Double scaling and master curve to predict Kts for elliptically notched orthotropic plates from Kts in circularly notched isotropic plates

Accepted Manuscript Double scaling and master curve to predict Kts for elliptically notched orthotropic plates from Kts in circularly notched isotropic plates Nando Troyani, Milagros Sánchez PII:

S1359-8368(18)31931-0

DOI:

https://doi.org/10.1016/j.compositesb.2018.11.129

Reference:

JCOMB 6314

To appear in:

Composites Part B

Received Date: 20 June 2018 Revised Date:

15 November 2018

Accepted Date: 28 November 2018

Please cite this article as: Troyani N, Sánchez M, Double scaling and master curve to predict Kts for elliptically notched orthotropic plates from Kts in circularly notched isotropic plates, Composites Part B (2018), doi: https://doi.org/10.1016/j.compositesb.2018.11.129. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT Double scaling and Master Curve to Predict Kts for Elliptically Notched Orthotropic Plates from Kts in Circularly Notched Isotropic Plates Nando Troyani, 2Milagros Sánchez, 1 Departamento de Ingeniería Mecánica, Escuela de Ingeniería y Ciencias Aplicadas, Universidad de Oriente, Barcelona, Venezuela, [email protected], +584148068943, fax +582812697785 2 Escuela de Ingeniería Mecánica, Universidad Experimental Antonio José de Sucre, Puerto Ordaz, Venezuela, [email protected].

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Abstract. The key role played by the so-called stress concentration factors (symbolically usually referred to as Kts) while performing either analysis or design in both mechanical engineering and structural engineering is a well-proven fact; hence, the corresponding accuracy and ease of determination in their estimation have critical implications related to engineering matters. In a previous work, we showed that starting with the Kts for rectangular isotropic plates with circular holes the Kts for rectangular orthotropic plates with elliptic holes can be easily and accurately predicted, using a double scaling procedure (a geometric scaling and a material scaling) together with a basic Kt curve employed as a master curve. In the present work we investigate, in an engineering heuristic manner, whether the stated double scaling procedure exhibits a more general structure so that it can accurately be applied to different plate stress raiser geometries. Specifically, we examine if starting with the known Kts for rectangular isotropic plates with circular notches, used as a master curve, the Kts for orthotropic plates with elliptic notches can be easily and accurately predicted as well. Using a simple hand calculator, we show that the resulting proposed set of predicting mathematical expressions makes virtually unnecessary the use of complex numerical programs (FEM based for example) to determine the corresponding Kts, producing at the same time accurate results with a maximum average recorded error of 2.59 %. Additionally, given that a Kt predictive procedure for holed plates was presented in our previous work, and for notched plates in this work, we found necessary first to examine the hypothesis that suggests using Kts from holed plates as good approximations for Kts for notched ones and vice versa. We found this hypothesis not to be valid for either isotropic or orthotropic plates. In the course of this work it was found, rather significantly, that the well-known double square root material contribution to Kts is valid for both holed plates as well as notched ones.

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Keywords Anisotropy; stress concentration; notched plates; master curve; material scaling; geometric scaling; Analytical modeling; Finite element analysis

1. Introduction

1.1 General aspects

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As is well known, smooth (mathematically speaking) geometric discontinuities (such as shoulders, holes, and notches) in loaded mechanical or structural members will lead to the so-called stress concentrations. In order to produce a member that is capable of fulfilling the strength and reliability requirements in each case the analyst or designer needs to take into consideration various aspects directly related to the stated requirements; one such essential consideration is the so-called stress concentration factor usually represented as Kt.

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Because of their importance, holes and notches of different geometrical forms have been given significant attention over the years. Relevant results pertaining to stress concentration in isotropic semiinfinite plates and finite-width plates, both with circular notches, have been presented in the literature [1-5]. The finite-width plates Kts constitute the starting point of the procedure presented in this investigation. Kts, in fact, play a fundamental designing role in those cases where the prevalent operating conditions induce brittle behavior such as low operating temperatures and/or fatigue conditions in isotropic mechanical or structural member realms [6-7]. They also play a fundamental role when studying fatigue failure and damage in holed and notched orthotropic plates [8-22]. Kts (with two variants, Ktn and Ktg, referred as net section Kt and gross section Kt, respectively) for isotropic components are usually extracted from charts presented in extensive and rather comprehensive compilations [23-24] that are not available for anisotropic members. In the orthotropic context, Kts for notched members are presented in a variety of published works either in specific numerical form or chart forms [25-30].

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Given the relative complexity of the related mathematical problems, Kts, for the most part, are determined either employing numerical schemes (the FEM usually being the method of choice) or easy-to-apply predicting procedures [26-31] when available, which are preferred because of substantial time and cost reductions.

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In this investigation, we examine the validity of extending the Kt estimating procedure previously proposed by us in [31], that predicts Kts for rectangular orthotropic plates with elliptic holes starting with the values corresponding to isotropic plates with circular holes, to predict Kts for orthotropic plates with elliptic notches starting with the respective values for circular notches in isotropic plates. The stated applicability is demonstrated herein with a high degree of accuracy by the usage of the three main concepts presented and employed in the stated reference [31], namely, geometric scaling, material scaling, and a master curve. The geometry defining the parameters for the elliptically notched rectangular plate and corresponding applied boundary conditions regarding this work are indicated in Figure 1.

