Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Green's functions for the isotropic planar relaxed micromorphic model -- concentrated force and concentrated couple

Panos Gourgiotis, Gianluca Rizzi, Peter Lewintan, Davide Bernardini, Adam Sky, Angela Madeo, Patrizio Neff

Abstract

We derive the Green's functions (concentrated force and couple in an infinite space) for the isotropic planar relaxed micromorphic model. Since the relaxed micromorphic model particularises into the microstretch, Cosserat (micropolar), couple-stress, and linear elasticity model for certain choices of material parameters, we recover the fundamental solutions in all these cases.

Green's functions for the isotropic planar relaxed micromorphic model -- concentrated force and concentrated couple

Abstract

We derive the Green's functions (concentrated force and couple in an infinite space) for the isotropic planar relaxed micromorphic model. Since the relaxed micromorphic model particularises into the microstretch, Cosserat (micropolar), couple-stress, and linear elasticity model for certain choices of material parameters, we recover the fundamental solutions in all these cases.
Paper Structure (42 sections, 161 equations, 9 figures)

This paper contains 42 sections, 161 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. The isotropic relaxed micromorphic model
  4. The isotropic relaxed micromorphic model in plane-strain
  5. Fundamental solutions for the relaxed micromorphic continuum under plane strain conditions
  6. Concentrated force: The Kelvin problem
  7. The relaxed micromorphic continuum with zero micro and macro Poisson's ratio
  8. The pure relaxed micromorphic continuum with symmetric force stress tensor
  9. Limiting cases
  10. Micro-stretch elasticity
  11. Cosserat (micropolar) elasticity
  12. Couple stress elasticity
  13. Classical linear elasticity ($L_{\rm c} \to 0$) - lower bound macroscopic stiffness
  14. Classical linear elasticity ($L_{\rm c} \to \infty$) - upper bound microscopic stiffness
  15. Concentrated couple
  16. ...and 27 more sections

Figures (9)

  • Figure 1: The stiffness of the relaxed micromorphic model (RMM) is bounded from above and below. Other generalized continua exhibit unbounded stiffness for small sizes. For large values of the characteristic length $L_{\rm c}$, linear elasticity with a micro elasticity tensor is recovered (one RVE) while linear elasticity with a macro elasticity tensor is obtained for small values of the characteristic length (many RVEs).
  • Figure 2: Inhomogeneous displacement solution for the concentrated couple. Circles are rotated and expanded by the deformation around zero.
  • Figure 3: $\lVert u \rVert$ behaves like $\frac{1}{r}$ in the relaxed micromorphic theory for the case of a concentrated couple.
  • Figure 4: Contours of the normalized displacements $\frac{u_i\, \mu_{\rm M}}{F}$ and micro-rotation $\frac{\vartheta_3\, \mu_{\rm M} \ell_2}{F}$ due to a concentrated unit line force ($F=1$) acting at the origin of relaxed micromorphic medium. The material is characterized by $g_1=1.2$, $g_2=3$, $g_3=5$ and $g_4=3$.
  • Figure 5: Variation of the normalized displacement $\frac{u_2\mu_{\rm M}}{F}$ and the normalized micro-rotation $\frac{\vartheta_3 \mu_{\rm M} \ell_2}{F}$ along the positive $x_1$-axis due to a concentrated unit line force ($F=1$) in various generalized continuum theories. The relaxed micromorphic material is characterized by $g_1=1.2$, $g_2=3$, $g_3=5$ and $g_4=3$.
  • ...and 4 more figures