Journal articles: 'Generalized critical plane criterion'

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Crăciun, Eduard Marius, Marin Marin, and Adrian Răbâea. "Anti-plane crack in human bone. I. Mathematical modelling." Analele Universitatii "Ovidius" Constanta - Seria Matematica 26, no. 1 (March 1, 2018): 81–90. http://dx.doi.org/10.2478/auom-2018-0004.

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Abstract We consider an anti-plane crack in a bone, considered as an initially deformed orthotropic, linear elastic composite material. Elastic incremental fields in the composite material are obtained following theories of Guz’s representation and of Riemann-Hilbert problem. Critical values of crack propagation angle are determined using Sih’s generalized criterion.

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Craciun, Eduard Marius, Tomasz Sadowski, Liviu Marsavina, and Adrian Rabaea. "Mathematical Aspects Regarding Cracks Behaviour in Wood Composites." Key Engineering Materials 601 (March 2014): 108–11. http://dx.doi.org/10.4028/www.scientific.net/kem.601.108.

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In this paper we study the interaction of two unequal collinear cracks in a pre-stressed fiber reinforced wood composite material in anti-plane mode of fracture. A mathematical model is associated to the mechanical problem. Starting from the boundary, constitutive and far field conditions we obtain the representation of the incremental displacement, stress and strain fields using two complex potentials. Using Sih s generalized fracture criterion we determine the critical incremental shear stress which produce crack propagation and the crack tips propagation direction in a particular case of a Pine wood composite.

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Loghman, A., and M. A. Wahab. "Loading and Unloading of Thick-Walled Cylindrical Pressure Vessels of Strain-Hardening Material." Journal of Pressure Vessel Technology 116, no. 2 (May 1, 1994): 105–9. http://dx.doi.org/10.1115/1.2929562.

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A thick-walled closed-end cylinder of isotropic, homogeneous and strain-hardening material is considered in this study. Loading is assumed to consist of a temperature gradient as well as an internal pressure. Unloading is completely elastic without considering a Bauschinger effect. A generalized plane strain case in which the material obeys Von Mises yield criterion is studied. Using the yield criterion, critical conditions for a wide range of loading combinations and thickness ratios are investigated. After the critical condition is established, load is increased beyond the critical values and calculations are made for plastic stresses and strains and progress of plastic zone using an incremental theory of plasticity. Residual stresses are obtained as the cylinder is unloaded from a 25 and 50-percent overstrained condition. Reverse yielding is not considered while the residual stresses at the onset of reverse yielding are calculated. Loading function is assumed to follow the stress-strain curve of SUS 304 at a constant temperature of 400°C, which is selected from the experimental work of earlier researchers.

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Kacker, Ruchin, and Shailendra Singh Bhadauria. "Investigations on critical anisotropic triaxiality at the crack tip under mixed-mode (I + II) fracture." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 233, no. 13 (February 6, 2019): 4688–706. http://dx.doi.org/10.1177/0954406219827337.

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In this analytical study, anisotropic triaxiality, at the yield loci of the crack tip under mixed-mode (I + II) loading, has been modeled. According to the well-known M-criterion the crack of any orientation from the loading axis, initiates at a polar angle at which criticality of triaxiality occurs. Thus, the critical values of triaxiality have been obtained by statistically evaluating the proposed model for various crack inclinations. The combination of six anisotropic constants classifies the five different yield conditions, which are useful in various alloys. Hill’s generalized anisotropic yield equation generates these yield conditions. The assumption of limited plastic deformation at the crack tip has been the basis of analysis. In particular, the principle of linear elastic fracture mechanics (considering small-scale yielding) holds well. Plastic zone shapes at the crack tip for various crack inclinations are plotted to supplement the results obtained from critical values of triaxiality. For both, plane stress and plane strain conditions, the analysis reveals the regions of the degree of anisotropy for particular Lankford’s coefficient and vice versa where the crack initiation angles show variations.

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Liao, Mengke, Yuanming Lai, and Chong Wang. "A strength criterion for frozen sodium sulfate saline soil." Canadian Geotechnical Journal 53, no. 7 (July 2016): 1176–85. http://dx.doi.org/10.1139/cgj-2015-0569.

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Salt content has been proven to be an important influencing factor on the mechanical properties of frozen saline soils, whose strength criterion is different from that of unfrozen saline soils or frozen soils without salts. In this paper, a series of conventional triaxial tests are carried out for frozen saline soils with sodium sulfate at a temperature of –6 °C. A strength criterion of the frozen saline soils, including the influence of salt content, is established by using the generalized nonlinear strength theory. Based on conventional triaxial test results, a modified hydrostatic pressure expression is proposed according to the critical strength function of the modified Cam clay model in the meridian plane. The influence of salt content on the shear strength of the frozen saline soils is investigated according to their freezing temperature curves as well as the formation of salt and ice crystals. The relationship between salt content and friction angle is analyzed. The proposed strength criterion of frozen saline soil is proved to be reliable by extrusion elongation triaxial tests. This criterion can reasonably reflect the major nonlinear strength characteristics of frozen saline soils, including the influences of change in salt content, pressure melting, and ice crushing.

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Chang, Ming-Fang, Cee Ing Teh, and LaiFa Cao. "Critical state strength parameters of saturated clays from the modified Cam clay model." Canadian Geotechnical Journal 36, no. 5 (November 23, 1999): 876–90. http://dx.doi.org/10.1139/t99-050.

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The evaluation of critical state strength parameters is important, especially with the introduction of limit state design. The modified Cam clay (MCC) model is often used, but it is suitable mainly for evaluating the critical state strength parameters from triaxial compression tests on isotropically consolidated soils. The initial stress condition of a natural soil is usually anisotropic, and the stress paths imposed by external loading could deviate from that of a simple triaxial compression. The use of MCC in practice deserves careful consideration. This paper describes a proposed extension of the MCC model for the evaluation of critical state strength parameters from undrained triaxial and plane strain tests on anisotropically consolidated clays. Using Lade's failure criterion and the plastic potential of MCC, a generalized yield surface is determined and the relationships between the critical state internal friction angles from various triaxial and plane strain tests are obtained. By relating the isotropic overconsolidation ratio to the conventional overconsolidation ratio (OCR), a procedure is suggested for the prediction of critical state undrained shear strength (su) for clays. The undrained strength normalized by the preconsolidation pressure is not a constant but a function of the angle of internal friction and the OCR. For overconsolidated clay, the predicted undrained strength ratio (su/σVO) for the triaxial compression condition is larger than that for the plane strain compression condition. Comparisons of predicted results with published data indicate that the procedure is applicable to various compression tests on normally and lightly to moderately overconsolidated clays. Key words: stress anisotropy, clay, constitutive model, critical state, laboratory test, shear strength parameters.

