An Euler-Bernoulli Beam Formulation in. Ordinary State-Based Peridynamic Framework

Transcription

1 An Euler-Bernoull Beam Formulaton n Ordnary State-Based Perdynamc Frameor Cagan Dyaroglu, Eran Oterus and Selda Oterus Abstract Department of Naval Archtecture, Ocean and Marne Engneerng Unversty of Strathclyde, Glasgo, UK Every obect n the orld has a 3-Dmensonal geometrcal shape and t s usually possble to model structures n a 3-Dmensonal fashon although ths approach can be computatonally expensve. In order to reduce computatonal tme, the 3-Dmensonal geometry can be smplfed as a beam, plate or shell type of structure dependng on the geometry and loadng. Ths smplfcaton should also be accurately reflected n the formulaton hch s used for the analyss. In ths study, such an approach s presented by developng an Euler-Bernoull beam formulaton thn ordnary-state based perdynamc frameor. The equaton of moton s obtaned by utlzng Euler-Lagrange equatons. The accuracy of the formulaton s valdated by consderng varous benchmar problems subected to dfferent loadng and dsplacement/rotaton boundary condtons. Introducton Every obect n the orld has a 3-Dmensonal geometrcal shape ncludng the graphene materal, hch s generally descrbed as a structure th a -Dmensonal geometrcal shape, snce t has slght avness n the thcness drecton. From a computatonal pont of ve, t s usually possble to model structures n a 3-Dmensonal fashon. Hoever, such an approach can be computatonally expensve especally consderng complex structures such as an aeroplane, shp, etc. Hence, n some cases, t s essental to mae reasonable assumptons, so that the 3-Dmensonal geometry can be smplfed as a beam, plate or shell type of structure. As a result, the computatonal tme can be sgnfcantly reduced. In order to represent such smplfcatons, the formulatons descrbng the problem of nterest should be modfed approprately hch s also true for perdynamcs (PD, a ne contnuum mechancs formulaton ntroduced by Sllng (000. As argued by dell Isola et al. (04a, 04b, 04c, 05, the orgns of PD go bac to Pola s contnuum formulaton. The orgnal PD formulaton as ntroduced for a 3-Dmensonal geometrcal confguraton and each materal pont has three translatonal degrees-of-freedom. As mentoned earler, for

2 smplfed geometres, t s necessary to modfy the formulaton to represent smplfed structural behavour accurately. O Grady and Foster (04a and O Grady and Foster (04b developed non-ordnary state-based PD formulatons for Euler-Bernoull beam and Krchoff-Love plate, respectvely. Moreover, Taylor and Stegmann (03 ntroduced a bond-based perdynamc plate model. Recently, Dyaroglu et al. (05 presented PD Tmosheno beam and Mndln plate formulatons by tang nto account transverse shear deformatons. These formulatons nclude not only the transverse deformaton as degree-offreedom, but also rotatons of the cross-secton. For slender beams, here the rato of length to thcness must be greater than 0,.e., L/ h> 0, transverse shear deformatons can be neglected and Euler-Bernoull beam formulaton can be used. By dong ths, t ll be possble to reduce the total number of degrees-of-freedom n the system by half. Hence, n ths study, a ne ordnary state-based perdynamc model s developed and valdated by consderng varous benchmar problems. The developed formulaton can be used for the analyss of complex systems shong slender beam behavour. Knematcs of Euler-Bernoull Beam n PD theory In order represent an Euler-Bernoull beam, t s suffcent to use a sngle ro of materal ponts along the beam axs, x, by usng a meshless dscretzaton as shon n Fgure. In ths partcular case, the shape of the horzon,.e. perdynamc nfluence doman, has a shape of a lne. Moreover, each materal pont has only one degree of freedom along the z-axs, hch s the transverse dsplacement,. Fgure. Knematcs of an Euler Bernoull beam n PD theory

