Coefficient of Linear Thermal Expansion (Coefficient of Linear Thermal Expansion)｜Basic Knowledge and Calculation Methods for Machine Design

 

Here.Materials for Machinery and Jigs　This is a characteristic that should be noted inList of coefficients of linear thermal expansion and basic knowledge and calculation methods for use in machine design."It is a note about.

 

In the past, I too have had the experience of worrying about the dimensional changes of components due to temperature changes.　　Thermal expansion can cause unexpected problems, especially in design.The extent to which thermal expansion is taken into account is also an important point.　The first time the company has been in the market for a new product, it has become

 

Many technical sites provide a list of expansion coefficients for each material and basic elongation formulas.　　However, I have felt that there is little information that consistently explains how this information relates to actual design issues, such as failure or loss of accuracy due to unexpected thermal stress, and how to take specific countermeasures. This article aims to fill that very gap.　　It systematically organizes information that tends to be discussed in fragments on other sites, and provides practical knowledge that goes one step further.

 

First, the course provides a solid understanding of the basics of expansion coefficients and how to calculate thermal deformation. Next, by comparing the differences in expansion coefficients among major materials, the course provides an insight into the selection of materials.　The course then delves deeply into specific design problems such as thermal stress and fit changes, and finally, specific design measures to avoid or allow for these problems are taught, providing a comprehensive overview of thermal expansion problems.

Basic knowledge of expansion coefficients and calculation methods

What is the Coefficient of Linear Thermal Expansion? Basics Explained

The coefficient of linear thermal expansion is a physical property that indicates the rate at which the length of a material changes when the temperature rises by 1°C (or 1 K)　It is.　　The symbol is represented by the Greek letter alpha (α) and the unit is "/°C" or "/K".

 

This phenomenon is attributed to the behavior of the atoms that make up the material.When heat is applied to a material, the vibrations of the atoms become more active and the average distance between atoms increases.　　The accumulation of these microscopic changes results in an increase in the overall dimensions of the material, which is the principle of thermal expansion.

 

Therefore, the stronger the bonding between atoms, the less likely a material is to undergo thermal expansion.　　For example, ceramics have a low coefficient of expansion due to the strong bonding between atoms, while resin materials tend to exhibit a much higher coefficient of expansion than metals due to the relatively weak bonding between molecules.

 

The coefficient of volume expansion (β) is an indicator of the rate of change in volume. For isotropic materials (materials whose physical properties do not change with direction), the relationship between the coefficient of volume expansion and the coefficient of linear thermal expansion is about three times greater (β ≈ 3α).In mechanical design, the coefficient of linear thermal expansion is considered the most important parameter, since it is primarily a matter of length and diameter variation.

 

 

Calculate dimensional elongation due to temperature change

A simple formula can be used to determine how much a material will expand or contract with changes in temperature.　　This calculation is the first step in designing for thermal expansion.

 

The basic formula for determining the amount of dimensional change (ΔL) is as follows

ΔL = α × L₀ × ΔT

• ΔL: Amount of change in length
• α: Coefficient of linear thermal expansion of material
• L₀: Original length before temperature change
• ΔT: Change in temperature

 

For example, consider the case of a 1000 mm long iron bar (α ≈ 12 × 10-⁶ /°C) that is heated from a reference temperature of 20°C to 100°C.　The temperature change (ΔT) in this case is 80°C.

ΔL = (12 × 10-⁶ /°C) × 1000mm × 80°C = 0.96mm

 

Thus, it can be seen that even a 1-meter rod will stretch by about 1 mm with a temperature rise of 80°C.For long parts and parts that require high precision, this amount of change is an error that can never be ignored, so quantitative evaluation at the design stage is essential.It is.

 

 

What is the relationship between thermal stress and Young's modulus?

Thermal expansion and shrinkage themselves do not generate stresses in the material.　　Thermal stress is an internal stress that occurs only when a material's attempt to freely expand or contract due to temperature change is prevented (restrained) by something external.

