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1. Micromechanics-based elasto-plastic–damage energy formulation for strain gradient solids with granular microstructure

   https://doi.org/10.1007/s00161-021-01023-1  

   Placidi, Luca; Barchiesi, Emilio; Misra, Anil; Timofeev, Dmitry (May 2021, Continuum Mechanics and Thermodynamics)

   null (Ed.)

   This paper is devoted to the development of a continuum theory for materials having granular microstructure and accounting for some dissipative phenomena like damage and plasticity. The continuum description is constructed by means of purely mechanical concepts, assuming expressions of elastic and dissipation energies as well as postulating a hemi-variational principle, without incorporating any additional postulate like flow rules. Granular micromechanics is connected kinematically to the continuum scale through Piola's ansatz. Mechanically meaningful objective kinematic descriptors aimed at accounting for grain-grain relative displacements in finite deformations are proposed. Karush-Kuhn-Tucker (KKT) type conditions, providing evolution equations for damage and plastic variables associated to grain-grain interactions, are derived solely from the fundamental postulates. Numerical experiments have been performed to investigate the applicability of the model. Cyclic loading-unloading histories have been considered to elucidate the material-hysteretic features of the continuum, which emerge from simple grain-grain interactions. We also assess the competition between damage and plasticity, each having an effect on the other. Further, the evolution of the load-free shape is shown not only to assess the plastic behavior, but also to make tangible the point that, in the proposed approach, plastic strain is found to be intrinsically compatible with the existence of a placement function. 

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2. Extended granular micromechanics approach: a micromorphic theory of degree n

   https://doi.org/10.1177/1081286519879479  

   Nejadsadeghi, Nima; Misra, Anil (October 2019, Mathematics and Mechanics of Solids)

   For many problems in science and engineering, it is necessary to describe the collective behavior of a very large number of grains. Complexity inherent in granular materials, whether due the variability of grain interactions or grain-scale morphological factors, requires modeling approaches that are both representative and tractable. In these cases, continuum modeling remains the most feasible approach; however, for such models to be representative, they must properly account for the granular nature of the material. The granular micromechanics approach has been shown to offer a way forward for linking the grain-scale behavior to the collective behavior of millions and billions of grains while keeping within the continuum framework. In this paper, an extended granular micromechanics approach is developed that leads to a micromorphic theory of degree n. This extended form aims at capturing the detailed grain-scale kinematics in disordered (mechanically or morphologically) granular media. To this end, additional continuum kinematic measures are introduced and related to the grain-pair relative motions. The need for enriched descriptions is justified through experimental measurements as well as results from simulations using discrete models. Stresses conjugate to the kinematic measures are then defined and related, through equivalence of deformation energy density, to forces conjugate to the measures of grain-pair relative motions. The kinetic energy density description for a continuum material point is also correspondingly enriched, and a variational approach is used to derive the governing equations of motion. By specifying a particular choice for degree n, abridged models of degrees 2 and 1 are derived, which are shown to further simplify to micro-polar or Cosserat-type and second-gradient models of granular materials. 

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3. Granular micromechanics‐based identification of isotropic strain gradient parameters for elastic geometrically nonlinear deformations

   https://doi.org/10.1002/zamm.202100059  

   Barchiesi, Emilio; Misra, Anil; Placidi, Luca; Turco, Emilio (May 2021, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik)

   null (Ed.)

   Although the primacy and utility of higher-gradient theories are being increasingly accepted, values of second gradient elastic parameters are not widely available due to lack of generally applicable methodologies. In this paper, we present such values for a second-gradient continuum. These values are obtained in the framework of finite deformations using granular micromechanics assumptions for materials that have granular textures at some ‘microscopic’ scale. The presented approach utilizes so-called Piola’s ansatz for discrete-continuum identification. As a fundamental quantity of this approach, an objective relative displacement between grain-pairs is obtained and deformation energy of grain-pair is defined in terms of this measure. Expressions for elastic constants of a macroscopically linear second gradient continuum are obtained in terms of the microscale grain-pair parameters. Finally, the main result is that the same coefficients, both in the 2D and in the 3D cases, have been identified in terms of Young’smodulus, of Poisson’s ratio and of a microstructural length. 

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4. A non-isothermal breakage-damage model for plastic-bonded granular materials incorporating temperature, pressure, and rate dependencies

   https://doi.org/10.1016/j.ijsolstr.2024.113085  

   Kokash, Yazeed; Regueiro, Richard; Miller, Nathan; Zhang, Yida (December 2024, International Journal of Solids and Structures)

   Plastic-bonded granular materials (PBM) are widely used in industrial sectors, including building construction, abrasive applications, and defense applications such as plastic-bonded explosives. The mechanical behavior of PBM is highly nonlinear, irreversible, rate dependent, and temperature sensitive governed by various micromechanical attributions such as grain crushing and binder damage. This paper presents a thermodynamically consistent, microstructure-informed constitutive model to capture these characteristic behaviors of PBM. Key features of the model include a breakage internal variable to upscale the grain-scale information to the continuum level and to predict grain size evolution under mechanical loading. In addition, a damage internal state variable is introduced to account for the damage, deterioration, and debonding of the binder matrix upon loading. Temperature is taken as a fundamental external state variable to handle non-isothermal loading paths. The proposed model is able to capture with good accuracy several important aspects of the mechanical properties of PBM, such as pressure-dependent elasticity, pressure-dependent yield strength, brittle-to-ductile transition, temperature dependency, and rate dependency in the post-yielding regime. The model is validated against multiple published datasets obtained from confined and unconfined compression tests, covering various PBM compositions, confining pressures, temperatures, and strain rates. 

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5. Revised identification of strain gradient elastic parameters

   Misra, A; Placidi, L; Rodriguez, C; La_Valle, G (June 2025, arxiv_25_c)

   The work reported in ``Granular micromechanics-based identification of isotropic strain gradient parameters for elastic geometrically nonlinear deformations" misidentified key terms in the grain-pair objective relative displacement when accounting for the second gradient of placement. In this paper, we correct that oversight by deriving a revised expression for the grain-pair objective relative displacement within the granular micromechanics framework. The amended terms, which resemble Christoffel symbols expressed in terms of strain gradients, modify the contributions of both the normal and tangential components to the strain energy and, consequently, alter the identified strain gradient elastic parameters. Importantly, the identification of the standard (first gradient) elastic tensor remains unchanged. This brief paper presents the corrected derivation, the resulting stiffness tensors for anisotropic strain gradient elasticity, and updated analytical expressions for the material parameters in both 2D and 3D isotropic settings. 

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