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y, Ey h

H m n L S a

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x, Ex

S

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b

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Figure 1. Geometrical parameters and boundary conditions utilized in the present work; the crosshatched region indicates the numerical domain for the FEM calculated reference values employed in this investigation.

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In Figure 1, the parameters L, H, a, b and S represent the half plate length, half plate width, elliptical notch major semi axis, elliptical notch minor semi axis, and uniform applied traction, respectively. The utilized boundary conditions for the FEM model were as follows: (i) load boundary conditions given by σxx(L,y)=S; (ii) the x and y directions displacement type boundary conditions, u(x,y) and v(x,y), given by: m boundary u(0,y)=0; n boundary v(x,0)=0, (iii) the remaining edge (h) is free. The boundary conditions for the n and h edges are exchanged in a specific section of this work wherein the exchange is required and clearly indicated. The basic definition to calculate Kts as given in [23,24] (1)

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=

=

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is used throughout this work, wherein σmax and σnom represent maximum stress and nominal stress, respectively. Herein the so-called gross section approach is used; accordingly, the nominal stress as used throughout will be the applied traction S at the ends of the plate, as shown in Figure1. From expression (1) and using standard notation for the gross section Kt, we will be using (2)

to calculate and represent the stress concentration factors exhibited through this report. As is well known, Ktgs are function of geometry only for the isotropic stress concentration cases, and a function of both geometric parameters and material parameters in anisotropic cases. For the latter case, symbolically this fact can be represented as ≡ ( , ) (3)

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The problem of a tension finite-width strip with opposite semi-circular notches in large isotropic plates has been treated and published in [1-5], among others, wherein Kt values in the form of a curve (Kts vs. a/H) is given corresponding to the case described in Figure 1 when a = b. However, in [4] it is shown that the value for Kt at a/H=0 published in five different papers are similar but somewhat different. As a result, in this work we chose to determine the circular notch finite-width isotropic plate Ktgs in the range a/H=0.05 to a/H= 0.6, using the FEM and employing the resulting curve as the master curve (MC) to be used in the predictive scheme proposed to estimate Ktgs for the following four cases.

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a) Ktgs for finite-width rectangular isotropic plates with opposite semi-elliptic notches for various a/b values from the Ktgs values corresponding to the MC curve for rectangular isotropic plates with opposite circular notches. b) Ktgs estimates for finite-width rectangular orthotropic plates with opposite semi-elliptic notches from the Ktgs values corresponding to isotropic plates with the same notch geometry. c) Ktgs estimates for finite-width rectangular orthotropic plates with opposite semi-elliptic notches of a given orthotropic material from existing Ktgs for the same geometry from a different orthotropic material. d) Ktgs estimates for finite-width rectangular orthotropic plates with opposite semi-elliptic notches of a given orthotropic material from existing Ktgs values corresponding to the MC for rectangular isotropic plates with opposite circular notches.

1.2 Analytical considerations

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As will be shown, the calculations can be performed with handheld calculators, and the resulting Ktgs estimates will be compared for accuracy purposes with FEM calculated Ktgs used as reference values.

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The general analytical material for this investigation has been presented in [32]. However, for completeness, we will briefly touch upon the salient features of the necessary background for this work. In this investigation, the problem being considered is restricted to two dimensions and hence valid for thin plate members. Under such conditions, it can be shown that the six original constitutive equations, in terms of the so-called engineering material constants under plane stress assumptions, reduce to +,

' ) − ), 0 1 ! & +, 0 2! ( & 0 0 2" $, "$= − ), ), & 0 3!" #!" &0 0 ( 0 . ,/ % (

(4)

where, εx, εy, and γxy represent strain components; σx, σy, and τxy represent stress components; and Ex, Ey, Gxy, υxy, and υyx represent the significant material properties of which only four are independent for our case.

ACCEPTED MANUSCRIPT It was shown in [32] that when an Airy-type stress function is incorporated in the equilibrium equations, the final distribution of stresses is defined by a characteristic equation. For sufficiently large plates subjected to in-plane loading, a fourth order characteristic algebraic equation results in conjugate roots that, for the case of orthotropic plates, take on the following form, 45 + 7. − 29!" : 4; − ) = 0. )

,

(5)

,

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Moreover, by further mathematical developments, an expression can be obtained that allows the computations of tangential stresses on the contour of a hole in an infinite plate. When this hole is elliptic in shape the expression to calculate the stress concentration factor is given by = 1 + ? − 29!" + 2@ , ) > . )

,

)

,

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=

(6)

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provided the applied tension load is aligned with the Ex material direction.