3 By usng the approach presented n Madenc and Oterus (04, the stran energy densty functon can be rtten n terms of mcro-potentals and for materal pont, t can be expressed as W ( PD = ( ω + ω here the mcro-potentals ω and ω are the functons of transverse dsplacements of materal ponts,.e. ω = ω,,... and ω = ω (, (,.... The total potental energy of the beam can be obtaned by summng potental energes of all materal ponts ncludng stran energy and energy due to external loads as U PD = n hch ω,,... + ω, (,... ( ˆb b ˆ s the body load th a unt of force/per unt volume and t may represent both the transverse load, p( x and the moment load change, m( x change should be converted nto a more convenent form of and max. mn. / x. The moment load m m Δ x, here m max. m represent maxmum and mnmum moment loads, respectvely, actng on a materal mn. volume. Smlarly, total netc energy of the beam can be obtaned by summng the netc energes of all materal ponts as T PD = ρ! (3 By usng Eqs. ( and (3, the Lagrangan of the system can be expressed as L = T PD U PD = ρ! ω (, (,... + ω, (,... (4 + ˆb Note that the Lagrangan s only a functon of transverse deflecton,. Hence, the Euler Lagrange equaton taes the form of 3

4 d dt L! ( L = 0 (5 Substtutng Equaton (4 nto Equaton (5 leads to ρ!! + ω ( ( ( ( + ω ( ( ( ( ˆb = 0 Moreover, the PD equaton of moton for an Euler-Bernoull beam can be expressed n a more compact form n terms of force denstes,!t ( and!t (, as ρ!! = (!t!t + ˆb (7 here the tlde sgn represents force denstes arsng from the bendng deformaton and they tae the form of (6!t = and ω ( ( = (!t = ω ( ( = ( ω ( ( ( ω ( ( ( (8a (8b Moreover, these force denstes can also be rtten n terms of stran energy denstes of materal ponts, and, n a PD bond as!t = W W = ω (9a th ( ( and!t = W th W = ( ( = ω ω ( ( (9b 4

5 The stran energy denstes for materal ponts, and, can also be expressed by utlzng the correspondng defnton from classcal contnuum mechancs as W = aκ (0a and W = aκ (0b here κ and ( κ represent the curvatures of materal ponts, and, respectvely, (Fgure and a s a PD parameter. The curvature functons for materal ponts, and, for a bond can be defned as κ and κ ( ( = d (a ( ( = d (b ( here d s a PD parameter and t ensures that the curvature, κ, has a dmenson of /length. Moreover, the summaton sgn ndce, the man materal pont and the ndce,, represents all materal ponts nsde the horzon of, represents all materal ponts nsde the horzon of the famly member materal pont here horzon defnes the nfluence doman of each materal pont. Moreover, dstances beteen materal ponts are defned as = x ( x and ( = x ( x. Note that Equatons (a,b correspond to the ( curvature defnton n classcal theory,.e. κ x = d dx. Substtutng Equatons (a,b nto Equatons (0a,b yelds the explct expressons of stran energy denstes as ( ( W = ad (a ( ( ( and 5

6 ( W = ad (b ( ( ( The PD force denstes can be rertten by substtutng Equatons (a,b nto Equatons (9a,b as!t ( = ad ( ( (3a and!t ( = ad ( ( (3b hch can also be expressed n terms of curvature functons as!t ( = ad κ (4a and!t ( = ad ( κ (4b Note that as n the classcal theory, the PD force denstes occur due to bendng deformaton and they are functons of curvatures, κ and ( κ, respectvely. As shon n Fgure, the force actng on the man materal pont s dfferent than the force actng on ts famly member,.e.!t (!t (. Ths s because the force functon!t ( s based on the dsplacements of materal ponts hch are nsde the horzon of the man materal pont and, on the contrary, the force functon!t ( s based on the dsplacements of materal ponts, hch are nsde the horzon of the famly member materal pont. Therefore, the equaton of moton of materal pont gven n Equaton (7 s based on ordnary state-based Perdynamc theory and t can be rertten n an open form as ρ!! = ad = = ( ( ( = + ˆb (5 6

