// Blog August 4, 2023

Thermomechanical Analysis in Metallurgy

Figure 1. Copper sheet metal. 


Thermomechanical analysis (TMA) is a vertical dilatometry technique that measures thermal expansion behavior. The use of thermomechanical analysis in metallurgy is applied to characterizing the coefficient of thermal expansion (CTE), phase distribution, softening point, sintering behavior, and other critical dimension-related properties.  CTE is a function of metals’ crystal structure, so by tuning phase distribution, the CTE can also be tuned. 


1. Coefficient of Thermal Expansion (CTE)

The coefficient of thermal expansion (CTE) indicates how much a material expands or shrinks when its temperature changes. The CTE of metals is essential for various applications in engineering design, such as expansion joints, heat sink design, railroad track buckling, and in bimetallic strip thermometers.

Solid materials can have thermal expansion anisotropy, so thermal expansion is usually measured along the principal axes of the material. The coefficient of linear expansion (α) is defined as the ratio of the change in length to the original length per unit temperature change. The following equation can calculate linear CTE:

CTE is usually expressed in ppm/°C in engineering applications, while in physics applications, it’s expressed in units of K-1. A directly analogous method calculates the volumetric thermal expansion coefficient (β), usually reported for liquids and gases.

The linear and volumetric CTE values for some pure substances are reported in Table 1.

Table 1. Typical linear (α) CTE values of some materials. Note that polymers like ABS and Nylon have CTE values that vary by grade, batch, and thermal history.


α (ppm/°C)













Quartz, fused





Understanding CTE is critical for most engineering fields. Thermomechanical analysis in metallurgy therefore sees wide applications. For example:

  • In civil engineering, understanding thermal expansion allows the incorporation of flexible fittings and joints to accommodate thermal expansion, preventing incidents like the 2018 West Onondaga Street railway bridge partial collapse and improving safety.
  • In public utility engineering, understanding thermal expansion allows engineers to design enough slack in power lines to prevent infrastructure failure during cold weather.
  • In residential engineering, flexible seals and joints are installed around materials of dissimilar thermal expansion coefficients: for example, glass is sealed into windows with a flexible polymer to prevent thermal strain from causing cracking of the windowpanes. The metal frames of windows, likewise, are sealed with flexible sealants.
  • In electronics, thermal management is employed to prevent thermal strain causing delamination and component failures
  • In chemical engineering, expansion loops are installed in process piping to prevent thermal strain causing failure of process equipment and thermal expansion is accounted for in setting maximum fill levels for storage tanks and reactors.


In metallic materials, CTE is a function of the phase distribution of the metal or alloy. Because it is a function of the phase distribution within the material, the observed CTE can be used to understand a metal sample’s phase distribution and thermal history, and conversely, the CTE can be tuned with heat treatments designed to induce a desired phase distribution.

2. Thermomechanical Analysis (TMA) of Metals

TMA is a method that measures how the sample’s size changes with temperature under a fixed load. A typical TMA device has a furnace, a sample holder, a probe, a displacement sensor, a force controller, and a temperature controller.

A diagram illustrating compression testing configuration of a differential TMA apparatus.
Figure 2. Differential TMA Compression Testing Configuration.

The most common testing configuration for thermomechanical analysis in metallurgy is the compression testing method. In compression testing, the sample is placed on the holder and heated or cooled by the furnace at a set rate. The probe touches the sample with force controlled by the force-controlling device (typically either an electromagnet, a spring, or a system of weights) and moves in response to the expansion of the sample. TMA can be run in either a differential or single-rod format. Differential TMA offers higher sensitivity and more accurate subtraction of sample holder and instrument effects. Single-rod TMA, by contrast, typically offers a wider dynamic range. Figure 2 illustrates a typical differential TMA setup. A hemispherical (point-contact) rod tip is used for hard materials like metals and ceramics, which improves contact stability but increases pressure on the sample. Figure 3 shows the two most commonly used rod types for compression testing.

A drawing showing the difference between a hemispherical TMA rod tip and a flat TMA rod tip.
Figure 3. Hemispherical and Flat TMA Detector Rods.

The tensile loading method is the other testing configuration commonly used to characterize metal samples by TMA, which is particularly useful for measuring metal foils. The sample is suspended between two metal clamps in a tensile loading configuration. A tension force is applied, and then the length is observed as a function of temperature. Figure 4 illustrates this testing configuration.

