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Spectra and pseudospectra in the evaluation of material stability in phase field schemes

Michele Benzi, Daniele La Pegna, Paolo Maria Mariano

Abstract

We consider the dynamics of bodies with "active" microstructure described by vector-valued phase fields. For waves with time-varying amplitude, the associated evolution equation involves a matrix that can be non-normal, depending on the constitutive choices adopted for the microstructural actions associated with the considered phase field. The occurrence of non-normality requires to look at the pseudospectrum of the considered matrix, namely the set of all possible eigenvalues of matrices in a $\varepsilon$-neighborhood of the matrix itself, because the eigenvalues of non-normal matrices can be very sensitive to small perturbations and therefore the spectral analysis alone would not be sufficient to distinguish with certainty between table and unstable behavior. We develop the relevant analyses in the case of quasicrystals for which the values of some constitutive parameters are not known or are uncertain from an experimental point of view, a circumstance suggesting parametric analyses. We find circumstances in which the pseudospectra obtained by means of the so-called structured perturbations predict instability when, instead, the spectral analysis indicates stability.

Spectra and pseudospectra in the evaluation of material stability in phase field schemes

Abstract

We consider the dynamics of bodies with "active" microstructure described by vector-valued phase fields. For waves with time-varying amplitude, the associated evolution equation involves a matrix that can be non-normal, depending on the constitutive choices adopted for the microstructural actions associated with the considered phase field. The occurrence of non-normality requires to look at the pseudospectrum of the considered matrix, namely the set of all possible eigenvalues of matrices in a -neighborhood of the matrix itself, because the eigenvalues of non-normal matrices can be very sensitive to small perturbations and therefore the spectral analysis alone would not be sufficient to distinguish with certainty between table and unstable behavior. We develop the relevant analyses in the case of quasicrystals for which the values of some constitutive parameters are not known or are uncertain from an experimental point of view, a circumstance suggesting parametric analyses. We find circumstances in which the pseudospectra obtained by means of the so-called structured perturbations predict instability when, instead, the spectral analysis indicates stability.

Paper Structure

This paper contains 23 sections, 72 equations, 8 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Basic fields
  3. Invariance and balance
  4. Observers and their changes
  5. External power and invariance requirement: consequences
  6. Constitutive structures
  7. Waves with time-dependent amplitude
  8. In the absence of a self-action
  9. In the presence of a conservative self-action
  10. A purely dissipative self-action
  11. Simultaneous presence of conservative and dissipative components in the self-action
  12. Pseudospectrum: general properties
  13. Reasons for calling upon the pseudospectrum
  14. Scalar measures of departure from normality
  15. Definitions and a few properties
  16. ...and 8 more sections

Figures (8)

  • Figure 1: Complex ${ \if@compatibility \mathchar"0122 {} \mathchar"0122 }$-pseudospectrum of $A$ for ${ \if@compatibility \mathchar"011F {} \mathchar"011F }=0.1$ GPa and ${ \if@compatibility \mathchar"011E {} \mathchar"011E }=19$. The dashed line is the imaginary axis.
  • Figure 2: Dissipative self-action: structured pseudospectrum of $A$ as ${ \if@compatibility \mathchar"011F {} \mathchar"011F }$ varies. Superposition of eigenvalues of $400$ matrices $A+E$, where $E$ is a matrix with random entries and the structure given in \ref{['Perturbations E well scaled']}.
  • Figure 3: Dissipative self-action: structured pseudospectrum of $A$ as ${ \if@compatibility \mathchar"011E {} \mathchar"011E }$ varies.
  • Figure 4: Complex ${ \if@compatibility \mathchar"0122 {} \mathchar"0122 }$-pseudospectrum of $A$ for ${ \if@compatibility \mathchar"011F {} \mathchar"011F }=1.5$ GPa, $k_0=0.001$ GPa/m$^2$, ${ \if@compatibility \mathchar"011E {} \mathchar"011E }=17$. The dashed line is the imaginary axis.
  • Figure 5: Simultaneous presence of conservative and dissipative components of $\mathbf{z}$: structured pseudospectrum of $A$ as a function of $k_0$.
  • ...and 3 more figures

Theorems & Definitions (4)

  • Definition 1: First version
  • Definition 2: Second version
  • Definition 3: Third version
  • Definition 4