7 here the summaton functons for materal ponts, and materal ponts nsde ther horzons, δ and δ. nvolve all famly member Fgure. Force functons of a PD Euler Bernoull beam In order to prove the valdty of Perdynamc equaton of moton (EOM gven n Equaton (5, t s essental to chec f ts classcal counterpart can be recovered n the lmt of horzon szes approachng to zero,.e. δ 0 and δ 0. Therefore, the transverse dsplacements, and ( and terms as, can be expressed n terms of ther man materal pont s dsplacements,.e. (, respectvely, by usng Taylor seres expansons hle gnorng the hgher order ( = + sgn, x ( x x +, xx (6a and ( = + sgn, x ( x x +, xx (6b Substtutng Equatons (6a,b n PD EOM,.e. Equaton (5, results n + ρ!! = ad,xx +,xx +,xx,xx + ˆb = = here the summaton sgns can ether nvolve all the famly members of the man materal pont nsde the left part of the horzon or rght part of the horzon. Agan, f Taylor seres = = (7 7

8 expanson s used for the famly member materal pont by dsregardng the hgher order terms as = +,, sgn xx xx, xxx ( x x + (8, xxxx and substtutng Equaton (8 nto Equaton (7 results n ( ( ρ!! = ad (,xxx (,xxxx + ( ( +,xxx (,xxxx = = = ( ( +ad (,xxx (,xxxx + ( ( + +,xxx (,xxxx + ˆb = = = (9 After performng some algebrac manpulatons, the fnal form of PD EOM can be obtaned as ρ!! = ad (,xxxx = 4 ( = + ˆb (0 here s replaced th. Moreover, the nfntesmal volumes, and ( can be expressed for D beam element as ( = AΔ (( and = AΔ, here Δ (( and Δ approach to dfferental dstances,.e. (( d the summaton terms n Equaton (0 nto ntegratons results n δ δ,xxxx Δ and Δ d. Convertng ρ!! = A ad 4 d d + ˆb ( δ δ Performng the ntegratons n Equaton ( yelds the PD EOM as ρ!! + A ad δ 4 x 4 = ˆb ( Note that, the PD EOM, gven n Equaton (, has the same form as ts classcal counterpart for an Euler Bernoull beam theory,.e. 8

9 ρ!! + EI A 4 x 4 = p m x (3 As mentoned earler, the body load, ˆb may represent both the transverse load, p, and the moment change, ( m m Δ x, actng on a materal volume. Therefore, t can be max. mn. / concluded that the proposed netc energy, T, and stran energy densty, W, expressons gven n Equatons (3 and (a,b, are sutable for representaton of Euler Bernoull beam problem. Fnally, equatng the coeffcents of the unnon functon,, n the PD EOM to the coeffcents of that n the classcal equaton yelds the relatonshps beteen the PD parameters, a and d, and the Young s modulus, E, and the moment of the nerta, I, as EI a = (4 3 Adδ The body load can be expressed as ˆ m b= p x (5 In order to obtan a complete PD formulaton, the Perdynamc materal parameter, d, must also be determned. For ths purpose, the curvature of a materal pont s compared th ts classcal counterpart under a smple loadng condton, hch can be chosen as a constant curvature, ζ. Fgure 3 shos such a loadng condton for a beam th a length of δ. Fgure 3. A beam subected to constant curvature In classcal beam theory, the constant curvature loadng s defned as κ = ζ = ddx (6 Equaton (6 can be solved for the specfed boundary condtons hch are 9

10 ( δ = = 0 (7 ( δ Thus, the transverse dsplacement of any pont on the beam axs can be defned as ( x ζx ζδ = for δ x δ (8 Here, the coordnate axs, x, s located at the centre of the beam and the man materal pont,, s also at the centre th ts horzon completely embedded nsde the beam as shon n Fgure 3. Hence, the dsplacement functons for materal ponts, and ts famly member pont, can be expressed th the help of Equaton (8, as ( ζδ = and ζ ζδ = (9 here x = = s used. Thus, substtutng Equaton (9 nto Equaton (a gves the ( curvature of materal pont as ζ κ = d ( (30 = Convertng summaton term nto ntegraton hle transformng materal volume as = AΔ Ad results n ( ( ( δ ζ κ( = d Ad (3 δ Performng ntegraton and equatng Equaton (3 to the constant curvature, ζ, lead to the Perdynamc materal parameter, d, as d = (3 Aδ Moreover, the PD parameter, a can also be expressed n a more convenent form by substtutng Equaton (3 nto Equaton (4 as EI a = (33 A After substtutng Equaton (33 nto Equaton (0a, the stran energy densty functon of PD theory becomes 0

