Materials formula

Linear Thermal Expansion Calculator

Calculate free linear expansion from length, temperature change, and a user-supplied coefficient. Solve ΔL, final length, α, or ΔT without a material database.

Instant result

Result

—

Enter values to calculate.

Inputs—

Mode—

Formula—

Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance

Input interpretation: Enter values to calculate.
Result: —
Assurance: Engineering
Declared partition coverage: PASS · 5/5 declared partitions (forward, final-length, coefficient, temperature, invalid-domain) · Matrix
Known limitations: Free one-dimensional expansion; no material lookup, constraints, or α(T) integration.
Core CVP does not include live graph, viewport, or pointer interaction.
Model: Constant-coefficient free linear thermal expansion, with inverse solves for α and ΔT.
Scope: Constant mean linear expansion coefficient over the entered temperature interval
Verification: Engine tested · Source checked · v1.0.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance· View Manifest · CVP overview · Specification
Versions: Calculation 1.0.0 · CVP protocol 1.0.0-proposed
CVP identity: 0/0 property · digest 3511c5999fb0
Legacy regression: 7/7 tests · Production surface contract 6/6
Trust layers: Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference: O1 model · O2 expected_values · O2 numerical_behavior
Interfaces: PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 1/1 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
Supplemental domain review: Not performed
Named expert review: Not performed
CVP suite: 5/5 golden · 2/2 CVP boundary · 7/7 invalid · 2/2 cross-interface · 6/6 CVP contract · Manifest
Sources: ISO 80000-5:2019 — Quantities and units — Thermodynamics — Thermodynamic quantities and units
ASTM E228-22 — Linear Thermal Expansion of Solid Materials — Scope and significance
NIST — Effects of Temperature, Humidity, and Pressure — Thermal expansion
Sources
Evidence: 4 legacy golden · 3 legacy boundary · legacy regression suite · 5/5 oracle-backed golden · 7/7 invalid · Artifact integrity PASS
Semantic contract: PASS

Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

Length changeΔL = α L₀ ΔT

Final lengthLf = L₀(1 + αΔT)

Mean coefficientα = ΔL / (L₀ ΔT)

iα must be supplied for the actual material, direction, condition, and temperature interval. The seed intentionally contains no material-property lookup table.

How to use

Choose the quantity to solve

Solve length change, final length, mean coefficient, or temperature change from the same model.

Enter the measured property

Use a mean linear expansion coefficient applicable to the material and temperature interval.

Check the model boundary

The result is free expansion. Constraints, thermal stress, phase changes, and temperature-dependent integration are outside this model.

Example calculations

Common configurations with formula and result.

Two-metre member heated by 50 K

L₀=2 m, α=12×10⁻⁶ K⁻¹, ΔT=50 K

ΔL = 12×10⁻⁶ × 2 × 50

ΔL=0.0012 m; Lf=2.0012 m

Cooling uses a signed temperature change

L₀=2 m, α=12×10⁻⁶ K⁻¹, ΔT=−50 K

ΔL = α L₀ ΔT

ΔL=−0.0012 m

Linear Thermal Expansion calculator specification

Version 1.0.0 · Engine tested

Calculation status

Engine tested 7/7 tests · Production surface contract 6/6
Named expert review Not performed
Calculation version 1.0.0

Review policy · Evidence

Definition: For constant mean linear expansion coefficient α and free one-dimensional expansion, ΔL = α L₀ ΔT and Lf = L₀ + ΔL.
What it calculates: Constant-coefficient free linear thermal expansion, with inverse solves for α and ΔT.
Inputs: mode
length_m
alpha_per_K
delta_temperature_K
delta_length_m
Outputs: delta_length_m
final_length_m
thermal_strain
alpha_per_K
delta_temperature_K
Formula: ΔL = α L₀ ΔT
Assumptions: Constant mean linear expansion coefficient over the entered temperature interval
Free, unconstrained expansion
One-dimensional isotropic response along the measured direction
No material-property database or automatic coefficient selection
Not thermal stress, volumetric expansion, phase transformation, creep, or temperature-dependent integration
Units: m
K
1/K
Boundary conditions: length_m ≤ 0 or alpha_per_K ≤ 0 → VALUE_MUST_BE_POSITIVE
coefficient mode with ΔT = 0 → DIVISION_BY_ZERO
solved α ≤ 0 → INVALID_MODEL_DOMAIN
final length ≤ 0 or non-finite → RESULT_OUT_OF_RANGE
unknown mode → INVALID_MODE
Example: length_m=2, alpha_per_K=12e-6, delta_temperature_K=50 → delta_length_m=0.0012
Validation cases: 3 published on this page · 7/7 tests · Production surface contract 6/6 · View evidence
mode=delta_length, length_m=2, alpha_per_K=12e-6, delta_temperature_K=50 → delta_length_m=0.0012; final_length_m=2.0012
mode=coefficient, length_m=2, delta_length_m=0.0012, delta_temperature_K=50 → alpha_per_K=12e-6
mode=temperature_change, length_m=2, delta_length_m=-0.0012, alpha_per_K=12e-6 → delta_temperature_K=-50
Sources: ISO 80000-5:2019 — Quantities and units — Thermodynamics — Thermodynamic quantities and units
Supports: Names, symbols, definitions, and units for thermodynamic quantities
ASTM E228-22 — Linear Thermal Expansion of Solid Materials — Scope and significance
Supports: Linear expansion uses measured length change and temperature change; coefficient applicability depends on the material and interval
NIST — Effects of Temperature, Humidity, and Pressure — Thermal expansion
Supports: ΔL/L is related to α and ΔT
Calculation version: 1.0.0

Background

Interpretation and common distinctions.

Calculate free linear thermal expansion from a user-supplied coefficient. This is the first Materials formula-first seed: the formula is canonical, while material-property selection stays explicit and traceable.

Supported and not supported

Supported — constant mean α · signed heating or cooling · solve ΔL, Lf, α, or ΔT · API materials.thermal.linear_expansion

Not supported — named-material lookup, constrained thermal stress, anisotropic tensors, phase changes, creep, volumetric expansion, or integrating α(T)

Agent / API notes

Capability id: materials.thermal.linear_expansion · tool id: linear-thermal-expansion · pin 1.0.0.

{ "mode": "delta_length", "length_m": 2, "alpha_per_K": 0.000012, "delta_temperature_K": 50 }

Modes: delta_length, final_length, coefficient, temperature_change. Canonical units are metre, kelvin difference, and inverse kelvin.

The Materials L1 remains planned. This Human Entry is intentionally hosted at /calc/physics/linear-thermal-expansion until a separate L1 promotion review.

Other calculators in this family: Young's modulus, Thermal resistance, Temperature conversion .

Frequently asked questions

Key distinctions behind the calculation.

Does this calculator select a coefficient for a named material?

No. This formula-first seed requires α as an input. Use a traceable specification or measurement for the actual material, direction, condition, and temperature interval.

Can I use Celsius temperature differences?

Yes. A temperature difference of 1 °C equals 1 K. Enter the signed difference, not an absolute Celsius temperature.

Does it calculate thermal stress?

No. This is free expansion. A constrained member needs a separate thermoelastic model, boundary conditions, and appropriate elastic properties.

Where does the calculation run?

Locally in the browser. REST and MCP call the same canonical materials engine.