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Alinaghizadeh, Farhad, and Mahmoud Shariati. "Buckling Analysis of Variable Thickness Radially Functionally Graded Annular Sector Plates Resting on Two-Parameter Elastic Foundations by the GDQ Method." International Journal of Applied Mechanics 07, no. 06 (December 2015): 1550083. http://dx.doi.org/10.1142/s1758825115500830.

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In this paper, buckling analysis of thick radially functionally graded circular/annular sector plates with variable thickness resting on two-parameter elastic foundations is studied. The material properties vary along radial direction according to either an exponential or a power-law distribution. The stability equations are derived using the adjacent equilibrium criterion and are based on a higher order shear deformation theory. The generalized differential quadrature method is employed to discretize the stability equations and convert them into a system of algebraic eigenvalue problem. The formulation and method of solution are validated by performing comparison studies with the available results in the open literature. Then, the effects of power-law index, boundary conditions, thickness variation and coefficients of foundation on the critical buckling load of the circular/annular sector plates subjected to different types of in-plane compressions or in-plane shear are investigated in detail.

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Roy, A. K., and S. W. Tsai. "Design of Thick Composite Cylinders." Journal of Pressure Vessel Technology 110, no. 3 (August 1, 1988): 255–62. http://dx.doi.org/10.1115/1.3265597.

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A simple and efficient design method for thick composite cylinders is presented. Micromechanics and macromechanics are integrated by simple relations and the integrated micro and macromechanics approach has been adopted to enable the designers to instantly study the sensitivity of the micromechanical variables on the final design. The stress analysis is based on 3-dimensional elasticity by considering the cylinder in the state of generalized plane strain. The analysis for both open-ended (pipes) and closed-ended (pressure vessels) cylinders subjected to internal and external pressures and axial load is presented. The failure of the cylinders is predicted by using a 3-dimensional quadratic failure criterion. A degradation model is used to calculate burst pressures and the calculated burst pressures agree very well with the available experimental results, for both thin and thick cylinders. In optimizing multilayer cylinders, the 3-D quadratic criterion enables one to obtain the optimal layer sequence very easily. It is found that the layer sequence is very critical in optimizing, in particular, thick cylinders. In addition, the design parameters and material use efficiency of multilayer closed cylinders subjected to internal pressure have also been studied.

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Nakhmanson, G. S., V. S. Kostennikov, and D. A. Nitsak. "Significance Level and the Strength of Test Shift Through the Finite Rate of the Transient Response." Ural Radio Engineering Journal 6, no. 4 (2022): 378–89. http://dx.doi.org/10.15826/urej.2022.6.4.002.

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A significance level and the strength of test shifts are studied then the error function as a transient response model of logical gate is used for testing of statistical hypothesis with Neumann-Pearson criterion. Shifts are rated for Rayleigh accumulated distributions. The generalized signalto-noise ratio is used as one of the variable parameters. The paper reveals that the finite slope of the transient response curve of logical gate results in a type I error increase. The significance level lock leads to bottom of critical region increase and the strength of test decrease in return. The twofold increase of the slope of the transient response curve leads to both the type I error and the bottom of critical region and the strength of test shifts decrease about 12 dB. The twofold increase of generalized signal-to-noise ratio causes the rise of the maximum displacement of a test power functions more than 3 dB up. As this takes place the extremum abscissa is diminished.

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Craciun, Eduard Marius, Adrian Carabineanu, and Niculae Peride. "Fracture Analysis of an Oblique Crack Propagation in a Pre-Stressed Glass-Epoxy Composite." Key Engineering Materials 385-387 (July 2008): 733–36. http://dx.doi.org/10.4028/www.scientific.net/kem.385-387.733.

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We consider a pre-stressed material containing an oblique crack of a length 2a situated in x1x3 plane. We supposed that the material is unbounded and the crack faces are acted by constant normal incremental stresses p. The initial applied stress ◦ is in direction of the crack. Our first aim is to determine the elastic state produced in the body using Guz’s representation theorem. Our second aim is to determine the critical values of the incremental stresses and the direction of crack propagation. To do this, we use Sih’s generalised fracture criterion for an orthotropic elastic composite. In the last part we find the critical values of the initial applied stress ◦ for which the phenomenon of resonance can appear in a pre-stressed Glass-Epoxy composite.

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NIKOL’SKIY, Aleksey A. "Generalized Stability Criteria for Special Linear Automatic Control Systems with a Time Delay." Elektrichestvo 11, no. 11 (2020): 38–46. http://dx.doi.org/10.24160/0013-5380-2020-11-38-46.

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Matters concerned with stability of special linear automatic control systems with a time delay are considered. The determining feature of such systems is that the control loop contains a link of pure (or transport) delay by time T (or for the path S) of the signal at its output with respect to the signal at its input. Apart from time delay links, these systems also contain other linear elements. Such systems include both longstudied systems for transporting various materials and relatively new selflearning repetitive control systems for accurately reproducing cyclically repeated motions or other signals with a period T (or S). The stability of special linear systems can be studied with fully resting on the Nyquist frequency criterion. According to the Nyquist frequency criterion, the stability of a closedloop system is judged by analyzing the frequency transfer function (FTF) loci of the openloop system that includes the controller and the controlled object with respect to the critical point (1, j0) on the complex plane. However, in the majority of cases, the presence of transcendental links results in that the FTF loci of the openloop system have the shape of a star with infinitely long rays, which makes it difficult to interpret the stability of systems with a time delay according to the Nyquist criterion. In such cases, it is more convenient to use a group of graphic criteria, the application of which makes it possible to establish sufficient stability conditions without examining the entire openloop system (the controller plus the object) by analyzing the relative position of the controlled object frequency response loci (obtained without taking into account the controller properties) and a certain stability region boundary, which is determined by the controller properties (obtained without taking into account the controlled object properties). To date, stability region boundaries have already been found for many types of specific structures of systems with a time delay. The problem is that an individually shaped stability region boundary has to be found for each specific type of controller. It is shown, based on a structural transposition of systems with a time delay, that their stability can be studied by using almost any stability boundaries that were previously proposed for some specific types of transcendental controllers. By generalizing the criteria it will be possible not only to estimate for the first time the stability of systems with a time delay in combination with a linear part of an arbitrary kind, but also to expand the variety of applied types of transcendental controllers.

