scholarly journals The generalized criterion of multiaxial random fatigue based on conception proposed by prof. Macha

2019 ◽  
Vol 300 ◽  
pp. 15001
Author(s):  
Tadeusz Łagoda ◽  
Marta Kurek ◽  
Karolina Łagoda
Keyword(s):  
Shear Stress ◽  
Complex Number ◽  
Mathematical Problem ◽  
Equivalent Stress ◽  
Complex Stress ◽  
Complex Numbers ◽  
Pure Bending ◽  
Pure Torsion ◽  
Special Cases ◽  
Random Fatigue

This criterion has been repeatedly verified, analyzed and special cases of this criterion reducing complex stress to equivalent uniaxial were taken into account. Since both normal and shear stress are vectors, we encounter the mathematical problem of adding these vectors, and the question arises how to understand the obtained equivalent stress, because two perpendicular vectors are added with weighting factors. Therefore, in this work it was proposed to adopt a system of complex numbers. Normal stress was defined as the real part and shear stress as imaginary part. As a result, on the basis of the defined complex number and basing on pure bending and pure torsion after transformations, the expression for equivalent stress was identical to the previously proposed criteria defined on the basis of the concept of prof. Macha.

scholarly journals Formulation of Strain Fatigue Criterion Based on Complex Numbers

Materials ◽  
2021 ◽  
Vol 14 (5) ◽  
pp. 1227
Author(s):  
Tadeusz Łagoda ◽  
Karolina Głowacka ◽  
Marta Kurek ◽  
Dariusz Skibicki
Keyword(s):  
Shear Strain ◽  
Mathematical Problem ◽  
Equivalent Strain ◽  
Multiaxial Fatigue ◽  
Complex Numbers ◽  
Normal Strain ◽  
Pure Bending ◽  
Different Types ◽  
Fatigue Criterion ◽  
Physical Quantities

In the case of many low-cycle multiaxial fatigue criteria, we encounter a mathematical problem of adding vectors of normal and shear strains. Typically, the problem of defining an equivalent strain is solved by weighting factors. Unfortunately, this ignores the fact that these vectors represent other physical quantities: the normal strain is a longitudinal strain, and the shear strain is a rotation angle. Therefore, the goal of the present work was to propose a method of combining different types of strains by adopting a system of complex numbers. The normal strain was defined as the real part and the shear strain was defined as the imaginary part. Using this approach, simple load states, such as pure bending and pure torsion, have been transformed into an expression for equivalent strain identical to the previously proposed criteria defined by Macha.

scholarly journals Winding Numbers, Unwinding Numbers, and the Lambert W Function

2021 ◽  
Author(s):  
A. F. Beardon
Keyword(s):  
Complex Plane ◽  
Complex Number ◽  
Lambert W Function ◽  
Complex Numbers ◽  
Winding Number ◽  
Line Integral ◽  
Complex Functions

AbstractThe unwinding number of a complex number was introduced to process automatic computations involving complex numbers and multi-valued complex functions, and has been successfully applied to computations involving branches of the Lambert W function. In this partly expository note we discuss the unwinding number from a purely topological perspective, and link it to the classical winding number of a curve in the complex plane. We also use the unwinding number to give a representation of the branches $$W_k$$ W k of the Lambert W function as a line integral.

Fatigue Tolerance of Aged Asphalt Binders Modified with Softeners

2021 ◽  
pp. 036119812110255
Author(s):  
Javier J. García Mainieri ◽  
Punit Singhvi ◽  
Hasan Ozer ◽  
Brajendra K. Sharma ◽  
Imad L. Al-Qadi
Keyword(s):  
Shear Stress ◽  
Heavy Traffic ◽  
Asphalt Binder ◽  
Fatigue Cracking ◽  
Data Interpretation ◽  
Complex Stress ◽  
Current Data ◽  
Asphalt Binders ◽  
Failure Patterns ◽  
Peak Shear Stress

Fatigue cracking caused by repeated heavy traffic loading is a critical distress in asphalt concrete pavements and is significantly affected by the selected binder. In recent years, the growing use of recycled asphalt materials has increased the need for the production of softer asphalt binder. Various modifiers/additives are marketed to adjust the grade and/or enhance the binder performance at high and low temperatures. The modifiers are expected to alter the rheological and chemical characteristics of binders and, therefore, their performance. In this study, the damage characteristics of modified and unmodified binders, at standard long-term and extended aging conditions, were tested using the linear amplitude sweep (LAS) test. Current data-interpretation methods for LAS measurements (including AASHTO TP 101-12, T 391-20, and recent literature) showed inconsistent results for modified binders. An alternative method to interpret LAS results was developed in this study. The method considers the data until peak shear-stress is reached because complex stress states and failure patterns are observed in the specimens after that point. The proposed parameter (Δ| G*|peak τ) quantifies the reduction in complex shear modulus measured at the peak shear-stress. The parameter successfully captures the effect of aging and modification of binders.

