Abstract
We develop a learning strategy to infer the constitutive relation for the stress of polymeric flows with memory. We make no assumptions regarding the functional form of the constitutive relations, except that they should be expressible in differential form as a function of the local stress- and strain-rate tensors. In particular, we use a Gaussian Process regression to infer the constitutive relations from stress trajectories generated from small-scale (fixed strain-rate) microscopic polymer simulations. For simplicity, a Hookean dumbbell representation is used as a microscopic model, but the method itself can be generalized to incorporate more realistic descriptions. The learned constitutive relation is then used to perform macroscopic flow simulations, allowing us to update the stress distribution in the fluid in a manner that accounts for the microscopic polymer dynamics. The results using the learned constitutive relation are in excellent agreement with full Multi-Scale Simulations, which directly couple micro/macro degrees of freedom, as well as the exact analytical solution given by the Maxwell constitutive relation. We are able to fully capture the history dependence of the flow, as well as the elastic effects in the fluid. We expect the proposed learning/simulation approach to be used not only to study the dynamics of entangled polymer flows, but also for the complex dynamics of other Soft Matter systems, which possess a similar hierarchy of length- and time-scales.
Highlights
Polymeric materials are ubiquitous in our modern industrial societies, having transformed our food, infrastructure, and modes of transportation
We will consider two basic flow problems in order to validate our proposed learning strategy: (1) simpleshear flow and (2) pressure driven flow. The flow in both systems is effectively one-dimensional, but a complete description of their dynamics requires that we account for all components of the stress, and their coupling to the flow. If this approach is to be applied to general geometries, it should be capable of learning the appropriate form of the constitutive relation for the stress without any simplifying assumptions
As we have chosen a system of noninteracting Hookean dumbbells for our microscopic polymer model, we are able to compare our results with the exact analytical constitutive equation, Eq (13)
Summary
Polymeric materials are ubiquitous in our modern industrial societies, having transformed our food, infrastructure, and modes of transportation. One of the preferred manufacturing methods for polymeric materials is polymer melt processing, where a molten polymer is extruded or molded into the desired shape, before allowing it to cool and solidify [2]. To accomplish these requirements, we need to understand the macroscopic flow behavior of the polymer process, and the microscopic dynamics of the polymer chains, in order to reliably control the resultant properties of the product. Any function of the system can be expressed as an interpolation over the (disordered) fluid particles, which serve as interpolation points. We adopt a Gaussian interpolating kernel r2
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