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Let us remark that any Ktgs predictive procedure must comply with the requirements implied in expression (3), that is, both geometric aspects and material properties aspects need to be satisfied in order for a Ktgs for notched orthotropic plate’s predictive procedure to be acceptable. These issues are examined in this work, and we show that both aspects of the predictive procedure are complied with significant precision given the accuracy of the resulting calculated Ktgs when compared with FEM calculated Ktgs. 2. Relationships based on geometric considerations and relationships based on material properties considerations

(A)BCCDE

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We begin by observing that the obvious difference between an isotropic rectangular plate with opposed elliptic notches and a rectangular isotropic plate with opposed circular notches is of a geometric nature. Also, the former´s plate geometry can be obtained from a vertical stretching (a > b) or vertical reduction (a < b) of the notch contour, in effect, a geometric scaling of the geometric circular notches in the plate of Figure 1. The amount of stretching (or shortening) is given by a constant C, DF GHFI

= J (A)FDKFLC=K GH

FI

(7)

The constant C is named the ellipticity ratio and is given by C=a/b (7b) Given that this is the only difference between the plates, we postulate accordingly that the Ktgs values for the elliptic notch case must reflect a proportional magnification (or proportional reduction) with respect to the Ktgs for circular notches in direct relation to the mentioned stretching or reduction; that is, a geometric scaling as was done in [31]. Accordingly, based on the relative values of the Ktgs at

ACCEPTED MANUSCRIPT a/H=0 for different a/b values, the expression that describes such a relation for finite-width rectangular notched plates is, BCCDE,GH FI,=/N

= 1+JO

FDKF,GH FI,=/N

− 1P,

(8)

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where, Ktg ellip,notch,a/H represents the elliptic notch Ktg values at given values of a/H; likewise, Ktg circ,notch,a/H represents the circular notch Ktgs values at the same a/H value, as given in the MC. In the results section of this work, the validity of equation (8) will be tested by the accuracy it exhibits in predicting Ktg values for diverse values of the ellipticity constant C using the C=1 curve as the reference MC. We underline that equation (8) is of a geometric nature since no material properties appear in it.

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Having established the geometric contribution to the Ktgs for the case of the isotropic notched plates, we now proceed to examine the material properties contribution to the Kts in orthotropic notched plates. It has already been mentioned that when an Airy-type stress function is incorporated into the relevant mathematical structure of the anisotropic plane problem [32], expression (5) above results; and that, by further analyses in the context of large plates, the expression for the stress concentration factor for a large orthotropic plate with an elliptical hole is obtained and given by expression (6). This expression exhibits a component made up of only the material properties of the plate. Therefore, given all the above considerations, we postulate that for the case of notched orthotropic plates the material properties provide the same form contribution to the final Ktgs values as they do in the case of the holed orthotropic plates; that is, the double square root part of equation (6) is valid for holed plates as well as for notched ones. The validity of this postulate is significantly and extensively tested in the results section of this work.

)QK

I;,GH F ,=/N

= 1 + R( )STUV(,GH

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With these considerations in mind, the specific form taken by the stated postulate, for the relation that holds for the Ktgs between two notched plates with the same geometry but different orthotropic materials, is also the same relation that holds between the Ktgs for two holed plates with the same notch geometric parameters but different orthotropic materials. Accordingly, as per [31], the stated relation is, F ,=/N

− 1Wμ(Orth2/Orth1)

(9)

wherein µ (a material scaling constant) is defined as the orthotropic-orthotropic property material ratio between material Orth2 and material Orth1 and is given by

μ (]

2/] ℎ1)

= _?

)

. ,

− 29!" + 2@ ` ) )

,

STUV;

/ _?

)

. ,

− 29!" + 2@ ` ) )

,

STUV(

(10)

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μ (]

/a )

= _?

)

. ,

− 29!" + 2@ ` )

),

STUV

/2

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Furthermore, recognizing that when the following limits are taken: Ey→Ex=E, νxy→ νyx= ν, and Gxy→E/2(1+ν) in the denominator (orthotropic material Orth1→isotropic); the usual value of 2 for the outer square root is recovered and equation (10) becomes

(11)

in which case µ represents a limiting form of (10) as an orthotropic-isotropic material ratio. Correspondingly equation (9) becomes W

QK I,GH F ,=/N

W

= 1 + OR

bcd,GH F ,=/N

− 1P μ(Orth/Iso)

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(12)

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Accordingly, equation (12) would represent a relation between the Ktgs of an isotropic notched plate and the corresponding Ktgs for an orthotropic notched plate, both with the same elliptic notch geometry. Further, by substituting equation (8) in equation (12), equation (13) below is obtained R

W

BCCDE,GH F ,QK I,=/N

= 1 + OJ(R

W

hiTh,jdUhV,bcd,=/N

− 1)P μ(Orth/Iso)

(13)

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Equation (13) represents the expression that will be used in this work to obtain, in a predictive fashion, Ktgs for elliptically notched orthotropic plates from the Ktgs for circularly notched isotropic plates. Let us underline that in equation (13) there are two different scalings implied,

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a) A geometric scaling from a Ktg for an isotropic plate with a circular notch to the corresponding Ktg for an isotropic plate with an elliptical notch. b) A material scaling from a Ktg for an isotropic plate with an elliptical notch to a corresponding Ktg for an orthotropic plate with an elliptical notch. The usage and accuracy of all the predictive equations presented in this section will be exhibited in the “Isotropic Ktgs and Orthotropic Ktgs Predictive Results for Notched plates” section of this work. Throughout this investigation, the results exhibiting the Ktg estimates are applicable in the so-called long length member regime a concept concomitantly related to two additional concepts namely, transition length and short member length regime as [33-34]. In this work, four different orthotropic materials are used to calculate the stress concentration factors that are presented as exhibits of the schemes proposed herein. Their corresponding properties are given in table 1 below,