11 W EI A κ = (34 hch s equvalent to the classcal theory s stran energy densty expresson. Calculatons For the Near Surface Materal Ponts Materal ponts stffnesses n a beam are effected from the free surfaces or materal nterfaces because the Perdynamc materal parameter, d, s derved under the assumpton that the man materal pont,, has a horzon hch s completely embedded nsde the beam body. On the other hand, there s no need for a correcton for the bendng bond constant, a. In Euler Bernoull beam theory, the curvatures and relevant force densty functons of materal ponts hch are close to the free surface are calculated numercally n a slghtly dfferent form than the gven curvature equatons,.e. Equatons (a and (b as ell as the force densty equatons,.e. Equatons (3a and (3b. The ne forms of these equatons are ntroduced th the reduced horzon szes as explaned n Appendx. Snce, the horzon sze s usually chosen as free surface. Boundary Condtons δ = 3.05Δ x, t s truncated at the frst three materal ponts near the As explaned n Oterus et al. (05 and Madenc and Oterus (06, the dsplacement boundary condtons n PD theory can be mposed through a nonzero volume of fcttous boundary layer, R c, as shon n Fgure 4. The sze of ths layer s equvalent to the horzon. An external load, such as a moment or a transverse load, can be appled n the form of body loads through a layer thn the actual materal, R. The sze of ths layer can be chosen as the same sze as the dscretzaton sze. Fgure 4. Applcaton of boundary condtons n perdynamcs The applcaton of boundary condtons n Euler Bernoull beam theory s slghtly complcated snce the theory tself only contan dsplacement degrees of freedom rather than rotatons. In ths regard, applcaton of dfferent types of boundary condtons s explaned n detal belo.

12 Clamped boundary condton In order to mplement clamped boundary condton, a fcttous boundary layer s ntroduced outsde the actual materal doman. The sze of ths layer can be equvalent to the horzon sze of δ = 3.05Δ x. In classcal beam theory, clamped boundary condton mposes zero dsplacement and zero slope on the boundary, as shon n Fgure 5. In PD formulaton of Euler Bernoull beam, ths condton s acheved by enforcng mrror mage of the dsplacement feld for the frst to nodes n the actual doman th respect to the frst adacent materal pont hch s fxed. Fgure 5 shos the Euler Bernoull beam and ts dscretzaton th ncremental volumes. The red dotted lne shos the deformed form of the beam axs. The dsplacements for the materal ponts n the boundary regon should be specfed as =, = x ( x and = 0 (35 ( x ( x + + x Fgure 5. Clamped boundary condton Smply supported boundary condton In order to apply smply supported boundary condton, a fcttous boundary layer s ntroduced outsde the actual materal doman. The sze of ths layer can be equvalent to the horzon sze of δ = δ3 =.05Δ x. In classcal beam theory, smply supported boundary condton mposes zero dsplacement and curvature on the boundary, as shon n Fgure 6. In PD formulaton of Euler Bernoull beam, ths condton s acheved by enforcng negatve mrror mage of the dsplacement feld for the frst to nodes n the actual doman th respect to the support pont. Fgure 6 shos the Euler Bernoull beam and ts PD dscretzaton th ncremental volumes. The dotted red lne shos the deformed form of the beam axs. The dsplacements for the materal ponts n the boundary regon should be specfed as

13 = and = x ( x (36 ( x ( x + + Free boundary condton Fgure 6. Smply supported boundary condton In order to mplement free boundary condton, a fcttous boundary layer s ntroduced outsde the actual materal doman. The sze of ths layer can be equvalent to the horzon sze of δ = 3.05Δ x. In classcal beam theory, free boundary condton mposes zero curvature on the boundary, as shon n Fgure 7. In PD formulaton of Euler Bernoull beam, ths condton s acheved by freeng boundary ponts. Fgure 7 shos the Euler Bernoull beam and ts dscretzaton th ncremental volumes. Agan, the dotted red lne shos the deformed form of the beam axs and there s no mposed dsplacements for the materal ponts n the boundary regon. Fgure 7. Free boundary condton 3