scholarly journals Simulation of human bone implant duralium material with variation loading using Ansys software

2018 ◽  
Vol 204 ◽  
pp. 07020
Author(s):  
Didin Mujahidin ◽  
Poppy Puspitasari ◽  
Djoko Kustono
Keyword(s):  
Stainless Steel ◽  
Shear Stress ◽  
Equivalent Stress ◽  
Titanium Implants ◽  
Material Strength ◽  
Body Parts ◽  
Ansys Software ◽  
Implant Materials ◽  
Broken Bones ◽  
Material Hardness

Bone implants are a tool used as a support of body parts, and bone support in cases of fractures. Scaffold, plate, bone screw, and some other tools can be used in combination to support and fill the connection between broken bones before the tissue grows. The most commonly used implant materials are Titanium, Stainless steel and ceramics, which are very common in the use of medical devices. Biocompatible materials are taken into consideration when planning a medical device. This research intended to know the durability of duralumin material as the latest implant material, as the development and breakthrough in health world. The research methodology used in this study was the optimization in Ansys software 18.1. The implants were designed, the material strength was determined and then given imposition with 6 variations (450 N, 550 N, 650 N, 750 N, 850 N and 950 N). The optimization was a method that identified mat erial strength including Equivalent Stress, Shear Stress and Total Deformation of duralumin material as implant materials with loading variations. Based on the results of the research, the duralumin material had a equivalent stress of 475,700 Pa which was higher than 950000 Pa for ZnO-Al2O3 implants, while the duralumin shear stress of 1084500 Pa was higher than 313720 Pa for ZnO-Al2O3 implants. When compared with titanium implants, the highest equivalent stress of 150000 Pa duralumin material had a higher compression stress than titanium. The highest shear stress of titanium 4358.1 Pa means an implant with a higher shear duralumin material of titanium. Whereas if it was compared to stainless steel with voltage press 564000000 Pa, then the duralumin’s pressure was getting lower. Material hardness affects resistance to wear and tear. Duralumin material hardness was lower than Titanium and ZnO-Al2O3, so total Duralumin deformation (elasticity) was higher than Titanium and ZnO-Al2O3.

scholarly journals Exact solutions for unsteady axial Couette flow of a fractional Maxwell fluid due to an accelerated shear

2011 ◽  
Vol 16 (2) ◽  
pp. 135-151 ◽  
Author(s):  
Muhammad Athar ◽  
Corina Fetecau ◽  
Muhammad Kamran ◽  
Ahmad Sohail ◽  
Muhammad Imran
Keyword(s):  
Shear Stress ◽  
Positive Real Number ◽  
Fractional Integration ◽  
Maxwell Fluid ◽  
Unsteady Motion ◽  
Positive Real ◽  
Special Cases ◽  
Multiple Integrals ◽  
Similar Solutions ◽  
R Functions

The velocity field and the adequate shear stress corresponding to the flow of a fractional Maxwell fluid (FMF) between two infinite coaxial cylinders, are determined by means of the Laplace and finite Hankel transforms. The motion is produced by the inner cylinder that at time t = 0+ applies a shear stress fta (a ≥ 0) to the fluid. The solutions that have been obtained, presented under series form in terms of the generalized G and R functions, satisfy all imposed initial and boundary conditions. Similar solutions for ordinary Maxwell and Newtonian fluids are obtained as special cases of general solutions. The unsteady solutions corresponding to a = 1, 2, 3, ... can be written as simple or multiple integrals of similar solutions for a = 0 and we extend this for any positive real number a expressing in fractional integration. Furthermore, for a = 0, 1 and 2, the solutions corresponding to Maxwell fluid compared graphically with the solutions obtained in [1–3], earlier by a different technique. For a = 0 and 1 the unsteady motion of a Maxwell fluid, as well as that of a Newtonian fluid ultimately becomes steady and the required time to reach the steady-state is graphically established. Finally a comparison between the motions of FMF and Maxwell fluid is underlined by graphical illustrations.

scholarly journals A study on damage to mechanical seat cushion made from different materials of extension frame

2018 ◽  
Vol 7 (3.3) ◽  
pp. 315
Author(s):  
Jae Won Kim ◽  
Jae Ung Cho ◽  
Chan Ki Cho ◽  
Jin Oh Kim
Keyword(s):  
Shear Stress ◽  
Statistical Analysis ◽  
Material Property ◽  
Equivalent Stress ◽  
Damage Prediction ◽  
Seat Cushion ◽  
Seat Cushions

