Peridynamics for Bending of Beams and Plates with Transverse Shear Deformation

Transcription

1 Peridynamic for Bending of Beam Plate with Tranvere Shear Deformation C. Diyaroglu*, E. Oterku*, S. Oterku** E. Madenci** * Department of Naval Architecture, Ocean Marine Engineering Univerity of Strathclyde, Glagow, United Kingdom ** Department of Aeropace Mechanical Engineering Univerity of Arizona, Tucon, United State of America Progreive failure analyi of tructure i till a major challenge. There exit variou predictive technique to tackle thi challenge by uing both claical (local) nonlocal theorie. Peridynamic (PD) theory (nonlocal) i very uitable for thi challenge, but computationally cotly with repect to the finite element method. When analyzing complex tructure, it i neceary to utilize tructural idealization to make the computation feaible. Therefore, thi tudy preent the PD equation of motion for tructural idealization a beam plate while accounting for tranvere hear deformation. Alo, their PD diperion relation are preented compared with thoe of claical theory.. Introduction Peridynamic (PD) theory wa originally introduced for the olution of deformation field equation (Silling, 000) without any tructural idealization. It atifie all the fundamental balance law of claical (local) continuum mechanic; however, it i different in the ene that it i a nonlocal continuum theory it introduce an internal length parameter into the field equation. Thi internal length parameter define the aociation among the material point within a finite ditance through micropotential. Removal of micropotential between the material point allow damage initiation growth through a ingle critical failure parameter regardle of the mixed-mode loading condition. The creation of a new (crack) urface i baed on a local damage meaure. The local damage i defined a the ratio of broken interaction to the total number of interaction at a material point. Finite Element Analyi (FEA) with traditional element uffer from the following hortcoming: () The interface between diimilar material i aumed to have zero thickne without any pecific material propertie; however, it preent a weak link it i uually the location of failure. Therefore, it fail to appropriately model the interface between diimilar material. () Failure i a dynamic proce, it require remehing. It i computationally cotly, the crack growth i guided baed on the linear elatic fracture mechanic (LEFM) concept. It break down when multiple complex crack growth pattern develop. () Stre train field are dicontinuou, meh refinement doe not necearily enure accurate tre field near geometric material dicontinuitie. (4) Finally, crack nucleation i not reolved. The analyi alway require a pre-exiting crack. Keyword: Peridynamic, Timohenko Beam, Mindlin Plate, Tranvere Shear Deformation, Diperion Relationhip

2 In order to remedy or remove thee hortcoming, Coheive Zone Element (CZE) extended Finite Element (XFEM) were developed; however, CZE require a priori knowledge of the crack path. In a complex analyi, it i not practical the reult are dependent on the meh (tructured or untructured). Furthermore, the reult are enitive to the trength parameter in the traction-eparation law of the coheive zone model. Determination of thee parameter poe additional uncertaintie. Although XFEM removed uch uncertaintie, it till require an external criteria for crack propagation. Thu, the reult depend on the criteria employed in the analyi. It alo break down when multiple complex crack growth pattern develop. The PD theory overcome the weaknee of the exiting method, it i capable of identifying all of the failure mode without implifying aumption. The PD methodology effectively predict complex failure in complex tructure under general loading condition. Damage i inherently calculated in a PD analyi without pecial procedure, making progreive failure analyi more practical. An extenive literature urvey on PD i given in a recently publihed textbook by Madenci Oterku (04). A comparion tudy between peridynamic, CZE, XFEM technique i given by Agwai et al. (0). They howed that the crack peed obtained from all three approache are on the ame order; however, the fracture path obtained by uing peridynamic are cloer to experimental reult with repect to other two technique. Another advantage of PD i it length-cale parameter, which doe not exit in claical continuum mechanic. Such a length-cale parameter give PD a nonlocal character. Hence, it allow the capture of phyical phenomena not only at the macro-cale, but alo at variou other cale. Thi characteritic can be etablihed through the PD diperion relation. The claical theory i only valid for a pecial cae of a long wavelength limit; however, the PD how diperion behavior imilar to that oberved in real material. Hence, it i proven to be acceptable to perform multi-cale analyi imulation. Although peridynamic i a powerful technique in failure analyi ha an internal length cale, it i uually computationally more expenive, epecially with repect to finite element analyi. The computational time can be ignificantly reduced by uing parallel computing either by uing a CPU (Central Proceing Unit) /or GPU - baed (Graphic Proceing Unit) architecture for which PD equation of motion are very uitable. However, modeling very large detailed tructure uch a aeropace marine vehicle can till be computationally deming. Hence, in uch cae it i neceary to reduce computational time through tructural idealization. Taylor Steigmann (0) propoed a peridynamic plate model baed on bondbaed formulation by uing an aymptotic analyi. The formulation i capable of capturing outof-plane deformation for thin plate. Moreover, O Grady Foter (04a,b) developed a nonordinary tate-baed peridynamic model for Euler-Bernoulli beam Kirchhoff-Love plate formulation by diregarding the tranvere hear deformation. Therefore, the focu of thi tudy i preent a new PD formulation for thin or thick beam plate by taking into account tranvere hear deformation, i.e. a Timohenko beam Mindlin plate, repectively, baed on an original (bond-baed) PD formulation. Moreover, PD diperion relation are obtained compared againt thoe from claical theory.

