Differential Implementation of the Viscoelastic Response of a Curing Thermoset Matrix for Composites Processing

The ability to manufacture thermoset matrix composite materials into large and complex structures can be significantly enhanced by modeling the behavior of the system during the process. As a result there has been much research on all aspects of the cure of these materials. A particularly important aspect is the development of mechanical properties in the thermoset matrix as it evolves from a low molecular weight material into a fully cross-linked solid. The behavior is generally acknowledged to be viscoelastic, and as both temperature and degree of cure vary with time, the characterization and representation of the behavior is both critical and complex. Many approaches have been suggested and tried, ranging from 1D or 2D implementations of simple linear elastic cure hardening responses (which have been shown to be essentially pseudo-viscoelastic formulations) through to more sophisticated representations of viscoelastic behavior as Prony series of Maxwell elements coded in 3D hereditary integral FE implementations. In this paper we present a differential approach for the viscoelastic representation of a curing thermoset matrix composite undergoing an arbitrary temperature cycle by noting that the viscoelastic response can be represented very well by a Prony series. For this case, we show that a differential approach is equivalent to an integral formulation, but appears to have some significant benefits in terms of extension to more general descriptions (e.g., thermo-viscoelastic behavior), ease of coding and implementation, and perhaps most importantly, computer runtimes. Rather than using a differential approach where the order of the governing differential equation grows very fast with the number of springs or dashpots, we use the stresses in the individual Maxwell elements to capture the complete history of the material and allow for a much simpler formulation. A 1D formulation of this differential approach, including thermo-viscoelasticity, is developed, and results and benchmarks are presented.