Shrinkage stress and cuspal deflection in MOD restorations: analytical solutions and design guidelines

Objective: This paper aimed to derive analytical solutions for the shrinkage stress and cuspal deflection in model Class-II mesial-occlusal-distal (MOD) resin-composite restorations to better understand their dependence on geometrical and material parameters. Based on the stress solutions, it was shown how design curves could be obtained to guide the selection of dimensions and materials for the preparation and restoration of this class of cavities. Methods: The cavity wall was considered as a cantilevered beam while the resin composite was modeled as Winkler's elastic foundation with closely-spaced linear springs. Further, a mathematical model that took into account the combined effect of material properties, sample geometry and compliance of the surrounding constraint was employed to relate the shrinkage stress at the “tooth-composite” interface to the local compliance of the cavity wall. Exact analytical solutions were obtained for cuspal deflection and shrinkage stress along the cavity wall by solving the resulting differential equation, which had the same form as that for a beam on elastic foundation with a distributed load. To quantify the shrinkage stress at the cavity floor, the resin composite was assumed to be a beam, fixed at both ends and loaded with a uniformly distributed load that approximated the shrinkage stress. The analytical solutions thus obtained were compared with results from finite element analysis (FEA). Results: The analytical solution for cuspal deflection contains a dimensionless parameter, γ, which represents the stiffness of the cavity wall relative to that of the cured resin composite. For the same shrinkage strain, cuspal deflection increases with reducing γ, i.e. reducing stiffness of the cavity wall or increasing stiffness of the composite. For the same γ, cuspal deflection increases proportionally with shrinkage strain. Shrinkage stress along the cavity wall is maximum at the cavity corner and reduces towards the occlusal surface; the maximum value depends only on Young's modulus and the shrinkage strain of the resin composite. For low values of γ, the interfacial stress at the occlusal surface can become compressive. The interfacial stress at the cavity floor can be much higher than that along the cavity wall, increasing exponentially with the resin composite's thickness. The analytical solutions agree well with FEA predictions. Significance: When validated, the analytical solutions and design curves presented in this study can provide useful guidelines for choosing appropriate dimensions of cavity preparations and resin composite materials with suitable mechanical properties for Class-II MOD restorations to help avoid tooth fracture and interfacial debonding caused by polymerization shrinkage.