ACCEPTED MANUSCRIPT Table 1. Materials used in this work and their respective properties

90.9E3

Ey [MPa] 6.0E4 1.2E5 132.0E3

υxy 0.071 0.142 0.22

Gxy [MPa] 7.03E3 7.03E3 44.8E3

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Material Orth 0˚ Material Orth 90˚ Material (M3) [0]n B/Al unidirectional laminate Material (M4) homogenized [04/±45]s Graphite/Epoxy laminate

Ex [MPa] 1.2E5 6.0E4 210.0E3

18.9E3

0.58

14.6E3

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Note: materials Orth 0˚ and Orth 90˚ are in fact the same material, the first is rotated 0˚ with respect to the tension load direction, and the second is rotated 90˚ with respect to the tensional load direction [32].

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3. A numerical and conceptual comparison between Ktgs for holed plates and Ktgs for notched plates, both in the isotropic case and in the orthotropic case Given that it has been suggested in both engineering classrooms and in published works [24,26] that there is a reasonable degree of equivalency or relation between the Ktgs values corresponding to plates with holes and the Ktgs values corresponding to notched plates (both with the same type of geometric stress raiser discontinuity parameters), and given the nature of this work together with [31], we found essential to examine the validity of using such approximating procedures; that is, to use one geometry (hole) Ktg value as a valid approximation for the other geometry (notch) Ktg value or vice versa.

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By examining Figure 1, and more specifically the crosshatched area, it becomes clear that from a conceptual point of view the two cases are different in a fundamental sense; that is, the boundary conditions are different for the hole case as compared to those for the notch case. In the former, boundary h is a symmetry boundary, and boundary n is a free boundary; in the latter, the opposite is true, while the other two boundaries keep their mathematical nature in both cases. These observations have two significant consequences:

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1. Under the indicated applied traction, the displacement fields and, consequently the corresponding strain fields and stress fields are different for the two cases being considered. 2. As a result, the numerical values of the corresponding Ktgs may be significantly different as it will be shown herein. Figure 2 exhibits the Ktgs FEM performed calculations for plates made of isotropic materials for the three indicated values of the ellipticity ratio a/b, wherein a comparison is made at each value of a/b of the Ktgs corresponding to the two geometries being considered: notches vs. holes.

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14 12

1. Ktg semi elliptic notch Iso, a/b=0.4

10

2. Ktg elliptic hole Iso, a/b=0.4

8 6

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3. Ktg semi elliptic notch Iso, a/b=2.0

4. Ktg elliptic hole Iso, a/b=2.0

4 2

5. Ktg semi elliptic notch Iso, a/b=4.0

0

6. Ktg elliptic hole Iso, a/b=4.0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 a/H

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Ktg

Figure 2. Comparisons of Ktgs in terms of notch vs. hole geometries for the isotropic plates case. Table 2 below shows both the maximum and the average difference (%) from Figure 2 at the markers positions between each pair of the two geometries being considered in terms of the TSCFs in the gross section version. In all the reported cases the values are based on the absolute value of the difference so as to exclude potentially misleading sign difference cancelations.

Maximum difference (%) 59.33 29.13 17.77

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1-2 3-4 5-6

Average difference (%) based on notch values 28.66 9.82 6.91

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Table 2. Average and maximum difference between notch Ktgs and hole Ktgs in elliptical geometry stress raisers based considerations for isotropic materials

As can be verified from any of the major references [23-24], the case for circular notch geometry (for clarity the curves for a/b=1.0 were omitted from Figure 2) exhibits the same trends as indicated in Figure 2 between hole Ktgs and notches Ktgs. Only when a/H is very small is the approximation Ktg-hole Ktg-notch valid, as is the case in Figure 2 for a/b=0.4. Furthermore, note that even the region where the approximation may be considered valid changes as the ellipticity ratio changes. The validity of the notch/hole Ktgs analogy has also been examined for orthotropic materials. Figure 3 shows the FEM performed calculations for the two cases being considered, notch vs. hole, for plates of material M4 (homogenized properties of a Graphite/Epoxy laminate), for two different values of the ellipticity constant C, a/b= 2.0 and a/b=4.0.

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14.0

1. Ktg semi elliptic notch (a/b=2), M4 2. Ktg elliptic hole (a/b=2), M4 3. Ktg semi elliptic notch, (a/b=4), M4 4. Ktg elliptic hole (a/b=4), M4

12.0 10.0 8.0 6.0 4.0 2.0 0.0 0.1

0.2

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0.4

0.5

0.6 0.7 a/H

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Ktg

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Figure 3. Comparison of Ktgs for the two cases considered in this section for an orthotropic material. Table 3 shows both the maximum and average difference (%) at the markers positions between the two geometries being considered in terms of the TSCFs in the gross section version Ktg. In all the cases reported the values are based on the absolute value of the difference so as to exclude, as was done previously, misleading opposing sign cancelations. Table 3. Average and maximum difference between notch Ktgs and hole Ktgs in elliptical geometry stress raisers for orthotropic materials

1-2 3-4

Average difference (%) based on notch values 7.94 6.91

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Maximum difference (%) 21.37 17.77

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In our view, approximating Ktgs for one of the considered geometries (notches) based on the other geometry being considered (holes), for the same ellipticity ratio a/b, including a/b=1, may result in rather significant errors that can hardly be considered acceptable. In the concluding section, a general reference is made to the magnitudes of these differences (for plates made of isotropic materials) in regards to fatigue life considerations. 4. Isotropic Ktgs and Orthotropic Ktgs Predictive Results for Notched plates Whenever numerical Ktg reference values were needed in this work and presented in the charts that follow, they were obtained using a commercial FEM package (ANSYS®) used in both industry and academia; selecting for the calculations the quadratic quadrilateral plane element corresponding to the assumption of plane stress prevailing conditions, applicable to thin plates.

