Probability Density Function Of Stress Research Articles

Sinus Hemodynamics Variation with Tilted Transcatheter Aortic Valve Deployments.

Leaflet thrombosis is a complication associated with transcatheter aortic valve (TAV) replacement (TAVR) correlated with sinus flow stasis. Sinus hemodynamics are important because they dictate shear stress and washout necessary to avoid stasis on TAV leaflets. Sinus flow is controlled by TAV axial deployment position but little is known regarding TAV axis misalignment effect. This study aims to elucidate TAV angular misalignment with respect to aortic root axis effect on sinus flow stasis potentially leading to leaflet thrombosis. Sinus hemodynamics were assessed in vitro using particle-image velocimetry in three different angular misalignments with respect to aorta axis: untilted, tilted away from the sinus and tilted towards sinus. A 26mm Edwards SAPIEN3 was implanted in a 3D printed model of an anatomically realistic aortic root. TAV hemodynamics, sinus vortex tracking, leaflet shear stress probability density functions, and sinus blood time to washout were calculated. While pressure gradients differed insignificantly, blood velocity and vorticity decreased significantly in both tilted cases sinuses. Shear stress probability near the leaflet decreases with tilt indicating stasis. TAV tilted away from the sinus is the most unfavorable scenario with poor washout. TAV axial misalignment adds to factors list that could influence leaflet thrombosis risk through modifying sinus hemodynamics and washout.

Sinus Hemodynamics in Representative Stenotic Native Bicuspid and Tricuspid Aortic Valves: An In-Vitro Study

(1) The study’s objective is to assess sinus hemodynamics differences between stenotic native bicuspid aortic valve (BAV) and native tricuspid aortic valve (TrAV) sinuses in order to assess sinus flow shear and vorticity dynamics in these common pathological states of the aortic valve. (2) Representative patient-specific aortic roots with BAV and TrAV were selected, segmented, and 3D printed. The flow dynamics within the sinus were assessed in-vitro using particle image velocimetry in a left heart simulator at physiological pressure and flow conditions. Hemodynamic data calculations, vortex tracking, shear stress probability density functions and sinus washout calculations based on Lagrangian particle tracking were performed. (3) (a) At peak systole, velocity and vorticity in BAV reach 0.67 ± 0.02 m/s and 374 ± 5 s−1 versus 0.49 ± 0.03 m/s and 293 ± 3 s−1 in TrAV; (b) Aortic sinus vortex is slower to form but conserved in BAV sinus; (c) BAV shear stresses exceed those of TrAV (1.05 Pa versus 0.8 Pa); (d) Complete TrAV washout was achieved after 1.5 cycles while it was not for BAV. (4) In conclusion, sinus hemodynamics dependence on the different native aortic valve types and sinus morphologies was clearly highlighted in this study.

Impact of patient-specific morphologies on sinus flow stasis in transcatheter aortic valve replacement: An in vitro study

ObjectiveThe goal of this study is to evaluate how sinus flow patterns after transcatheter aortic valve replacement in realistic representative patient roots vary. Sinus flow can affect transcatheter aortic valve operation and likely leaflet thrombosis occurrence due to stasis and poor washout. How the interaction between transcatheter aortic valve and representative patient aortic roots affects sinus hemodynamics is important to establish for future individualization of transcatheter aortic valve replacement therapy. MethodsTwo representative patient aortic roots were selected, segmented and 3-dimensional printed followed by deployment of Medtronic CoreValve (Medtronic Inc, Irvine, Calif) and Edwards SAPIEN (Edwards Lifesciences, Irvine Calif) transcatheter aortic valves. Sinus hemodynamics were assessed in vitro using high spatio-temporal resolution particle-image-velocimetry. Detailed sinus vortex tracking, shear stress probability density functions, and sinus washout were evaluated and assessed as a function of valve type and representative patient morphology as independent case studies. ResultsPeak velocity in the sinus with SAPIEN valve was approximately 3 times higher than with CoreValve for both models (0.30 ± 0.02 m/s and 0.34 ± 0.041 m/s vs 0.13 ± 0.01 m/s and 0.10 ± 0.02 m/s) (P < .01). Between representative patient models, vorticity magnitudes were significantly different (75 ± 1.1 s−1, 77 ± 3.2 s−1, 109 ± 2.3 s−1, and 250 ± 4.1 s−1) (P < .01) regardless of valve type. Sinus blood washout characteristic as a function of cardiac cycles was strongly both patient related and valve specific. Fluid dynamics favored shear stresses and washout characteristics due to a smaller sinus and sinotubular junction, further amplified by the SAPIEN valve. ConclusionsSinus flow dynamics are highly sensitive to aortic root characteristics and transcatheter aortic valve aortic root interaction. Differences in sinus-flow washout and stasis regions between representative patient models may be reflected in different risks of leaflet thrombosis or valve degeneration.

