UNIAXIAL COMPRESSION TEST AND STRESS WAVE PROPAGATION MODELLING USING SPH

Transcription

1 Fifth International Conference on CFD in the Proce Indutrie CSIRO, Melbourne, Autralia - December UNIAXIAL COMPRESSION TEST AND STRESS WAVE PROPAGATION MODELLING USING SPH Rajarhi DAS and Paul W. CLEARY CSIRO Mathematical and Information Science, Clayton, Victoria 8, AUSTRALIA ABSTRACT The paper demontrate the application of Smoothed Particle Hydrodynamic (SPH) for modelling compreion and tre wave propagation in elatic olid. Thi i illutrated uing a laboratory cale uniaxial compreion tet under different loading condition. To validate the SPH baed approach, the reult are compared to matching reult uing the Finite Element Method. The olution predicted by Smoothed Particle Hydrodynamic are found to agree well. Thi paper illutrate the potential of Smoothed Particle Hydrodynamic for accurate and efficient modelling of olid material that are ubjected to compreion, and of the reulting elatic wave propagation. INTRODUCTION The uniaxial tet i the mot widely ued mechanical teting proce for characteriing olid material behaviour. It i a imple and veratile method for determining material propertie in almot all kind of material application (Bradley et al., ). Uniaxial tet are alo employed to tudy localied deformation behaviour, uch a train ditribution around a notch in a pecimen (Nawrocki et al., 998). In thi tet, a tandard pecimen i gripped between the jaw of a tenile teting machine. One end of the pecimen i puhed/pulled by a moving piton and the other end i held under a fixed jaw. The compreion or elongation in the pecimen i meaured by extenometer. Computational modelling offer an improved way of undertanding the deformation behaviour, which in turn can ait in determining the tet parameter (Li and Wang, ). For example, in the cae of a planar tenion tet, the pecimen mut be in plain train condition during deformation. Thi i enured by a pecimen of high apect ratio and wide jaw grip to prevent lateral contraction. A priori modelling of the tet can help deign the optimum pecimen dimenion and type of apparatu to be ued to enure a plain train condition. Modelling can alo be ued to guide the intrumentation proce. An initial analyi can determine the area of large train in flexible material, hence can warrant location of meaurement and type of intrumentation required. For example, video extenometry technique i ueful for meauring large train and capturing necking procee (intead of uing conventional mechanical extenometer). In thi work we ue a meh-free method Smoothed Particle Hydrodynamic (SPH) (Monaghan, 99) to model uniaxial teting of laboratory cale pecimen. Numerical modelling of fracture with traditional mehbaed technique (e.g. FEM and BEM) require a very fine meh to model the theoretically ingular tre field in the neighbourhood of damage location (e.g. crack tip). One common approach to imulate crack propagation with FEM i to releae the adjacent node of the element repreenting damaged area (Aliabadi and Rooke, 99). Furthermore, a the damage evolve, the tructure need to be re-mehed to take into account the localied change in geometry. Automated re-mehing can lead to meh ditortion and inaccurate reult, epecially in gradient computation (Fernandez-Mendez et al., ). Due to the meh-free nature of SPH, the fracturing proce and the aociated change in tructural configuration can be eaily handled without the need to re-meh. SPH ue a particle baed Lagrangian approach, in which the frame of reference i attached to the moving particle. Thi give the ability to track hitory dependent propertie of the material (Cleary et al., ). In general a fracturing proce i driven by the tre-train hitory in the material. Traditional Eulerian method experience difficultie in capturing the tre-train hitory on a particle by particle bai and predicting the evolution of damage in the pecimen. The hitory tracking ability of SPH can be utilied to monitor the damage initiation and crack propagation, thu providing failure hitory of the pecimen. The dynamic of damage evolution can thu be explicitly included in the analyi. The SPH method ha been uccefully applied to model different type of metal forming procee involving large deformation (Cleary et al., ). In thi work, the aim i to etablih SPH a an effective and accurate computational method to predict tranient tre field in an elatic olid, which can then be extended to model brittle fracture under compreive load. In the initial tranient tate of uniaxial compreive loading, the elatic tre wave propagation through the pecimen affect it repone. It i important to analye tranient tructural behaviour in that the initial fluctuating tre field can trigger generation and propagation of localied damage/flaw, which can then play a critical role in initiating the fracture failure of the pecimen. The tre wave generation and propagation through the pecimen a a reult of contact excitation by a piton i therefore invetigated. The ability of SPH to model tre wave i illutrated uing a imple problem of modelling a uniaxial teter. We alo ue the fundamentally different and widely ued Finite Element Method for modelling the ame example o a to compare and correlate the FE reult with the SPH olution.