 

For example, consider a rod with both ends completely fixed.　　As the temperature rises, the bar tries to stretch, but since both ends are fixed, it cannot stretch.　　This is the same condition as if the bar were forced to compress to the extent that it is elongated by heat.　　In this case, the resistance of a material to deformation is indicated byYoung's modulus (modulus of longitudinal elasticity, E)　Compressive stress is generated internally in proportion to the

 

The thermal stress (σ) that occurs when fully constrained can be calculated using the following equation

σ = E × α × ΔT

• σ: Thermal stress
• E: Young's modulus
• α: Coefficient of linear thermal expansion
• ΔT: Change in temperature

 

If a bar of carbon steel (E ≈ 206 GPa, α ≈ 12 × 10-⁶ /°C) is fixed at both ends and the temperature is increased by 60°C, the stress generated is206 GPa × (12 × 10-⁶ /°C) × 60°C ≈ 148.3 MPaThis value is so large that it approaches the yield point of the material.　　This is a large value, approaching the yield point of the material, and requires extreme care in design because it can cause plastic deformation or failure of the part.

 

 

List of coefficients for each main material

When designing, it is important to have a detailed understanding of the expansion coefficient of the materials used.　　Here we compile a more detailed database of linear thermal expansion coefficients for typical materials frequently used in mechanical design.

*Since aluminum is frequently used in equipment design, a separateDetailed values for the thermal expansion of aluminum.　notes.

 

 

Warping and buckling of members due to heat

Heat can cause deformation such as "warping" and "buckling" in thin plate-like parts.　This phenomenon occurs when the thermal stress generated inside a component exceeds the rigidity of the component itself.

 

 

Mechanism of buckling

Buckling occurs when the entire member is uniformly heated but its thermal expansion is constrained from the surroundings.　　Compressive thermal stresses are generated inside the member, and when these compressive forces exceed the limit, the member escapes laterally and bends.

 

To prevent these deformations, it is effective to increase the rigidity of the member at the design stage.　　Specifically, ribs (reinforcing bones) and beads (convex reinforcement lines) can be added to the plates to strengthen the cross-sectional shape and improve resistance to warping and buckling.

 

 

Occurrence of warping to be considered in design

Warpage occurs mainly when there is a temperature difference between the front and back of a member or when there is localized heating.　Here are a few situations where the designer should be especially careful.

 

 

Warpage due to cutting

Machining heat is a major cause of warpage, especially when cutting thin parts.Frictional heat from the blade cutting the material causes a local increase in temperature near the machined surface.　This heat produces a non-uniform temperature distribution inside the material, which causes residual stresses after cooling.　　This heat produces a non-uniform temperature distribution inside the material, which after cooling becomes residual stress and causes warpage.　　Countermeasures include optimizing machining conditions, such as adjusting cutting speed and depth of cut to suppress the heat generated, and using water-soluble cutting fluids that have a high cooling effect.In the design stage, it is effective to eliminate extremely thin sections and make the wall thickness uniform, or to add reinforcing ribs in anticipation of deformation.It is.

 

 

Warpage due to welding

Welding is one of the processing methods most prone to heat-induced warpage because it locally heats the metal at a high enough temperature to melt it.　　After the weld is heated rapidly, it cools and shrinks as the heat is transferred to the surrounding base metal.　　This non-uniform shrinkage creates large tensile stresses that distort the entire member.　　There are a wide range of countermeasures that can be taken,In the design phase, minimize the number of welding points and place welds in symmetrical locations on the structure.The following are some examples.　　It is also important to take measures in construction, such as avoiding heat concentration by devising the welding sequence (using the stepping stone method) and preventing deformation by firmly restraining the member with a jig.

 

 

Warping due to environmental heat

Control panels and equipment housings installed outdoors can also warp due to radiant heat from sunlight.　　In particular, if only one side of an enclosure is exposed to direct sunlight, that side will become hotter, causing warpage due to the temperature difference between the inside and outside.　　As a countermeasure, choose a shady location for the enclosure,Or, basically, avoid direct sunlight by installing heat shields or eaves.It is.　　Also, the color of the enclosure can be white or other highly reflective color,Proper placement of ventilation fans to dissipate internal heat is also effective in reducing temperature riseIt is.

