TY - CHAP

T1 - Limits of distributed dislocations in geometric and constitutive paradigms

AU - Epstein, Marcelo

AU - Kupferman, Raz

AU - Maor, Cy

PY - 2020

Y1 - 2020

N2 - The 1950s foundational literature on rational mechanics exhibits two somewhat distinct paradigms to the representation of continuous distributions of defects in solids. In one paradigm, the fundamental objects are geometric structures on the body manifold, e.g., an affine connection and a Riemannian metric, which represent its internal microstructure. In the other paradigm, the fundamental object is the constitutive relation; if the constitutive relations satisfy a property of material uniformity, then it induces certain geometric structures on the manifold. In this paper, we first review these paradigms, and show that they are equivalent if the constitutive model has a discrete symmetry group (otherwise, they are still consistent; however, the geometric paradigm contains more information). We then consider bodies with continuously distributed edge dislocations, and show, in both paradigms, how they can be obtained as homogenization limits of bodies with finitely many dislocations as the number of dislocations tends to infinity. Homogenization in the geometric paradigm amounts to a convergence of manifolds; in the constitutive paradigm it amounts to a Γ-convergence of energy functionals. We show that these two homogenization theories are consistent, and even identical in the case of constitutive relations having discrete symmetries.

AB - The 1950s foundational literature on rational mechanics exhibits two somewhat distinct paradigms to the representation of continuous distributions of defects in solids. In one paradigm, the fundamental objects are geometric structures on the body manifold, e.g., an affine connection and a Riemannian metric, which represent its internal microstructure. In the other paradigm, the fundamental object is the constitutive relation; if the constitutive relations satisfy a property of material uniformity, then it induces certain geometric structures on the manifold. In this paper, we first review these paradigms, and show that they are equivalent if the constitutive model has a discrete symmetry group (otherwise, they are still consistent; however, the geometric paradigm contains more information). We then consider bodies with continuously distributed edge dislocations, and show, in both paradigms, how they can be obtained as homogenization limits of bodies with finitely many dislocations as the number of dislocations tends to infinity. Homogenization in the geometric paradigm amounts to a convergence of manifolds; in the constitutive paradigm it amounts to a Γ-convergence of energy functionals. We show that these two homogenization theories are consistent, and even identical in the case of constitutive relations having discrete symmetries.

U2 - 10.1007/978-3-030-42683-5_8

DO - 10.1007/978-3-030-42683-5_8

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SN - 978-3-030-42682-8; 978-3-030-42685-9; 978-3-030-42683-5

T3 - Advances in Mechanics and Mathematics

SP - 349

EP - 380

BT - Geometric continuum mechanics

PB - Birkhäuser

ER -