3 The following ection preent the PD kinematic for a Timohenko beam a Mindlin plate, the correponding PD equation of motion a well a the PD material parameter. They alo decribe the procedure to determine the urface correction factor for thee parameter the application of the boundary condition determination of the critical curvature critical hear angle in term of the fracture mechanic parameter. Finally, the correponding diperion relation are derived compared with the claical theory. The numerical reult etablih the validity of the preent formulation by conidering imple benchmark problem.. Peridynamic kinematic At any intant of time, every point in the beam or plate denote the out-of-plane deflection rotation of a material particle, thee infinitely many material point (particle) contitute the beam or the plate. In the undeformed tate of the body, each material point i identified by it coordinate, x ( k ) with ( k,,..., ), i aociated with an incremental volume, V ( k ), a ma denity of ( x ( k )). According to the PD theory introduced by Silling (000), the motion of a body i analyzed by conidering the pair-wie interaction between material point x ( k ) x( j). The interaction between the material point i precribed through a micropotential that depend on the deformation contitutive propertie of the material. Alo, a material point i only influenced by the other material point within a neighborhood defined by it horizon,. The micropotential are zero for material point outide it horizon. Each material point can be ubjected to precribed body load, diplacement, or velocity, reulting in motion deformation... Beam kinematic A hown in Fig., the tranvere hear angle, ( j) ( k ), of material point j k can be expreed a w w gn x x ( j) ( k) ( j) ( j) ( j) ( k ) ( j)( k) w w gn x x ( j) ( k) ( k ) ( k ) ( j) ( k ) ( j)( k) (a) (b) in which w ( j), ( j) w ( k ), ( k ) repreent the out-of-plane deflection rotation of material point j k, repectively. The ditance between the material point j k i pecified a x x. ( j)( k ) ( j) ( k) Conidering the material point k a the point of interet, the tranvere hear angle, ( k)( j), ariing from the interaction between material point j k can be defined a the average of the tranvere hear angle at thee material point in the form

4 w w gn x x ( j) ( k ) ( j) ( k ) ( k )( j) ( j) ( k ) ( j)( k) () The curvature between the material point j k can be defined a ( k)( j) ( j)( k) ( j) ( k) () Figure. Original deformed configuration of a Timohenko beam. When conidering the material point j a the point of interet, the tranvere hear angle curvature for the interaction between the material point j k become w w gn x x ( k ) ( j) ( k ) ( j) ( j)( k ) ( j) ( k ) ( j)( k) or ( j)( k) ( k)( j ) (4a) ( j)( k) ( j)( k) ( k) ( j) or ( j)( k) ( k)( j ) (4b).. Plate kinematic A illutrated in Fig., ( j) ( k ) repreent the rotation with repect to the line of action between the material point j k. Conidering the material point k a the point of interet, the curvature, ( k)( j), with repect to the line of action between the material point j k can be defined a 4

5 ( k)( j) ( j) ( k) (5) ( j)( k) Through coordinate tranformation, the rotation curvature with repect to the line of action between the material point j k can be decompoed a co in (6a) ( j) x( j) y( j) co in (6b) ( k) x( k) y( k ) x( j) x( k ) y( j) y( k ) ( k)( j) co in x( j) x ( k ) y( j) y ( k ) (7) in which x co ( j) x( k) ( j)( k) y in ( j) y( k) ( j)( k), with ( j)( k) repreenting the ditance between material point j k. The lope with repect to the line of action between the material point j k can be expreed a ( k)( j) w w ( j) ( k) (8) ( j)( k) in which w ( j) w ( k ) repreent the out-of-plane deflection at material point j k. A ketched in Fig., the tranvere hear angle at material point j k can be expreed a (9a) ( j) ( k)( j) ( j) (9b) ( k) ( k)( j) ( k) 5

6 Figure. Original deformed configuration of a Mindlin plate. Conidering the material point k a the point of interet, the tranvere hear angle, ( k )( j ), between material point j k can be defined a the average of the hear angle at thee material point in the form ( k)( j) w w (0a) ( j) ( k ) ( j) ( k ) ( j)( k) or ( k)( j) x( j) co y( j) in x( k ) co y( k ) in w w ( j) ( k) (0b) ( j)( k) When conidering the material point j a the point of interet, the tranvere hear angle curvature for the interaction between the material point j k become ( j)( k) w w ( k ) ( j) ( k) ( j) ( j)( k) (a) ( j)( k) ( k) ( j) (b) ( j)( k) It i worth noting that ( )( ) ( )( ) j k k j ( j)( k) ( k)( j ). 6

7 . Peridynamic equation of motion The PD equation of motion at material point k can be derived by applying the principle of virtual work t ( T U) dt 0 () t0 where T U repreent the total kinetic potential energie in the beam or plate. Thi principle i atified by olving for the Lagrange equation d L L 0, () dt q( k) q( k) where the vector q ( k ) include the independent field variable (out-of-plane deflection rotation), the Lagrangian L i defined a L T U. (4).. Beam equation of motion The total kinetic energy of the ytem due to bending tranvere hear deformation can be written a I T w( k ) ( k ) V( k ) & & k A (5) in which V ( k ) repreent the infiniteimally mall incremental volume of material point k the dot (.) above a parameter denote differentiation with repect to time. The parameter, I, A correpond to ma denity of the material, the moment of inertia, the cro ectional area of the beam, repectively. The total potential energy of the ytem can be obtained by umming the micropotential, w (k)( j) ( (k)( j) ) w ˆ (k)( j) ( (k)( j) ), between material point ariing from bending tranvere hear deformation U V b V k j w w (k)( j) (k)( j) ( j)(k) ( j)(k) ( j) (k) (k) (k) ˆ ˆ ˆ V b w V k j w w (k)( j) (k)( j) ( j)(k) ( j)(k) ( j) (k) (k) (k) (6) 7

