A New Constitutive Equation to Include the Mullins' Effect

Rubber is an indispensible component of many downhole oilfield tools due to its versatile sealing capabilities. When calculating the deformation and sealing force of rubber parts with finite element analysis (FEA) or some established equations, rubber is usually assumed purely elastic as a first approximation. In reality, rubber shows not only strong viscoelastic behavior but also strain-induced stress softening phenomenon (or Mullins’ effect). The Mullins’ effect can be attributed to breakage of molecular chains and slippage and breakage of molecular chain-filler bonding upon deformation.

For rubber parts, the force of the first loading-unloading cycle will be higher than that of the subsequent loading-unloading cycles owing to the Mullins’ effect. The first loading-unloading cycle of many dynamic applications is probably not of interest and the subsequent stable loading cycles are more important. However, for applications in which evaluation of the assembly/setting force is required - such as the packoff of a sealing element of a downhole packer - the stress-strain curve of the first loading cycle needs to be investigated. Unfortunately, most of the constitutive equations for rubber are not developed to tackle the first loading-unloading cycle and the Mullins’ effect.

In this paper, a new constitutive equation for rubber that includes both the Mullins’ effect and the viscoelastic effect is proposed. This new constitutive equation generalizes the Gent model to encompass the stress softening phenomenon. A technique to obtain the model constants and coefficients in the Prony series from cyclic tests at several different maximum strain levels is discussed. Tensile, planar, biaxial, and compression cyclic test data of several rubber compounds are fitted with this new constitutive equation.