Stress Transmission and Failure in Disordered Porous Media.

By means of extensive lattice-element simulations, we investigate stress transmission and its relation with failure properties in increasingly disordered porous systems. We observe a non-Gaussian broadening of stress probability density functions under tensile loading with increasing porosity and disorder, revealing a gradual transition from a state governed by single-pore stress concentration to a state controlled by multipore interactions and metric disorder. This effect is captured by the excess kurtosis of stress distributions and shown to be nicely correlated with the second moment of local porosity fluctuations, which appears thus as a (dis)order parameter for the system. By generating statistical ensembles of porous textures with varying porosity and disorder, we derive a general expression for the fracture stress as a decreasing function of porosity and disorder. Focusing on critical sites where the local stress is above the global fracture threshold, we also analyze the transition to failure in terms of a coarse-graining length. These findings provide a general framework which can also be more generally applied to multiphase and structural heterogeneous materials.

Intergranular stress distributions in polycrystalline aggregates of irradiated stainless steel

In order to predict InterGranular Stress Corrosion Cracking (IGSCC) of post-irradiated austenitic stainless steel in Light Water Reactor (LWR) environment, reliable predictions of intergranular stresses are required. Finite elements simulations have been performed on realistic polycrystalline aggregate with recently proposed physically-based crystal plasticity constitutive equations validated for neutron-irradiated austenitic stainless steel. Intergranular normal stress probability density functions are found with respect to plastic strain and irradiation level, for uniaxial loading conditions. In addition, plastic slip activity jumps at grain boundaries are also presented. Intergranular normal stress distributions describe, from a statistical point of view, the potential increase of intergranular stress with respect to the macroscopic stress due to grain-grain interactions. The distributions are shown to be well described by a master curve once rescaled by the macroscopic stress, in the range of irradiation level and strain considered in this study. The upper tail of this master curve is shown to be insensitive to free surface effect, which is relevant for IGSCC predictions, and also relatively insensitive to small perturbations in crystallographic texture, but sensitive to grain shapes.

Dynamic Reliability Analysis Method of Degraded Mechanical Components Based on Process Probability Density Function of Stress

It is necessary to develop dynamic reliability models when considering strength degradation of mechanical components. Instant probability density function (IPDF) of stress and process probability density function (PPDF) of stress, which are obtained via different statistical methods, are defined, respectively. In practical engineering, the probability density function (PDF) for the usage of mechanical components is mostly PPDF, such as the PDF acquired via the rain flow counting method. For the convenience of application, IPDF is always approximated by PPDF when using the existing dynamic reliability models. However, it may cause errors in the reliability calculation due to the approximation of IPDF by PPDF. Therefore, dynamic reliability models directly based on PPDF of stress are developed in this paper. Furthermore, the proposed models can be used for reliability assessment in the case of small amount of stress process samples by employing the fuzzy set theory. In addition, the mechanical components in solar array of satellites are chosen as representative examples to illustrate the proposed models. The results show that errors are caused because of the approximation of IPDF by PPDF and the proposed models are accurate in the reliability computation.

A reliability analysis method for fatigue design

Structural design against fatigue is a complex task due to the significant number of uncertainties that are inherent to the fatigue phenomenon. In this context, the safety margin and the influence of each design parameter on structural reliability are extremely valuable knowledge for the robust design of structures subjected to fatigue loadings. The Stress–stRength approach is a well established probabilistic method for assessing the failure probability of already designed structures using a load-time history whatever mechanical behavior. It consists of the comparison of two Probability Density Functions (PDFs), the Stress (S) and the stRength (R) of the structure.This widely used engineering approach is very convenient to use but presents some weaknesses that are underlined in this paper through an illustrative case study. First, the failure probability is very sensitive to the PDFs selected for S and R. Second, the influence of each random variable on reliability cannot be determined since the uncertain parameters characterizing geometry, material properties and loads are gathered in the Stress PDF. This paper proposes a more general and robust approach that is able to accurately assess the failure probability and determine importance factors of each random variable for potential time-demanding mechanical models, such as those encountered in industry. An application provided by the engine manufacturer Snecma of SAFRAN Group shows the applicability in an industrial context.