Tensile loading configuration of a TMA apparatus.
Figure 4. Tensile Loading Method.


Regardless of the test configuration and rod format, the expansion of the sample causes motion in the probe. The motion of the probe is detected by the displacement sensor (typically a linear-variable differential transducer, or LVDT). The signal is saved by a computer and graphed against the temperature to get a TMA curve. The TMA curve shows how the sample length varies with temperature. CTE is then calculated from the slope of the curve.

TMA has several advantages for measuring CTE in metals:

  • Wide temperature range, allowing observation of CTE at conditions relevant to the application.
  • Versatility of sample format (soft and hard solids, films, etc)
  • Different modes, allowing observation of other properties (softening point, deflection temperature, etc)
  • Variable force, allowing testing of softer samples and optimizing test conditions
  • Variable probe configurations (hemispherical, flat, penetration, etc) for more reliable performance in different test methods and sample formats.


Most metals show almost constant CTE across broad temperature ranges. Metals also typically have a low CTE – with the alkali metals, magnesium, manganese, aluminum, copper, silver, tin, and zinc being notable exceptions to the rule. Aluminum, copper, and zinc alloys can also have large CTE. It’s generally accepted that metals with a face-centered cubic (FCC) structure will have larger CTE than metals with a body-centered cubic structure (BCP) or hexagonal close-packed (HCP) structure, for example.

Some metals may show nonlinear or anisotropic behavior at higher temperatures, after mechanical treatments, or under different loading conditions. Thermal expansion anisotropy can be measured in TMA simply by preparing a sample with the axis of the cylinder parallel to the principal axis of interest. This is especially useful for characterizing laminate structures, as these tend to be highly anisotropic.

Figure 5 shows an example TMA thermogram of an aluminum foil, measured with the tensile loading method, measured by Rigaku using the Rigaku TMA 8311.

A TMA curve of aluminum, summarized in Table 2
Figure 5. TMA measurement result of 10 µm Al foil.

Table 2 shows the CTE values measured with the Rigaku TMA 8311. Samples as thin as 10µm can be accurately measured using the tensile loading attachment of the TMA 8311.


Table 2. Comparison of Literature and Observed CTE of Aluminum Measured With the Tensile Loading Attachment

Temperature (°C)

CTE Literature Value (x10-5 K-1)

Observed CTE (x10-5 K-1)

Expansion (%)

Bias (%)
































Other important dilatometric properties that can be measured by TMA include:

  • Softening point
  • Phase transformation temperatures
  • Annealing behavior
  • Sintering behavior

3. Rigaku TMA 8311

Figure 6. Rigaku TMA 8311


The Rigaku TMA 8311 (Figure 6) is the ideal vertical dilatometry system for characterizing metals’ thermal expansion, sintering, and phase behavior. Some of the most essential features of this device are:

  • High-sensitivity, high-precision measurement by the differential method
  • Flexible handling of various sample sizes
  • Wide range of heating and cooling rates
  • Simple sample setting mechanism
  • Different measurement methods
    • Compression loading method
    • Tensile loading method
    • Penetration method
    • High-sensitivity differential penetration method
    • Dynamic TMA
  • Measurement temperature range:
    • Standard model: Ambient to 1100°C
    • High-temperature model: Ambient to 1500°C
    • Low-temperature model: -150 to 600°C

For more information on the Rigaku TMA, or to request a quote, check out the Rigaku page here.


Thermal expansion characterization is critical to safe engineering in various industries and applications. Thermomechanical analysis in metallurgy can be used to understand phase distribution and thermal history and to engineer heat treatment, annealing, and sintering processes. The Rigaku TMA 8311 is ideal for characterizing CTE, softening point, annealing, and sintering behavior of metals and alloys.

About the Author

  Hamidreza Parsimehr worked in the Thermal Analysis Labs division of C-Therm Technologies. He holds an MSc in Polymer Chemistry from the Iran Polymer and Petrochemical Institute
Photo of Sarah Ackermann, Laboratory Services Manager at Thermal Analysis Labs Sarah Ackermann is our laboratory services manager. She has extensive experience in thermal analysis and materials characterization and has been helping clients with their thermal testing problems for over five years. She holds a MSc in Chemistry from the University of New Brunswick and is an award-winning scientist who has published in journals such as the ASME Journal of Heat and Mass Transfer, the International Journal of Powder Metallurgy, and Canadian Journal of Chemical Engineering


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