Crack-tip stress evaluation of multi-scale Griffith crack subjected to
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1 Crack-tip stress evaluation of multi-scale Griffith crack subjecte to tensile loaing by using periynamics Xiao-Wei Jiang, Hai Wang* School of Aeronautics an Astronautics, Shanghai Jiao Tong University, Shanghai 4, China ABSTRAT Crack-tip stress evaluation has always been a problem in the frame of classical elasticity theory. Periynamics has been shown to have great avantages in ealing with crack problems. In the present stuy, we present a periynamic crack-tip stress evaluation metho for multi-scale Griffith crack subject to tensile loaing. The bon-base periynamics is use to calculate the isplacement fiel. Non-local eformation graient efinition from non-orinary state-base periynamics is use for stress calculation. Besies, a scale factor is introuce for evaluating crack-tip stress of multiscale Griffith crack. Numerical results compare with Eringen s results show that this periynamic crack-tip stress evaluation metho is vali for multi-scale cracks, an with the change of istance of material points, the evaluate crack-tip stress tens to be stable. Keywors: Crack-tip stress; Periynamics; Non-local; Multi-scale. Introuction Crack-tip stress evaluation has always been a problem in the frame of classical elasticity theory. Accoring to the results of linear elastic fracture mechanics, the cracktip stress will increase to infinite. In orer to solve this crack-tip stress singularity problem, Eringen et al. [977] propose the non-local elasticity an presente results of finite crack-tip stress. The non-local elasticity abanons the assumption that stress at
2 a point is only relate to the strain at that point. Instea, it uses a non-local stress-strain relationship, where stress at a point is relate to the strain both at that point an its surrouning horizon. Non-local elasticity has been wiely use for crack-tip stress evaluation. Zhou et al. [999] etermine the state of stress in a plate with a Griffith crack subject to the antiplane shear by using the non-local theory. Tovo et al. [7] aresse the problem of stress singularities at the tip of sharp V-notches by means of a non-local implicit graient approach. Ghosh et al. [3] evelope an integral type non-local continuum moel for epoxy from phonon ispersion ata, which can be use to regularize the stress fiel at crack tips an molecular efect cores. Jamia et al. [4] consiere the problem of a mixe-moe crack embee in an infinite meium mae of a functionally grae magneto-electro-elastic material (FGMEEM) with the crack surfaces subjecte to magneto-electro-mechanical loaings an non-local theory of elasticity is applie to obtain the governing magneto-electro-elastic equations. Silling [] erive the periynamic theory (PD) for analysis of iscontinuous problems. Periynamics has been shown to have great avantages in the simulation of crack propagation. Kilic an Maenci [9] employe the periynamic theory to preict crack growth patterns in quenche glass plates. Silling et al. [] propose a conition for the emergence of a iscontinuity in an elastic periynamic boy, resulting in a material stability conition for crack nucleation. Ghajari et al. [4] propose a new material moel for the ynamic crack propagation analysis of anisotropic materials within the framework of bon-base periynamic theory. Lee an Hong [6] presente periynamic simulation on crack branching an curving in a pre-exisitng center-notche brittle polymer. De Meo et al. [6] presente a numerical Multiphysics periynamic framework for the moelling of asorbe-hyrogen stresscorrosion cracking (SCC). The periynamic theory uses integration, instea of ifferentiation, to compute the force on a material point. The material point within a finite horizon can interact with each other, therefore the periynamic theory can be categorize as a non-local theory [Silling, ; Breitenfel et al., 4].