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WANG, XINGYUAN, QINGYONG LIANG, and JUAN MENG. "DYNAMIC ANALYSIS OF THE CAROTID–KUNDALINI MAP." Modern Physics Letters B 22, no. 04 (February 10, 2008): 243–62. http://dx.doi.org/10.1142/s0217984908014717.

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The nature of the fixed points of the Carotid–Kundalini (C–K) map was studied and the boundary equation of the first bifurcation of the C–K map in the parameter plane is presented. Using the quantitative criterion and rule of chaotic system, the paper reveals the general features of the C–K Map transforming from regularity to chaos. The following conclusions are obtained: (i) chaotic patterns of the C–K map may emerge out of double-periodic bifurcation; (ii) the chaotic crisis phenomena are found. At the same time, the authors analyzed the orbit of critical point of the complex C–K Map and put forward the definition of Mandelbrot–Julia set of the complex C–K Map. The authors generalized the Welstead and Cromer's periodic scanning technique and using this technology constructed a series of the Mandelbrot–Julia sets of the complex C–K Map. Based on the experimental mathematics method of combining the theory of analytic function of one complex variable with computer aided drawing, we investigated the symmetry of the Mandelbrot–Julia set and studied the topological inflexibility of distribution of the periodic region in the Mandelbrot set, and found that the Mandelbrot set contains abundant information of the structure of Julia sets by finding the whole portray of Julia sets based on Mandelbrot set qualitatively.

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Andrianopoulos, Nikos P., and Aggelos Pikrakis. "On the effect of hydrostatic stress on fatigue crack propagation." International Journal of Structural Integrity 8, no. 2 (April 10, 2017): 240–55. http://dx.doi.org/10.1108/ijsi-06-2016-0021.

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Purpose The purpose of this paper is to study mutual interaction between von Mises equivalent and hydrostatic stresses at the crack tip area of an elastoplastic material in order to obtain critical conditions for crack propagation under fatigue loading. Design/methodology/approach A5083-H111 aluminum alloy is used to obtain a Chaboche-type constitutive equation, which is introduced in a commercial finite elements package to evaluate stress distribution at crack tip area. A simplified three-dimensional (generalized plane strain) grid is used, resulting in fast and accurate results. Numerical simulations are performed to connect crack propagation rate with various combinations of fatigue stress amplitude, initial crack length and number of loading cycles. Distance between characteristic points of stresses distribution in the crack tip area are compared to experimental fatigue crack growth rates in order to assess the validity of the present approach. Findings It is found that saturation of plastic strains, i.e. maximization of von Mises equivalent stress, is a prerequisite for hydrostatic stress to take a critical-maximum value, outside the plastically saturated zone. At the point of maximum hydrostatic stress brittle fracture is initiated, driving to separation of the ligament up to crack tip, without formation of new plastic strains. The length of this ligament is defined as crack propagation step, showing good agreement with experimental data. Originality/value The present approach seems to constitute a reasonable and adequate method for the description of fatigue crack propagation in terms of continuum mechanics, not necessitating microscopic considerations or empirical criteria lacking theoretical or physical basis. In addition, it liberates from the notion of stress intensity factors, strongly disputed beyond linear elasticity. Improved constitutive equations and numerical models are expected to drive in a complete fatigue failure criterion similar to those of static loading.

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Bilyk, Serhii, Artem Bilyk, and Vitalii Tonkacheiev. "The stability of low-pitched von Mises trusses with horizontal elastic supports." Strength of Materials and Theory of Structures, no. 108 (May 30, 2022): 131–44. http://dx.doi.org/10.32347/2410-2547.2022.108.131-144.

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Abstract. Purpose. The work’s aim is to study the horizontal supports stiffness impact, which simulate the conditions for supporting the domes upper tier on the von Mises trusses' stability. Methodology. A three-hinged truss' deformed scheme under applying a concentrated vertical load in the ridge joint was considered. An analytical method was used to obtain a generalized equation for the three-hinged trusses' stability criterion to determine the critical load depending on the design system's parameters such as the rods' inclination angle, the rods' stiffness, and the horizontal elastic supports stiffness. A two transcendental equations' system for the dependence of the load on vertical and horizontal displacements taking into account the rods' compression was obtained. Such equations' system's analytical solution through a generalized parameter - a variable rods' angle tangent, made it possible to obtain one equation for the dependence of the vertical load on the vertical and symmetric horizontal supports' displacement. The truss' stability numerical studies were carried out depending on the structure's design geometry. Findings. An analytical expression of the dependence for the load on the structure, which was reduced to the rod’s stiffness depending on the rods’ angle to the horizontal stiffness of the supports, was obtained. The low-pitched double-rod three-hinged trusses' nonlinear deformation nature depending on the elastic supports' stiffness and the rods' angle was confirmed. It was found that with the two-rod low-pitched three-hinged systems’ nonlinear deformation nature the ridge joint's snap-through effect takes place. It was found that the relative reduced critical load value decreases along with the rods' inclination angle decrease depending on the horizontal supports' stiffness. Scientific innovation. On the theoretical studies basis of the three-hinged two-rod low-pitched trusses with elastic horizontal supports deformed scheme a generalized analytical equation for the such systems' solution was obtained. The generalized analytical solution models the dome system annular elements stiffness through the horizontal supports' stiffness and determines the general lower tier elements stiffness effect on the dome uppermost tier structural system stability. Practical value. The obtained analytical equation makes it possible to determine the dome annular elements rational design parameters to ensure the upper tier stability.