 

 

Expansion coefficients and thermal issues for design

Combination of different materials and cautions

In mechanical design, there are many situations where different types of materials are used in combination.　　However, care must be taken when joining materials with significantly different coefficients of linear thermal expansion, as temperature changes can cause serious problems.

 

For example, consider a case where an aluminum alloy plate (α ≈ 23 × 10-⁶ /K) is fastened with steel bolts (α ≈ 12 × 10-⁶ /K).　　As the temperature rises from the state of being tightened at room temperature, the aluminum plate attempts to expand approximately twice as much as the steel bolt.　　However, since the movement is restrained by the bolt, strong compressive stresses are generated inside the plate, and conversely, additional tensile stresses are generated in the bolt.

 

If this stress becomes excessive, the plate may be plastically deformed or the bolt may stretch beyond its yield point.　Conversely, when the temperature drops, the plate shrinks more significantly than the bolt, causing the bolt to lose its fastening force and possibly cause loosening.　When designing for precision, bolt fastening is based on the axial force (pressing force) obtained by the set tightening torque and theNeed to know the margin axial force of the bolt itself and assemble it　will be.

 

To avoid such problems,Basically, materials with similar coefficients of expansion should be combined as much as possible.It is.　　If it is absolutely necessary to combine dissimilar materials, some device is required to allow for relative displacement due to thermal expansion, such as designing clearances and using long holes, as described below.

 

 

Thermal effects on fitting tolerances

The design of the "fit" combining the shaft and hole is usually based on room temperature, such as 20°C (68°F).　　However, if the actual operating temperature is significantly different from room temperature, thermal expansion may change the "interference" or "clearance" and the product may not function as designed.

 

Particular attention is required when combining materials with different expansion coefficients.　　A typical example is a "clamping fit" in which a steel bearing (α ≈ 12 × 10-⁶ /K) is press-fitted into an aluminum housing (α ≈ 23 × 10-⁶ /K).

 

Even if the proper interference is maintained at room temperature, when the machine is operated and the temperature rises, the housing expands more than the bearing.　　This causes the initial interference to decrease and, in the worst case, a clearance to form. This condition causes the outer ring of the bearing to slip inside the housing, a phenomenon known as "creep," which can lead to wear, abnormal vibration, and component damage.

 

To prevent such problems,It is essential to assume both the minimum and maximum temperatures at which the machine will operate, and to confirm by calculation that the tightness and clearance will remain within the allowable range at each temperature.So, if you are visiting this article, if the issue is thermal expansion countermeasures for peripheral parts related to bearings,Correct installation of the bellows　We recommend that you check the

 

 

Failure due to insufficient clearance

The most basic and effective method to prevent stresses and component interference due to thermal expansion is to provide adequate clearance　It is　　It is based on the concept of intentionally "letting go" of thermal expansion rather than suppressing its movement with force.

 

If there is insufficient clearance, when the temperature rises, the expanded parts have nowhere to go and press hard against adjacent parts.　　This can result in unexpectedly high stresses, causing deformation and breakage of the parts or even malfunctioning of the entire machine.

 

One specific design technique is to use long holes (slots) instead of perfect circles for bolt fixing holes.　　The important point in this process is to clearly separate the "reference" and "escape" points.　　First, one reference fixing point for positioning the entire assembly is firmly fixed with a perfect circle hole.　　Then, the other fixing point should be a long hole to allow for thermal expansion from this reference point.

 

The orientation of the long hole should basically be aligned with the radial direction from the reference point to the center of the hole, i.e., the direction in which the member will extend.　　This allows the member to expand and contract smoothly in each direction from the reference point.　　The length of the long hole should be set to sufficiently absorb the amount of elongation (ΔL) calculated using the maximum expected temperature change and the distance from the reference point.Application of this concept effectively prevents thermal stresses caused by restraints　It is.

 

 

Thermal deformation that reduces machine accuracy

In machines that require high precision, such as machine tools, semiconductor manufacturing equipment, and CMMs, dimensional change due to thermal expansion, or "thermal displacement," is one of the most serious error factors that affect their performance.