ACCEPTED MANUSCRIPT As a measure of errors or deviations in the comparisons presented in this work we use the same measuring expressions developed and utilized in [31]. Consequently, the reported errors e are calculated averages of percent error values using the following expressions = 100 ∗ R∑GDs( m (

no,D

−

Bp Dq= Br,D )/

u)n,D

−

Bp Dq= Br,D )/

no,D W/t

(14)

= 100 ∗ R∑GDs( m (

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or, u)n,D W/t

(15)

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n: number of calculations for each curve.

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In expressions (14-15) the absolute values of the differences were used to avoid alternating signs in the summations, possibly producing misleading lower non-representative errors of the calculations performed in this work. 5. Ktgs for isotropic plates with elliptic notches for various a/b values from the Ktg values from the MC curve, geometric scaling only

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Figure 4 exhibits the estimated Ktgs for opposed elliptic notches in isotropic finite-width plates, for three values of a/b (2,4,6) as predicted by using equation (8) based on the MC (curve 1), together with their corresponding FEM estimates used for comparison.

1. Ktg semicircular notch, iso, FEM 2. a/b = 2.0, iso, equation estimate 3. a/b = 2.0, iso, FEM

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18 Ktg 16 14 12 10 8 6 4 2 0

0

0.1

0.2

4. a/b = 4.0, iso, equation estimate 5. a/b = 4.0, iso, FEM 6. a/b = 6.0, iso, equation estimate 7. a/b = 6.0, iso, FEM

0.3

0.4

0.5

0.6 0.7 a/H

Figure 4. Comparison of Ktgs estimated from MC (curve 1) and FEM calculated ones for three different a/b values.

ACCEPTED MANUSCRIPT The average errors (percent) corresponding to the results presented in Figure 4, at the a/H calculated points, using the FEM calculated values Ktgs as the reference values are as follows: 1.56 % between curves 2 and 3, 2.57 % between curves 4 and 5 and 2.59 % between curves 6 and 7.

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6. Ktg estimates for orthotropic plates with opposed elliptic notches from the Ktg values corresponding to isotropic plates with the same geometry, material scaling only Figure 5 shows the estimated Ktgs for notched plates (a/b=2) made of two different orthotropic materials, M3 and M4. They were obtained using equation (12) starting with the Ktgs corresponding to C=2 notched isotropic curve as the MC. This last curve was first obtained using equation (8).

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For a reference comparison for the orthotropic results, the chart includes the approximated Ktgs values obtained using the FEM method.

9.0 8.0 Ktg 7.0 6.0 5.0 4.0 3.0 2.0 1.0 0.0

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1. Ktg, iso, a/b=2, FEM 2. Ktg, ellip notch, M3, a/b=2, from Ktg for a/b=2, iso 3. Ktg FEM, M3, a/b=2

0.0

0.1

0.2

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4. Ktg, ellip notch, M4, a/b=2, from Ktg for a/b=2 iso 5. Ktg FEM, M4, a/b=2

0.3

0.4

0.5

0.6 0.7 a/H

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Figure 5. Using equation (12) to estimate Ktgs for notched orthotropic plates from notched isotropic plates MC (curve 1) with the same notch geometry.

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Table 4 presents the appropriate µ value utilized in equation (12) for each case and the corresponding average errors in the estimates of the Ktgs exhibited in Figure 5. Table 4. Average of the absolute errors calculated at each a/H point, using the FEM Ktgs calculations as the reference. case

Average error (%) between estimated Ktgs and FEM calculated Ktgs a/b = 2, M3, estimated from 1.06 (curves 2 and 3) a/b = 2, Iso a/b = 4, M4, estimated from 0.99 (curves 4 and 5) a/b = 2, Iso

µ 1.3 1.55

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The example treated in this subsection shows that isotropic Ktgs curves, as well as the one corresponding to C=1, can be used as MCs to accurately estimate Ktgs for plates with other C values.

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7. Ktg estimates for plates with opposite semi-elliptic notches of a given orthotropic material from existing Ktgs for the same geometry from a different orthotropic material, material scaling only

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Figure 6 shows the estimated Ktgs for elliptically notched plates (a/b=6) made of two different orthotropic materials, Orth 0˚and Orth 90˚. First, the Orth 0˚ and Orth 90˚ corresponding Ktgs were determined using the FEM method, and afterward, the Ktgs were each estimated from the other one using equation (9).