8 in which b (k) ˆb (k) repreent the body moment body force at material point k. The independent variable are out-of-plane deflection rotation of the material point, w (k) (k). Hence, the reulting Euler-Lagrange equation can be expreed a d L L dt w w ( k) ( k) 0 (7a) d L L dt ( k) ( k) 0 (7b) Uing the Lagrangian definition L T U performing differentiation yield the following equation of motion ˆ ( k )( j) ˆ ( j)( k) w ˆ ( k ) ( j)( k ) f( k)( j) ( j)( k) f( j)( k) V( j) b( k) 0 (8a) j w( k) w( k) I A ( k )( j) ( j)( k ) ( k ) ( j)( k ) f( k )( j) f( j)( k ) V( j) j ( k) ( k) ˆ ( k )( j) ˆ ( j)( k ) ( j)( k ) f( k )( j) f( j)( k ) V( j) b( k ) 0 j ( k) ( k) (8b) ˆ k ˆ j in which f ( )( j ), f ( )( k ), f ( k)( j), f ( j)( k) are defined a fˆ wˆ wˆ ˆ ( k )( j ) ( k )( j ) ( )( ) ( )( ) ( )( ), j k j k k j f( j)( k ) ( j)( k ) ( k )( j) ( j)( k ) ( j)( k ) (9a,b) f w w ( k )( j ) ( k )( j ) ( )( ) ( )( ) ( )( ), j k j k k j f( j)( k ) ( j)( k ) ( k )( j) ( j)( k ) ( j)( k ) (0a,b) They repreent the peridynamic interaction force between material point j k ariing from tranvere hear deformation bending. For a linear material behavior, they can alo be defined in the form 8

9 ˆ fˆ c f c (a,b) ( k)( j) ( k)( j) ( j)( k) ( j)( k) f c f c (a,b) ( k)( j) b ( k)( j) ( j)( k) b ( j)( k) in which c c b are the peridynamic material parameter aociated with the tranvere hear deformation bending of the beam, repectively. Invoking Eq. (-4) ubtituting for the peridynamic force from Eq. () () in Eq. (8) reult in w( j) w( k ) ( j) ( k ) w( ) gn ( ) ( ) ˆ k c x j x k V( j) b( k ) (a) j ( j)( k) I w w c V c x x V b A ( ) ( ) j k ( j ) ( k ) gn ( j ) ( k ) (b) ( k ) b ( j) ( j) ( k ) ( j)( k ) ( j) ( k) j ( j)( k ) j ( j)( k) A mentioned earlier, if the peridynamic interaction are limited within the horizon, then thee equation can alo be written in an integral form a,,,, w x t w x t x t x t wx, t c gn ˆ x x dv bx, t (4a) H I x, t x, t, b A x t c dv H w x, t w x, t x, t x, t c gn x x dv b x, t H (4b) A derived in Appendix A, the PD material contant c c b can be expreed in term of the hear Young moduli, G E, a c kg A E I A c b 4 k G A (5a,b) 9

10 in which k i the hear correction factor, i equal to 5 / 6 for rectangular cro ection... Plate equation of motion The total kinetic energy of the ytem due to bending tranvere hear deformation can be expreed a T u v w V ( k ) ( k ) ( k) ( k) (6) k Invoking the repreentation of the in-plane diplacement component in term of rotation a u( k) zx ( k) v( k) z y( k) performing integration through the thickne of the plate reult in T w z z dz A or h ( k ) x( k ) y( k ) ( k ) (7a) k h h h T h w( k ) x( k ) y( k ) A( k ) k (7b) in which A ( k ) repreent the infiniteimally mall incremental area of each material point h repreent the thickne of the plate. The total potential energy of the plate can be obtained by umming the micropotential, w, between material point ariing from bending tranvere w (k)( j) (k)( j) ˆ (k)( j) (k)( j) hear deformation b U k j w w V V h bˆ ˆ ˆ V w V k j w w h (k) (k)( j) (k)( j) ( j)(k) ( j)(k) ( j) (k) (k) (k) (k)( j) (k)( j) ( j)(k) ( j)(k) ( j) (k) (k) (8) in which b (k) ˆb (k) repreent the reultant body moment body force at material point k. The independent variable are out-of-plane deflection rotation of the material point, w ( k ) ( k ). Hence, the reulting Euler-Lagrange equation can be expreed a 0

11 d L L dt w w ( k) ( k) 0 (9a) d L L dt ( k) ( k) 0 with xy, (9b) Uing the Lagrangian definition L T U, performing the differentiation while invoking Eq. (6, 7) (0, ), ubtituting from Eq. (9-) yield the following equation of motion w( j) w( k ) x( j) x( k ) y( j) y( k ) hw ˆ ( k ) c co in V( j) b( k ) (0a) j ( j)( k) h x( j) x( k ) y( j) y( k ) x( k ) cb co in co V ( j) j ( j)( k ) ( j)( k ) w w c V b ( j) ( k ) x( j) x( k ) y( j) y( k ) ( j)( k ) co in co ( j) x( k ) j ( j)( k) (0b) h x( j) x( k ) y( j) y( k ) y( k ) cb co in in V ( j) j ( j)( k ) ( j)( k ) w w c V b ( j) ( k ) x( j) x( k ) y( j) y( k ) ( j)( k ) co in in ( j) y( k ) j ( j)( k) (0c) Note that the peridynamic interaction are limited within the horizon of material point. A derived in Appendix A, the PD material contant c c b can be expreed in term of the hear Young moduli, G E, a c 9E 4 k c b E h k (a,b) in which k i the hear correction factor that can be choen baed on the frequency of the lowet thickne hear mode a The PD material parameter c b k,. c are determined for a material point whoe horizon i completely embedded in the material. For thee material point, both claical peridynamic

