Rb-Ao4^3^ RIA-77-U1052 .JJ ADrt OVSLX'L.

COMPUTER PROGRAM FOR ANALYSIS

OF STRESS-STRAIN DATA, REVISED

TECHNICAL - LIBRARY

RALPH PAPIRNO ENGINEERING MECHANICS DIVISION

April 1977

Approved for public release; distribution unlimited.

ARMY MATERIALS AND MECHANICS RESEARCH CENTER Watertown, Massachusetts 02172

The findings in this report are not to be construed as an official

Department of the Army position, unless so designated by other authorized documents.

Mention of any trade names or manufacturers in this report

shall not be construed as advertising nor as an official

indorsement or approval of such products or companies by the United States Government.

DISPOSITION INSTRUCTIONS

Destroy this report when it is no longer needed. Do not return it to the originator.

UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (Whtn Dal* Enured)

REPORT DOCUMENTATION PAGE 1. REPORT NUMBER

AMMRC TR 77-11

2. GOVT ACCESSION NO

4. TITLE (mtd Submit)

COMPUTER PROGRAM FOR ANALYSIS OF STRESS-STRAIN DATA, REVISED

READ INSTRUCTIONS BEFORE COMPLETING FORM

3. RECIPIENT'S CATALOG NUMBER

S. TYPE OF REPORT A PERIOD COVERED

Final Report S. PERFORMING ORG. REPORT NUMBER

7. AUTHORf..-

Ralph Papimo

S. CONTRACT OR GRANT NUMBERCO

*. PERFORMING ORGANIZATION NAME AND ADDRESS

Army Materials and Mechanics Research Center Watertovm, Massachusetts 02172 DRXMR-TE

10. PROGRAM ELEMENT. PROJECT. TASK AREA • WORK UNIT NUMBERS

D/A Project: 1T162105AH84 &MCMS Code: 612105,H840011 ^gencv Accession: DA OC4687

II. CONTROLLING OFFICE NAME AND ADDRESS

U. S. Army Materiel Development and Readiness Command, Alexandria, Virginia 22333

12. REPORT DATE

April 1977 13. NUMBER OF PAGES

22 U. MONITORING AGENCY NAME » AODRESSf// dillermnl from Conlrolllng OKict) IS. SECURITY CLASS, (ol Ihlt rmpotl)

Unclassified IS«. DECLASSIFI CATION/DOWN GRADING

SCHEDULE

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Approved for public release; distribution unlimited,

17. DISTRIBUTION STATEMENT (ol Iht mbalrmcl mnfred In Block 20. II dllltrtnl from Rtporl)

II. SUPPLEMENTARY NOTES

19. KEY WORDS CConfinu* on reverse efde il necmssmry end Idenllly by block number)

Curve-fitting Stress-strain diagrams Computer analysis

20. ABSTRACT (Continue on reverie elde II neceteery end Idenlily by block number)

(SEE REVERSE SIDE)

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Block No. 20

ABSTRACT

A computer program, previously developed for fitting analytic curves to experimental stress-strain data and described in AMMRC TR 76-12 and its Supplement, has been revised. The program determines the three param- eters required for the Ramberg-Osgood equation and two additional constants for a power law approximation for data beyond the Ramberg-Osgood secant yield stress. The revisions include a routine to arrange the input data in increasing order of stress-strain magnitude; a new method for tangent modulus calculation; a data extrapolation routine for the case where the last input data point is a small increment below the secant yield stress; additional statistical calculations to indicate goodness-of-fit of the data; and output values expressed in S-I metric units as well as customary English units. A number of nonessential analyses and their associated outputs have been eliminated. User instructions, samples of the output,

and a program listing are included.

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CONTENTS

Page

INTRODUCTION 1

DATA ORDERING 1

DATA EXTRAPOLATION FOR RAMBERG-OSGOOD ANALYSES 2

Data Test 2

Extrapolation Procedure 2

TANGENT MODULUS 3

MISCELLANEOUS REVISIONS

Metric Units 6

Header Card 6

Optional Table 6

Additional Parameters 6

APPENDIX A. SAMPLES OF OUTPUT 7

APPENDIX B. PROGRAM LISTING 8

INTRODUCTION

This supplement describes a revised version of the computer program, now designated as ARPY 5, which was developed to analyze stress-strain data.* The revisions to AMMRC TR 76-12 are as follows.