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Lee, Seung-Joon, George T. Yates, and T. Yaotsu Wu. "Experiments and analyses of upstream-advancing solitary waves generated by moving disturbances." Journal of Fluid Mechanics 199 (February 1989): 569–93. http://dx.doi.org/10.1017/s0022112089000492.

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In this joint theoretical, numerical and experimental study, we investigate the phenomenon of forced generation of nonlinear waves by disturbances moving steadily with a transcritical velocity through a layer of shallow water. The plane motion considered here is modelled by the generalized Boussinesq equations and the forced Korteweg-de Vries (fKdV) equation, both of which admit two types of forcing agencies in the form of an external surface pressure and a bottom topography. Numerical results are obtained using both theoretical models for the two types of forcings. These results illustrate that within a transcritical speed range, a succession of solitary waves are generated, periodically and indefinitely, to form a procession advancing upstream of the disturbance, while a train of weakly nonlinear and weakly dispersive waves develops downstream of an ever elongating stretch of a uniformly depressed water surface immediately behind the disturbance. This is a beautiful example showing that the response of a dynamic system to steady forcing need not asymptotically tend to a steady state, but can be conspicuously periodic, after an impulsive start, when the system is being forced at resonance.A series of laboratory experiments was conducted with a cambered bottom topography impulsively started from rest to a constant transcritical velocity U, the corresponding depth Froude number F = U/(gh0)½ (g being the gravitational constant and h0 the original uniform water depth) being nearly the critical value of unity. For the two types of forcing, the generalized Boussinesq model indicates that the surface pressure can be more effective in generating the precursor solitary waves than the submerged topography of the same normalized spatial distribution. However, according to the fKdV model, these two types of forcing are entirely equivalent. Besides these and some other rather refined differences, a broad agreement is found between theory and experiment, both in respect of the amplitudes and phases of the waves generated, when the speed is nearly critical (0.9 < F < 1.1) and when the forcing is sufficiently weak (the topography-height to water-depth ratio less than 0.15) to avoid breaking. Experimentally, wave breaking was observed to occur in the precursor solitary waves at low supercritical speeds (about 1.1 < F < 1.2) and in the first few trailing waves at high subcritical speeds (about 0.8 < F < 0.9), when sufficiently forced. For still lower subcritical speeds, the trailing waves behaved more like sinusoidal waves as found in the classical case and the forward-running solitary waves, while still experimentally discernible and numerically predicted for 0.6 > F > 0.2, finally disappear at F ≈ 0.2. In the other direction, as the Froude number is increased beyond F ≈ 1.2, the precursor soliton phenomenon was found also to evanesce as no finite-amplitude solitary waves can outrun, nor can any two-dimensional waves continue to follow, the rapidly moving disturbance. In this supercritical range and for asymptotically large times, all the effects remain only local to the disturbance. Thus, the criterion of the fascinating phenomenon of the generation of precursor solitons is ascertained.

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Karolczuk, Aleksander, and Ewald Macha. "Critical Planes in Multiaxial Fatigue." Materials Science Forum 482 (April 2005): 109–14. http://dx.doi.org/10.4028/www.scientific.net/msf.482.109.

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The paper includes a review of literature on the multiaxial fatigue failure criteria based on the critical plane concept. The criteria were divided into three groups according to the distinguished fatigue damage parameter used in the criterion, i.e. (i) stress, (ii) strain and (iii) strain energy density criteria. Each criterion was described mainly by the applied the critical plane position. The multiaxial fatigue criteria based on two critical planes seem to be the most promising. These two critical planes are determined by different fatigue damage mechanisms (shear and tensile mechanisms).

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Jin, Dan, Hai Ling Fu, and Wei Lin. "Fatigue Life Prediction Based on FS Criterion for Multi-Axial Loading." Key Engineering Materials 452-453 (November 2010): 605–8. http://dx.doi.org/10.4028/www.scientific.net/kem.452-453.605.

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The Fatemi–Socie criterion is evaluated using the combined axial torsion fatigue testing results obtained from extensive experiments on thin-walled tubular specimens made from 304 stainless steel. The Fatemi–Socie criterion combines the maximum shear strain amplitude with a consideration of the normal stress on the critical plane. Rainflow cycle counting and Morrow’s plastic work interaction rule are used to calculate fatigue damage. The fatigue life prediction is conducted by using the maximum shear strain plane as the critical plane by considering the weight function and the maximum damage plane as the critical plane. It is concluded that the results gained by two approaches are both acceptable for the case examined.

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Zhang, Jia-Liang, De-Guang Shang, Yu-Juan Sun, and Xiao-Wei Wang. "Multiaxial high-cycle fatigue life prediction model based on the critical plane approach considering mean stress effects." International Journal of Damage Mechanics 27, no. 1 (July 11, 2016): 32–46. http://dx.doi.org/10.1177/1056789516659331.

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The aim of this paper is to propose a modified multiaxial high-cycle fatigue criterion based on the critical plane approach. The proposed criterion contains three parameters, that is, shear stress amplitude, normal stress amplitude and mean normal stress. In order to take into account the mean shear stress effects, the critical plane is determined by the maximum shear stress. In the proposed multiaxial fatigue criterion, the influence of mean normal stress on fatigue damage is also considered. Based on the proposed criterion, the multiaxial fatigue life is predicted, and the results showed a good agreement with experimental data obtained from some literatures.

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NERLI, A., S. CAMARRI, and M. V. SALVETTI. "A conditional stability criterion based on generalized energies." Journal of Fluid Mechanics 581 (May 22, 2007): 277–86. http://dx.doi.org/10.1017/s002211200700609x.

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An energy criterion for conditional stability is proposed, based on the definition of generalized energies, obtained through a perturbation of the classical L2 (kinetic) energy. This perturbation is such that the contribution of the linear term in the perturbation equation to the generalized energy time derivative is negative definite. A critical amplitude threshold is then obtained by imposing the monotonic decay of the generalized energy. The capabilities of the procedure are appraised through the application to three different low-dimensional models. The effects of different choices in the construction of the generalized energy on the prediction of the critical amplitude threshold in the subcritical regime are also discussed.