3 Since periynamics uses the similar non-local concept as Eringen s non-local elasticity oes, an periynamics brings out great avantageous in ealing with crack problems, a motivation is to use periynamics to evaluate the crack-tip stress as Eringen s non-local elasticity coul o. In the present paper, we focus our attention on the periynamic evaluation of crack-tip stress for multi-scale Griffith crack subjecte to tensile loaing. Firstly, in Section, we present a periynamic crack-tip stress evaluation metho for multi-scale Griffith crack subjecte to tensile loaing. Then, in Section 3, we illustrate that this periynamic crack-tip stress evaluation metho is vali for micro-scale Griffith crack uner tensile loaing. Finally, in Section 4, we exten this periynamic crack-tip stress evaluation metho to macro-scale Griffith crack uner tensile loaing an show that the periynamic evaluation result of crack-tip stress tens to be stable.. Periynamic crack-tip stress evaluation metho for multi-scale Griffith crack A periynamic crack-tip stress evaluation metho for multi-scale Griffith crack subjecte to tensile loaing is presente here. This evaluation metho has three steps: () Calculate the isplacement fiel using bon-base periynamics; () Calculate the crack-tip stress base on the non-local efinition of eformation graient from non-orinary state-base periynamics (NOSB PD) [Silling et al., 7]; (3) Evaluate the crack-tip stress of multi-scale Griffith crack through multiplying the result from step () by a scale factor. The scale factor in step (3) will be given in Section.3, an the reason for introucing this factor will be iscusse in Section Bon-base periynamic theory
4 Fig.. Periynamic notations In the present paper, bon-base periynamics propose by Silling [] was use to calculate the isplacement fiel. As shown by Silling an Askari [5], the acceleration of any material point at x in in reference configuration at time t is foun from where u an ( x) u( x, t) f ( u( x, t) u( x, t), x x) V b( x, t) () u H are isplacements at x an x x, respectively, H is the horizon zone, as shown in Fig., an ξ x x, η u u. ( x) is the ensity, an f is a pairwise force function efine as w f( η, ξ) ( η, ξ) ξ, η η () where w is a scalar name as micropotential an a material is sai to be microelastic if Eq. () is satisfie. Besies, there exists a scalar-value function wˆ( y, ξ) w( η, ξ) ξ, η y η ξ (3) Combining Eq. () an (3) an ifferentiating the latter with respect to the components of η leas to
5 ξ η f( η, ξ) f ( ξ η, ξ) ξ, η ξ η (4) From Eq. () an (3) to Eq. (4), the etaile erivation was omitte by Silling an Askari [5]. For better unerstaning of the irection of pairwise force, here we supplement the etaile erivation: ˆ ˆ(, ) (, ) w (, ) w w y y ( y, ) ξ f η ξ η ξ ξ η η y η (5) wy ˆ(, ξ) f ( y, ξ) f ( ξ η, ξ ) (6) y y ξ η ( x x u u ) ( x x u u ) (7) y y / ( x x u u ) ( x x u u ) ( x x u u ) y ( u u) / η y y ( x x u u ) ( x x u u ) ( x x u u ) ( u u) x x u u + ( x x u u) ( x x u u) ξ η = = ξ η = x + x u u ξ η ( x x u u) ( x x u u) ξ η (8) From Eq. (6) an (8), we can easily get Eq. (4). An Eq. (8) also illustrates that the irection of pairwise force is parallel to the eforme bon. For homogeneous elastic isotropic material, the scalar bon force only epens on the bon stretch, efine by s ξ η ξ ξ f in Eq. (4) (9) f cs () Comparing the periynamic strain energy ensity to the classical theory of elasticity strain energy ensity uner isotropic extension, the spring constant be expresse as c in Eq. () coul