12 Strain Energy Denitie (SED) are equivalent. However, the PD material parameter c b c require correction if the material point i cloe to boundarie. The correction of the material parameter i achieved by numerically integrating the SED at each material point inide the body for imple loading condition comparing them to their counterpart obtained from claical olution. SED i compoed of bending tranvere hear deformation. Therefore, the parameter c b i corrected by g b baed on the SED due to the bending, c by g due to the tranvere hear deformation. Their derivation i explicitly decribed in Appendix B. 4. Boundary condition Unlike the local theory, the boundary condition are impoed through a nonzero volume of fictitiou boundary layer. Thi neceity arie becaue the PD field equation do not contain any patial derivative; therefore, contraint condition are, in general, not neceary for the olution of an integro-differential equation. However, uch condition can be impoed by precribing contraint through a fictitiou boundary layer. In order to apply a diplacement or rotation contraint, a fictitiou boundary layer, R c, i introduced outide the actual material, a hown in Fig.. The ize of thi layer i equivalent to the horizon. An external load, uch a a moment or a tranvere load, can be applied in the form of body load through a layer within the actual material, R. Thi layer can have a thickne of a ingle layer of material point if the dicretization i done by uing a mehle approach (Madenci Oterku, 04). (a) (b) Figure. Application of boundary condition in peridynamic: (a) beam (b) plate. 5. Peridynamic diperion relation Peridynamic diperion relation are compared againt thoe of the claical Timohenko Mindlin plate theorie. In the derivation of thee diperion relation, the wave number, the wave frequency phae velocity of the wave are denoted by,,, repectively. The relationhip among thee parameter i. The compreional hear wave peed are defined by E G, repectively, where G E are the hear Young c

13 moduli of the material. relationhip. The wave number i related to the half-wavelength,, by the 5. Beam diperion relation Diperion relation are determined by conidering a wave propagating in the x-direction. Therefore, wave olution for material point located at x x can be expreed a i x t, w x, t w e 0 i x t, x, t e 0 i x t gn( x x) w x t w0e (a) i x t gn( x x) x t 0e (b) in which i the phae difference between material point located at x x, 0 w 0 repreent the amplitude of thee wave. Subtituting thee wave olution into the PD equation of motion given in Eq. (4) lead to a homogeneou et of equation for 0 w 0. For a nontrivial olution to exit, the determinant of their coefficient matrix mut vanih, reulting in the wave diperion relation a Ac B Ac B Ac B Ac B Ac B b I A 0 () where the term B i with i,, are explicitly given in Appendix C; they are dependent on the phae difference the horizon. A explained in Appendix C, for long wavelength (or mall wave number, ), the reulting wave diperion relation i the ame a that of the claical theory (Rei, 978; Amirkulova, 0). A expected, both theorie yield the ame relationhip for long wavelength. For pecified value of E 00 GPa, 7850 kg/m, k 5 / 6, h 7 0 m, 0., finite 8 horizon ize, 0 m, the evaluation of the determinant without any implification lead to the variation of the wave frequency,, a a function of the wave number,, for the firt econd mode a hown in Fig. 4. Although both mode involve a combination of tranvere angular diplacement, the firt mode i dominated by tranvere motion the econd mode i dominated by angular motion (Rei, 978). For both mode, the PD wave diperion level off a the wave number increae which i a well-known behavior oberved in experimental tudie (Eringen, 97; Weckner Silling, 0). However, the wave frequency alway increae linearly according to the claical theory (CT).

14 (a) (b) Figure 4. Comparion of PD CT wave frequency diperion: (a) firt mode (b) econd mode. The variation of the normalized phae peed, / k a a function of wave number,, for the firt mode i hown in Fig. 5a. A hown in Fig. 5a, both PD CT predict zero peed in the limit a approache zero. The phae peed of the CT reache a contant value cloe to the hear wave peed of a bar for hort wavelength (or relatively large wave number). For the econd mode, both theorie predict comparable reult for the long wavelength (or relatively mall wave number) while claical phae peed reache the compreional wave peed, c, of a bar for hort wavelength (or relatively large wave number), a depicted in Fig. 5b. However, the phae peed decreae a the wave number increae according to the PD theory a oberved in real material. (a) (b) Figure 5. Comparion of PD CT wave peed diperion: (a) firt mode (b) econd mode. 5.. Plate diperion relation A in the cae of a beam, the diperion relation for a plate can be obtained by conidering a wave propagating in the x-direction. Therefore, wave olution for material point located at x x can be expreed a 4

15 x, 0 i x t, i x t, t x0e x x, i x t, t e x, w t w e x y in which x0 y0 i x t co w x t w e (4a) y 0 i x t co t e x (4b) x0 i x t co t e x (4c) y0 co i the phae difference between material point located at x x, w 0, y0 repreent the amplitude of thee wave. Subtituting thee wave olution into the PD equation of motion given in Eq. (0) lead to a homogeneou et of equation for x0, y0, w 0. For a nontrivial olution to exit, the determinant of their coefficient matrix mut vanih, reulting in the wave diperion relation a c M h c M c M h cm 6 cbm 4 cm 7 cbm 5 cm 8 0 h cm0 cbm 5 cm 8 cbm 9 cm (5) where the term M i with i,..., are explicitly given in Appendix C; they are dependent on the phae difference the horizon. A explained in Appendix C, for long wavelength (or mall wave number, ), the reulting wave diperion relation i the ame a that of the CT (Soedel, 004). Thu, both theorie give the ame relationhip for long wavelength. For pecified value of E 00 GPa, 7850 kg/m, 5 h m, k 5 (6 ), /, 8 finite horizon ize, 0 m, the evaluation of the determinant without any implification lead to the variation of the wave frequency,, a a function of the wave number,. Figure 6 how comparion of nondimenionalized phae peed ( / k ) a a function of wave number / ( / ) for the firt three mode; lowet flexural mode, thickne-hear mode, thickne-twit mode. A oberved in Fig. 6a, both the claical PD theorie etimate zero peed in the limit a wave number,, approache zero, wherea the claical theory phae peed nearly approache the Rayleigh urface wave peed, 0.974, for Poion ratio of 0.0 a wave number,, increae (Stephen, 997). In Fig. 6b 6c, both theorie etimate comparable reult for the long wavelength (or relatively mall ). However, PD theory capture the feature of real material that phae velocity decreae a the wave number increae. Alo comparion of wave frequency diperion for increaing wave number are hown in Fig. 7. A a characteritic of real material, diperion curve of the peridynamic theory for all mode level