14 Numercal Soluton Method In ths secton, the numercal soluton procedure for the EOM of Euler Bernoull beam theory, gven n Equaton (5, s presented for the problems n statc equlbrum condton. In ths regard, the acceleraton term at the left hand sde of Eq. (5 s elmnated and rearrangng the terms results n ad ( ˆ = b ( ( ( ( (37 Eq. (37 can also be rtten n a matrx form as [ K]{ U} = { b} (38 here [ K ], { U } and { b } represent stffness matrx, dsplacement and body force vectors, respectvely. The stffness matrx ncludes the Perdynamc parameters, a and d, the reference length of each bond and the materal pont s volume as ell as the volume and the surface correcton parameters. The unnon dsplacement vector can be determned after mposng the boundary condtons. In order to mpose specfed boundary constrants, the master slave condton method can be utlzed. In ths method, dsplacement matrx can be expressed as { U} = [ T]{ Uˆ } (39 here [T] represents transformaton matrx and { U ˆ } s the reduced dsplacement matrx th only master nodes. For example, to mpose the condtons, = 5 = 4 (40 hch can be used to defne a clamped boundary, the transformaton and the reduced dsplacement vectors can be expressed as 4

15 = n n and { U} = [ T]{ Uˆ } (4 In Equaton (4, and are the slave nodes. Next, the equaton of moton can be rertten as T T [ T] [ K][ T]{ Uˆ } [ T] { b} = (4 Solvng Equaton (4 leads to the unnon reduced dsplacement vector hch nvolves only master nodes. Numercal Results Clamped free beam problem The clamped free beam s subected to a pont load of shon n Fgure 8. The length of the beam s P = 50 N, from the rght end as L = m, th a cross-sectonal area of A = m. Its Young s modulus s specfed as E = 00 GPa. Only a sngle ro of materal (collocaton ponts are necessary to dscretze the beam. The dstance beteen materal ponts s sze of Δ x = 0.0 m. Fcttous regons are created at the left and rght edges th a δ = 3.05Δ x. The loadng s mposed on only one materal pont, hch s denoted by yello colour n Fgure 8, th a body load of b = P AΔ x. Fgure 8. Clamped free beam 5

16 The Perdynamc soluton of the transverse dsplacement,, s compared th the fnte element (FE method by usng the beam element BEAM3, hch s sutable for slender beams, neglects shear deformaton and s avalable n the commercal softare, ANSYS. As depcted n Fgure 9, the PD and the FE solutons agree ell th each other. Ths verfes that the PD equaton of moton can accurately capture the deformaton behavour of an Euler- Bernoull beam for clamped-free boundary condtons. Clamped clamped beam problem Fgure 9. Dsplacement results of clamped free beam A clamped clamped beam s subected to a pont load of shon n Fgure 0. The length of the beam s P = 50 N, from ts center as L = m, th a cross-sectonal area of A = m. Its Young s modulus s specfed as E = 00 GPa. Only a sngle ro of materal (collocaton ponts are necessary to dscretze the beam. The dstance beteen materal ponts s sze of Δ x = 0.0 m. Fcttous regons are created at the left and rght edges th a δ = 3.05Δ x. The loadng s mposed on to materal ponts, hch are denoted by yello colour n Fgure 0, as a body load of b = P AΔx n order to eep the symmetry. 6

17 Fgure 0. Clamped clamped beam The Perdynamc soluton of the transverse dsplacement,, s agan compared th the FE method results. As depcted n Fgure, the PD theory and the FE method results agree ell th each other. Ths verfes that the proposed PD equaton of moton can accurately capture the deformaton behavour of an Euler-Bernoull beam for clamped-clamped boundary condtons. Fgure. Dsplacement results of clamped clamped beam Smply supported smply supported beam problem A smply supported smply supported beam s subected to a pont load of ts center as shon n Fgure. The length of the beam s area of P = 50 N, from L = m, th a cross-sectonal A = m. Its Young s modulus s specfed as E = 00 GPa. Only a sngle ro of materal (collocaton ponts are necessary to dscretze the beam. The dstance beteen materal ponts s Δ x = 0.0 m. Fcttous regons are created at the left and rght edges th a sze of δ = δ3 =.05Δ x. The loadng s appled to to materal ponts, hch 7

