Calculating Surface Strain Field Distribution across Deflecting Circuit Board Assemblies
Surface strain fields across deflecting circuit boards scale with distance from the neutral axis and concentrate sharply at stiff component perimeters.
Kirchhoff-Love Plate Theory
Kinematic Model
Thin structure deformation analysis employs simplified kinematic assumptions to calculate bending behavior without modeling three-dimensional volume elements. Under the Kirchhoff-Love plate theory, a straight line normal to the plate mid-surface remains straight and normal to that surface after deformation.
Governing Equation
The mathematical formulation relates the out-of-plane deflection to the applied transverse load through a fourth-order partial differential equation that incorporates the flexural rigidity of the plate. It assumes that the thickness of the plate is far smaller than its other dimensions, allowing the stress through the thickness to be ignored. Solving this equation yields the bending moments and curvature across the plate, from which designers calculate the localized tensile and compressive stresses.
Assumed Simplification
Neglecting transverse shear deformation simplifies the mathematical complexity but restricts the validity of the model to very thin plates. When the thickness-to-span ratio exceeds one-twentieth, the assumption of zero transverse shear strain leads to a severe underestimation of the actual deflection. For thicker plates, or for composite materials with low shear stiffness, engineers must abandon this simplified framework in favor of more advanced models that incorporate shear deformation.
Laminate Application
Composite laminate analysis utilizes this simplified theory to model the stiffness of multi-layered printed circuit boards during uniform thermal expansion. By assuming perfect bonding between the laminate layers, the theory allows the calculation of an equivalent stiffness matrix that captures the bending-stretching coupling of the assembly. This homogeneous representation simplifies the modeling of large-scale board assemblies while maintaining sufficient accuracy for evaluating macro-level warpage during manufacturing.