16 off a the wave number increae exceed a value of (Silling, 000). Thu, PD theory capture the experimentally oberved feature of real material, which are alway diperive a a reult of long-range force. (a) (b) (c) Figure 6. Comparion of wave peed diperion: (a) lowet flexural mode, (b) thicknehear mode, (c) thickne-twit mode. (a) (b) 6

17 (c) Figure 7. Comparion of wave frequency diperion: (a) lowet flexural mode, (b) thickne-hear mode, (c) thickne-twit mode, 6. Critical curvature critical angle In order to include failure in the material repone, the repone function in the governing equation for the plate can be modified through a hitory-dependent calar value function, H ( x x, t ) a j k j k fˆ c H x x, t (6a) ( k)( j) ( k)( j), j k f c H x x t (6b) ( k)( j) b ( k )( j) It i defined a, j k H x x t c if x x, t ' x x, t ' k j j k k j j k c 0 otherwie (7) Critical curvature angle value can be expreed in term of the critical energy releae rate of the material. In order to find thee relationhip, the total train energy required to remove all of the interaction acro a newly created crack urface, A, hown in Fig. 8, mut be determined equated to the correponding critical energy releae rate value. 7

18 Figure 8. Interaction between two material point whoe line of action croe the crack urface. The total bending train energy required to remove all of the interaction acro the new crack urface A i W c x x V V (8) K J c b b c ( j ) ( k ) ( k ) ( j ) k j c The total bending train energy, W b, can be equated to the mode-i critical energy releae rate, G, in order to determine the value of the bending critical curvature a Ic G Ic c V V K J x x b c ( j ) ( k ) ( k ) ( j ) k j A (9) Baed on the expreion derived by Silling Akari (005) Madenci Oterku (0) for the critical energy releae rate, it i evident that K J x k j x ( ) ( ) V V j k ( k ) ( j ) 4 A h (40) Finally, the critical curvature can be expreed a c 4G Ic 4 b ch (4) Similarly, the total hear train energy required to remove all of the interaction acro the urface A i W c x x V V (4) K J c c ( j ) ( k ) ( k ) ( j ) k j 8

19 Thi total hear train energy, G IIIc c W, can be equated to the mode-iii critical energy releae rate,, in order to determine the value of the critical hear angle a G IIIc c V V K J x x c ( j ) ( k ) ( k ) ( j ) k j A (4) By uing the relationhip given in Eq. (40), the critical hear angle can be obtained a c 4G IIIc 4 ch (44) 7. Reult A part of the numerical reult, imple tatic loading condition are conidered firt to compare the PD prediction with the analytical olution. A plate with a center crack under bending i conidered next. In order to obtain the tatic olution, the adaptive dynamic relaxation technique given by Kilic Madenci (00) i ued, the horizon ize i choen a.05 x where x i the uniform grid pacing. 7.. Timohenko beam ubjected to pure bending tranvere loading The length of the beam i L m, with a cro ectional area of A 0. 0.m. It Young modulu i pecified a E 00GPa. Only a ingle row of material (collocation) point are neceary to dicretize the beam. The ditance between material point i x 0.0m. The left edge i contrained by introducing a fictitiou region with a ize of. The beam i firt ubjected to bending moment then tranvere loading, a hown in Fig The loading i applied to a ingle material point at the right end of the bar a a body load of 9 9 b.0 N/m for bending bˆ 50 N/m for the tranvere loading, correponding to an applied moment of 5 M.0 Nm a tranvere load of P N, repectively. 9

20 (a) (b) Figure 9. (a) Timohenko beam ubjected to pure bending (b) it dicretization. (a) 0

21 (b) Figure 0. (a) Timohenko beam ubjected to tranvere loading (b) it dicretization. The analytical olution for the tranvere diplacement the rotation due to pure bending are given a Mx w (45a) EI Mx (45b) EI A depicted in Fig., the PD analytical olution are in good agreement. (a) (b) Figure. Variation of (a) rotation (b) tranvere diplacement along a Timohenko beam ubjected to pure bending loading. Under the tranvere loading cae, the analytical olution for the tranvere diplacement the rotation are given a

22 w Px P x kga EI Lx (46a) P Lx x (46b) EI A hown in Fig., the PD the analytical olution alo agree well with each other. Thi verifie that the PD equation of motion accurately capture the deformation behavior of a Timohenko beam. (a) (b) Figure. Variation of (a) rotation (b) tranvere diplacement along a Timohenko beam ubjected to tranvere force loading. 7.. Mindlin plate ubjected to pure bending tranvere force loading A hown in Fig. 4, the length width of the plate i LW m with a thickne of h 0. m. The Young modulu of the plate i pecified a E 00GPa. Only a ingle row of material (collocation) point in the thickne direction i neceary to dicretize the domain. The ditance between material point i x 0.0m. The left edge i contrained by introducing a fictitiou region with a ize of x. The loading i applied to a ingle row of material point at 8 the right end of the plate a a reultant body load of bx. 0 N/m for bending ˆ 8 b 5 0 N/m for the tranvere loading.

23 (a) (b) Figure. (a) Mindlin plate ubjected to pure bending loading (b) it dicretization.