18 are denoted by yello colour n Fgure, th a body load of the symmetry. b = P AΔx n order to eep Fgure. Smply supported smply supported beam The Perdynamc soluton of the transverse dsplacement,, s compared th the FE method results. As depcted n Fgure 3, the PD and the FE method results agree ell th each other. Fgure 3. Dsplacement results of smply supported smply supported beam Conclusons In ths study, a ne ordnary state-based perdynamc formulaton for Euler-Bernoll beam s presented. The equaton of moton s obtaned by usng the Euler-Lagrange equaton. The relatonshps beteen perdynamc parameters and relevant parameters n the classcal theory are establshed by utlzng Taylor expanson for a specal case of horzon sze convergng to zero. The man advantage of the developed formulaton s the reducton of number of degrees 8

19 of freedom for each materal pont by half th respect to Tmosheno beam formulaton. Applcaton of boundary condtons n perdynamcs s also dfferent from classcal theory. Elegant ays of applyng dfferent types of boundary condtons ncludng clamped, smply supported and free edge boundary condtons are explaned. arous benchmar cases are consdered to demonstrate the accuracy of the current formulaton and boundary condtons. Remarable agreement beteen perdynamc and fnte element results are observed. References dell Isola, F., Andreaus, U., Cazzan, A., Perugo, U., Placd, L., Ruta, G. and Scerrato, D. On a Debated Prncple of Lagrange Analytcal Mechancs and on Its Multple Applcatons, The Complete Wors of Gabrola Pola: ol. I, Chapter, Advanced Structured Materals, ol. 38, 04a, pp dell Isola, F., Andreaus, U., Placd, L. and Scerrato, D., About the Fundamental Equatons of the Moton of Bodes Whatsoever, As Consdered Follong the Natural Ther Form and Consttuton, Memor of Sr Doctor Gabro Pola, The Complete Wors of Gabro Pola: ol. I, Chapter, Advanced Structured Materals, ol. 38, 04b, pp dell Isola, F., Andreaus, U. and Placd, L., A Stll Topcal Contrbuton of Gabro Pola to Contnuum Mechancs: The Creaton of Per-dynamcs, Non-local and Hgher Gradent Contnuum Mechancs, The Complete Wors of Gabro Pola, ol. I, Chapter 5, Advanced Structured Materals, ol. 38, 04c, pp dell Isola, F., Andreaus, U. and Placd, L., At The Orgns and In the anguard of Perdynamcs, Non-local and Hgher-Gradent Contnuum Mechancs: An Underestmated and Stll Topcal Contrbuton of Gabro Pola, Mathematcs and Mechancs of Solds, ol. 0(8, 05, pp Dyaroglu, C., Oterus, E., Oterus, S. and Madenc, E., Perdynamcs for Bendng of Beams and Plates th Transverse Shear Deformaton, Internatonal Journal of Solds and Structures, ols , 05, pp Madenc, E. and Oterus, E., Perdynamc Theory and Its Applcatons, Sprnger Ne Yor, Ne Yor, 04. Madenc, E. and Oterus, S., Ordnary State-based Perdynamcs for Plastc Deformaton Accordng to von Mses Yeld Crtera th Isotropc Hardenng, Journal of the Mechancs and Physcs of Solds, ol. 86, 06, pp

20 O Grady, J., and Foster, J., Perdynamc Beams: A Non-ordnary, State-based Model, Internatonal Journal of Solds and Structures, ol. 5, No. 8, 04, pp O Grady, J. and Foster, J., 04, Perdynamc Plates and Flat Shells: A Non-ordnary, Statebased Model, Internatonal Journal of Solds and Structures, ol. 5, No. 5, pp Oterus, S., Madenc, E. and Aga, A., Perdynamc Thermal Dffuson, Journal of Computatonal Physcs, ol. 65, 04, pp Sllng, S. A., Reformulaton of Elastcty Theory for Dscontnutes and Long-range Forces, Journal of the Mechancs and Physcs of Solds, ol. 48, 000, pp Taylor, M., and Stegmann, D.J., A To-dmensonal Perdynamc Model for Thn Plates, Mathematcs and Mechancs of Solds, ol. 0, No. 8, 05, pp Appendx Curvature and force densty calculatons for the frst materal pont near the free surface For the frst materal pont,, near the free surface, the curvature can be calculated from κ ( = d Δx Δx (A here d s the modfed Perdynamc materal parameter and Δx s the dstance beteen the materal ponts. The materal ponts + and ++ are shon n Fgure A. In Equaton (A, the horzon sze s assumed as δ =.05Δ x and t s used only f the materal pont s the frst pont near the free surface. Note that Equaton (A s obtaned by usng a fnte dfference formula for the second dervate snce the curvature, κ, s the second dervatve of the transverse dsplacement,. Next, the force densty functon can be obtaned by substtutng Equaton (A nto Equaton (4a as 4 ad t = + + ( ++ ( Δx Δx (A n hch materal ponts volumes can tae the form of ( A x + = ++ = Δ. 0