1. Inclusion of a subroutine to arrange the input data in increasing order of stress-strain magnitude. (In the original version the program was aborted if the input data were not ordered.)

2. Inclusion of a data extrapolation routine for the case where the stress- strain values of the last data point are just below those of the S^ yield values. (In the original version if the last data point was not equal to or larger than Si, the Ramberg-Osgood secant yield stress, the calculations were aborted.)

3. Inclusion of a new method for tangent modulus calculations.

4. Inclusion of computations for the correlation coefficient.

5. Inclusion in the output of the stress-strain parameters expressed in S-I metric units.

6. Elimination of all plastic strain component analyses.

7. Miscellaneous revisions: Elimination of optional Table 2 (tangent modu- lus) and Table 3 (plastic strain parameters). Inclusion of S-I metric unit values in optional Table I (comparison of input data with computed stress values). Compu- tations of the elastic-plastic values of Poisson's ratio have also been eliminated.

Appropriate changes in the output formats have been made to accommodate the various revisions. There have also been some small changes made in notation, in- ternal computations, input, or output. A few revisions have been made in the input data requirements but these are limited to data required on the header card.

Descriptions of the revisions are given in this report. Samples of the printed output and listings of the revised main program and the subroutines are given in appendixes.

DATA ORDERING

The routines for obtaining the Ramberg-Osgood Sj and m parameters require that the data be in increasing order of stress-strain magnitude. The program contains a test for data order. If one or more data cards are not in the required order, a subroutine, called ORDER, orders the data. The subroutine contains a provision for aborting the ordering process if the number of iterations exceeds the number of data points. When this occurs the calculations for the given data set are aborted and a diagnostic message appears in the printout.

*PAPIRNO, R. Computer Analysis of Stress-Strain Data: Program Description and User Instructions. Army Materials and Mechanics Research Center. AMMRC TR 76-12, April 1976.

DATA EXTRAPOLATION FOR RAMBERG-OSGOOD ANALYSES

In the original version of the program, the stress value of the last input data point S(N) must equal or exceed the secant yield strength Si, or the Ramberg- Osgood and subsequent analyses are aborted. Let a test stress S(T) be defined by

S(T) = 0.7E/E(N) CD

where E(N) is the strain value associated with S(N). Now let a test increment be defined by

AS = S(N) - S(T).

Data Test

(2)

If AS is negative or zero, the last data point will have a stress value equal to or greater than S^ If AS is positive, the case shown in Figure 1, then S(N) will always be less than Sj. In the revised program an additional test is done if the latter situation occurs. The ratio S(T)/S(N} is calculated. If the value of the ratio is 0.95 or greater, an additional data point is extrapolated; if the ratio value is less than 0.95, the R-0 calculations are aborted. The value 0.95 was chosen arbitrarily to be the criterion of a last data point which was close to the Sj stress.

Extrapolation Procedure

S(N)

E(N) STRAIN

Figure 1. Identification of stress and strain values used in the program for a test with the extrapolation criterion.

A curve of the strain-hardening type,

S = AeP, (3)

is fitted to the last four data points of the set using a logarithmic least-squares procedure. The stress-strain coordinates of the intersection of this curve and the secant modulus line, 0.7E, are now cal- culated. This point is designated S^, e^ in Figure 2. The strain difference between e^ and E(N), designated Ae in Figure 2, is determined.

Now 2Ae is added to E(N) for the extrapolated value of strain. The extrapolated value of stress is then calculated from Eq. 3, using the extrapolated value of strain. This procedure assures that the extrapolated data point will always be in excess of the secant yield stress.

The program now treats the extrapolated point as if it were the last data input point for R-0 computations. However, a diagnostic message appears in the output printout advising the user that the ex- trapolation procedure has been employed. The message also contains the stress and strain values of the

SIN)

Figure 2. Extrapolation procedure outline.

Q'^/ &v Extrapolated l)V / i i point

S(N+1), E(N+1)

1. Find A and p of Eq.3 by least squares using last 4 data points.

STRAIN

2. Find S. , e., and Ae.

3. LetE(N+l)=E(N)+2Ae.

4. Find S(N+1) from Eq.3 using e=E(N+l).

E(N)

extrapolated point. No entries are made for the extrapolated point in the op- tional output table; a message is printed to this effect at the end of the table.

