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.
Highlights
− Value of a vector in the {x,y,z} coordinate frame sym( ) The symmetric part of the tensor X t Time TCM Tensor-based constitutive model transversely isotropic linear elastic (TILE) Transversely isotropic linear elastic tr() Trace operator u Pore pressure uic Uniaxial stress tensor associated with the ith principal direction of the stress tensor associated with the cth inter-particle contact V (Un-subscripted) specific volume V (Subscripted with a direction) directional specific volume VREP Representative small volume W Work per unit particle volume x,y,z Cartesian coordinates
This paper proposes new continuum descriptions of stress, strain, and related quantities that are intermediate in detail between tensor mechanics and particle mechanics
Terzaghi [72, 73] proposed a first modification, the well-established Principle of Effective Stress, which implies that the symmetric stress tensor ′ that is effective in terms of the stress–strain–strength behaviours of fully saturated, uncemented granular soils is related to the total (Cauchy) stress tensor σ by:
Summary
− Value of a vector in the {x,y,z} coordinate frame (element dimensions of length) sym( ) The symmetric part of the tensor X t Time TCM Tensor-based constitutive model TILE Transversely isotropic linear elastic tr() Trace operator u Pore pressure uic Uniaxial stress tensor associated with the ith principal direction of the stress tensor associated with the cth inter-particle contact V (Un-subscripted) specific volume (dimensionless) V (Subscripted with a direction) directional specific volume (dimensionless) VREP Representative small volume (units of volume) W Work per unit particle volume x,y,z Cartesian coordinates (units of length). Notation Matrices, and tensors expressible as matrices, are denoted using bold type. Effective stresses are denoted using the prime symbol, which is used for functions and parameters that relate to effective stress. Rates are denoted using an overdot, equivalent to differentiation with respect to time t, so that: dq q. Quantities with subscript ψ or θ, such as ′ or ′ , represent functions of direction ψ or polar angle θ respectively, while the corresponding unsubscripted quantity would be independent of direction
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