 

For example, the temperature of ball screws used in machine tools rises due to heat from the motor and friction heat.　　Since the expansion coefficient of a steel ball screw is approximately 12 × 10-⁶/℃, a slight increase of 1℃ in the temperature of a 1m-long ball screw will cause it to expand by 12µm (0.012mm) in the axial direction.　　This is an error that cannot be ignored for modern machine tools that perform machining with micron-level precision.

 

Similarly, if the spindle of a machine is elongated due to heat from rotation, the tool tip position will be misaligned.　The machine bed and column themselves can also be deformed by ambient temperature changes and internal heat sources, causing the geometric accuracy of the entire machine to be off.

 

Like this,The accumulation of minute thermal displacements generated by individual parts can create large positioning errors in the machine as a whole and significantly degrade product quality.　It is a

 

 

Specific measures to address thermal expansion issues

Addressing the various problems caused by thermal expansion requires a multifaceted approach at each stage of design, rather than relying on a single method.

 

 

1. measures by material selection

The first basic step is to select appropriate materials.　　When combining multiple components, selecting materials with linear thermal expansion coefficients as close as possible will minimize relative displacement and stress.　　In addition, if high dimensional stability is required, consider the use of low thermal expansion materials as described below.

 

 

(2) Structural innovations

Structural innovations to allow for thermal displacement or minimize its effects are also effective.　　In addition to the aforementioned clearance and long-hole designs, high-precision machines canThermally symmetric structure.is sometimes employed.　　This is a design concept in which heat sources and structures are symmetrically arranged so that thermal deformation occurs in a uniform and predictable manner and complex deformations such as torsion are suppressed.

 

 

3. temperature control and cooling

Another common method is to forcibly cool parts that generate heat in order to suppress the temperature rise itself.　　One example is a mechanism that circulates cooling oil inside the spindle or ball screw of a machine tool.　　Environmental control, such as maintaining a constant room temperature in the factory where the machine is installed, is also essential to stabilize accuracy.

 

 

Thermal displacement compensation function

In recent years, more advanced measures have been taken in high-precision machine tools. Intelligent functions have been put to practical use that use information from temperature sensors attached to various parts of the machine to predict the amount of thermal displacement in real time, and automatically correct the position to cancel out the displacement.

 

 

Advanced design considering expansion coefficient

Design innovations to allow for thermal displacement (in the case of piping)

As mentioned above,One of the most effective approaches to preventing stress development due to thermal expansion is to incorporate a design that structurally permits (allows) displacement rather than constraining it with force　It is.　　This concept is extremely useful especially for structures exposed to large temperature changes and when combining dissimilar materials.

 

A typical method is to use long or large clearance holes at bolted fastening points.　　This allows parts such as plates to slide with respect to the bolted fastening as temperature changes, preventing internal stress buildup.　U-bolts to fasten piping also demonstrate this effect.

 

Another possible design is to absorb thermal expansion throughout the structure.　　For example, a U-shaped loop in the middle of a long pipeline (expansion loop(U-shape) is a method of providing a U-shaped section of piping.　　When pipes stretch due to heat, this U-shaped section absorbs the overall expansion and contraction, preventing excessive loads on the pipes and connecting equipment.

 

It is useful for both piping knowledge and machine structure (easy to visualize) TechnoFlex Co.telescopic site　It is.　You may want to keep this in mind as an idea for thermal expansion countermeasures.

 

Furthermore, a sliding support structure with one end fixed and the other end supported by rollers or other means is another classic and reliable method used in large structures such as bridges.　　Thus, the key to a reliable structure is to anticipate thermal movement and incorporate in the design a degree of freedom that does not impede such movement.

 

 

Application Examples of Low Thermal Expansion Materials

In fields where dimensional stability on the nanometer scale is required, such as exposure devices for manufacturing semiconductors and ultra-precision measuring instruments, it is difficult to respond with general materials and thermal displacement compensation technology alone.　　SuchIn situations where extremely high precision is required, it is essential to use "low thermal expansion materials" whose dimensions hardly change at all depending on temperature.　The first two are the following.