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40 Ktg 35 30 25 20 15 10 5 0

1. Ktg ellip notch, a/b=6, Orth 0˚, FEM 2. Ktg, ellip notch, a/b=6, Orth 0˚, estimated from Orth 90˚

0

0.1

0.2

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3. Ktg ellip notch, a/b=6, Orth 90˚, FEM

0.3

0.4

0.5

0.6 0.7 a/H

4. Ktg, ellip notch, Orth 90˚, a/b=6, estimated from Orth 0˚

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Figure 6. Usage of equation (9) to estimate Ktgs for notched orthotropic plates from notched plates of a different orthotropic material with the same notch geometry.

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Table 5 shows the appropriate µ value (mutually inverse values) in each case to be utilized in equation (9) and the corresponding average errors in the Ktgs estimates exhibited in Figure 6. Table 5. Average of the absolute errors calculated at each a/H point, using the FEM Ktgs calculations as the reference. case a/b = 6, Orth 0˚, estimated from Orth 90˚ a/b = 6, Orth 90˚, estimated from Orth 0˚

Average error (%) between estimated Ktgs and FEM calculated Ktgs 1.38 (curves 1 and 2) 1.43 (curves 3 and 4)

µ 1.43 0.7

ACCEPTED MANUSCRIPT The rather low average errors reported in Table 5 indicate the significant achievable accuracy with which the proposed predictive equations produce the calculated Ktgs reported in this work for orthotropic plates of the type considered herein. Consequently, it is concluded that the postulate made in the introductory section, regarding the material properties contribution in the square root expression form in equation (6), to the Ktgs in the case of notched plates is validated in an engineering sense.

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In our view, the above statement is especially confirmed in this subsection since the two different plates considered have exactly the same geometry (hence the same geometric contribution to the respective Ktgs at each a/H value), and that they differ only in the material mechanical properties contribution leading to the stated two µ values indicated in Table 5. These µ values were calculated from equation (10) and used in equation (9), resulting in the Ktgs accuracy exhibited in the reciprocal estimates shown in Figure 6.

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Let us emphasize that we find the validity of the postulate that the well known double square root (equation 6) provides the same valid material contributions to the Kts for both notched plates and holed ones rather significant. It does remain to be verifyed if it is also valid for other smooth geometrical discontinuities in rectangular orthotropic plates.

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8. Simultaneous use of geometric scaling and material scaling to obtain estimates for Ktgs for orthotropic plates with semi-elliptic notches from the semi-circular notches Ktgs in the isotropic MC curve

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In order to illustrate both the material scaling and the geometric scaling structure embedded in equation (13), and its full capabilities, the results in this subsection are presented wherein two different ellipticity ratio values were used (4 and 6), and two different materials (Orth 0˚ and Orth 90˚) were used as well. Figure 7 shows the estimated Ktgs for elliptically notched plates made of two different orthotropic materials, Orth 0˚ and Orth 90˚, both for an ellipticity ratio of a/b=4 (C=4).

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25.0 Ktg 22.5 20.0 17.5 15.0 12.5 10.0 7.5 5.0 2.5 0.0

1. FEM Ktg semicircular notch, iso, FEM 2. Ktg, ellip notch, Orth 0˚, a/b=4, from Ktg for semicircular notch Iso

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3. Ktg ellip notch, Orth 0˚, a/b=4, FEM 4. Ktg, ellip notch, Orth 90˚, a/b=4, from Ktg for semicircular notch Iso 5. Ktg ellip notch, Orth 90˚, a/b=4FEM

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0.6 0.7 a/H

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Figure 7. Usage of equation (13) to estimate Ktgs for notched orthotropic plates (a/b=4) for two orthotropic materials from the MC (curve 1). Table 6 presents the appropriate µ value in each case as was used in equation (13) and the corresponding average errors in the Ktgs estimates exhibited in Figure 7.

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Table 6. Average of the absolute errors calculated at each a/H point, using the FEM Ktgs calculations as the reference.

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a/b = 4, Orth 0˚, estimated from MC a/b = 4, Orth 90˚, estimated from MC

Average error (%) between estimated Ktgs and FEM calculated Ktgs 1.80 (curves 2 and 3) 2.27 (curves 4 and 5)

µ 2.23 1.56

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Figure 8 shows the estimated Ktgs for elliptically notched plates made of the same two materials used for the results shown in Figure 7 (Orth 0˚ and Orth 90˚), however in order to explore the generality of equation (13), also used in this instance, the ellipticity ratio has been changed to a/b=6 (C=6), keeping as the starting Ktg values those from the isotropic MC.

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1. FEM Ktg semicircular notch, iso, FEM 2. Ktg, ellip notch, Orth 0˚, a/b=6, from Ktg for semicircular notch Iso 3. Ktg ellip notch, Orth 0˚, a/b=6, FEM

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Ktg

40 35 30 25 20 15 10 5 0

4. Ktg, ellip notch, Orth 90˚, a/b=6, from Ktg for semicircular notch Iso

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5. Ktg ellip notch, Orth 90˚, a/b=6, FEM

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Figure 8. Usage of equation (13) to estimate Ktgs for notched orthotropic plates (a/b=6) for two orthotropic materials from the MC (curve 1). Table 7 presents the appropriate µ value in each case as was used in equation (13) and the corresponding average errors in the Ktgs estimates exhibited in Figure 8. Table 7. Average of the absolute errors calculated at each a/H point, using the FEM Ktgs calculations as the reference. Average error (%) between estimated Ktgs and FEM calculated Ktgs 1.39 (curves 2 and 3)

