Non-local constitutive relations and the quasi-homogeneous approximation

2005 ◽  
Vol 342 (5-6) ◽  
pp. 363-367 ◽  
Author(s):  
Robert W. Schoonover ◽  
Joseph M. Rutherford ◽  
Ole Keller ◽  
P. Scott Carney
Keyword(s):  
Constitutive Relations ◽  
Non Local ◽  
Homogeneous Approximation

scholarly journals Non-Local Buckling Analysis of Functionally Graded Nanoporous Metal Foam Nanoplates

Coatings ◽  
2018 ◽  
Vol 8 (11) ◽  
pp. 389 ◽  
Author(s):  
Yanqing Wang ◽  
Zhiyuan Zhang
Keyword(s):  
Metal Foam ◽  
Plate Theory ◽  
Constitutive Relations ◽  
Equations Of Motion ◽  
Local Buckling ◽  
Functionally Graded ◽  
Small Scale ◽  
Refined Plate Theory ◽  
Nanoporous Metal ◽  
Non Local

In this study, the buckling of functionally graded (FG) nanoporous metal foam nanoplates is investigated by combining the refined plate theory with the non-local elasticity theory. The refined plate theory takes into account transverse shear strains which vary quadratically through the thickness without considering the shear correction factor. Based on Eringen’s non-local differential constitutive relations, the equations of motion are derived from Hamilton’s principle. The analytical solutions for the buckling of FG nanoporous metal foam nanoplates are obtained via Navier’s method. Moreover, the effects of porosity distributions, porosity coefficient, small scale parameter, axial compression ratio, mode number, aspect ratio and length-to-thickness ratio on the buckling loads are discussed. In order to verify the validity of present analysis, the analytical results have been compared with other previous studies.

Study on Non-Local Damage Constitutive Model of Concrete under Freeze-Thaw Action

2010 ◽  
Vol 452-453 ◽  
pp. 133-136 ◽  
Author(s):  
Zong Min Liu ◽  
Ji Ze Mao ◽  
Hai Yan Song
Keyword(s):  
Constitutive Model ◽  
Internal State ◽  
Constitutive Relations ◽  
Irreversible Thermodynamics ◽  
Local Damage ◽  
Variable Theory ◽  
Freeze Thaw ◽  
Adjacent Point ◽  
Damage Constitutive Model ◽  
Non Local

Concrete is multi-phase composites. Due to the inhomogeneity of mechanical properties and complexity of physical properties, constitutive relations of concrete are more complicated. Starting from irreversible thermodynamics theory, internal state variable theory and nonlocal field theory, non-local damage constitutive model of concrete under freeze-thaw action is established in this paper. In the model, non-local influence functions are discussed which are used to describe interplay of damage between adjacent point.

scholarly journals A non-local asymptotic theory for thin elastic plates

2017 ◽  
Vol 473 (2203) ◽  
pp. 20170249 ◽  
Author(s):  
R. Chebakov ◽  
J. Kaplunov ◽  
G. A. Rogerson
Keyword(s):  
Boundary Layers ◽  
Constitutive Relations ◽  
Plate Thickness ◽  
Three Dimensional ◽  
Low Frequency ◽  
Elastic Plates ◽  
Local Behaviour ◽  
Elasticity Equations ◽  
Plate Equations ◽  
Non Local

The three-dimensional dynamic non-local elasticity equations for a thin plate are subject to asymptotic analysis assuming the plate thickness to be much greater than a typical microscale size. The integral constitutive relations, incorporating the variation of an exponential non-local kernel across the thickness, are adopted. Long-wave low-frequency approximations are derived for both bending and extensional motions. Boundary layers specific for non-local behaviour are revealed near the plate faces. It is established that the effect of the boundary layers leads to the first-order corrections to the bending and extensional stiffness in the classical two-dimensional plate equations.