6 8 c () 4 where is the bulk moulus an is the raius of the horizon, H. As the periynamic governing equation () is in ynamic form, the aaptive ynamic relaxation (ADR) metho propose by Kilic an Maenci [] is use for static problem. Accoring to the ADR metho, Eq. () at the n th iteration can be rewritten as U X t + c U X t = D F U U X X () n n n n n - n n n (, ) (, ) (,,, ) where D is the fictitious iagonal ensity matrix an c is the amping coefficient which can be expresse by n n T n n n T n c (( U ) K U ) / (( U ) U ) (3) in which n K is the iagonal local stiffness matrix, which is given as K ( F / F / ) / ( tu ) (4) n n n n/ ii i ii i ii i where n F i is the value of force vector n F at material point x, which inclues both the periynamic force state vector an external forces. An ii is the iagonal elements of D which shoul be large enough for numerical convergence. By utilizing central-ifference explicit integration, isplacements an velocities for the next time step can be obtaine. an n n/ n n/ (( c t) U td F ) n U = (5) ( c t) n n n/ U = U tu (6) To start the iteration process, we assume that can be starte by U an U, so the integration td F / U = (7)
7 . Stress calculation in non-orinary state-base periynamics As the stress concept is not obvious in bon-base periynamics, some efinitions from NOSB PD is use here to get the calculate value of crack-tip stress. The most important efinition use is the non-local efinition of eformation graient F [Warren et al., 9] n F( xk ) wxi xk ( Yxi xk ( xi xk ) Vi K ( x k ) (8) i n K( x ) wx x (( x x ) ( x x )) V (9) k i k i k i k i i where K is the shape tensor an wx x is the influence function efine by wx x x x () After the non-local eformation graient is got, stress can be easily calculate by using the knowlege from nonlinear continuum mechanics, as shown by Fan an Li [6] where E is the green strain tensor an ( T E F F I ) () I is the ientity matrix. Green strain is energy conjugate with secon Piola-Kirchhoff stress tensor PK σ σ PK CE () where C is the stiffness tensor, an for D isotropic problems, C is a tensor. where E is elastic moulus an is Poisson ratio. E E E C (3) E E E
8 Normally, we nee the Cauchy stress tensor σ, an PK T σ J F σf (4) Transfer Eq. (4) an we can get PK T σ Fσ F (5) J.3 The scale factor for evaluating crack-tip stress of multi-scale Griffith crack Here we irectly give this scale factor. The reason for the usage of the scale factor will be iscusse in Section 3.3. crt ( / ) (6) / PD crt where crt is the evaluate crack-tip stress an PD crt is the calculate crack-tip stress got from Eq. (5). is the istance of the material point of periynamic moel, an is the atomic istance as escribe by Eringen et al. [977]. 3. Crack-tip stress evaluation of micro-scale Griffith crack subjecte to tensile loaing In the current section, firstly we set the istance of material point in periynamic moel equals the atomic istance, an the scale factor in Eq. (6) is. We will show that at this circumstance, the evaluate crack-tip stress is vali compare with Eringen s results [977]. Seconly, we vary the istance of material point in periynamic moel, an we will see that by using the scale factor in Eq. (6), a stable an vali crack-tip stress can also be evaluate. Finally, a few iscussions will be given about the usage of this scale factor. 3.
9 Fig.. A typical Griffith crack problem For a typical Griffith crack subject to tensile loaing, as shown in Fig., Eringen [977] gave the finite crack-tip stress istribution for / l, 4 an. An an analytical crack-tip stress was given crt.73( l / ) p (7) / where l is the crack length, is the atomic istance, an p is the external loa pressure. Setting the istance of material point equals atomic istance, we use periynamics to calculate the stress fiel of a micro-scale Griffith crack uner tensile loaing. The resulting yy / p istribution is shown in Fig. 3.
