What is the coefficient of linear expansion?

It is the fractional length change per 1°C temperature change, with units of 1/°C. Steel is about 12×10⁻⁶/°C and aluminum is about 23×10⁻⁶/°C, so aluminum expands roughly twice as much as steel.

Why does thermal stress occur?

When thermal expansion is constrained, the material develops internal stress because it cannot freely expand or contract. Under full constraint, σ = E·α·ΔT can produce large compressive or tensile stress.

Why use expansion joints for thermal expansion?

In pipes and bridges, constraining thermal expansion can generate large thermal stresses and damage the structure. Expansion joints absorb the displacement and greatly reduce thermal stress.

How are linear and volumetric expansion related?

For isotropic materials, the volumetric expansion coefficient is approximately β≈3α. The volume change is ΔV≈V₀(3α)ΔT because expansion occurs by the same fraction in all three directions.

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Thermodynamics / Mechanics of Materials

Thermal Expansion & Thermal Stress Calculator

Calculate and visualize length/volume change due to temperature and thermal stress generated under constraint in real time. Directly applicable to bridge, pipeline, and electronic device design — core CAE knowledge.

Material Selection

Material

Parameters

Linear expansion coefficient α

ppm/°C

×10⁻⁶/°C

Young's Modulus E

GPa

Member Length L₀

Temperature difference ΔT

°C

Thermal Expansion & Thermal Stress Formulas

$\Delta L = L_0 \alpha \Delta T$
$\varepsilon_{th} = \alpha \Delta T$ (thermal strain)
$\sigma_{th} = E\alpha\Delta T$ (under full constraint)

Results

1.20

Elongation ΔL (mm)

1200

Thermal Strain ε (με)

240

Thermal stress σ (MPa)

—

Constraint force F (kN/m²)

Expansion View

Temperature-Stress Curve

Material Comparison

Expansion

💬 Ask the Professor

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I often see gaps in bridges. Are those there because of thermal expansion?

🎓

Exactly. They are expansion joints. For example, a 1 km steel bridge (α=12×10⁻⁶/°C) with a 50°C seasonal temperature difference changes length by ΔL=1000×12×10⁻⁶×50=0.6 m. If that movement were fully constrained, σ=200GPa×12×10⁻⁶×50=120 MPa of thermal stress would develop, so the joint absorbs the movement instead.

🙋

What happens when an aluminum part is joined to steel and the temperature changes?

🎓

That is the bimetal effect, once widely used in temperature switches. Aluminum has α≈23×10⁻⁶/°C, almost twice steel's α≈12×10⁻⁶/°C. At a bonded interface, each material tries to expand by a different amount, producing bending deformation and interface stress. Automotive aluminum parts joined to steel members must account for this thermal-deformation mismatch.

🙋

Invar has α=1.2×10⁻⁶/°C and barely expands. How is that possible?

🎓

Invar is an iron-64% / nickel-36% alloy whose magnetic effects nearly cancel normal thermal expansion. Charles Édouard Guillaume discovered it in 1896 and later received the Nobel Prize in Physics. It is used in precision instruments, clock springs, and low-expansion electronic substrates. Its expansion is less than one tenth that of ordinary steel.

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What is calculated in CAE thermal-stress analysis?

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A typical workflow has two stages: thermal FEM analysis to obtain the temperature field, then structural analysis using thermal strain. First, heat-conduction analysis computes nodal temperatures. Next, the structural model applies thermal strain ε=αΔT as an initial strain and computes stress. This is routine in fatigue-life evaluation for solder joints, engine parts, and reactor vessels.

Frequently Asked Questions

Can a design reduce thermal stress to zero?

It can be minimized by avoiding full constraint, using free ends or expansion joints, selecting materials with similar expansion coefficients, or reducing the temperature change through insulation or cooling. Exactly zero is an idealization; practical design aims to keep thermal stress below the allowable limit.

Does the linear expansion coefficient α depend on temperature?

Yes. For many metals, α tends to increase as temperature rises. High-accuracy calculations treat α as a temperature-dependent function α(T), while engineering estimates often use a constant value. FEM analysis can include temperature-dependent material properties.

Why can rails buckle?

In summer, rails heat up and try to expand, but because they are constrained by supports and joints, large compressive stress can develop and cause buckling. Continuous welded rails are installed with initial tensile stress so that stress is near zero at a neutral temperature, typically around 30°C.

How are volumetric expansion coefficient β and linear expansion coefficient α related?

For isotropic materials, β≈3α is a good approximation. More exactly, $(1+\alpha\Delta T)^3-1 \approx 3\alpha\Delta T$ for small changes. When ΔT is large, second- and third-order terms can matter, but for temperature changes of a few hundred degrees, the approximation is usually adequate.

List of Major Material Linear Expansion Coefficients and Young's Moduli

Material	α (×10⁻⁶/°C)	E (GPa)	Fully constrained σ/ΔT (MPa/°C)	Typical uses
Steel	12	200	2.4	Bridges and building frames
Aluminum	23	70	1.61	Aircraft and automotive parts
Copper	17	130	2.21	Heat exchangers and electrical parts
Titanium	8.6	110	0.95	Aerospace, medical, and chemical equipment
Stainless (SUS304)	17.3	193	3.34	Chemical equipment and kitchens
Invar	1.2	148	0.18	Precision instruments and measuring devices
Glass	8.5	72	0.61	Windows and optics
Concrete	10	30	0.30	Building structures
CFRP (fiber direction)	0-2	70-200	~0.1	Aviation and sports equipment

What is Thermal Expansion Simulator?

Thermal Expansion Simulator is a fundamental topic in engineering and applied physics. This interactive simulator lets you explore the key behaviors and relationships by directly manipulating parameters and observing real-time results.

By combining numerical computation with visual feedback, the simulator bridges the gap between abstract theory and physical intuition — making it an effective learning tool for students and a rapid-verification tool for practicing engineers.

Physical Model & Key Equations

The simulator is based on the governing equations behind Thermal Expansion & Thermal Stress Calculator. Understanding these equations is key to interpreting the results correctly.

Each parameter in the equations corresponds to a slider in the control panel. Moving a slider changes the equation's solution in real time, helping you build a direct connection between mathematical expressions and physical behavior.

Real-World Applications

Engineering Design: The concepts behind Thermal Expansion & Thermal Stress Calculator are applied across mechanical, structural, electrical, and fluid engineering disciplines. This tool provides a quick way to estimate design parameters and sensitivity before committing to full CAE analysis.

Education & Research: Widely used in engineering curricula to connect theory with numerical computation. Also serves as a first-pass validation tool in research settings.

CAE Workflow Integration: Before running finite element (FEM) or computational fluid dynamics (CFD) simulations, engineers use simplified models like this to establish physical scale, identify dominant parameters, and define realistic boundary conditions.

Common Misconceptions and Points of Caution

Model assumptions: The mathematical model used here relies on simplifying assumptions such as linearity, homogeneity, and isotropy. Always verify that your real system satisfies these assumptions before applying results directly to design decisions.

Units and scale: Many calculation errors arise from unit conversion mistakes or order-of-magnitude errors. Pay close attention to the units shown next to each parameter input.

Validating results: Always sanity-check simulator output against physical intuition or hand calculations. If a result seems unexpected, review your input parameters or verify with an independent method.