TANGENT MODULUS

If the Ramberg-Osgood expression is differentiated, the resulting tangent modulus expression is given by

dS/de = Et = E/[l + (3m/7)CS/S1)m"1].

The corresponding expression for the strain hardening relation is

dS/de = E., Ape p-1

Expressing Eq. 5 in terms of stress,

Et = Ap (S^A)^ (S/SJ^

where q = (p-1)/p.

(4)

C5)

(6)

When the value of E^ for stresses up to Sj is found from Eq. 4 and for stresses beyond Sj from Eq. 6, there is likely to be a discontinuity at the S]^ stress value. This discontinuity can have two forms as shown in graphs in Figure 3. Such a discontinuity leads to ambiguous results in calculations involving the tangent modulus for stress values at S^ (for example, in plastic buckling).

AtS,: R-0 too flat and/or S-H too steep

STRESS

Eq. 4

AtS,: R-0 too steep and/or S-H too flat

Figure 3. Two possible types of tangent modulus

discontinuities at the secant yield stress (schematic)

STRESS

Note, however, that it is possible for the two tangent modulus curves to intersect, thereby eliminating the discontinuity. The stress value at the inter- section has here been designated as the equitan stress Seqt. This stress value becomes one additional parameter to describe the stress-strain properties.

The equitan stress is found in the program by solving Eq. 4 and Eq. 6 simul- taneously, using an iterative procedure. This is done by a subroutine called ISOTAN. At the equitan stress.

E/[l + (3m/7)(S/S1)m ] = Ap(S1/A)

q(S/S1)Cl. (7)

Rearranging Eq. 7 results in

(S/S^ + (3m/7)(S/S1)q+m"1 - E/[Ap(S1/A)

q] = 0. (8)

The S value in Eq. 8 is the desired equitan stress.

It is possible for there to be no real solution for Eq. 8 or for solutions which are not useful for computation. The program contains tests which identify these cases and diagnostic messages appear in the printout. The nonsolution and nonuseful solutions for which tests are done are as follows.

No solution can be found in 100 iterations.

2. The computations diverge, resulting in numbers too large for the computer. (A running size-check of key calculated quantities is made in a subroutine called S1ZECK.)

3. The equation has an imaginary solution.

4. The solution stress value is larger than the stress of the last data point,

5. The solution stress value is less than the proportional limit.

Shown in Figure 4 are the results of a computation for a titanium alloy. In this case the equitan stress is less than the secant yield stress. The curve- fitting parameters for the stress-strain data are as follows:

E = 15.2 x io6 psi (104.8 GPa)

Sx = 165.57 ksi (1141 MPa)

m = 25.07

p = 0.1683

A = 333.6 ksi (2300 MPa)

The computed value of the equitan stress is 162.36 ksi (1119 MPa).

3.0

Z5

2.0

a o

L5

Figure 4. Computed tangent modulus values for a

titanium alloy using analytical expressions derived

from differentiating the Ramberg-Osgood and strain

hardening equations.

Referring to Figure 4, note that there is a discontinuity in the tangent mod- ulus if it is computed from Eq. 4 to Sj and from Eq. 6 beyond Sj at the values designated B and C in the figure. The discontinuity disappears when Eq. 4 is used for stress values to the equitan stress (A in the figure) and Eq. 6 thereafter. It should be noted, however, that the stress-strain values associated with the tangent modulus would be computed from the R-0 curve to Sx and from the S-H curve beyond S^

Applying the procedure results in a rough approximation of the tangent modu- lus and is suggested only for economy of storage of parameters. If a better approximation is required, then other more accurate methods should be used which operate directly on the stress-strain data. For the latter case the stress-strain data may also have to be stored.

t

MISCELLANEOUS REVISIONS

Metric Units

The output parameters are given in both Standard English units and S-I metric units. For the S-I units, the modulus of elasticity is given in GPa and all other stress values in MPa.