Zum Materialgesetz eines elastischen Mediums mit Momentenspannungen

1965 ◽  
Vol 20 (3) ◽  
pp. 336-359 ◽  
Author(s):  
F. Hehl ◽  
E. Kröner
Keyword(s):  
Constitutive Relations ◽  
Short Review ◽  
Local Effect ◽  
Classical Elasticity ◽  
Couple Stresses ◽  
State Of Stress ◽  
Order Of Magnitude ◽  
Non Local ◽  
Peierls Model ◽  
Edge Dislocations

If through an element of area of a continuum there acts not only a force but also a couple, we have to introduce besides the force-stresses the so-called couple-stresses. In this article we emphasize the importance of couple-stresses in dislocated solids.—§ 2 gives a short review of the present state of the theory of couple-stresses. In classical elasticity couple-stresses are to be interpreted as a non-local effect intimately connected with the range of the atomic forces. The couplestresses are of a higher order in this range than force-stresses and can therefore usually be neglected.In the field theory of dislocations couple-stresses generally are of the same order of magnitude as force-stresses, however. Hence they cause considerable effects. In § 3 we determine the macroscopic observable couple-stresses of homogeneously distributed screw and edge dislocations through averaging over their microscopic fluctuating stress field. With the PEIERLS model we show in § 4 that the core of a dislocation produces an asymmetric state of stress and for that reason also couple-stresses, which are negligibly small under certain circumstances. Introducing a simple polycrystal model we derive in § 5 the constitutive relations for couple-stresses and dislocation density in an isotropic form. The results are discussed in § 6.

Toward High-Resolution Low-Voltage SEM

1988 ◽  
Vol 46 ◽  
pp. 218-219
Author(s):  
Zhifeng Shao
Keyword(s):  
Low Voltage ◽  
Spherical Aberration ◽  
Chromatic Aberration ◽  
Limiting Factor ◽  
Operating Voltage ◽  
Probe Radius ◽  
Non Local ◽  
Electron Microscopes ◽  
Image Position ◽  
Scanning Electron Microscopes

Recently, low voltage (≤5kV) scanning electron microscopes have become popular because of their unprecedented advantages, such as minimized charging effects and smaller specimen damage, etc. Perhaps the most important advantage of LVSEM is that they may be able to provide ultrahigh resolution since the interaction volume decreases when electron energy is reduced. It is obvious that no matter how low the operating voltage is, the resolution is always poorer than the probe radius. To achieve 10Å resolution at 5kV (including non-local effects), we would require a probe radius of 5∽6 Å. At low voltages, we can no longer ignore the effects of chromatic aberration because of the increased ratio δV/V. The 3rd order spherical aberration is another major limiting factor. The optimized aperture should be calculated as

Chromatic aberration and low-voltage SEM

1987 ◽  
Vol 45 ◽  
pp. 546-547
Author(s):  
Zhifeng Shao ◽  
A.V. Crewe
Keyword(s):  
Electron Energy ◽  
Low Voltage ◽  
Spherical Aberration ◽  
Chromatic Aberration ◽  
Primary Electron ◽  
Probe Size ◽  
Non Local ◽  
Optical Treatment ◽  
Electron Microscopes ◽  
Scanning Electron Microscopes

For scanning electron microscopes, it is plausible that by lowering the primary electron energy, one can decrease the volume of interaction and improve resolution. As shown by Crewe /1/, at V0 =5kV a 10Å resolution (including non-local effects) is possible. To achieve this, we would need a probe size about 5Å. However, at low voltages, the chromatic aberration becomes the major concern even for field emission sources. In this case, δV/V = 0.1 V/5kV = 2x10-5. As a rough estimate, it has been shown that /2/ the chromatic aberration δC should be less than ⅓ of δ0 the probe size determined by diffraction and spherical aberration in order to neglect its effect. But this did not take into account the distribution of electron energy. We will show that by using a wave optical treatment, the tolerance on the chromatic aberration is much larger than we expected.

Sign in / Sign up

Export Citation Format

Share Document