scholarly journals Hadamard’s variational formula in terms of stress and strain tensors

2022 ◽  
Vol 12 (1) ◽  
Author(s):  
Björn Gustafsson ◽  
Ahmed Sebbar
Keyword(s):  
Green Function ◽  
Stress Tensor ◽  
Strain Tensor ◽  
Variational Formula ◽  
Scalar Fields ◽  
Action Functional ◽  
Lagrangian Action ◽  
The Green Function ◽  
Strain Tensors ◽  
Hadamard Variational Formula

AbstractStarting from a Lagrangian action functional for two scalar fields we construct, by variational methods, the Laplacian Green function for a bounded domain and an appropriate stress tensor. By a further variation, imposed by a given vector field, we arrive at an interior version of the Hadamard variational formula, previously considered by P. Garabedian. It gives the variation of the Green function in terms of a pairing between the stress tensor and a strain tensor in the interior of the domain, this contrasting the classical Hadamard formula which is expressed as a pure boundary variation.

scholarly journals The axiomatic introduction of arbitrary strain tensors by Hans Richter – a commented translation of ‘Strain tensor, strain deviator and stress tensor for finite deformations’

2020 ◽  
Vol 25 (5) ◽  
pp. 1060-1080
Author(s):  
Patrizio Neff ◽  
Kai Graban ◽  
Eva Schweickert ◽  
Robert J. Martin
Keyword(s):  
Stress Tensor ◽  
Strain Tensor ◽  
Finite Deformations ◽  
German Version ◽  
Strain Deviator ◽  
Strain Tensors

We provide a faithful translation of Hans Richter’s important 1949 paper ‘Verzerrungstensor, Verzerrungsdeviator und Spannungstensor bei endlichen Formänderungen’ from its original German version into English, complemented by an introduction summarizing Richter’s achievements.

scholarly journals Verification of Continuum Mechanics Predictions with Experimental Mechanics

Materials ◽  
2019 ◽  
Vol 13 (1) ◽  
pp. 77
Author(s):  
Cesar A. Sciammarella ◽  
Luciano Lamberti ◽  
Federico M. Sciammarella
Keyword(s):  
Stress Tensor ◽  
Continuum Mechanics ◽  
Strain Tensor ◽  
Scalar Fields ◽  
Identification Problem ◽  
Theoretical Concept ◽  
Cauchy Stress ◽  
Cauchy Stress Tensor ◽  
Mathematical Functions ◽  
Eulerian Derivative

The general goal of the study is to connect theoretical predictions of continuum mechanics with actual experimental observations that support these predictions. The representative volume element (RVE) bridges the theoretical concept of continuum with the actual discontinuous structure of matter. This paper presents an experimental verification of the RVE concept. Foundations of continuum kinematics as well as mathematical functions relating displacement vectorial fields to the recording of these fields by a light sensor in the form of gray-level scalar fields are reviewed. The Eulerian derivative field tensors are related to the deformation of the continuum: the Euler–Almansi tensor is extracted, and its properties are discussed. The compatibility between the Euler–Almansi tensor and the Cauchy stress tensor is analyzed. In order to verify the concept of the RVE, a multiscale analysis of an Al–SiC composite material is carried out. Furthermore, it is proven that the Euler–Almansi strain tensor and the Cauchy stress tensor are conjugate in the Hill–Mandel sense by solving an identification problem of the constitutive model of urethane rubber.

Exponentially Convergent Approximation to the Elliptic Solution Operator

2006 ◽  
Vol 6 (4) ◽  
pp. 386-404 ◽  
Author(s):  
Ivan. P. Gavrilyuk ◽  
V.L. Makarov ◽  
V.B. Vasylyk
Keyword(s):  
Green Function ◽  
Boundary Value Problem ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Solution Operator ◽  
Hyperbolic Operator ◽  
Elliptic Solution ◽  
Elliptic Boundary Value ◽  
Sine Family ◽  
The Green Function

AbstractWe develop an accurate approximation of the normalized hyperbolic operator sine family generated by a strongly positive operator A in a Banach space X which represents the solution operator for the elliptic boundary value problem. The solution of the corresponding inhomogeneous boundary value problem is found through the solution operator and the Green function. Starting with the Dunford — Cauchy representation for the normalized hyperbolic operator sine family and for the Green function, we then discretize the integrals involved by the exponentially convergent Sinc quadratures involving a short sum of resolvents of A. Our algorithm inherits a two-level parallelism with respect to both the computation of resolvents and the treatment of different values of the spatial variable x ∈ [0, 1].

The Green function method for the two-surface problem

1970 ◽  
Vol 8 (13) ◽  
pp. 1069-1071 ◽  
Author(s):  
F. Flores ◽  
F. Garcia-Moliner ◽  
J. Rubio
Keyword(s):  
Green Function ◽  
Function Method ◽  
Green Function Method ◽  
The Green Function

Electrostatic oscillations in cold inhomogeneous plasma I. Differential equation approach

1971 ◽  
Vol 5 (2) ◽  
pp. 239-263 ◽  
Author(s):  
Z. Sedláček
Keyword(s):  
Differential Equation ◽  
Green Function ◽  
Normal Mode Analysis ◽  
Inhomogeneous Plasma ◽  
Mode Analysis ◽  
Complex Frequency ◽  
Plasma Oscillations ◽  
The Laplace Transform ◽  
Collective Oscillations ◽  
The Green Function

Small amplitude electrostatic oscillations in a cold plasma with continuously varying density have been investigated. The problem is the same as that treated by Barston (1964) but instead of his normal-mode analysis we employ the Laplace transform approach to solve the corresponding initial-value problem. We construct the Green function of the differential equation of the problem to show that there are branch-point singularities on the real axis of the complex frequency-plane, which correspond to the singularities of the Barston eigenmodes and which, asymptotically, give rise to non-collective oscillations with position-dependent frequency and damping proportional to negative powers of time. In addition we find an infinity of new singularities (simple poles) of the analytic continuation of the Green function into the lower half of the complex frequency-plane whose position is independent of the spatial co-ordinate so that they represent collective, exponentially damped modes of plasma oscillations. Thus, although there may be no discrete spectrum, in a more general sense a dispersion relation does exist but must be interpreted in the same way as in the case of Landau damping of hot plasma oscillations.

Sign in / Sign up

Export Citation Format

Share Document