Header Card

kn input Poisson's ratio is no longer required. The following changes from the information given in Table 1 of the Supplement to AMMRC TR 76-12 (page 4) are required

Columns 1-43 No change Columns 44-50 Specimen Area, Format F7.4 Columns 51-62 Date of Test, Format 2A6 Columns 63-65 Output Option, Format 13; enter 1 in column 65

for optional table. Leave blank for standard output.

Optional Table

Calculations of point-for-point tangent modulus and elastic-plastic Poisson's ratio have been eliminated as well as calculations of plastic components of strain. These data were printed in optional Tables 2 and 3 of the original program and these have been eliminated. In the single optional table in the revised program, a point-by-point comparison is given of the actual input data stress values with those calculated from the approximate analytic expressions. The values are ex- pressed in both Standard English units and S-I metric units.

Additional Parameters

The maximum strain of the data is printed out. Computations of the correla- tion coefficient are made for three separate data regions: elastic region; from the proportional limit to the secant yield stress; the strain hardening region. The values appear in the printout and serve, with the standard error of estimate, as an indicator of the goodness-of-fit of the analytic values to the test data.

APPENDIX A. SAMPLES OF OUTPUT

1. Sample of printed output of stress-strain parameters and statistical data for a specimen of ESR 4340 steel tested in tension.

2. Sample of tabular data for the same specimen in which the observed stress values are compared with those computed from the Ramberg-Osgood and strain- hardening analytic approximations. The columns labeled "Observed Strain" and "Observed Stress" are the input test data values.

IIALYTIC 4PPR3XIMATI0N 3F STREGS-STS*IN PROPERTIES RAMBERG-OSGOOD ANO STRAIN HARDENINC PARAMETERS

OSTAINEO FROM ANALYSES OF EXPERIMENTAL DATA

EPS=SIG/E«(3/7 IIS1/E ItSIG/Sl l»»H RAMBERG-CSG00D EQUATION

STRAIN HAHDENING LAW SIG:A(EPSI»»P

PEC NO ESR1C1

APPENDIX B. PROGRAM LISTING

The prograii. is written in FORTRAN IV and has been run on a UN.TVAC 1106 com- puter. The listing includes the main program and five subroutines. The number of punch-cards for the program, including comment cards, is as follows:

Main program: Subroutine ORDER Subroutine LSTSQ Subroutine RJA75 Subroutine ISOTAN Subroutine SIZECK

472 cards 28 cards 20 cards 28 cards 83 cards 13 cards

TOTAL: 644 cards

arcPfis MAIN COMMON S

1C

11 c

12

13

9 CONTINUE IFtNSUT.CO.DIGO TO IP

C ORDERING OF DATA MOPDER^D CALL ORDE^tN.^fE.EEiMORDIR) IF(MCRDER.E3.0J 60 TO 10 EN=E(N» NEND=0 GO TO 51 SFL-rrL*it;DO MVrO SEOT=D NRUN=P NRUN=NRUN*1

ELASTIC DATA ANALYSIS 1=0 1=1*1 IF( SI D.GT.SPL JGO TO 13 IFJI.EQ.NJGO TO 13 GO TO 12

J=JFL-1 IF(J.GT.2JC0 TO m EN-EfN|

C INSUFICIENT ELASTIC DATA NEN0=1 GO TO 51

m DO 15 1=1.J XfII=E(II

15 Y(I)=SII) LT = 1 LJ=J CALL LSTSOtX.Y.SL.CF.LI.LJI EMOD=SL IF(NRUN.E3.2IGC TO 1C S10F = 0F E10F=-S10F/EM0D PE0F=E10F*(10.«»S.) FS0F=S10F E0FF=E10F SO TO 17

1G S20F=0F r20F=-S20F/EMOD E3FF=E2CF PS0F=S10F*S20r E120F=E10F*E20F PE:0F=E1Z0F«{10. ♦*£.!

