E/nu basis and derives G. The two material-selection accounts must remain distinct.

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.

First, the reference calculates the torsion constant of the hollow circular section. The exact-point case uses the modulus at that point directly. The temperature case interpolates linearly between adjacent points. Neither case uses the base modulus.

For a circular shaft in pure torsion, tip rotation is TL/(GJ). The same formula with the base modulus gives the rotation from an incorrect fallback. This rotation differs from the 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 case record describes these checks:

The tip rotation with the exact-point modulus matches the hand value.
The tip rotation with the interpolated modulus matches the hand value.
The tip rotation with the base modulus misses the targets by 37.5% and 40.625%, far outside the tolerance.
Separate tests establish that material selection uses the modulus at an exact point. Interpolation uses and records the two adjacent points.
Temperature-based interpolation stops with an error at or beyond the stored range ends. It never extrapolates. Exact-ID selection remains separate.
A missing, non-positive, non-finite or wrongly dimensioned modulus stops selection. Duplicate temperatures and conflicting selectors also stop selection.
Selection never falls back to the base modulus. When no temperature basis is named, selection uses the base modulus.
The torsion comparison is independent of selection code. By itself, that comparison does not establish selection behavior.

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

The base modulus differs substantially from both selected values. A program that incorrectly used the base modulus would miss each target by more than a third.

4.Recorded command.

This command belongs to the linked case record and its recorded execution path. This website update did not rerun it. A historical command or result does not establish current product support.

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.