 

 

Invar Super Invar

Super Invar is a special alloy made by blending nickel and other elements with iron, and exhibits an extremely low coefficient of linear thermal expansion near room temperature.　　Super Invar, in particular, boasts an astonishingly low coefficient of thermal expansion, less than 1/100th that of iron.　　These alloys are widely used for parts that require absolute dimensional stability, such as frames for precision instruments, measuring standards, and housings for laser equipment.

 

 

Low thermal expansion ceramics

Certain ceramics, such as cordierite, can achieve a near-zero coefficient of thermal expansion in certain temperature ranges.　　Cordierite developed by Kyocera is used in state-of-the-art semiconductor lithography equipment and is a fundamental technology that supports nano-level processing precision.

 

 

CFRP (Carbon Fiber Reinforced Plastic)

CFRP has the unique property of being able to control the coefficient of thermal expansion by devising the orientation of the fibers.　　By combining carbon fiber with a negative coefficient of expansion and resin with a positive coefficient of expansion and optimizing the stacking angles, it is possible to achieve zero expansion as a whole. Combined with the advantages of light weight and high rigidity, carbon fiber is used for structural components of space satellites and parts of high-precision optical equipment.

 

 

Expansion joints to absorb large displacements

Structures ranging in length from tens of meters to several kilometers, such as chemical plant piping, district heating and cooling pipelines, or large ducts, can expand and contract considerably due to heat. To absorb such large displacements throughout the structure, specialized mechanical elements called "expansion joints" are used.

 

There are various types of expansion joints, but the most widely used is the "bellows type," made of metal and shaped like the bellows of an accordion.　　The bellows expands and contracts to absorb the expansion and contraction of the entire piping system.　　There are a wide variety of bellows types, ranging from simple single bellows types to double bellows types that can accommodate larger displacements, and universal bellows types that can absorb angular displacements.

 

Other types include the "sleeve type," in which a sleeve-shaped part expands and contracts by sliding on a packing part, and a form that combines a ball joint to accommodate three-dimensional movement.　In mechanical elements, the "Bellows couplingsThese are represented by such items as

 

When selecting an expansion joint, it is necessary to comprehensively consider the pressure and temperature of the fluid flowing inside, corrosion resistance, durability, as well as the amount of displacement to be absorbed, to select the best type for the system.

 

 

Improved design through understanding of optimal expansion coefficients

• Thermal expansion is a physical phenomenon caused by vibrations of atoms due to temperature changes
• Coefficient of linear thermal expansion (α) is a physical property that indicates the rate of change in length per 1°C of temperature
• The amount of dimensional change is ΔL = α × L₀ × ΔT can be calculated with the formula
• When thermal expansion is constrained σ = E × α × ΔT Thermal stresses expressed as
• The coefficient of expansion varies greatly depending on the material, and tends to decrease in the order of resin > aluminum > copper > iron > ceramics
• Uneven heating and restraint can cause warping and buckling of members
• Combining materials with different expansion coefficients increases the risk of failure due to stress at the interface
• Failure to take operating temperatures into account can cause changes in the "interference" or "clearance" of the fit, resulting in malfunctions.
• Clearance design to absorb thermal expansion is a basic measure
• In precision machinery, the slightest thermal displacement can significantly reduce machining accuracy
• Thermal countermeasures include multifaceted approaches such as material selection, structural innovations, cooling, and thermal displacement compensation.
• Designs that intentionally "release" displacement, such as long holes and sliding structures, are effective
• Invar, low thermal expansion ceramics, and CFRP are used in the ultra-precision field
• Expansion joints such as bellows type are used for large expansion and contraction such as piping
• Systematic understanding of thermal issues and their preventive reflection in design leads to the development of high quality products.

 

That's it.

 

Summary Article

Material Selection Guide for Jigs｜Optimal Selection by Application

  This section notes "How to select the best material for each jig application" as a guide to jig material selection. In the design of jigs, the selection of appropriate materials is an important factor that determines product quality and production efficiency ...

See more.