24 (a) (b) Figure 4. (a) Mindlin plate ubjected to tranvere force loading (b) it dicretization. The peridynamic olution of the tranvere diplacement the x-direction rotation for bending moment tranvere loading cae are compared with finite element (FE) olution by uing a hell element, which i uitable for thick hell tructure, available in commercial oftware, ANSYS. A depicted in Fig. 5 6, the PD the FE olution agree well with each other. Thi verifie that the PD equation of motion given in Eq. (0a-c) can accurately capture the deformation behavior of a Mindlin plate. 4

25 (a) (b) Figure 5. Variation of (a) rotation (b) tranvere diplacement along a Mindlin plate ubjected to pure bending loading. (a) (b) Figure 6. Variation of (a) rotation (b) tranvere diplacement along a Mindlin plate ubjected to tranvere force loading. 7.. Mindlin plate with central crack Crack growth in a quare plate with an initial central crack aligned with the x-axi, a hown in Fig. 7, i analyzed. The length width of the plate are LW m with a thickne of h 0. m. Plate thickne to crack length ratio i h a 0.5, which ha the propertie of a thick plate, where a i the initial crack length. The Young modulu of the plate i pecified a E.7 GPa the hear modulu i G.GPa. Only a ingle row of material (collocation) point in the thickne direction i neceary to dicretize the domain. The ditance between material point i x 0 m. The horizon ize i choen a.05 x. 5

26 (a) (b) Figure 7. (a) Mindlin plate with a central crack ubjected to pure bending loading (b) it dicretization. The material i choen a polymethyl-methacrylate (PMMA), which how a brittle fracture behavior. Mode-I fracture toughne of thi material i given a.mpa m (Ayatollahi Aliha, 009) Mode-III fracture toughne i given a 7.684MPa m (Farhad Flueler, 998). The critical energy releae rate of mode-i mode-iii can be found from G Ic KIc E G IIIc KIIIc (47) G 6

27 In order to how imple mode-i crack growth, a bending moment loading i applied through a ingle row of material point at the horizontal boundary region of the plate. Small increment of reultant body loading of 50 N/m are induced in order to achieve table crack b y growth. Under the applied uniform bending, the crack tart to grow at the end of nearly time tep, a expected, it propagate toward the edge of the plate, a hown in Fig. 8a-d. (a) (b) (c) (d) Figure 8. Crack evolution at (a) th, (b) th, (c) th, (d) th time tep. 8. Final remark Thi tudy preented the PD equation of motion for a Timohenko beam Mindlin plate. PD diperion relationhip were alo obtained it wa oberved that they are imilar to the one oberved in real-material, which cannot be predicted by uing claical theory. After 7

28 etablihing the validity of the PD prediction, the expreion for critical curvature hear angle value in term of mode-i mode-iii critical energy releae rate of the material were alo utilized to predict crack growth in a plate under pure bending. Reference A. Agwai, I. Guven E. Madenci, 0, Predicting crack propagation with peridynamic: a comparative tudy, Int. J. Fract., Vol. 7, pp F. A. Amirkulova, 0, Diperion Relation for Elatic Wave in Plate Rod, M.Sc Thei, Rutger, The State Univerity of New Jerey. M. R. Ayatollahi M. R. M. Aliha, 009, Analyi of a new pecimen for mixed mode fracture tet on brittle material, Eng. Fract. Mech., Vol. 76, pp A. C. Eringen, 97, Linear theory of nonlocal elaticity diperion of plane wave, Int. J. Eng. Sci., Vol. 0 (5), pp M. Farhad P. Flueler, 998, Invetigation of mode III fracture toughne uing an anticlatic plate bending method, Eng. Fract. Mech., Vol. 60, pp B. Kilic E. Madenci, 00, An adaptive dynamic relaxation method for quai-tatic imulation by uing peridynamic theory, Theor. Appl. Fract. Mec., Vol. 5, pp E. Madenci E. Oterku, 04, Peridynamic Theory It Application, Springer, New York. J. O Grady J. Foter, 04a, Peridynamic beam: A non-ordinary, tate-baed model, Int. J. Solid Struct., Vol. 5, pp J. O Grady J. Foter, 04b, Peridynamic plate flat hell: A non-ordinary, tate-baed model, Int. J. Solid Struct., Vol. 5, pp M. de A. e S. Rei, 978, Wave Propagation in Elatic Beam Rod, Ph.D. Diertation, Maachuett Intitute of Technology. S. A. Silling, 000, Reformulation of elaticity theory for dicontinuitie long-range force, J. Mech. Phy. Solid, Vol. 48, pp S. A. Silling E. Akari, 005, A mehfree method baed on the peridynamic model of olid mechanic, Comput. Struct., Vol. 8, pp W. Soedel, 004, Vibration of Shell Plate (rd ed.). New York : Marcel Dekker. M. Taylor D. J. Steigmann, 0, A two-dimenional peridynamic model for thin plate, Math. & Mech. of Solid, pp.. 8

