Representation of stress and strain in granular materials using functions of direction
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ORIGINAL PAPER
Representation of stress and strain in granular materials using functions of direction E. T. R. Dean1 Received: 5 May 2019 © The Author(s) 2020
Abstract This paper proposes a new way of describing effective stress in granular materials, in which stress is represented by a continuous function of direction in physical space. The proposal provides a rigorous approach to the task of upscaling from particle mechanics to continuum mechanics, but is simplified compared to a full discrete element analysis. It leads to an alternative framework of stress–strain constitutive modelling of granular materials that in particular considers directional dependency. The continuous function also contains more information that the corresponding tensor, and thereby provides space for storing information about history and memory. A work-conjugate set of geometric rates representing strain-rates is calculated, and the fundamental principles of local action, determinism, frame indifference, and rigid transformation indifference are shown to apply. A new principle of freedom from tensor constraint is proposed. Existing thermo-mechanics of granular media is extended to apply for the proposed functions, and a new method is described by which strain-rate equations can be used in large-deformations modelling. The new features are illustrated and explored using simple linear elastic models, producing new results for Poisson’s ratio and elastic modulus. Ways of using the new framework to model elastoplasticity including critical states are also discussed. Keywords Anisotropy · Continuum mechanics · Critical states · Discrete element method · Elasticity · Elasto-plasticity · Orientational averaging · Particle mechanics · Spin · Thermo-mechanics Abbreviations Keyboard symbols a Area a, b Quantities illustrating linearity (Eq. 12) A,B,C (Subscripted) functions of the directions identified by the subscripts A–F Parameters in elasticity matrix (Appendices 5 and 6 and Table 2) d Infinitesimal of DEM discrete element method det() Determinant e Void ratio E Function whose directional average is zero (Eq. 14) E Receipt (Eq. 58) E Young’s modulus (Eq. 95) 𝐅c Vector of inter-particle force at c-th contact (Eq. 17) * E. T. R. Dean [email protected] 1
Caribbean Geotechnical Design Limited, Ammanford, UK
F Deformation gradient tensor FCM Functions-based constitutive model FILE Fully isotropic linear elastic g Geometric factor associated with the bunching effect (dimensionless) H Helmholtz free energy per unit particle volume (units of stress) I 3 × 3 identity tensor j, k Dimensionless constants K Modulus (units of stress) 𝐋c Branch vector for cth contact (units of length) M Number (dimensionless) 𝐦𝜁 Unit material vector in direction ζ at the reference configuration (Appendix 3) n Element of unit vector n Dimensionless constant (Appendix 5) 𝐧𝜓 Unit vector in direction ψ N Number of contacts p/ Mean normal effective stress P–U Parameters in elasticity matrix (Table 2) Q A general quantity or function Q
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