Tensor Representation
Mathematical arrays describe the mechanical responsiveness of non-isotropic materials to applied forces. The anisotropic compliance matrix defines the relationship between strain and stress for PCB substrates where properties vary along the horizontal and vertical planes of the laminate. It captures the directional dependence of the material stiffness.
Directional Sensitivity
Woven glass reinforcement causes printed circuit boards to exhibit different expansion and contraction behaviors in the warp and weft directions. The anisotropic compliance matrix provides the coefficients necessary to model these variations in finite element analysis software. These coefficients are required when calculating the reliability of through hole vias or surface mount solder joints under thermal cycling.
Stress Calculation
Structural engineering of complex stackups relies on the inverse of the stiffness matrix to determine deformation. Use of the anisotropic compliance matrix allows designers to predict how a board will warp during the curing of solder mask or the cooling from reflow temperatures. This matrix contains up to twenty one independent constants for a fully general material, though many PCB laminates are treated as orthotropic to simplify the equations to nine constants.
Accurate values for each constant come from ultrasonic testing or static loading of material samples. Modeling without these directional terms leads to errors in displacement predictions and miscalculation of copper trace fatigue. Accuracy depends on these inputs.