21 Fgure A. Frst man materal pont near the free surface In order to determne the modfed Perdynamc parameter,.e. d, the beam can be subected to a constant curvature loadng, ζ, as shon n Fgure A. In ths case, Equaton (6 can be solved for the dfferent boundary condtons by mposng the values of Thus, the transverse dsplacement of any pont on the beam axs can be calculated as ζ x ( x = ζδ x for 0 x Δ x (A3 ( Δx 0 = =. From Equaton (A3, the dsplacement functons for the materal pont, as ell as ts famly member ponts + and ++, can be expressed as ( Δx ( x = ζ Δ, + ( ( Δx 3ζ = and 0 ( ++ = (A4 Substtutng Equaton (A4 nto Equaton (A leads to the curvature of materal pont as κ = ζdaδ x (A5 Equatng Equaton (A5 to the constant curvature value, ζ, results n modfed Perdynamc parameter, d, as d = A Δ x (A6 As a summary, for the frst materal pont near the free surface, the curvature and the force densty should be calculated from Equatons (A and (A, respectvely, hle usng Equaton (A6 as a modfed Perdynamc materal parameter, d. Moreover, the horzon sze should be assumed as δ =.05Δ x for ths materal pont. Curvature and force densty calculatons for the second materal pont near the free surface

22 For the second materal pont,, near the free surface, the curvature can be calculated from Equaton (a. Hoever, the Perdynamc materal parameter d should be replaced th d hch s the modfed Perdynamc parameter snce the horzon sze s chosen as δ =.05Δ x as shon n Fgure A. Thus, Equaton (a taes the form of κ ( ( = d (A7 ( The force densty functon can be obtaned by substtutng Equaton (A7 nto Equaton (4a as!t ( = ad ( (A8 Fgure A. Second man materal pont near the free surface The beam s agan subected to a constant curvature loadng, ζ, shon n Fgure A, n order to determne the modfed Perdynamc parameter,.e. d. In ths case, Equaton (6 can be solved for the boundary condtons defned as = =. Follong smlar procedures ( δ ( 0 δ explaned earler, the modfed Perdynamc parameter, d, can be calculated as d = (A9 Aδ As a summary, for the second materal pont near the free surface, the curvature and the force densty can be calculated from Equatons (A7 and (A8, respectvely, hle usng Equaton (A9 as a modfed Perdynamc parameter, d. Moreover, the horzon sze should be assumed as δ =.05Δ x for ths materal pont. Curvature and force densty calculatons for the thrd materal pont near the free surface

23 Fnally, the curvature for the thrd materal pont,, near the free surface can be calculated from Equaton (a. The modfed Perdynamc parameter, d 3, can be used for the chosen horzon sze of δ 3 =.05Δ x as shon n Fgure A3. Thus, Equaton (a taes the form of κ ( ( = d3 (A0 ( The force densty functon can be obtaned by substtutng Equaton (A0 nto Equaton (4a as!t ( = ad 3 ( (A Fgure A3. Thrd man materal pont near the free surface The modfed Perdynamc parameter, d 3, can be obtaned by applyng a constant curvature loadng, ζ, to the beam as shon n Fgure A3. In ths case, Equaton (6 can be solved for the boundary condtons of ( δ = = 0. Follong smlar procedures as explaned δ 3 3 earler, the modfed Perdynamc parameter, d 3, can be calculated as d 3 = (A Aδ 3 As a summary, for the thrd materal pont near the free surface the curvature and the force densty can be calculated from Equatons (A0 and (A, respectvely, hle usng Equaton (A as a modfed Perdynamc parameter, d 3. Moreover, the horzon sze should be assumed as δ 3 =.05Δ x for ths materal pont. 3

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