17 DO 13 1=1.N

18 ECIimi l-EOFT C 7-0 ANALYSIS

EN=E (NJ SCT1=,7»EM0D SCT2r,85»E:M0D CHK1=Z(M)»SCT1 IF(CHK1.GE.S(NI IGO TC 19 CHK3=CHK1/S(NJ IF{CHK3.3C.,95IGO TO tOD

C LAST DATA POINT STRESS VALUE LESS THAN SI NEND=2 GO TO 33

C EXTRAPOLATION ROUTINE

23 JP=JS1*1 XNUM=»L0Gt7.»lEM0D»E2-S2)/(3.»Sl») XDEN=ALOG(S2/S1 I XM=XNUM/XOEM SPL=S1*J7.*EMCD/I3.*S1*{20.«*6.tll**ll' ITJMR'JN.Ea.UGO TO 11 IF

35 DRlrORl-MSBARl-SdM"?. OVl=SQRT(OTl/JI COR=l.-OTl/ORl ir(COR.LT.D) GO TO 3G C0Rl:r3RT(C0RI

36 irtNCNO.LT.t) ) GO TO «t ? DO 37 I:JPLtJSl Z=C3(I-1J/Sl BE-L (I I»EKCD/S1 CALL RJA75(2.XMt33) csm=z*si CV(I)=GCSfIl DT3rDT3*0VII1**2.

tO SBAR3rSBAR3+S(I)/JSH 00 HI I=JP.N

HI DR3=DR3*(SBAR3-5(I)»**2. 0V3=53RTt0T3/J3H) C0R=1.-0T3/DR3 ir(COR.LT.a) GO T0 H3 C0R3rS0RT(CCRI GO T'' m

C EaUITAN STRESS COMPUTATION C NO R-C DATA OR NO S-H DATA » NO TQUITAN STRESS COMPUTATION 12 JTCL'Trl

GO TO 18 H3 JTCUTrD

SE3T:0. SEGTMrO. XMT:XM-1. XNT=CP-1.I/P CW=EMOD/(A*P*(Sl/A)••XNT) KU-V wn. CALL ISCTAN'( WtXMTf XNTtCWiKWI ir(KW.LE.lOn) GO TO H£ IF(KW.LE.20C1 GO TO ^5 ir(KW,LE.3001 GO TC tU

C NO SOLUTION CAN BE FOUND FOR I! JT0UT=2 GO TO 43

C MAGNITUDE OF kt JTCUT:3

GO TO 4 3 C IMAGINARY SOLUTION 15 JTOUT = '»

GO TO HB

iCTAN EQUATION

VARIABLES IN IS^TAN COMPUTATION TOO URGE

FOP ISOTAN EOLATION

12

ts IFISE3T,LE.SPLI GO TO «t7 IFCSE3T.LE.S1) GO TC ««3 E:EQT=(SEaT/AI«»(l./PI IF(EE9T.LT.EN) GO TO t| 3

C 117

v^I';r : r!4

51

57

EL!

333 70 71 72 73 71 1DD

101

102

:RIN■' ' •:.>

PRINT CO "C P.7IV' PRINT PR IN

r r-.

it CO

re ^c -o 5 7

-UP

IF JNl'.CQ.D I CC TO 53 PRINT 110 ir

1C3

10««

105

105

107

103

103

1111

111

112

113

11«4

115

11C

117

113

113

12C

121

F0?MAT(/iT25.3(5(1H*)tSX).//^T32l,♦♦•♦• NO ANALYSIS UA IE DATA FOR SPCC.NO. •»A5t••*•«••f//tT25t•THE DATA CARD 23Z ARRANGED IN INCREASING ORDER OF STRESS MAGNITUDE.'. 3H« LEY J I

FORMAT!/.T25i3J5(lH*)fSXJt//tT27t»»»*»« INSUFFICIENT 3 1LASTIC REGION WERE INPUT TC THE PRCGRAK. •♦•♦*• »/»T2«l f 2ST SQUARES ANALYSIS COULD NOT 3E MADE AND THE CALCULAT

3CRTE0.,t//»T25t8(5«1H* »t6X M FORMAT!TlQt•••♦••THERE WERE INSUFFICIENT DATA FOR A V5

1YCISM FORMAT! 1H+.T71 I'TOC FEW DATA POINTS IN THE KNEE OF THE

l««**«t/l FORMAT! 1H«-.T71.'STRESS VALUE OF LAST DATA POINT LESS ^

!

