Edge stabilization and consistent tying of constituents at Neumann boundaries in multi-constituent mixture models

Harishanker Gajendran, Richard B. Hall, Arif Masud

Research output: Contribution to journalArticlepeer-review


A mixture-theory-based model for multi-constituent solids is presented where each constituent is governed by its own balance laws and constitutive equations. Interactive forces between constituents that emanate from maximization of entropy production inequality provide the coupling between constituent-specific balance laws and constitutive models. The deformation of multi-constituent mixtures at the Neumann boundaries requires imposing inter-constituent coupling constraints such that the constituents deform in a self-consistent fashion. A set of boundary conditions is presented that accounts for the non-zero applied tractions, and a variationally consistent method is developed to enforce inter-constituent constraints at Neumann boundaries in the finite deformation context. The new method finds roots in a local multiscale decomposition of the deformation map at the Neumann boundary. Locally satisfying the Lagrange multiplier field and subsequent modeling of the fine scales via edge bubble functions result in closed-form expressions for a generalized penalty tensor and a weighted numerical flux that are free from tunable parameters. The key novelty is that the consistently derived constituent coupling parameters evolve with material and geometric nonlinearity, thereby resulting in optimal enforcement of inter-constituent constraints. Various benchmark problems are presented to validate the method and show its range of application.

Original languageEnglish (US)
Pages (from-to)1142-1172
Number of pages31
JournalInternational Journal for Numerical Methods in Engineering
Issue number12
StatePublished - Jun 22 2017


  • composite materials
  • coupled deformation of constituents
  • finite strains
  • interfaces and interphases
  • mixture theory
  • stabilized methods

ASJC Scopus subject areas

  • Numerical Analysis
  • Engineering(all)
  • Applied Mathematics


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