Silicon Die Stress Transduction Mechanics under Packaging Temperature Changes
Packaging thermal expansion mismatch generates die stress that converts to electrical offset drift via piezoresistive coupling and polymer viscoelastic creep.
Maxwell-Weichert Model
Viscoelastic Representation
Rheological modeling uses a parallel assembly of Maxwell units to represent the multi-stage stress relaxation of viscoelastic materials. This maxwell-weichert model couples a spring and a dashpot in series within each individual unit to capture different relaxation times. The combined behavior allows precise modeling of materials that do not exhibit a single characteristic relaxation time.
Relaxation Spectrum
Viscoelastic behavior changes over time as different internal molecular mechanisms respond to applied strain. By utilizing several Maxwell elements in parallel, the system establishes a discrete relaxation spectrum that matches observed experimental data. Each element possesses a distinct relaxation time determined by the ratio of viscosity to elasticity.
Stress Response
Material testing shows that a sudden step in strain produces a peak initial stress followed by a decay to a plateau value. The model reproduces this decay curve by summing the exponential decay responses of the parallel Maxwell branches. This mathematical representation provides high accuracy when simulating polymer behavior in pressure sensors and load cells.
Accurate predictions are necessary because polymer seals and diaphragms suffer from drift due to viscoelastic relaxation under constant mechanical load. The elastic spring in parallel with the Maxwell elements ensures that the model can represent a non-zero long-term equilibrium stress.
Material Parameterization
Instrument manufacturers calibrate the model parameters using dynamic mechanical thermal analysis. The measured storage and loss moduli determine the spring constants and viscosity values across a range of frequencies. These parameters are verified by comparing simulated stress relaxation curves with laboratory sensor test results.