29 N. G. Stephen, 997, Mindlin Plate Theory: Bet Shear Coefficient Higher Spectra Validity, J. Sound Vib., Vol. 0, pp O. Weckner S. A. Silling, 0, Determination of nonlocal contitutive equation from phonon diperion relation, Int. J. Mult. Comp. Eng., Vol. 9 (6), pp A. Timohenko beam Appendix A: PD material parameter A the horizon ize approache zero, PD equation mut recover their claical counterpart. Therefore, the out-of-plane deflection tranvere hear angle at material point j can be expreed, uing Taylor erie expanion diregarding higher order term, a w w w gn x x w gn x x ( j) ( k ) ( k ), x ( j)( k ) ( j) ( k ) ( k ), xx ( j)( k ) ( j) ( k ) ( k ), x ( j)( k ) ( j) ( k ) ( k ), xx ( j)( k ) (A.a,b) Subtituting from Eq. (A.a,b) into Eq. (a,b) performing the algebraic manipulation lead to ( k ) ( k ), xx ( k ), x ( j)( k ) ( j) ( k ) j w c w V bˆ (A.a) I A ( k ) cb ( k ), xx ( j)( k) V( j) c w( k), x ( k) ( k), xx ( j)( k) ( j)( k) V( j) b( k) j j 4 (A.b) Expreing the infiniteimal incremental volume, V( j) A ( j)( k), with ( j)( k) repreenting the pacing between two conecutive material point, converting the ummation to an integration a ( j)( k) approache zero, i.e., ( j)( k) d, yield, xx, x 0 w c w Ad bˆ (A.a) cb, xx Ad c w, x, xx Ad b I A (A.b) Performing the integration reult in 9

30 w Ac,, ˆ w xx x b (A.4a) I A cb c, xx A c w, x b A 8 (A.4b) Note that thee PD equation of motion have the ame form a thoe of the claical Timohenko beam equation, xx, x w k G w bˆ (A.5a) I E I, xx k Gw, x b (A.5b) A A where the hear correction factor, k, can be taken a 5 / 6 for rectangular cro ection. Finally, equating the coefficient of the independent variable in the PD equation of motion to thoe of the claical equation yield the relationhip between the PD material contant, c c b, the hear Young moduli, G E, a c kg A c b E I k G A 4 A (A.6a,b) A. Mindlin plate A the horizon ize approache zero, PD equation mut recover their claical counterpart. Therefore, out-of-plane deflection rotation at material point j can be expreed, uing Taylor erie expanion diregarding higher order term, a w( j) w( k ) w( k ), x( j)( k) co w( k), y( j)( k) in w( k), xx( j)( k) co (A.7a) w( k ), xy( j)( k ) co in w( k ), yy( j)( k ) in ( j) ( k ) ( k ), x( j)( k ) co ( k ), y( j)( k) in ( k), xx( j)( k) co ( k ), xy( j)( k ) co in ( k ), yy( j)( k) in (A.7b) 0

31 Subtituting from Eq. (A.7a,b) into Eq. (0a-c) performing the algebraic manipulation lead to hw c,,,, ˆ h w xx w yy x x y y b (A.8a) h 6 0 x h cb c x, xx x, yy y, xy ch w, x x bx (A.8b) h 6 0 y h cb c y, yy y, xx x, xy ch w, y y by (A.8c) The claical counterpart of the PD equation of motion for a Mindlin plate can be written a hw hk G w, xx w,yy x, x y,y b ˆ (A.9a) h D D x D x, xx x,yy y,xy k hg x w, x bx (A.9b) h y D y, yy D y, xx D x,xy k hg y w, y by (A.9c) where k i a correction coefficient that i introduced to account for the fact that the hear tree are not contant over the thickne. The parameter D G are the flexural rigidity hear modulu, repectively, which are defined a Eh D G E (A.0a,b) The correction factor can be choen baed on the frequency of the lowet thickne hear mode a k (A.) Finally, equating equation of the peridynamic claical theorie reveal the relationhip between the peridynamic material contant, c c b, Young modulu, E, a well a contraint on the value of the Poion ratio a

32 c 9E 4 k c b E h k (A.a,b). Appendix B: Surface correction B. Timohenko beam SED due to bending in the claical theory can be expreed a W CM b EI x k A (B.) where,x. It counterpart in PD theory can be expreed in dicretized form a cb W x V (B.) PD b k k j j k j j Conidering pure bending loading, the PD SED become cb W x V (B.) PD b k j k j j Hence, the urface correction factor for pure bending can be defined a g x b k PD Wb k CM Wb x k (B.4) x On the other h, SED due to tranvere hear deformation in the claical theory can be written a k CM W x Gk (B.5) where w,x. It counterpart in the PD theory can be expreed in dicretized form a c W x V (B.6) PD k k j j k j j

33 Conidering pure hear loading along the x-direction, PD SED become c W x V (B.7) PD k j k j j Similar to pure bending, the correction factor for pure hear loading can be obtained a g x k PD W k CM W x k (B.8) x The urface correction factor for material point j material point k can have different value; therefore, the urface correction factor for an interaction between material point j material point k can be taken a their average g b g k b j gb x g x (B.9a) k j g x g x (B.9b) B. Mindlin plate SED due to bending in the claical theory can be expreed a k CM W x D b x x y y xy (B.0) where x x, x, y y,y, xy x,y y,x. It counterpart in the PD theory can be expreed in dicretized form a cb W x V (B.) PD b k k j j k j j Conidering pure bending loading along the x- y-direction, i.e. M x M 0, M 0 (B.) y xy reult in curvature

34 0, 0 (B.) x y xy The claical PD SED for thi loading condition become k CM Wb x D x x y y (B.4a) cb co in W x V (B.4b) PD b k x y j k j j Hence, the urface correction factor for pure bending can be defined a g x b k PD Wb k CM Wb x k (B.5) x On the other h, SED due to tranvere hear deformation in the claical theory can be written a k CM W x Gk h x y (B.6) where x w, x x y w,y y. It counterpart in the PD theory can be expreed in dicretized form a c W x V (B.7) PD k k j j k j j Applying pure hear loading along the x-direction, i.e. Q 0 Q 0 (B.8) x reult in hear angle y 0 0. (B.9) x y Claical PD SED for thi loading condition become k x CM W x Gk h (B.0a) 4