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1.79 (curves 4 and 5)

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a/b = 6, Orth 0˚, estimated from MC a/b = 6, Orth 90˚, estimated from MC

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Comparison with published works

In this section we compare diverse Kts published representative procedures with the presently proposed predicting procedure and compare the corresponding generality and accuracy proposed herein, underlining at the same time the importance that any predictive procedure must comply properly and accurately with the geometric contribution and the material contribution to stress concentration to produce accurate Kt estimates. In order to be valid, procedures to be acceptable must first and foremost comply with accepted standards of accuracy and only then exhibit ease of calculations. We find important to reiterate that although we are unaware of known existing statements regarding the effect of inaccuracies in the determination of TSCFs on errors in fatigue life estimates (or designs against fatigue) for orthotropic loaded members, it is conjectured that there might be a possibly large sensitivity dependency to this inaccuracy issue that should be investigated for this class of materials. This issue has been investigated for Kts in isotropic contexts [35] where the following statement is

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made: “An error in these tables of, for example, 10 %, could lead to a factor of 5 or greater in the estimated fatigue life”. An observation related by the strongly non-linear relationship between number of load cycles and fatigue strength in S-n curves. Consequently, 10 % errors in Kts prediction estimates are objectively considered excessive in isotropic applications. As a result, given the growing usage and the complexities associated with designs and analyses of mechanical and structural components made of anisotropic materials, a lower upper bound of error acceptability must be used. Pending a formal study leading to a setting of such bound we conjecture that it should not be larger than 5 %. The recorded maximum error during the calculations reported in this work was 2.59 %, significantly lower than the stated 10 % in [35], in fact, approximately four times lower. Moreover, significantly lower than some double-digit errors reported in [26] for orthotropic applications.

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1. In reference [27] Kts for elliptically notched plates are obtained using the FEM and reported in the form of tables and fitted curves for diverse materials. No reference values were reported and hence no errors are reported either. The results presented are limited in that they only apply to the specific materials utilized in the numerical calculations. 2. In reference [37] An empirical predicting method to determine the stress concentrations factors for isotropic and orthotropic plates and cylinders with a circular hole was proposed and the results agreed to a certain extent with the FEM simulations. These results were in turn used as reference values in reference [26]. This case illustrates the perils of ignoring the overall geometry of the member in the determination of Kts as will be shown below. 3. In reference [26] a formula is proposed to predict stress concentration factors for orthotropic material components in a single step procedure for an unusual variety of geometries even including a 3-D geometry in one case. The formula being,

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v = 1 + R w − 1W 2

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Where, Ktg stands for orthotropic SCF estimate, w stands for isotropic SCF and ς for half the double square root in equation (6). Unfortunately, all the results are shown in tabular form as opposed to chart form, a form which would have otherwise visually shown the accuracy or lack thereof of the proposed scheme. We will now briefly, examine just a few of the cases presented in [26] and proceed to explain why, in our view, they result in such high percentage errors in the predicted Kts values; many in excess of 10% and some even larger than 20 %. It will also help explain the reason why we chose not to attempt to employ our predictive procedure in all the cases treated in [26]. 3.1 Table 1 [26] represents the isotropic starting Ktgs reference for the above equation and Table 2 shows the results accordingly generated. Table 1 contains 35 numbers. Our procedure (Eq. 13) would require only 7 in that structure or, alternatively, just 3 or four coefficients for an approximating polynomial for the MC, obtainable from EXCELL® and easily programmable in a handheld calculator. In fact, a single MC would suffice for all practical possibilities not for just five a/b values as in [26] as a result of the proposed geometric scaling which is given by the constant C in Eq. 13.

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3.2 In addition, as indicated in sections 5 and 7 above, our proposed scheme is more general than the one proposed in [26], for the specific geometry treated in that ours include possibilities such as: 3.2.1 Determination of Ktgs for notched isotropic plates, for all practical values of the ellipticity ratio a/b, from the MC. In fact, it would be a cursory matter to verify that any of the curves in Figure 4 could be used as a MC to obtain all the other curves in the chart with notable accuracy. 3.2.2 Determination of Ktgs for notched orthotropic plates, from a curve containing the Ktgs for notched plates of a different orthotropic material, as accurately exhibited in the results of figure 6. 3.2.3 In Table 4 [26] the results for Kts prediction for rectangular plates with centric circular holes are presented. 7 out of 20 (35%) reported Kt values exceed the already large 10 % set by the coauthors. Using the above proposed 5% threshold would show that 14 out of 20 values exceed the reference FEM calculated values, with a maximum error of 33.01 %. 3.2.4 Table 5 [26] presents the results for Kts prediction for rectangular plates with eccentric circular holes. In this case 11 out of 15 estimates exceed the 10 % error threshold, that is, 73 % of the cases. Applying the more conservative and appropriate threshold of 5 % none of the estimates in this table meet this accuracy criterion. As the plates width/hole diameter ratio becomes smaller (gradually deviating from the large plate regime, implicitly embedded in the above equation) the Kts estimates significantly deteriorate. This example represents a case where geometry is not accounted for properly by the utilized estimating scheme. 3.2.5 Table 6 [26] presents the results for Kts prediction for rectangular plates with centric circular holes subjected to biaxial loading. This case corresponds to an uncharted territory. It is rather difficult, to say the least, to account with a single simple formula as per above to deal with all the following issues: 3.2.5.1 Short length regime in one direction only. Definitions and usage for the concepts of short member regime, long member regime and transition length, Lt, can be found in [33-34]. These concepts strongly depend on the orthotropy direction, as shown in the figure below reproduced from [38] for the following orthotropic material: Ex = 1.176x104 MPa, Ey = 0. 588 x104 MPa, Gxy = 0.0686 x104 MPa, 9xy = 0.071.