Influence of an anisotropic slip-length boundary condition on turbulent channel flow

The effects of an anisotropic Navier slip-length boundary condition on turbulent channel flow are investigated parametrically by direct numerical simulations. The slip-length boundary condition is made direction dependent by specifying the value of the slip length independently for the streamwise and spanwise direction. The change in drag is mapped versus a wide range of streamwise and spanwise slip-length combinations at two different friction Reynolds numbers, \documentclass[12pt]{minimal}\begin{document}$Re_{\tau _0}=180$\end{document}Reτ0=180 and \documentclass[12pt]{minimal}\begin{document}$Re_{\tau _0}=360$\end{document}Reτ0=360. For moderate slip lengths both drag-reducing and drag-increasing slip-length combinations are found. The percentage drag increase saturates at approximately 60% for high spanwise slip. Once a threshold value for the streamwise slip length is exceeded, drag is reduced in all cases irrespective of the value of the spanwise slip length. The Reynolds number appears to have only little influence on the change in drag for the moderate Reynolds numbers studied here. A detailed comparison with the implicit theoretical formula of Fukagata et al. [Phys. Fluids 18, 051703 (2006)], which relates the change in drag with the streamwise and spanwise slip length, has been made. In general, this formula gives a fair representation of the change in drag; a modified version of this relation is presented, which improves the prediction for the change in drag for small slip length values and reduces the number of free parameters contained in the model. The effects of the slip-length boundary condition on the flow are further investigated using mean flow and turbulence statistics. For drag-neutral slip-length combinations the level of turbulent fluctuations is approximately unchanged. The presence of a slip-length boundary condition affects both the level of wall-shear stress fluctuations and the degree of intermittency of the wall-shear stress probability density function. The correlation statistics of the velocity field show that a high spanwise slip length causes a disruption of the near-wall streaks, while high streamwise slip favours an increasing streak regularity.

The frequency domain approach in virtual fatigue estimation of non-linear systems: The problem of non-Gaussian states of stress

The use of the frequency domain approach in the virtual estimation of mechanical component fatigue life under random loads is related to two conditions regarding the dynamic behaviour of components and the state of stress. The mechanical system must have linear behaviour and the probability density function of stress must be Gaussian, respectively. Obviously, these conditions are not independent, because there is a close tie between the transformations induced by the system to the random inputs and stress distribution. The rigorous procedure for the extension of these hypotheses is not available and only approximated approaches can be used: normally these are based on a corrective coefficient to the narrow-band formula. The main goal of this report is to suggest a separation of the effects on the corrective coefficient. In this manner, the global coefficient can be seen as the product between a partial coefficient related only to the wide-band effects of stress power spectral density function and another one dependent on non-normality indices of stress probability density function. A meaningful application has been investigated to validate the practical employment of this approach. By this example the authors also defined an original analytical expression of a corrective coefficient for Gaussian damage; however, the formulation has to be improved by other applications, because its validity is tested only on a too much limited domain of Kurtosis values. Moreover, the authors suggest that a modal approach to the stress recovery procedure of a flexible body might be an interesting way to the rapid identification of non-Gaussianity indices in the analysis of frequency and time domain dynamics. For this reason, they believe that the investigation of tying the stress non-Gaussianity to the non-Gaussianity of the component modal coordinates to be useful.

Probabilistic elasto-plasticity: formulation in 1D

A second-order exact expression for the evolution of probability density function of stress is derived for general, one-dimensional (1-D) elastic–plastic constitutive rate equations with uncertain material parameters. The Eulerian–Lagrangian (EL) form of Fokker–Planck–Kolmogorov (FPK) equation is used for this purpose. It is also shown that by using EL form of FPK, the so called “closure problem” associated with regular perturbation methods used so far, is resolved too. The use of EL form of FPK also replaces repetitive and computationally expensive deterministic elastic–plastic computations associated with Monte Carlo technique. The derived general expressions are specialized to the particular cases of point location scale linear elastic and elastic–plastic constitutive equations, related to associated Drucker–Prager with linear hardening. In a companion paper, the solution of FPK equations for 1D is presented, discussed and illustrated through a number of examples.