10 . yy / p Y/l X/l -.3 Fig. 3. Periynamic yy / p istribution of Griffith crack subject to tensile loaing, / l Comparing with Eringen s results, the crack-tip yy / p istribution for / l, 4, an are shown in Fig. 4. It can be seen from the figure that the periynamic crack-tip stress istribution fits the Eringen s results well. The crack-tip stress istribution becomes more an more sharp with the increase of the crack length, an the stress concentration also becomes more an more serious at the crack-tip σ yy /p X/l Eringen [977] Periyanmics (a) / l
11 5 4 3 σ yy /p X/l Eringen [977] Periyanmics (b) / l σ yy /p Eringen [977] Periyanmics X/l (c) / l Fig. 4. Crack-tip yy / p istribution of Griffith crack subject to tensile loaing, / ( / ) Crack-tip stress evaluation results of micro-scale Griffith crack subject to tensile loaing are given in Table. Six kins of micro-scale Griffith crack with ifferent / l are given. Two kins of materials are iscusse. From the table, we can see that compare with Eringen s results, the evaluate periynamic crack-tip results are within
12 the acceptable error when / ( / ). For ifferent material, with the same / l, crack-tip stress is nearly the same. It may be worth noting that the atomic istance for ifferent material is ifferent. So for the same crack length, the crack-tip stress of ifferent material is ifferent. This phenomenon seems obvious, as ifferent material has ifferent elastic moulus. But we shoul note that this ifference iscusse here is from the point of atomic istance of material, which may reveals that the crack-tip stress evaluation problem is an interesting multi-scale problem, especially for macro-scale crack. Table Crack-tip stress evaluation of micro-scale Griffith crack subject to tensile loaing, p MPa, / ( / ) / l Periynamics (MPa) Eringen (MPa) Steel ( =.48Å) Error iamon ( =.54Å) Error % % % % % % % % % % % % 3. Changing the istance of material point In the previous section, we have shown that when setting / ( / ), the evaluate crack-tip stress is acceptable. Here we want to show that with the change of, the evaluate crack-tip stress are also acceptable an tren to be stable. As presente in Table, by using Eq. (6), the crack-tip stress of a Griffith crack ( l/ ) subjecte to tensile loaing is evaluate through multiplying the periynamic result by a scale factor ( / ) /. Comparing the results with Eringen s analytical result in Table, we can see that after multiplying the scale factor ( / ) /
13 , the evaluate crack-tip stresses are all within acceptable error. Besies, from Fig. 5, we can see that with the ecrease of /, the error trens to be stable. Table Crack-tip stress evaluation of micro-scale Griffith crack subject to tensile loaing, p MPa, l/, =.48Å / PD result (MPa) Evaluate result (MPa) Error % % % % % % 6.% 4.%.% Error.% -.% -4.% /(/ ) Fig. 5. Tren of the error with varying / for a micro-scale Griffith crack subject to tensile loaing, p MPa, l/, =.48Å 3.3 Discussion on the usage of scale factor ( / ) / From the previous results, we have shown that with the introucing of a scale factor ( / ) / in Eq. (6), the evaluate crack-tip stress of micro-scale Griffith crack subject to tensile loaing is vali at any /. This is especially important for crack-
14 tip stress evaluation of macro-scale Griffith crack, since for a macro-scale crack, we are unable to set the istance of material point to be equal to atomic istance. The above iscussion is purely from the point of eveloping a vali periynamic evaluation metho of crack-tip stress. From another perspective, this introucing of a scale factor ( / ) / may also reveals that the crack-tip stress evaluation problem coul be an interesting multi-scale problem, an this may be the reason why this problem can t be solve in the frame of classical elasticity. 4. Crack-tip stress evaluation of macro-scale Griffith crack subjecte to tensile loaing The periynamic crack-tip stress evaluation metho presente in Section has been prove to be vali for micro-scale Griffith crack subjecte to tensile loaing in Section 3. In the current section, we want to exten this metho to macro-scale Griffith crack subjecte to tensile loaing. For a macro-scale Griffith crack with crack length l mm, an atomic istance of material.48å, the evaluate crack-tip stresses are presente in Table 3 with the changing of /. From Fig. 6, we can easily see that with the ecrease of /, the evaluate crack-tip stress trens to be stable.
