Mathematical Model
Mathematical representations of the time-dependent relaxation of polymers allow for the simulation of viscoelastic behavior in printed circuit board materials. The prony series uses a sum of decaying exponential terms to describe how the stress in a material decreases over time when held at a constant strain. This model is essential for accurate finite element analysis of the warping and stress development that occurs during the reflow process.
Relaxation Modulus
Coefficients within the series represent the different timescales over which the resin chains reorganize and dissipate energy. By fitting experimental data from dynamic mechanical analysis to a prony series, engineers can predict the modulus of the material at any temperature and time. This allows for a more realistic assessment of the board’s structural integrity than a simple elastic model would provide.
Simulation Efficiency
Computational tools use these series because they allow for the efficient calculation of history-dependent stresses without storing the entire strain history of every element. Each term in the series acts as an internal state variable that is updated at every time step. This method reduces the time required to simulate complex thermal cycles and provides a clear picture of the permanent deformation remaining after the board cools and the manufacturing process is complete.
Such precision is necessary for high-reliability applications where small dimensional changes can impact the performance of sensitive high-frequency circuits.