Probabilistic elasto-plasticity: solution and verification in 1D

In this paper, a solution is presented for evolution of probability density function (PDF) of elastic–plastic stress–strain relationship for material models with uncertain parameters. Developments in this paper are based on already derived general formulation presented in the companion paper. The solution presented is then specialized to a specific Drucker–Prager elastic–plastic material model. Three numerical problems are used to illustrate the developed solution. The stress–strain response (1D) is given as a PDF of stress as a function of strain. The presentation of the stress–strain response through the PDF differs significantly from the traditional presentation of such results, which are represented by a single, unique curve in stress–strain space. In addition to that the numerical solutions are verified against closed form solutions where available (elastic). In cases where the closed form solution does not exist (elastic–plastic), Monte Carlo simulations are used for verification.

Strength and failure of cemented granular matter

Cemented granular materials (CGMs) consist of densely packed solid particles and a pore-filling solid matrix sticking to the particles. We use a sub-particle lattice discretization method to investigate the particle-scale origins of strength and failure properties of CGMs. We show that jamming of the particles leads to highly inhomogeneous stress fields. The stress probability density functions are increasingly wider for a decreasing matrix volume fraction, the stresses being more and more concentrated in the interparticle contact zones with an exponential distribution as in cohesionless granular media. Under uniaxial loading, pronounced asymmetry can occur between tension and compression both in strength and in the initial stiffness as a result of the presence of bare contacts (with no matrix interposed) between the particles. Damage growth is analyzed by considering the evolution of stiffness degradation and the number of broken bonds in the particle phase. A brutal degradation appears in tension as a consequence of brittle fracture in contrast to the more progressive nature of damage growth in compression. We also carry out a detailed parametric study in order to assess the combined influence of the matrix volume fraction and particle-matrix adherence. Three regimes of crack propagation can be distinguished corresponding to no particle damage, particle abrasion and particle fragmentation, respectively. We find that particle damage scales well with the relative toughness of the particle-matrix interface with respect to the particle toughness. This relative toughness is a function of both matrix volume fraction and particle-matrix adherence and it appears therefore to be the unique control parameter governing transition from soft to hard behavior.

Predicting fatigue lives for bi-modal stress spectral densities

Fatigue failures of components subject to random stress histories with bi-modal spectral densities are investigated. Such stress spectral densities are often experienced by chassis and suspension components of road vehicles. Some of the methods for predicting fatigue damage using frequency-domain statistics of random stress processes are reviewed. A new ‘bi-modal’ model is devised for calculating the probability density function of stress ranges in the case of a spectral density with two peaks. The method has the attraction of converging to the same results as the well known Rayleigh distribution for narrow band processes, if either of the spectral peaks is small. A component in a heavy road vehicle suspension is used as a case study. Sensitivity analyses are conducted to compare the new method with various other frequency-domain approaches, and with the widely accepted rain-flow method of cycle counting in the time domain.

Some stress-strength reliability models

This paper presents reliability expressions for five stress-strength models of parallel systems with non-identical units. Models are developed for the following systems: 1. (1) when system stress probability density function is exponentially distributed 2. (2) the stress probability density function follows gamma distribution 3. (3) system stresss probability density function can be described by beta distribution 4. (4) stress probability density function is given by a power series 5. (5) and finally, when system stress probability density function follows chi-square distribution. In these models, the unit strength probability density function is exponentially distributed with constant failure rate.

Expected fatigue life of bridges under traffic load

Random vibrations theory is applied to study the response characteristics of elastic damped multi-degrees-of-freedom structures subjected to traffic loads. Four kinds of randomness are included in the non stationary process modeling the crossings: the weight of the vehicles, the excentricity of the passing lane, the initial dynamic conditions and disturbances due to road surface uneveness. Modal analysis and impulse response functions are employied to derive the non stationary probability density function of stresses and displacements of the structure.