Matching Thermal Expansion in Assemblies

Materials grow and shrink with temperature

Most solids expand when heated and contract when cooled. The amount is described by the coefficient of thermal expansion, usually written as a change in length per unit length for each degree of temperature change. For a free part, expansion is harmless; it simply gets a little longer. The trouble appears when two materials with different coefficients are bonded together, since each wants to change size by a different amount but the bond forces them to move together.

The result is internal stress. The material that wants to expand more is held back, and the one that wants to expand less is pulled along. Stress builds up in both and concentrates at the interface and near edges and corners.

A simple hypothetical example

The basic relation for a free bar is that the change in length equals the coefficient times the original length times the temperature change. Suppose, purely as a hypothetical illustration, a metal plate and a ceramic plate are both 100 mm long, and the temperature rises by 100 degrees.

• The metal has a hypothetical coefficient of 10 per million per degree. Its free growth is 100 mm times 0.00001 times 100, which equals 0.1 mm.
• The ceramic has a hypothetical coefficient of 4 per million per degree. Its free growth is 100 mm times 0.000004 times 100, which equals 0.04 mm.

The difference is 0.06 mm. That sounds tiny, but if the two are rigidly bonded, this gap must be absorbed by elastic strain over a short distance near the bond. For a stiff, brittle ceramic, even small strains can reach the level where cracks begin, especially after repeated cycles. The actual coefficients for real materials vary with grade and temperature, so these numbers only show the arithmetic.

Where mismatch causes trouble

• Coatings on substrates. A hard film on a metal part may crack or peel when heated or cooled. Mismatch adds to the residual stress already present from deposition, which is why coating adhesion and substrate preparation are so important.
• Ceramic-to-metal joints. Brazed or soldered joints between a ceramic and a metal are common in electronics and sensors. On cooling from the joining temperature, the metal contracts more and may crack the ceramic or the joint.
• Chips and substrates. Semiconductor devices attached to substrates see repeated heating and cooling. Solder joints can fatigue and crack after many cycles.
• Composite layups. Layers of different fibers and matrices can warp or delaminate when heated. Composites with carefully chosen fiber directions can be designed to have near-zero expansion in some directions.
• Thermal barrier coatings. Ceramic layers on hot metal parts need a structure that tolerates the expansion mismatch with the metal beneath, one reason these coatings are built with a strain-tolerant microstructure.
• Heat spreaders. Highly conductive materials used with semiconductors must also have compatible expansion, or the chip may be stressed.

Why temperature gradients make it worse

Even a single material can crack if parts of it are at different temperatures, because hotter regions expand against cooler ones. This overlaps with thermal shock, where fast temperature change creates gradients. Mismatch between materials adds to the effect; the two issues often act together in real parts.

Ways to manage it

Engineers do not usually try to eliminate mismatch entirely. They reduce it or give the assembly room to accommodate it.

1. Choose compatible materials. Select pairs with similar coefficients. Some alloys are designed to match common ceramics and glasses for sealing. Some silicon carbide, aluminum nitride and similar ceramics are chosen partly because their expansion is close to that of semiconductors.
2. Use composites with tuned expansion. Metal matrix composites with ceramic or diamond reinforcement can be built to have a coefficient between those of the metal and the device, combining high conductivity with a closer match. Heat spreader materials such as metal-diamond composites are an example.
3. Grade the transition. An intermediate layer, or several, with expansion values between the extremes spreads the mismatch across steps. Graded coatings and compliant interlayers follow this idea.
4. Add compliance. Soft solders, thicker bond lines, flexible adhesives, bellows, spring mounts and slip joints let parts move relative to each other. The cost is lower stiffness or thermal performance.
5. Control geometry. Rounded corners, thinner layers and smaller bonded areas reduce stress. Large flat joints are the most exposed; splitting them into segments can help.
6. Limit the temperature swing. Slower heating and cooling rates reduce gradients, and operating the assembly in a narrower range reduces cycling damage.
7. Put brittle materials in compression. Ceramics tolerate compression much better than tension. A design where the metal shrinks onto the ceramic can keep it in compression across the working range, though this requires careful analysis.

Checking the design

Calculations with simple formulas provide a first estimate, but real assemblies are complicated, so designers often use finite element analysis and then confirm with thermal cycling tests. After a failure, failure analysis of cracked parts can show whether cracks start at the interface or at an edge, which points to the mechanism. Nondestructive evaluation, such as ultrasonic scanning, can detect delamination in bonded layers before it becomes visible.

A word on uncertainty

Expansion coefficients change with temperature, with the processing history of the material and, in anisotropic materials, with direction. Single-number values from a table are approximations. For a critical joint, it is wise to use data for the specific grade, or to measure it, and to include a safety margin since fatigue from repeated cycles can cause failure at stresses that look safe in a single heating.