Lesson 1.4: Thermal Stresses and Strains

Learn thermal stress analysis through heated piston-cylinder systems, covering thermal expansion, constrained deformation, and stress development in mechatronic applications.

🎯 Learning Objectives

By the end of this lesson, you will be able to:

1. Calculate thermal expansion and contraction in heated components
2. Analyze thermal stress development in constrained systems
3. Apply thermal stress principles to piston-cylinder interfaces
4. Design for thermal expansion in mechatronic systems

🔧 Real-World System Problem: Heated Piston-Cylinder Interface

In 3D printer extruder heads, injection molding systems, and heated actuators, piston-cylinder interfaces operate at elevated temperatures. Understanding thermal stress is crucial for preventing seizure, excessive wear, and component failure.

System Description

Heated Extruder Head Components:

• Aluminum Cylinder (heated to 200°C for plastic melting)
• Steel Piston (reciprocates within heated cylinder)
• Heating Elements (maintain precise temperature)
• Temperature Sensors (monitor thermal conditions)

The Thermal Challenge

During heating from room temperature (20°C) to operating temperature (200°C):

Engineering Question: How do we calculate the thermal stresses and clearance changes when an aluminum cylinder and steel piston are heated from 20°C to 200°C?

Why Thermal Analysis Matters in Mechatronics

Consequences of Poor Thermal Design:

• Seized actuators (thermal expansion eliminates clearances)
• Reduced precision (thermal growth affects positioning)
• Component failure (excessive thermal stress)
• Performance drift (thermal effects change system behavior)

Benefits of Proper Thermal Design:

• Reliable operation across temperature ranges
• Maintained precision through thermal compensation
• Extended component life with controlled stress levels
• Predictable performance in varying thermal environments

📚 Fundamental Theory: Thermal Expansion and Stress

Linear Thermal Expansion

When materials are heated, they expand according to:

🌡️ Thermal Expansion Formula

Where:

• = Thermal expansion (m)
• = Coefficient of thermal expansion (1/°C)
• = Original length (m)
• = Temperature change (°C)

Physical Meaning: Materials expand linearly with temperature increase. Different materials have different expansion coefficients, leading to differential expansion in assemblies.

Thermal Strain

The thermal strain (dimensionless) is:

📏 Thermal Strain Formula

Where:

• = Thermal strain (dimensionless)
• = Thermal expansion (m)
• = Original length (m)
• = Coefficient of thermal expansion (1/°C)
• = Temperature change (°C)

Physical Meaning: Thermal strain is the ratio of thermal expansion to original length, representing the fractional change in dimensions due to temperature change.

• Material Properties
• Temperature Effects
• Stress Calculation

Aluminum 6061:

• = 23.6 × 10⁻⁶ /°C
• Higher thermal expansion
• Common in heated housings

Steel 1045:

• = 12.0 × 10⁻⁶ /°C
• Lower thermal expansion
• Common in precision components

Expansion Ratio: Aluminum expands ~2× more than steel

Thermal Stress Development

🔒 Thermal Stress Formula

Fully constrained expansion:

Partially constrained expansion:

Where:

• = Thermal stress (Pa)
• = Young’s modulus (Pa)
• = Coefficient of thermal expansion (1/°C)
• = Temperature change (°C)

Physical Meaning: When thermal expansion is prevented, materials develop internal stress proportional to their stiffness and expansion tendency.

🔧 Application: Piston-Cylinder Thermal Analysis

Let’s analyze our heated extruder system step by step.

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System Parameters:

• Heated 3D printer extruder head (aluminum cylinder with steel piston)
• Aluminum Cylinder: Inner diameter 20.00 mm, Length 100 mm, Material: Al 6061 (α = 23.6 × 10⁻⁶ /°C, E = 70 GPa)
• Steel Piston: Outer diameter 19.95 mm, Length 80 mm, Material: Steel 1045 (α = 12.0 × 10⁻⁶ /°C, E = 200 GPa)
• Temperature change: ΔT = 180°C (from 20°C to 200°C)
• Cold clearance: 0.05 mm

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Step 1: Calculate Free Thermal Expansion

Click to reveal thermal expansion calculations

1. Aluminum cylinder inner diameter expansion:

   New inner diameter: 20.00 + 0.0849 = 20.0849 mm

2. Steel piston outer diameter expansion:

   New outer diameter: 19.95 + 0.0431 = 19.9931 mm

3. Calculate running clearance at operating temperature:

   The clearance increases from 0.05 mm (cold) to 0.092 mm (hot)

Step 2: Analyze Constrained Thermal Stress

Click to reveal thermal stress calculations

1. Thermal stress in constrained aluminum cylinder:

2. Thermal stress in constrained steel piston:

3. Assessment against yield strengths:

   • Aluminum: 297 MPa > 270 MPa (yield) → Would yield if constrained!
   • Steel: 432 MPa < 530 MPa (yield) → Would remain elastic
   • Conclusion: Never fully constrain aluminum during heating

Step 3: Design Recommendations

Click to reveal design optimization analysis

1. Optimal cold clearance calculation:

   Required hot clearance (minimum): 0.05 mm Differential expansion: 0.0849 - 0.0431 = 0.0418 mm Required cold clearance: 0.05 + 0.0418 = 0.092 mm

2. Thermal stress management:

   • Allow free expansion → Zero thermal stress
   • Design clearances for hot operation
   • Use materials with similar α when possible

3. Design validation:

   ✅ Hot clearance adequate (0.05 mm minimum) ✅ No thermal stress (free expansion) ✅ Reliable operation across temperature range

🎯 Advanced Analysis: Bi-Material System

Composite Rod Analysis

Consider a fixed aluminum rod with a steel sleeve heated together:

Setup:

• Both materials heat up by ΔT = 100°C
• Both want to expand, but by different amounts
• They’re bonded together (compatibility constraint)

Analysis:

Compatibility equation:

Force equilibrium: (internal forces balance)

Solution yields:

• Aluminum experiences compression (wants to expand more but is held back)
• Steel experiences tension (is pulled to expand more than it naturally would)

🛠️ Design Guidelines for Thermal Systems

Material Selection Strategy

• Similar α Values
• Expansion Joints
• Thermal Barriers

Strategy: Choose materials with similar thermal expansion coefficients

Examples:

• Steel-Iron combinations (α ≈ 12 × 10⁻⁶ /°C)
• Aluminum alloys with aluminum (α ≈ 23 × 10⁻⁶ /°C)

Benefit: Minimal differential expansion

Common Thermal Design Mistakes

📊 Thermal Analysis Summary

📋 Summary and Next Steps

In this unit, you learned to:

1. Calculate thermal expansion using δ = αLΔT
2. Analyze thermal stress in constrained systems (σ = EαΔT)
3. Design clearances for thermal operation
4. Prevent thermal stress through proper accommodation

Key Design Principle: Allow thermal expansion to prevent stress

Critical Formula: Thermal stress = E × α × ΔT (for fully constrained systems)

Coming Next: In Lesson 1.5, we’ll analyze torsional loading in a Geneva mechanism crankshaft, exploring how twisting forces create shear stresses and angular deformation in rotating mechatronic systems.

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