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Figure 9. Transition length for each of the two cases analyzed herein.

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3.2.5.2 Short length regime in both directions. 3.2.5.3 Same sign biaxiality loading with both equal magnitudes and different ones. 3.2.5.4 Opposite sign biaxiality loading with both equal magnitudes and different ones. 3.2.5.5 Additionally, it must be remembered that there is a directional strength dependency in orthotropy. 3.2.5.6 As an explicit example of these perils, the actual FEM Kt approximation for the following parameter values in [26] d/W = 0.75 and L/W = 2.5 is Kt = 6.67, reference [26] formula estimates a Kt of 7.6, an error of 13.9 %. To complicate matters even further, the maximum stress in this case does not take place on the hole contour; rather it takes place on the longitudinal edges of the plate closest to the hole. 3.3 Table 7 [26] shows the results of applying the same formula to orthotropic cylinders with circular holes. The two cases analyzed exhibit significant accuracy errors when viewed individually and by comparison. For the same geometry, material C1 produces two (out of 3) results above the 10 % (all positive) error threshold. Material C2, on the other hand, produces one small positive error and two larger negative ones. It is unavoidable to conclude that given the same geometry for both cases there is a notable problem associated with the way the proposed formula misses to properly account for the material contribution to the Kts. How is the analyst or the designer to know ahead of time when this shortcoming may be significant? By the way, as is well known, the material properties in isotropic cylinders and plates enter the solution to the Kt determination problems in different ways.

It must be said that today’s standards call for far more accurate thresholds to go with the rather accurate computational tools available to the designer and the analyst. Formulas to simplify Kts estimates are only competitive with those tools if they produce comparable or better accuracy estimates for Kts.

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In this work, we investigated the possibility to predict the stress concentration factor values for finitewidth rectangular plates made of both isotropic and orthotropic materials, both with opposing elliptical notches. The stated goal was accurately achieved using a double scaling procedure (both a geometric scaling and a material properties scaling) together with the corresponding values for circularly notched isotropic plates. As was done in a previous publication, these preliminarily calculated isotropic Ktgs values were considered a master curve and constituted the basis on which the mentioned predictions rest. These predictions rest as well on appropriately developed and presented equations that take into consideration both geometric aspects and material properties aspects of the notched finite-width plates in order to estimate the Ktgs values exhibited in this work. The accuracy exhibited by the proposed predictive procedure for a geometrically different type of orthotropic plates suggests that there is a significant degree of generality in the double scaling (geometric and material) procedure together with a master curve scheme to estimate Ktgs for orthotropic plates from Ktgs from isotropic plates. Generally speaking, the accuracy exhibited by the proposed estimating procedure for notched plates is very similar to the case when it is applied to the holed plates [31]. In the former, the maximum average error is reported to be 2.62 % whereas in the present work the maximum average error is 2.59 %. Each maximum value corresponds to specific cases in each work. Although only integer values of the ellipticity constant C were used in this work, non-integer values could be used as well with the same general reported degrees of accuracy shown herein in the Ktgs predictive subsections. We find important to reiterate that although we are unaware of known existing statements regarding the effect of inaccuracies in the determination of TSCFs on errors in fatigue life estimates (or designs against fatigue) for orthotropic loaded members, it is conjectured that there might be a possibly large sensitivity dependency to this inaccuracy issue that should be investigated for this class of materials. This issue has been investigated for Kts in isotropic contexts [35] where the following statement is made: “An error in these tables of, for example, 10 %, could lead to a factor of 5 or greater in the estimated fatigue life”. An observation related by the strongly non-linear relationship between number of load cycles and fatigue strength in S-n curves. The maximum error recorded during the calculations reported in this work was 2.59 %, significantly lower than the stated 10 % in [35], in fact, approximately four times lower. Moreover, significantly lower than some double-digit errors reported in [26] for orthotropic applications. Regarding accuracy, let us underline that the reported errors throughout this work, were determined with respect to FEM calculated values as references which by its very nature carry the method´s unavoidable numerical inaccuracies. Rather significantly, it was found in the course of this work that the well-known double square root material properties contribution to Kts is valid for both holed plates as well as notched ones. It does remain to be verifyed if it stays valid for other smooth geometrical discontinuities (other than circular and elliptic geometries) in rectangular orthotropic plates, and possibly and more generally in orthotropic plane problems of the nature treated herein. Finally, one salient feature of the proposed predicting scheme is that it can be used to reliable estimate Ktg values with significant degrees of accuracy for notched orthotropic plates using a simple handheld calculator, making it unnecessary to use complex and time-consuming computer numerical methods to perform such estimates.

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Funding: This work was supported by Consejo de Investigación, Universidad de Oriente, Venezuela. References

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