Strong ellipticity within the Toupin-Mindlin first strain gradient elasticity theory

Article Thermodynamics

Mathematical modeling of the elastic properties of cubic crystals at small scales based on the Toupin-Mindlin anisotropic first strain gradient elasticity

Markus Lazar et al.

Summary: In this work, a mathematical modeling of the elastic properties of cubic crystals with centrosymmetry at small scales is presented using the Toupin-Mindlin anisotropic first strain gradient elasticity theory. The study derived expressions for 11 gradient-elastic constants and calculated material parameters for four representative cubic materials. The positive definiteness of the strain energy density is examined leading to necessary and sufficient conditions for the elastic and gradient-elastic constants.

CONTINUUM MECHANICS AND THERMODYNAMICS (2022)

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Article Mechanics

Nonlinear strain gradient and micromorphic one-dimensional elastic continua: Comparison through strong ellipticity conditions

Victor A. Eremeyev et al.

Summary: This study discusses the strong ellipticity conditions for strain gradient and micromorphic continua, considering them as enhancements of nonlinearly elastic materials. It is found that the SE conditions for a micromorphic medium are more restrictive than for its gradient counterpart, and a violation of SE for the primary material can affect solutions within enhanced models.

MECHANICS RESEARCH COMMUNICATIONS (2022)

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Article Mechanics

Incompatible strain gradient elasticity of Mindlin type: screw and edge dislocations

Markus Lazar

Summary: This study resolves the long-standing problem of dislocations in Mindlin-type incompatible isotropic strain gradient elasticity theory. Exact analytical solutions for screw and edge dislocations are derived and numerical analysis is conducted using parameters obtained from first-principles calculations. The results show that the theory with three characteristic lengths provides a more realistic description of edge dislocations than the simplified incompatible strain gradient elasticity with only one characteristic length.

ACTA MECHANICA (2021)

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Article Mechanics

Strong ellipticity conditions and infinitesimal stability within nonlinear strain gradient elasticity

Victor A. Eremeyev

Summary: The study explores the connection between the strong ellipticity condition and infinitesimal instability in nonlinear strain gradient elasticity, establishing implications that simplify further stability analysis using the formulated SE conditions. The results have potential applications in analyzing solutions of homogenized models of beam-lattice materials at different scales.

MECHANICS RESEARCH COMMUNICATIONS (2021)

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Article Mechanics