122

123

12H

125

12S

127

123

123

130

131

132

FORMATIT50i♦•♦••♦ STASOA^O ENGLISH UNITS ♦••••»,//» 1T11 ttMGDULi;S,tT21.'S-INTCrT,.T30i,E:-INTCPT»iTi»2tfSrLf »r51f 'Sl'tTSS 2t •EXPNT'.TGSi '.l-PCT SY'tT77t» .2-PCT SY»iT33i*COEFF-A'i T10G»•EXPNT Z'.Tingt'SEOT'./t ITllt •MEGA-P3I'tT2«»i 'P SI't T3D •' MU-IN/IN' t m2 f » KS I • t T 51 f « KSI't TGlt • M B^TESi 'KSI ,.T80f,KSI,f TSPf^KSI' tTl02 » •?• tTHOi 'KSI* i/ I

FORMAT! Tll.F7.2f T22iF7.0tT3CtF7.0tT'4C.F7.2.T'O,F7.2tT53tF7.2i TG7. F 17.2tT78tF7.2fTB7tE3.«»fT2StF7.5»Tir3»F7,2 F

FO?MAT( 1 HPf TtG t'••♦♦• S-I INTERNATIONAL METRIC JNIT'; ♦••♦♦»,//t IT 11 t tMCiULUS,tT21 t'S-INTCFT' tTIO t'E-INTCFT • f Tt| 21'SPL f t T51. 'Sl'tTB? 2i*EXPNT,.T55.• .1-PCT SY»tT77.».2-PCT SY♦iT3Bf«COEFF-A•t TlODt'EXPNT 3,tT109i*SE(»T,»/i ««T13f ,3PA,tT2«tt •MPA•.T31•• MU-M/M» 114 2 f • MPA • # T 511 ♦ MPA « i^Sl i « M'i TG3. • 5MFA»tT8Ci ,MPA'tT90t»MPA»IT1C2i»P•?T1101•HPA*»/I

FORMAT! Tll.F7.2tT21iF7.2fT3C.F7.nfT33tF3.2.T«j3tF3.2fT5-itF7.2fTCG.F ie.2fT7 7tF8.2»T8 7fE3,'«tT5StF7.5»TlC7fF8.ZI

FORMAT! 1HC. T«40»»STANDARD TRROR OF ESTIMATE ANO CORRELATION COEFFIC lICNT't//tT2tl.'STANDARD ENGLISH UNITS ' t TGI »'S-I METRIC UNITS'tT39f,

T39t»S-H ZEl T5E 2C0RR, COCFF.'t/i TIS.'CLAST SEEffT27.,R 3ELAST SEE'tTGBt'R-O SEE* tTSOf »S-H SEE ' t T91 i'EL ASTIC t T1C1 i'R-C • i Tl '

7 NOUTrNOUT*! SA=S(I I E:A = r( u S3rSll*l I EO:Ef1*1 ) S{II=SB ZCDrEfl S(I+1l-ZH Ztl*l)=EA

H CONTINUE COUNT-COUN'Tt-1 IF ICOUNT.LT.NI GO TC 5 MORDER^l RETURN

5 Tr(NOUT.GT.C) GO TO 1 DC 6 Irl.N

G EZ

3 VrZ-OZ ;RV:FUNC< VJ IF (ABSCRV I.LE.RMIGC TC 6 KKrKK+1 IF (KK-10C J*«f«tf7

*i Ir(RV.LT.r. 130 TO 5 DZ=D7/1L. GO TO 3

5 Z = Z+0Z GO T0 3

E ZrV 7 RETURN

END 3r0^fIS ISOTAN

SUBROUTINE ISOT AN ( W t XN!T . X\'T t C W» KW I FUNC( 'JUW*»XNT*(3.*(XMT*1 .J/7J»W**(XNT + XMTI-CW L=D K\t-V. XCW=ALCG1L(3.»(XMT*3.1/7.1 RMzw/

7 RW=rUNC(WI IF( A9«;( RWJ.LC.RM >30 TO 1«« IF(KW.GT.IOOIGO TO 1H KW=KW*1 IF (RW.LT.UfGO TC 3 L = L*1 W-W-»I«L»3W WV = W CALL SIZECKCXNTtXMTtXCWtWVtKXI ir(Ky,uT.C)GO TC 12 GO TO 7

3 V=W-I*OW WV=V CALL SIZECK(XNT.XHTiXCWtWVfKXI

IF CKX.GT.O rec TC 12 Ry=FUNC

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DRXHR-AG Author

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