Elastic Eigenvalues Research Articles

A non-local model to analyse 2D and 3D microbuckling of long carbon fibre epoxy material

Compression failure is a mechanism of non-local ruin that depends on the composite structure (layer thickness, load gradient) which makes it a peculiarity. The mechanism has been described and modelled in the literature with very specific numerical tools and experiments that accounted for this effect. A new finite element model purely based on classical elements of ABAQUS® is developed in this article to predict microbuckling in a 2D or 3D structure. More precisely, a theoretical formulation is proposed which defines the material and structural properties (BNL model) of a numerical model, coupling a continuum media and beams (2D or 3D). The elastic eigenvalues of instability are derived from classical solver of ABAQUS® (subspace). It permits to predict the elastic local instability of unidirectional plies inside a structure in case of complex loads. The ‘structural effect’ explained in the literature is confirmed and new results are obtained. To predict the failure in compression under complex load, a nonlinear behaviour is developed and implemented in the model with the user subroutine (USDFLD). The quantification of the different parameters (plasticity, initial defects, structural effects) is obtained with the BNL model which can be used to study the strength of compression in complex composite structures.

Elastic transmission eigenvalues and their computation via the method of fundamental solutions

ABSTRACT A stabilized version of the fundamental solution method to catch ill-conditioning effects is investigated with focus on the computation of complex-valued elastic interior transmission eigenvalues in two dimensions for homogeneous and isotropic media. Its algorithm can be implemented very shortly and adopts to many similar partial differential equation-based eigenproblems as long as the underlying fundamental solution function can be easily generated. We develop a corroborative approximation analysis which also implicates new basic results for transmission eigenfunctions and present some numerical examples which together prove successful feasibility of our eigenvalue recovery approach.

Eigenvalues of linear viscoelastic systems

The calculation of eigenvalues of single- and multiple-degree-of-freedom linear viscoelastic systems is considered. The assumed viscoelastic forces depend on the past history of motion via convolution integrals over exponentially decaying kernel functions. Current methods to solve this type of problem normally use the state-space approach involving additional internal variables. Such approaches often increase the size of the eigenvalue problem to be solved and can become computationally expensive for large systems. Here an approximate non-state-space based approach is proposed for this type of problem. The proposed approximations are based on certain physical assumptions which simplify the underlying characteristic equation to be solved. Closed-form approximate expressions of the complex and real eigenvalues of the system are derived. These approximate expressions are obtained as functions of the elastic eigenvalues only. This enables one to approximately calculate the eigenvalues of complex viscoelastic systems by simple post-processing of the elastic (undamped) eigenvalues. Representative numerical examples are given to verify the accuracy of the derived expressions.

Influence of non-classical elastic-plastic constitutive features on shock wave existence and spectral solutions

T he influence of non-classical elastic-plastic constitutive features on dynamically moving discontinuities in stress, strain and material velocity is investigated. Non-classical behavior here includes non-normality of the plastic strain increment to the yield surface, plastic compressibility, pressure sensitivity of yield and dependence of the elastic moduli on plastic strain. D rugan and S hen's (1987) analysis of dynamically moving discontinuities with strain as well as stress jumps in classical materials is shown to be valid for a broad class of non-associative material models until deviation from normality exceeds a critical (non-infinitesimal) level. For these non-classical materials, an inequality that bounds the magnitude of the stress jump is derived, which is information not obtainable from a standard spectral analysis of a shock. For the special case of stress discontinuities with continuous strain or for quasi-static deformations, this inequality is shown to rule out jumps in specific projections of the stress tensor unless the non-normality is sufficiently large. These results invalidate a recent claim in the literature that an infinitesimal amount of non-normality permits moving surfaces of discontinuity in stress (with no strain jump) near the tip of a dynamically advancing crack tip. Using a very general plastic constitutive law that subsumes most non-classical (and classical) descriptions currently in use, a complete closed form solution is obtained for the plastic wave speeds and eigenvectors. A novel feature of the analysis is the clarity and completeness of the solutions. If the elastic part of the response is isotropic, one plastic wave speed equals the elastic shear wave speed, while the other two possible wave speeds depend in general on the stress and plastic strain within the shock transition layer. Concise necessary and sufficient conditions for real eigenvalues and for vanishing eigenvalues are derived. The real eigenvalues are classified by numerical sign and ordering relative to the elastic eigenvalues. The geometric multiplicity of plastic eigenvectors associated with elastic eigenvalues is shown to depend on the stress state within the shock transition layer. These solutions, several of which hold for arbitrary elastic anisotropy, are also applicable to acceleration waves and localization problems and to materials with dependence of the elastic moduli on plastic strain. Such elastic-plastic coupling is shown to imply a non-self-adjoint fourth order tangent stiffness tensor even if the plastic constitutive law is associative.

Spinodal temperatures for macroscopic density fluctuations in Pd-H. I. Eigenvalues of hydrogen density modes for a thin-disc geometry

The elastic eigenvalues associated with hydrogen density fluctuations near the critical point of a metal-hydrogen system have been evaluated approximately for the case of a thin-disc geometry, with a small but finite thickness.

A perturbation theory for Love waves in anelastic media

We give a systematic formulation and a rigorous justification of a perturbation technique for the computation of the eigenvalues and eigenfunctions of Love waves (and toroidal oscillations by an appropriate change for variables) in an anelastic medium with a constitutive law modelling geophysical media of current interest such as the Kelvin—Voigt Solid, the Maxwell Solid, the Standard Linear Solid, and the Standard Linear Solid with a continuous spectrum of relaxation times. We develop expressions relating the eigenvalues of eigenfunctions for Love waves in a continuously varying vertically stratified anelastic half-space to the corresponding elastic eigenvalues and eigenfunctions. Analytically, our correspondence principle has the form of a regular perturbation expansion in terms of a parameter for both the eigenvalues and eigenfunctions. The identification of ɛ is motivated by the dissipativity principle of viscoelasticity theory. Moreover, we show that our correspondence principle applies respectively only in the high and low frequency range for the Maxwell and Kelvin—Voigt Solids. Outside of the applicable range of frequencies, our correspondence principle yields no useful information. For the family of Standard Linear Solids it is uniformly applicable for all non-zero frequencies. We also derive an explicit formula to estimate the radius of convergence of our perturbation expansions. This estimate of the radius of convergence for each eigenvalue and eigenfunction is functionally defined by the constitutive model for the anelastic medium. The estimate is frequency dependent and depends on the separation distance between the eigenvalue and the remainder of the spectrum of the corresponding elastic problem.