Background/Objectives: : Automotive seat is a very important component to prevent accidents by reducing passenger’s tiredness, thus, this study worked on analyzing damage with different materials of extension frames of mechanical seat cushions.Methods/Statistical analysis: In this study, we performed an experiment on cushion extension frames by splitting it into two parts. We studied about the damage prediction of slave body for each material property of ABS, PP, PLA, and PA6.6. For analyzing the condition, we assigned the side part of the master body for fixed support, and we progressed on analysis by applying with 690N on the entire part of the slave body.Findings: This research worked on the study of damage to different materials of extension frames of seat cushions. After confirming the stress equivalence of the entire model for each material, PP showed the highest equivalent stress of 180.88MPa, and ABS showed the lowest equivalent stress of 151.73MPa. Overall, we could see that in the order of ABS, PA6.6, PLA, PP have a higher tendency to be broken. In addition, when confirming equivalent stress of master body depending on materials of slave body, PA6.6 showed the highest equivalent stress of 166.3MPa, and ABS showed the lowest equivalent stress of 124.06MPa. Overall, we could see that in the order of ABS, PP, PLA and PP6.6 have a higher tendency to be broken. In comparing shear stress on the gear part, which has the highest tendency to be broken in among the entire model, depending on the material of the slave body, PLA showed the greatest shear stress of 88.945MPa, and ABS showed the lowest shear stress of 69.766MPa.Improvements/Applications: This study worked for the improvements and applications of cushion extension frames as the securement of material by investigating these factors.  

Seismic interpretation theory for an elastic earth

1954 ◽  
Vol 224 (1156) ◽  
pp. 43-63 ◽  
Keyword(s):  
Shear Stress ◽  
Seismic Interpretation ◽  
Elastic Theory ◽  
Static Case ◽  
Surface Shear Stress ◽  
Elastic Parameters ◽  
Surface Shear ◽  
Special Cases ◽  
Present Solution ◽  
Ray Path

The seismic interpretation problem for an isotropic spherical earth is analyzed on the basis of elastic theory, under the assumption that the three independent elastic parameters are unknown continuous functions of the depth. It is shown that solutions for these functions may be obtained in the form of Taylor’s series. The problem is treated for three types of symmetrical excitation conditions on the free surface: (1) a shear source of type p rϕ only; (2) a pressure distribution with vanishing surface shear stress; (3) an excitation consisting of pressure in combination with surface shear stress of type p rθ . In each case the excitation functions are arbitrary functions of time. It is assumed that the associated components of surface displacement over the sphere are known from available observations, as functions of time. Thus, the complete information contained in seismic records is used in the proposed interpretation process, without need of selecting, identifying and assigning arrival times to specific events on the records. The two static elastic parameters may theoretically be determined from observations at a single frequency, including the frequency zero, or static case. The determination of the dynamic elastic parameter requires the use of at least two frequencies. Algebraic checks are obtained by comparing the general solutions with the corresponding results for two special cases in which the elastic parameters vary in a prescribed manner in the interior of the sphere. In both these cases treatment by the classical ray-path method of interpretation is excluded, because the wave velocity decreases with depth. Furthermore, the ray-path method (which is essentially a method of geometrical optics) would fail to distinguish between the two examples in any case, since the velocity function is the same in both, although the elastic parameters differ. In contrast to the valuable ray-path method, the analytical procedures in the present solution of the elastic problem are prohibitively cumbersome. Practical application of elastic theory to the direct interpretation of seismograms requires further development of the theory with probable utilization of modern high-speed computing methods.

Encircling Graphs

2017 ◽  
Author(s):  
Arthur Benjamin ◽  
Gary Chartrand ◽  
Ping Zhang
Keyword(s):  
Nineteenth Century ◽  
Real Number ◽  
Traveling Salesman Problem ◽  
Complex Number ◽  
Traveling Salesman ◽  
Complex Numbers ◽  
Real Numbers ◽  
Hamiltonian Graphs ◽  
The World ◽  
The Traveling Salesman Problem

This chapter considers Hamiltonian graphs, a class of graphs named for nineteenth-century physicist and mathematician Sir William Rowan Hamilton. In 1835 Hamilton discovered that complex numbers could be represented as ordered pairs of real numbers. That is, a complex number a + b i (where a and b are real numbers) could be treated as the ordered pair (a, b). Here the number i has the property that i² = -1. Consequently, while the equation x² = -1 has no real number solutions, this equation has two solutions that are complex numbers, namely i and -i. The chapter first examines Hamilton's icosian calculus and Icosian Game, which has a version called Traveller's Dodecahedron or Voyage Round the World, before concluding with an analysis of the Knight's Tour Puzzle, the conditions that make a given graph Hamiltonian, and the Traveling Salesman Problem.

Optimum Thickness Variation in a Curved Strip under Pure Bending

1972 ◽  
Vol 23 (1) ◽  
pp. 77-86
Author(s):  
D R Blackeby ◽  
E H Mansfield
Keyword(s):  
Yield Criterion ◽  
Equivalent Stress ◽  
Thickness Variation ◽  
Curved Beam ◽  
Optimum Thickness ◽  
Pure Bending

SummaryThis paper determines the thickness variation in an annular strip (a curved beam) under pure bending to give a constant equivalent stress based on the Mises-Hencky yield criterion. The stresses due to an applied shear are also determined.

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