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Ushaksaraei, R., and S. Pietruszczak. "Failure Criterion for Structural Masonry based on Critical Plane Approach." Journal of Engineering Mechanics 128, no. 7 (July 2002): 769–78. http://dx.doi.org/10.1061/(asce)0733-9399(2002)128:7(769).

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Carpinteri, Andrea, Andrea Spagnoli, and Sabrina Vantadori. "Multiaxial fatigue assessment using a simplified critical plane-based criterion." International Journal of Fatigue 33, no. 8 (August 2011): 969–76. http://dx.doi.org/10.1016/j.ijfatigue.2011.01.004.

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Miller, Evan. "A locally anisotropic regularity criterion for the Navier–Stokes equation in terms of vorticity." Proceedings of the American Mathematical Society, Series B 8, no. 6 (February 10, 2021): 60–74. http://dx.doi.org/10.1090/bproc/74.

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In this paper, we will prove a regularity criterion that guarantees solutions of the Navier–Stokes equation must remain smooth so long as the vorticity restricted to a plane remains bounded in the scale critical space L t 4 L x 2 L^4_t L^2_x , where the plane may vary in space and time as long as the gradient of the unit vector orthogonal to the plane remains bounded. This extends previous work by Chae and Choe that guaranteed that solutions of the Navier–Stokes equation must remain smooth as long as the vorticity restricted to a fixed plane remains bounded in a family of scale critical spaces. This regularity criterion also can be seen as interpolating between Chae and Choe’s regularity criterion in terms of two vorticity components and Beirão da Veiga and Berselli’s regularity criterion in terms of the gradient of vorticity direction. In physical terms, this regularity criterion is consistent with key aspects of the Kolmogorov theory of turbulence, because it requires that finite-time blowup for solutions of the Navier–Stokes equation must be fully three dimensional at all length scales.

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Zou, Jin-feng, and Jia-min Du. "A Numerical Approach for the Quasi-Plane Strain-Softening Problem of Cylindrical Cavity Expansion Based on the Hoek-Brown Failure Criterion." Mathematical Problems in Engineering 2016 (2016): 1–10. http://dx.doi.org/10.1155/2016/3698525.

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This paper focuses on a novel approach for the quasi-plane strain-softening problem of the cylindrical cavity expansion based on generalized Hoek-Brown failure criterion. Because the intermediate principal stress is deformation-dependent, the quasi-plane strain problem is defined to implement the numerical solution of the intermediate principal stress. This approach assumes that the initial total strain in axial direction is a nonzero constant (ε0) and the plastic strain in axial direction is not zero. Based on 3D failure criterion, the numerical solution of plastic strain is given. Solution of the intermediate principal stress can be derived by Hooke’s law. The radial and circumferential stress and strain considering the intermediate principal stress are obtained by the proposed approach of the intermediate principal stress, stress equilibrium equation, and generalized H-B failure criterion. The numerical results can be used for the solution of strain-softening surrounding rock. In additional, the validity and accuracy of the proposed approach are verified with the published results. At last, parametric studies are carried out using MATLAB programming to highlight the influences of the out-of-plane stress on the stress and displacement of surrounding rock.

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Ronchei, Camilla, Andrea Carpinteri, and Sabrina Vantadori. "Energy Concepts and Critical Plane for Fatigue Assessment of Ti-6Al-4V Notched Specimens." Applied Sciences 9, no. 10 (May 27, 2019): 2163. http://dx.doi.org/10.3390/app9102163.

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In the present paper, the fatigue life assessment of notched structural components is performed by applying a critical plane-based multiaxial fatigue criterion. Such a criterion is formulated by using the control volume concept related to the strain energy density criterion. The verification point is assumed to be at a given distance from the notch tip. Such a distance is taken as a function of the control volume radii around the notch tip under both Mode I and Mode III loading. The accuracy of the present criterion is evaluated through experimental data available in the literature, concerning titanium alloy notched specimens under uniaxial and multiaxial fatigue loading.

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Chu, C. C. "Fatigue Damage Calculation Using the Critical Plane Approach." Journal of Engineering Materials and Technology 117, no. 1 (January 1, 1995): 41–49. http://dx.doi.org/10.1115/1.2804370.

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The critical plane approach is applied to model material fatigue behavior under any constant amplitude proportional loading. The most critical plane and the largest damage parameter are determined in closed form for six damage criteria that have been proposed in the literature. The correct procedures of utilizing these closed-form solutions to construct the damage parameter versus fatigue life curve are outlined. It is shown that the common practice of characterizing the material’s fatigue behavior by plotting the damage parameter evaluated on the maximum shear plane against the observed fatigue life violates the principle of the critical plane approach, a problem which can arise during the calibration of any biaxial type of damage criterion. The study emphasizes that the critical plane approach should be consistently applied to both the initial calibration and the subsequent fatigue analysis.

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Hamdi, A., M. Naït Abdelaziz, N. Aït Hocine, P. Heuillet, and N. Benseddiq. "A new generalized fracture criterion of elastomers under quasi-static plane stress loadings." Polymer Testing 26, no. 7 (October 2007): 896–902. http://dx.doi.org/10.1016/j.polymertesting.2007.06.014.

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Kawa, Marek. "Failure Criterion for Brick Masonry: A Micro-Mechanics Approach." Studia Geotechnica et Mechanica 36, no. 3 (February 28, 2015): 37–48. http://dx.doi.org/10.2478/sgem-2014-0025.

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Abstract The paper deals with the formulation of failure criterion for an in-plane loaded masonry. Using micro-mechanics approach the strength estimation for masonry microstructure with constituents obeying the Drucker-Prager criterion is determined numerically. The procedure invokes lower bound analysis: for assumed stress fields constructed within masonry periodic cell critical load is obtained as a solution of constrained optimization problem. The analysis is carried out for many different loading conditions at different orientations of bed joints. The performance of the approach is verified against solutions obtained for corresponding layered and block microstructures, which provides the upper and lower strength bounds for masonry microstructure, respectively. Subsequently, a phenomenological anisotropic strength criterion for masonry microstructure is proposed. The criterion has a form of conjunction of Jaeger critical plane condition and Tsai-Wu criterion. The model proposed is identified based on the fitting of numerical results obtained from the microstructural analysis. Identified criterion is then verified against results obtained for different loading orientations. It appears that strength of masonry microstructure can be satisfactorily described by the criterion proposed.

