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Thermal Stress Due to Geometric and Material Nonlinearity

Key Takeaways

  • Complicated geometries, such as IC leads and solder balls, can experience stress concentration along curved regions of their surfaces.

  • When the CTE value is also nonlinear, such as near the glass transition temperature of the substrate, there is a large stress increase for a small increase in temperature.

  • The two effects combine to cause greater stress concentration near curved surfaces at high temperature.

High and low temperature in geometric and material nonlinearity with thermal stress

Thermal simulation results for a complex structure show the appearance of thermal stress between regions of high and low temperature.


PCBs and IC packages can have complicated geometries. On an IC package, the leads can be curved structures, such as in SOP or SOT packages. In addition, solder points can have odd curves as they bond to pads and leads in a PCB. The geometry of these systems is not terribly complicated, but evaluating thermal stress in these systems quickly and easily without a set of tensor equations is difficult without the right analysis tools.

When you add in the effect of a glass transition in a PCB substrate, the potential for thermal stress to cause fracture increases. Although theoretical methods can be used to analyze these structures, this is the domain of mathematicians. Engineers can stay productive and easily visualize their results when they have access to a 3D multiphysics field solver to run simulations directly from their design and layout data.

What Causes Thermal Stress?

The components in your PCB will heat up as they operate, and heat transfers elsewhere in your board. As components operate, different portions of the system will heat up and expand at different rates. There are a few important causes of thermal stress in a complex structure, all of which are made more severe in a structure with nonlinear material properties.

  • Thermal expansion. When a material heats up, it will expand. If the expansion is constrained in some way, the material will experience compressive stress. This stress can be uniformly distributed depending on how the material is constrained.

  • Thermal gradients. When a thermal gradient exists between different regions of a system, the material will experience different levels of stress in different regions. As a result, there is a strain field with some gradient.

  • Glass transitions. When a glass transition occurs, the material exhibits a large increase in the CTE value once the material’s temperature exceeds its glass transition temperature. In the presence of a temperature gradient, one portion of a system will experience more expansion than another portion of the system, creating a complex thermal stress field with gradients.

  • Inhomogeneous CTE and thermal conductivity values. The CTE and thermal conductivity values of a material or a complex system can be inhomogeneous or discontinuous. A CTE mismatch between copper and the PCB substrate is a primary cause of failure in blind or buried vias with high aspect ratio. In addition, different thermal conductivity values can cause large temperature gradients, which increase thermal stress between different regions of a system.

Any of these four effects can combine to produce a complex stress field in the system. The last point above, and the geometry of the structure, can also produce a complex stress field. It is important to note that these effects can act in synergy or in competition. For example, thermal stress produced along a gradient in an inhomogeneous structure might compensate thermal stress produced by a glass transition.

Once a glass transition occurs, the CTE value of the material becomes highly complex near the glass transition temperature. This means a small change in temperature will produce a large change in volume until temperature rises far past the glass transition temperature. An attempt to keep CTE value low and predictable is one reason for the development of high-Tg substrate laminates (e.g., high-Tg FR4).

A graph of glass transition in complex geometry problems

Glass transition in two types of FR4. The exact glass transition temperature is determined through interpolation.


There is plenty of mathematical research in this area, and some results have shown that a complex geometry with inhomogeneous material constants can exhibit regions with zero stress. It is incredibly hard to design for this condition without solving thermal stress tensor equations, which may rely on some approximation for your system.

Because of the difficulty in analyzing thermal behavior in these systems, designers and engineers need a tool that can accommodate complex geometric structures. In addition, airflow is common in electronics with cooling fans, or ones that are cooled via conduction, making these problems more complex. This is where a field solver is ideally suited to help solve these problems.

Analyzing Stress and Airflow With Complex Geometry

When you’ve brought a cooling fan into the system or you’re relying on natural convection, your field solver needs to run CFD simulations to examine how airflow aids heat transfer around a board. Moving heat away from components contributes to keeping them within their ideal operating temperatures, which also reduces thermal stress near complex geometry.

When looking at thermal expansion at high temperatures in curved structures with complex geometry, we have a complicated multiphysics problem that most often cannot be treated analytically. This means you need a 3D field solver to examine temperature rise throughout the structure. 3D field solvers with CFD codes can be used to examine how airflow in a board will influence the final temperature distribution, both for forced airflow and natural convection.

Vector heat map showing airflow and temperature around geometric and material nonlinearity

Airflow along a connector body in a 3D field solver. The airflow velocity and temperature distribution are shown in a vector heat map.


Once you add airflow into your simulation, you can examine the difference between the temperature distributions for the system with and without airflow. Further CFD simulations allow you to adjust your cooling strategy to ensure the board does not exhibit large thermal gradients or hot spots in the layout. You may need to take other measures to ensure a uniform low-temperature distribution, such as adding heatsinks or using an alternative substrate material.