35 c xco PD W x V (B.0b) k j k j j Similar reult can alo be obtained for pure hear loading along the y-direction a k y CM W x Gk h (B.a) c yin PD W x V (B.b) k j k j j Similar to pure bending, the correction factor for pure hear loading can be obtained a g x k PD W k CM W x k (B.) x The urface correction factor for material point j material point k can have different value; therefore, the urface correction factor for an interaction between material point j material point k can be taken a their average g b g k b j gb x g x (B.a) k j g x g x (B.b) C. Timohenko beam The term appearing in Eq. () are defined a Appendix C: Diperion relation 5

36 0 B co d (C.a) B 0 co d (C.b) 0 B iin d (C.c) A uggeted by Silling (000), in the limit of long wavelength (or mall ), thee integral can be analytically evaluated by conidering the firt three term of the Taylor erie expanion of the coine ine function 4 co( )... (C.a)! 4! 5 in( )... (C.b)! 5! With the evaluation of thee integral conidering the PD material parameter, the determinant from Eq. () can be expreed a k k mk mk pd pd k mk t 4t 576t 88t 560t k mk 4 mk mk k t 880t 9600t 8t (C.) in which i the Poion ratio. The nondimenional wave frequency wave number,, repectively, are defined a pd h pd h (C.4a,b) The non-dimenional geometric parameter, m t, are defined a Ah m I h t (C5.a,b) 6

37 Diregarding the higher order term of the horizon implifie the wave diperion relation for long wavelength limit (or mall ) a (C.6) pd pd k mk k A expected, thi equation recover the nondimenional wave diperion relation in the claical theory (CT) (Amirkulova, 0) for a long wavelength limit. It lead to four different value for wave frequency, which repreent two wave traveling to the right two wave traveling to the left of the beam. Therefore, there are two ditinct mode that can propagate in a Timohenko beam (Rei, 978). The firt mode yield zero frequency ( 0 ) when the wave number i equal to zero ( 0) k km k km 4 k ccm pd (C.7) the econd mode can be expreed a k km k km 4 k c cm pd (C.8) The wave diperion relation for long wavelength limit are hown in Fig. C. C. for the 7 pecified value of E 00 GPa, 7850 kg/m, k 5 / 6, h 0 m, 0., horizon 8 ize, 0 m. Figure C.. Claical wave frequency diperion for the firt econd mode (combine thee plot) 7

38 Note that the wave number of the econd mode i real for.6. Thi indicate that the firt mode i a propagating mode for any wave frequency, while the econd mode only propagate for wave frequencie.6 i exponentially attenuated for.6, a dicued by Rei (978). Therefore,.6 i the cut-off frequency for the econd mode. The variation of nondimenional phae peed, / k / E/ G function of the wave number are hown in Fig. C. for the firt econd mode., a a Figure C.. Claical wave peed diperion for the firt econd mode The phae peed, / k or / E/ G, increae. Hence, the phae peed for the firt mode, / k wave peed,, of the CT, ince, converge to unity a the wave number,, converge to the hear (C.9) k k Moreover, the phae peed for the econd mode, / E/ G compreional wave peed, c, of the CT, ince, converge to the E c G (C.0) C. Mindlin plate The term appearing in Eq. (5) are defined a 8

39 M M M M M i co e dv (C.a) 0 0 i co e co dv (C.b) 0 0 i co e in dv (C.c) 0 0 i co e 4 co dv (C.d) i co e in co dv (C.e) 0 0 i co co M 6 e dv (C.f) M M M 0 0 i co e 7 co dv (C.g) 0 0 i co e in co dv (C.h) i co e 9 in dv (C.i) 0 0 i co 0 in M e dv (C.j) M 0 0 i co e in dv (C.k) 0 0 in which dv denote the infiniteimal volume of a material point, which can be written in cylindrical coordinate a dv hdd. Evaluation of thee integral yield Beel function of the firt kind, J0() J(), Struve function, H0() H(). A indicated by Silling (000), in the limit of a long wavelength () or for a very mall wave number ( 0 ), the integral in Eq. (C.a-k) can be implified by uing the firt three term of the Taylor erie expanion for coine ine function co co 4 co co... (C.a)! 4! 9

40 co co 5 in( co ) co... (C.b)! 5! Subtituting the relationhip given in Eq. (C.a,b) into integration given in Eq. (C.a-k) olving the determinant equation given in Eq. (5), while ignoring higher order term of horizon ize, yield the diperion relationhip for a long wavelength limit in the PD theory k Gh h k Ghi 0 h k Ghi D k Gh 0 0 D h 0 0 k Gh (C.) with the contraint on Poion ratio ( ). Thi relationhip i equivalent to the diperion relationhip obtained from the claical theory (CT). Moreover, root of Eq. (C.) correpond to three different natural frequencie. Soedel (004) explained that the lowet of thee frequencie i the one that the tranvere deflection mode dominate other two are conidered a hear mode. Shown in Fig. C. are the nondimenionalized phae peed ( / ) diperion relationhip with the change of wave number ( h / ) for three different wave mode for the long wavelength limit while conidering the following propertie of a plate k 5 (6 ). Figure C.. Claical phae peed diperion wave frequency diperion. Variation imilar to thoe in Fig. C. comparion with Rayleigh-Lamb wave, which have the property of wave in a plate with infinite extent, can be een in Stephen (997). Alo, in Stephen (997), i named a the lowet flexural mode, a the thickne-hear mode, 40

41 a the thickne-twit mode. In Fig. C., when the wave number ( h / ) reache the value of two, which mean the wave length i now comparable with the thickne of a plate, all the phae peed become flat. Figure C. how the nondimenionalized wave frequency ( / h / ) diperion with the change of wave number ( h / ) for three wave mode of the CT or for the PD theory in the long wavelength limit. 4