15 Table 3 Crack-tip stress evaluation of macro-scale Griffith crack subject to tensile loaing, p MPa, l mm, =.48Å / PD result (MPa) Evaluate result (MPa) 4.3E E+6.E E+6.E E+6 6.7E E+6 5.4E E+6 4.3E E+6 σ crt 9.8x 6 9.7x 6 9.6x 6 9.5x 6 9.4x 6 9.3x 6 9.x 6 9.x 6 9.x 6 8.9x 6. 5.x -9.x -8.5x -8.x -8.5x -8 /(/ ) Fig. 6. Tren of the evaluate crack-tip stress with varying / for a macro-scale Griffith crack subjecte to tensile loaing, p MPa, l mm, =.48Å 5 Conclusion Crack-tip stress of multi-scale Griffith crack subjecte to tensile loaing was evaluate by using periynamics. A periynamic crack-tip stress evaluation metho was presente. Bon-base periynamics was use to calculate the isplacement fiel. The non-local eformation graient efinition from NOSB PD was use for the calculation of crack-tip stress. A scale factor ( / ) / was introuce for evaluating
16 crack-tip stress. Numerical results illustrate that this periynamic evaluation metho is vali for both micro-scale an macro-scale Griffith crack subjecte to tensile loaing. With the changing of /, the evaluate crack-tip stress tens to be stable. Acknowlegments The authors are inebte to Prof. Hui-Shen Shen of Shanghai Jiao Tong University for his consierable support. References Breitenfel, M. S., Geubelle, P. H., Weckner, O., an Silling, S. A. [4] Nonorinary state-base periynamic analysis of stationary crack problems, Computer Methos in Applie Mechanics an Engineering 7, De Meo, D., Diyaroglu, C., Zhu, N., Oterkus, E., an Siiq, M. A. [6] Moelling of stress-corrosion cracking by using periynamics, International Journal of Hyrogen Energy, 4(5), Eringen, A. C., Speziale, C. G., an Kim, B. S. [977] Crack-tip problem in non-local elasticity, Journal of the Mechanics an Physics of Solis 5(5), Fan, H., an Li, S. [6] Parallel periynamics SPH simulation of explosion inuce soil fragmentation by using OpenMP, Computational Particle Mechanics -3. Ghajari, M., Iannucci, L., an Curtis, P. [4] A periynamic material moel for the analysis of ynamic crack propagation in orthotropic meia, Computer Methos in Applie Mechanics an Engineering 76, Ghosh, S., Kumar, A., Sunararaghavan, V., an Waas, A. M. [3] Non-local moeling of epoxy using an atomistically-informe kernel, International Journal of Solis an Structures 5(9), Warren, T. L., Silling, S. A., Askari, A., Weckner, O., Epton, M. A., an Xu, J. [9] A non-orinary state-base periynamic metho to moel soli material
17 eformation an fracture, International Journal of Solis an Structures 46(5), Lee, J., an Hong, J. W. [6] Dynamic crack branching an curving in brittle polymers, International Journal of Solis an Structures, Silling, S. A. [] Reformulation of elasticity theory for iscontinuities an longrange forces, Journal of the Mechanics an Physics of Solis 48(), Silling, S. A., an Askari, E. [5] A meshfree metho base on the periynamic moel of soli mechanics, Computers an Structures 83(7), Silling, S. A., Epton, M., Weckner, O., Xu, J., an Askari, E. [7] Periynamic states an constitutive moeling, Journal of Elasticity 88(), Jamia, N., El-Borgi, S., Rekik, M., an Usman, S. [4] Investigation of the behavior of a mixe-moe crack in a functionally grae magneto electro-elastic material by use of the non-local theory, Theoretical an Applie Fracture Mechanics 74, 6-4. Kilic, B., an Maenci, E. [9] Preiction of crack paths in a quenche glass plate by using periynamic theory, International journal of fracture 56(), Kilic, B., Maenci, E. [] An aaptive ynamic relaxation metho for quasi-static simulations using the periynamic theory, Theoretical an Applie Fracture Mechanics 53(3), Kohlhoff, S., Gumbsch, P., an Fischmeister, H. F. [99] Crack propagation in bcc crystals stuie with a combine finite-element an atomistic moel, Philosophical Magazine A, 64(4), Silling, S. A., Weckner, O., Askari, E., an Bobaru, F. [] Crack nucleation in a periynamic soli, International Journal of Fracture 6(-), 9-7. Tovo, R., an Livieri, P. [7] An implicit graient application to fatigue of sharp notches an welments, Engineering Fracture Mechanics 74(4),
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