Topic

# Critical value

About: Critical value is a research topic. Over the lifetime, 3201 publications have been published within this topic receiving 71846 citations.

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TL;DR: In this paper, the authors investigated the general problem of crack extension in a combined stress field where a crack can grow in any arbitrary direction with reference to its original position, and showed that the direction of crack growth and fracture toughness for the mixed problem of Mode I and Mode II are governed by the critical value of the strain-energy-density factor.

Abstract: This paper deals with the general problem of crack extension in a combined stress field where a crack can grow in any arbitrary direction with reference to its original position. In a situation, when both of the stress-intensity factors,k
1,k
2 are present along the crack front, the crack may spread in any direction in a plane normal to the crack edge depending on the loading conditions. Preliminary results indicate that the direction of crack growth and fracture toughness for the mixed problem of Mode I and Mode II are governed by the critical value of the strain-energy-density factor,S
cr. The basic assumption is that crack initiation occurs when the interior minimum ofS reaches a critical value designatedS
cr. The strain-energy-density factorS represents the strength of the elastic energy field in the vicinity of the crack tip which is singular of the order of 1/r where the radial distancer is measured from the crack front. In the special case of Mode I crack extensionS
cr is related tok
1c alone asS
cr = (κ − 1)k
1
2
/8μ. In general,S takes the quadratic forma
1 1
k
1 + 2a
1 2
k
1
k
2 +a
2 2
k
2 whose critical value is assumed to be a material constant. The analytical predictions are in good agreement with experimental data on the problem of an inclined crack in plexiglass and aluminum alloy specimens. The result of this investigation provides a convenient procedure for determining the critical crack size that a structure will tolerate under mixed mode conditions for a given applied stress.

2,066 citations

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TL;DR: In this article, the critical layer thickness for growth of GexSi1−x strained layers on Si substrates is calculated for 0≤x≤1.0.

Abstract: A calculation of the critical layer thickness hc for growth of GexSi1−x strained layers on Si substrates is presented for 0≤x≤1.0. The present results are obtained assuming misfit dislocation generation is determined solely by energy balance. This approach differs therefore from previous theories (e.g., Matthews et al.), in which the absence of mechanical equilibrium for grown‐in threading dislocations determines the onset of the generation of interfacial misfit dislocations. It is assumed that interfacial misfit dislocations will be generated when the areal strain energy density of the film exceeds the energy density associated with the formation of a screw dislocation at a distance from the free surface equal to the film thickness h. For films thicker than this critical value, screw (and edge) dislocations will be generated at the film/substrate interface. Values obtained for the critical thickness versus lattice mismatch are in excellent agreement with measurements of hc for GexSi1−x strained layers on...

1,584 citations

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TL;DR: In this paper, the role of pertubative renormalization group (RG) approaches and self-consistent renormalized spin fluctuation (SCR-SF) theories to understand the quantum-classical crossover in the vicinity of the quantum critical point with generalization to the Kondo effect in heavy-fermion systems is discussed.

Abstract: We give a general introduction to quantum phase transitions in strongly-correlated electron systems. These transitions which occur at zero temperature when a non-thermal parameter $g$ like pressure, chemical composition or magnetic field is tuned to a critical value are characterized by a dynamic exponent $z$ related to the energy and length scales $\Delta$ and $\xi$. Simple arguments based on an expansion to first order in the effective interaction allow to define an upper-critical dimension $D_{C}=4$ (where $D=d+z$ and $d$ is the spatial dimension) below which mean-field description is no longer valid. We emphasize the role of pertubative renormalization group (RG) approaches and self-consistent renormalized spin fluctuation (SCR-SF) theories to understand the quantum-classical crossover in the vicinity of the quantum critical point with generalization to the Kondo effect in heavy-fermion systems. Finally we quote some recent inelastic neutron scattering experiments performed on heavy-fermions which lead to unusual scaling law in $\omega /T$ for the dynamical spin susceptibility revealing critical local modes beyond the itinerant magnetism scheme and mention new attempts to describe this local quantum critical point.

1,347 citations

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DuPont

^{1}TL;DR: In this paper, the effects of rubber particle size and rubber-matrix adhesion on notched impact toughness of nylon-rubber blends are analyzed. And the general condition for toughening is that the interparticle distance must be smaller than the critical value.

Abstract: The effects of rubber particle size and rubber-matrix adhesion on notched impact toughness of nylon-rubber blends are analysed. A sharp tough-brittle transition is found to occur at a critical particle size, when the rubber volume fraction and rubber-matrix adhesion are held constant. The critical particle size increases with increasing rubber volume fraction, given by d c = T c {( π 6Φ r ) 1 3 − 1} −1 , dc is the critical particle diameter, Tc the critical interparticle distance, and or the rubber volume fraction. The critical interparticle distance is a material property of the matrix, independent of rubber volume fraction and particle size. Thus, the general condition for toughening is that the interparticle distance must be smaller than the critical value. Van der Waals attraction gives sufficient adhesion for toughening. Interfacial chemical bonding is not necessary. Even if there is interfacial chemical bonding, a polymer-rubber blend will still be brittle, if the interparticle distance is greater than the critical value. The minimum adhesion required is about 1000 J m−2, typical for van der Waals adhesion. In contrast, chemical adhesion is typically 8000 J m−2. The present criterion for toughening is proposed to be valid for all polymer—rubber blends which dissipate the impact energy mainly by increased matrix yielding.

1,241 citations

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TL;DR: In this article, the influence of impact conditions and related phenomena on the coating quality was investigated through spray experiments, which were complemented by ballistic impact tests and explosive powder compaction.

Abstract: In cold spraying, bonding occurs when the impact velocities of particles exceed a critical value. This critical velocity depends not only on the type of spray material, but also on the powder quality, the particle size and the particle impact temperature. Bonding in cold spraying is associated with adiabatic shear instabilities caused by high strain rate deformation during impact. Numerical and experimental methods are developed to investigate the influence of impact conditions and related phenomena on the coating quality. For a quantitative analysis, the materials behaviour was investigated through spray experiments, which were complemented by ballistic impact tests and explosive powder compaction. In this way, impact dynamics, bonding mechanism and critical velocities are linked to develop a general formulation, incorporating material properties and particle size, for the prediction of required impact conditions for cold spray deposition.

923 citations