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28

Gu, Jiefei, Puhui Chen, Ke Li, and Lei Su. "A Macroscopic Strength Criterion for Isotropic Metals Based on the Concept of Fracture Plane." Metals 9, no. 6 (May 31, 2019): 634. http://dx.doi.org/10.3390/met9060634.

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Although the linear Mohr–Coulomb criterion is frequently applied to predict the failure of brittle materials such as cast iron, it can be used for ductile metals too. However, the criterion has some significant deficiencies which limit its predictive ability. In the present study, the underlying failure hypotheses of the linear Mohr–Coulomb criterion were thoroughly discussed. Based on Mohr’s physically meaningful concept of fracture plane, a macroscopic strength criterion was developed to explain the failure mechanism of isotropic metals. The failure function was expressed as a polynomial expansion in terms of the stresses acting on the fracture plane, and the quadratic approximation was employed to describe the non-linear behavior of the failure envelope. With an in-depth understanding of Mohr’s fracture plane concept, the failure angle was regarded as a generalized strength parameter in addition to the failure stress (i.e., the conventional basic strength). The undetermined coefficients of the non-linear failure function were calibrated by the strength parameters obtained from the common uniaxial tension and compression tests. Theoretical and experimental assessment for different types of isotropic metals validated the effectiveness of the proposed criterion in predicting material failure.

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29

Wu, Souheng. "A generalized criterion for rubber toughening: The critical matrix ligament thickness." Journal of Applied Polymer Science 35, no. 2 (February 5, 1988): 549–61. http://dx.doi.org/10.1002/app.1988.070350220.

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30

Alexandrov, Sergei, and Prashant Date. "A general method of stress analysis for a generalized linear yield criterion under plane strain and plane stress." Continuum Mechanics and Thermodynamics 31, no. 3 (December 14, 2018): 841–49. http://dx.doi.org/10.1007/s00161-018-0743-6.

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31

ZHANG, CHENG-CHENG, and WEI-XING YAO. "An improved multiaxial high-cycle fatigue criterion based on critical plane approach." Fatigue & Fracture of Engineering Materials & Structures 34, no. 5 (October 28, 2010): 337–44. http://dx.doi.org/10.1111/j.1460-2695.2010.01523.x.

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32

Carpinteri, Andrea, Andrea Spagnoli, Camilla Ronchei, Daniela Scorza, and Sabrina Vantadori. "Critical Plane Criterion for Fatigue Life Calculation: Time and Frequency Domain Formulations." Procedia Engineering 101 (2015): 518–23. http://dx.doi.org/10.1016/j.proeng.2015.02.062.

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33

Carpinteri, Andrea, Andrea Spagnoli, Camilla Ronchei, and Sabrina Vantadori. "A Strain-based Multiaxial Fatigue Criterion Connected to the Critical Plane Approach." Procedia Engineering 74 (2014): 317–20. http://dx.doi.org/10.1016/j.proeng.2014.06.270.

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34

Gabr Al Hagri. "Correspondence between two bicolored plane trees And generalized Tchebytchev polynomial." Albaydha University Journal 1, no. 2 (January 7, 2020): 200–213. http://dx.doi.org/10.56807/buj.v1i2.23.

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In this paper, we give some theoretical results, for the Tchebytchev polynomials of a tree. A Tchebytchev polynomial, any polynomial has at most two critical values. we consider two bicolored plane trees (two neighboring vertices are always indifferent colorings).We will introduce the composition and decomposition of two bicolored plane trees A and B denoted by (A o B) and we are interested in the relation between two bicolored plane trees and generalized Tchebytchev polynomials.

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35

Jiang, Xiaoyong, Zegen Wang, Lianyong Liu, and Zhiping Zhou. "The Determination of Reduction Ratio Factor in Homogeneous Soil-Slope with Finite Element Double Strength Reduction Method." Open Civil Engineering Journal 7, no. 1 (October 31, 2013): 205–9. http://dx.doi.org/10.2174/1874149501307010205.

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Even though there exist lots of documents on Finite Element Strength Reduction Method, cohesion and internal friction angle in these materials are reduced through the same reduction factor, which fails to reflect either of their safe reserving quality or the quality to precisely match slope failure process and mechanism. As is known, the exterior angle circumcircle yield criterion DP1 of the generalized Mises would cause larger slope stability factor in slope stability analysis, whereas inscribed circle yield criterion DP4 of the generalized Mises has higher precision for the plane strain problem. Since the ANSYS has only DP1 criterion, considering the DP4 yielding criterion’s higher precision than DP1 in plane strain, the author first converts DP4 to DP1 that can be accepted by ANSYS, and then conducts separate reductions to cohesion and internal friction angle with different reduction proportionality factors according to double reduction factor method, and analyzes the calculation results afterwards. The results after analysis show that when cohesion and internal friction angle are reduced by reduction proportionality factor K=1.75, the failure characteristic of slope and attenuation specification of strength parameter match well to the real situation Therefore, reduction proportionality factor K =1.75 of cohesion and internal friction angle is typically recommended in homogeneous soil-slope.

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36

Leng, Song, Li Hou, and Yang Duan. "An Approach for Stability Analysis of Gas/Fluid-Filled Cylindrical Hole in Laminated Materials." Mathematical Problems in Engineering 2020 (May 5, 2020): 1–15. http://dx.doi.org/10.1155/2020/6084972.

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An explicit analytical workflow for cylindrical hole stability analyses in general laminated materials that possess transversely isotropic (TI) anisotropy is presented. In this approach, the calculation of the distribution of the stresses around a cylindrical hole and the failure evaluation at the hole wall consider the effects of both material elasticity anisotropy and strength anisotropy caused by material laminated structures. Material strength anisotropy is assumed to be caused by the sliding of preexisting weakness planes oriented parallel to the isotropic plane of the material. The effect of anisotropy on strength is modeled by combining a shear failure criterion for the intact matrix and a weak plane failure criterion for the planes of weakness. We derive critical pressure solutions for the stability of the intact matrix around a hole filled with gas or fluid based on the Mohr–Coulomb failure criterion and Drucker–Prager failure criterion; either one of them can be combined with the weak plane failure criterion to give the solution for hole wall shear failure pressure. The solution for hole wall fracture initiation pressure is derived based on the tensile failure criterion. This approach can be applied to holes of arbitrary orientation in general laminated materials.

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37

Xu, Shen, Shun-Peng Zhu, Yong-Zhen Hao, and Ding Liao. "Critical plane–based multiaxial fatigue life prediction of turbine disk alloys by refining normal stress sensitivity." Journal of Strain Analysis for Engineering Design 53, no. 8 (June 20, 2018): 719–29. http://dx.doi.org/10.1177/0309324718779922.

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For engineering components subjected to complex multiaxial loadings, critical plane approaches like Fatemi–Socie criterion have been commonly utilized for life prediction of these components. Within the Fatemi–Socie criterion, the normal stress sensitivity parameter k is usually fitted from additional experimental data, which introduces inconvenience for practice especially under limited testing data conditions. In this regard, a simple critical plane–based damage parameter is put forward with no additional material constants, which attempts to provide a robust method for multiaxial fatigue analysis of turbine disk alloys. Using experimental datasets of TC4 and GH4169 alloys under different loadings, the proposed model provides better correlations with fatigue life of the two alloys than the models of Smith–Watson–Topper and Wang–Brown.

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38

Meng, Qingfeng, Xuyue Hu, Guanghui Chen, Peng Li, and Zhi Wang. "Estimation of the Critical Seismic Acceleration for Three-Dimensional Rock Slopes." Applied Sciences 11, no. 24 (December 7, 2021): 11625. http://dx.doi.org/10.3390/app112411625.

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An analytical approach for the estimating of critical seismic acceleration of rock slopes was proposed in this study. Based on the 3D horn failure model, the critical seismic acceleration coefficient of rock slopes was conducted with the modified Hoek–Brown (MHB) failure criterion in the framework of upper-bound theory for the first time. The nonlinear Hoek–Brown failure criterion is incorporated into the three-dimensional rotational failure mechanism, and a generalized tangent technique is introduced and employed to convert the nonlinear Hoek–Brown failure criterion into a linear criterion. The critical seismic acceleration coefficients obtained from this study were validated by the numerical simulation results based on finite element limit analysis. The agreement showed that the proposed method is effective. Finally, design charts were provided for exceptional cases for practical use in rock engineering.

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Chen, Dun, Dayan Wang, Wei Ma, Lele Lei, and Guoyu Li. "A strength criterion for frozen clay considering the influence of stress Lode angle." Canadian Geotechnical Journal 56, no. 11 (November 2019): 1557–72. http://dx.doi.org/10.1139/cgj-2018-0054.

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To investigate the influence of stress Lode angle on frozen soil, a series of directional shear tests was conducted on artificial frozen clay at three temperatures (–6, –10, and –15 °C) and five stress Lode angles (θσ = –30°, –16.1°, 0°, 16.1°, and 30°) using a hollow cylindrical apparatus. An elliptical function was proposed according to the strength envelope evaluation with the mean principal stress (p) in the p–q plane. In addition, generalized nonlinear strength theory (GNST) was introduced in the π plane to describe the evolution of the strength envelope with increasing mean principal stress. Then a strength criterion for frozen clay in three-dimensional principal stress space was proposed by combining strength functions in the p–q and π planes. The temperature effect was also introduced into the strength criterion. The proposed strength criterion can predict the multi-axial strength characteristics of frozen clay and reveal the influence of the stress Lode angle.

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40

Carpinteri, Andrea, Giovanni Fortese, Camilla Ronchei, Daniela Scorza, Sabrina Vantadori, and Filippo Berto. "Joined application of a multiaxial critical plane criterion and a strain energy density criterion in low-cycle fatigue." Frattura ed Integrità Strutturale 11, no. 41 (June 28, 2017): 66–70. http://dx.doi.org/10.3221/igf-esis.41.10.

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41

Fokkema, Jacob T., and Anton Ziolkowski. "The critical reflection theorem." GEOPHYSICS 52, no. 7 (July 1987): 965–72. http://dx.doi.org/10.1190/1.1442365.

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In predictive deconvolution of seismic data, it is assumed that the response of the earth is white. Any nonwhite components are presumed to be caused by the source wavelet or by unwanted multiples. We show that this whiteness assumption is invalid at precritical incidence. We consider plane waves incident on a layered acoustic half‐space. At exactly critical incidence at any interface in the half‐space, the lower layer acts similar to a rigid plate. The response of the half‐space is then all‐pass, or white. This result we call the critical reflection theorem. The response is also white if the waves are postcritically incident on the lower half‐space. In normal data processing these postcritical components are removed by muting. Thus the whiteness assumption is normally applied to exactly that part of the data where it is invalid. The demarcation between precritical and postcritical incidence can be exploited for the purposes of deconvolution, provided the data can be decomposed into plane waves. To develop this application, we consider the response of a point source in the uppermost layer of the layered half‐space, with a free surface above. The response is simply a superposition of the plane‐wave responses already studied, with complications introduced by the source and receiver ghosts and by multiples in the upper layer. At postcritical incidence the earth response is white for all plane‐wave components; the source spectrum may be estimated from the postcritical plane‐wave components after removing the effects of ghosts and multiples in the upper layer. If the source signature is already known, the demarcation criterion can be used to separate intrinsic absorption effects from attenuation effects caused by scattering.

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42

Profant, Tomáš, Jan Klusák, and Michal Kotoul. "Case Criterion of Crack Onset in Orthotropic Bi-Material Notches." Key Engineering Materials 465 (January 2011): 157–60. http://dx.doi.org/10.4028/www.scientific.net/kem.465.157.

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A bi-material notch composed of two orthotropic parts is considered. The stresses and displacements are expressed using the Stroh-Eshelby-Lekhnitskii formalism for plane elasticity. The potential direction of crack initiation is determined from the maximum mean value of the tangential stress or the local minimum of the mean value of the generalized strain energy density factor in both materials [1, 2]. The matched asymptotic procedure is introduced to derive the change of potential energy for the debonding crack and the crack initiated in the determined direction [3].

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43

Zhang, Y. Q., H. Hao, and M. H. Yu. "A Unified Characteristic Theory for Plastic Plane Stress and Strain Problems." Journal of Applied Mechanics 70, no. 5 (September 1, 2003): 649–54. http://dx.doi.org/10.1115/1.1602484.

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Based on the unified strength criterion, a characteristic theory for solving the plastic plane stress and plane strain problems of an ideal rigid-plastic body is established in this paper, which can be adapted for a wide variety of materials. Through this new theory, a suitable characteristic method for material of interest can be obtained and the relations among different sorts of characteristic methods can be revealed. Those characteristic methods on the basis of different strength criteria, such as Tresca, von Mises, Mohr-Coulomb, twin shear (TS) and generalized twin shear (GTS), are the special cases (Tresca, Mohr-Coulomb, TS, and GTS) or linear approximation (von Mises) of the proposed theory. Moreover, a series of new characteristic methods can be easily derived from it. Using the proposed theory, the influence of yield criterion on the limit analysis is analyzed. Two examples are given to illustrate the application of this theory.

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44

Carpinteri, Andrea, Andrea Spagnoli, and Sabrina Vantadori. "Reformulation in the frequency domain of a critical plane-based multiaxial fatigue criterion." International Journal of Fatigue 67 (October 2014): 55–61. http://dx.doi.org/10.1016/j.ijfatigue.2014.01.008.

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45

Liu, Yongming, Chao Zhang, and Yibing Xiang. "A critical plane-based fracture criterion for mixed-mode delamination in composite materials." Composites Part B: Engineering 82 (December 2015): 212–20. http://dx.doi.org/10.1016/j.compositesb.2015.08.017.

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46

Wang, Y., and J. Pan. "Analysis of Small Edge Cracks and Its Implications to Multiaxial Fatigue Theories." Journal of Pressure Vessel Technology 123, no. 1 (October 20, 2000): 2–9. http://dx.doi.org/10.1115/1.1342012.

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The near-tip fields of small edge (Case B) cracks in power-law hardening materials are investigated under generalized plane strain, mixed mode, and general yielding conditions by finite element analyses. The results of the J integral from the finite element analyses are used to correlate to a fatigue crack growth criterion for Case B cracks. The trend of constant J contours on the Γ-plane is compared reasonably well with those of the experimental results of constant fatigue life and constant fatigue crack growth rate under multiaxial loading conditions.

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Vaghefi, Reza, MR Hematiyan, and Ali Nayebi. "Three-dimensional thermo-mechanical analysis of continuous casting and comparison with two-dimensional models." Journal of Strain Analysis for Engineering Design 53, no. 6 (June 4, 2018): 421–34. http://dx.doi.org/10.1177/0309324718780131.

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In this study, a three-dimensional thermo-elasto-plastic model is developed for simulating a continuous casting process. The obtained results are compared with those from different two-dimensional analyses, which are based on plane stress, plane strain, and generalized plane strain assumptions. All analyses are carried out using the meshless local Petrov–Galerkin method. The effective heat capacity method is employed to simulate the phase change process. The von Mises yield criterion and elastic–perfectly-plastic model are used to simulate the stress state during the casting process; while, material parameters are assumed to be temperature-dependent. Based on the three-dimensional and two-dimensional models, numerical results are provided to determine the stress, displacement, and temperature fields induced in the cast material. It is observed that the present meshless local Petrov–Galerkin method is accurate in three-dimensional thermo-mechanical analysis of highly nonlinear phase change problems. Reasonable agreements are observed between the results obtained from the three-dimensional analysis with those retrieved by the generalized plane strain assumption. However, it is observed that the results obtained under plane stress/strain conditions have some significant differences with the results obtained from three-dimensional modeling of continuous casting.

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48

Stratula, Boris A., Nikolai G. Burago, Alexander D. Nikitin, and Ilia S. Nikitin. "Determination of critical plane and assessment of fatigue durability for multiaxial cyclic loading and arbitrary shift of phases." MATEC Web of Conferences 300 (2019): 16006. http://dx.doi.org/10.1051/matecconf/201930016006.

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An analytical solution for the well-known fatigue criterion proposed by Papadopoulos was obtained. An arbitrary shift of phases was taken into account. Comparison between analytical, numerical solutions with experimental data was made. Computation results based on FEM were compared with assessments by the mentioned above criterion.

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49

Huynh, Nam N., Cheng Lu, Liang Ying Si, and Kiet Tieu. "An Orientation-Dependent Failure Criterion for FCC Crystals." Key Engineering Materials 385-387 (July 2008): 801–4. http://dx.doi.org/10.4028/www.scientific.net/kem.385-387.801.

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The crystalline orientation significantly affects the fracture behavior of crystals. However, the orientation-dependent failure criterion is still lacking up to now. In this paper the failure criteria for different crystalline planes of aluminum have been developed. The critical normal stresses to separate two parallel crystallographic planes have been calculated based on Morse potential. The critical stresses on four different planes ({100}, {111}, {110} and {120}) were obtained. It has been found that plane {120} had the minimum critical normal stress. The developed failure criteria have been applied in the crystal plasticity finite element method (CPFEM) model to simulate the uniaxial tensile deformation of single crystal aluminum with a notch. The lattice orientation evolution during deformation has been predicted by the CPFEM model. Elements at notch tip reaching predefined orientation-dependent failure criterion were removed from the mesh so that the crack growing could be determined explicitly without any path assumption.

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50

Hao, Meng-Fei, Shun-Peng Zhu, and Ding Liao. "New strain energy-based critical plane approach for multiaxial fatigue life prediction." Journal of Strain Analysis for Engineering Design 54, no. 5-6 (July 2019): 310–19. http://dx.doi.org/10.1177/0309324719873251.

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Based on critical plane approach, this article develops a new damage parameter through combing the equivalent strain energy aspect for multiaxial fatigue analysis, which includes no additional fitted parameters and overcomes the deficiency of using only equivalent stress/strain criterion separately under multiaxial loadings. Then, experimental data of GH4169, TC4, Al 7050-T7451 alloys under different loading conditions are applied for model validation and comparison with other four models. Results indicate that the proposed damage parameter yields better multiaxial fatigue life predictions than others.

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Journal articles on the topic 'Generalized critical plane criterion'

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