2 SMOOTHED PARTICLE HYDRODYNAMICS (SPH) METHOD A brief ummary of the SPH method i preented here. SPH ha been extended to modelling of a variety of olid deformation problem (Liberky and Petchek, 99, Wingate and Fiher, 99, Gray et al.,, and more recently by Cleary et al., ). The interpolated value of a function A at any poition r can be expreed uing SPH moothing a: Ab A ( r) = mb W ( r rb, h) () b ρb where m b and r b are the ma and the denity of particle b and the um i over all particle b within a radiu h of r. Here W(r,h) i a C pline baed interpolation or moothing kernel with radiu h, that approximate the hape of a Gauian function, but ha compact upport. The gradient of the function A i given by differentiating the interpolation equation () to give: A A, ρ b ( r) = mb W ( r rb h) b b () Uing thee interpolation formulae and uitable finite difference approximation for econd order derivative, one i able to convert parabolic partial differential equation into ordinary differential equation for the motion of the particle and the rate of change of their propertie. We now preent the differential equation governing the tructural repone and their SPH approximation. Continuity Equation From Monaghan (99), our preferred form of the SPH continuity equation i: d ρ a = mb ( va vb ) W () ab dt b where ρ a i the denity of particle a with velocity v a and m b i the ma of particle b. We denote the poition vector from particle b to particle a by r = r r, and let ( r h) Wab = W ab, be the interpolation kernel with moothing length h evaluated for the ditance r ab. Thi form of the continuity equation i Galilean invariant (ince the poition and velocitie appear only a difference), ha good numerical conervation propertie, and i not affected by denity dicontinuitie or free urface. Momentum Equation The SPH momentum equation ued for the elato-platic deformation of the olid i: i dv = dt ρ ab σ i + g () j where v i the velocity, g denote the body force, and σ i the tre tenor which can be written a: σ = P δ + S () where P i the preure and S i the deviatoric tre. Auming Hooke law with hear modulu µ, the evolution equation for the deviatoric tre S (Gray et al., ) i: where ds dt.. ik = µ ε δ ε + S Ω jk a + Ω ik b S kj () and v v +. i j = j i ε i j v v Ω = (8) j i i the rotation tenor. The following equation of tate i ued: ( ρ ρ ) () P = c (9) where ρ i the reference denity. The ubcript in the above ymbol refer to the olid tate. The bulk modulu i K = ρ and the Poion ratio ν i: c ν = ( K / µ ) ( K / µ + ) TEST CONFIGURATION SPECIFICATION () The uniaxial teter i modelled with SPH via an axially loaded rectangular pecimen of width 8 mm and height mm, a hown in Figure. The pecimen i made of an elatic material of bulk modulu of. GPa, hear modulu of. GPa and denity of kg/m. It i held fixed at the bottom by a rigid plate, and the load i applied through a piton on the top, ee Figure. Thi imulate the requiite boundary condition for a typical uniaxial compreion tet. The tre wave propagation in the pecimen wa analyed uing Smoothed Particle Hydrodynamic (SPH). The pecimen domain wa dicretied with particle of reolution mm, giving a total of, particle in the two dimenional imulation. The partial differential equation governing the tre and diplacement field in elatic olid (Equation -) were ued to evaluate the tructural repone. A cubic kernel interpolation function wa ued for thi problem. Baed on the material propertie, the time tep wa. µ. An Improved Euler explicit integration cheme wa ued for the time tepping. V Moving piton Specimen Fixed end plate Figure : Uniaxial configuration with uniform velocity loading. DIFFERENT LOADING CONDITIONS In a uniaxial tet, the load on the pecimen i uually applied o a to produce a uniform train rate. The movement of the piton i hydraulically controlled to

3 enure that the loaded end i deformed at the required contant rate, thu cauing a uniform longitudinal train in the pecimen. Thi i termed a uniform velocity loading, alo known a contant train rate loading. Thi type of loading i particularly uitable where the poibility of acceleration or deceleration of the loading piton can caue udden fluctuation in the applied preure. The motion of the piton at a pecified contant rate prevent any accelerated motion and enure a uniform rate of deformation of the loaded face of the pecimen. Another common loading mechanim i uniform preure loading, where a contant force i applied on the piton, which in turn i tranmitted to the tet pecimen. Thi exert a uniform preure on the pecimen at the loaded urface. In thi work, we conider only the uniform velocity loading condition. In thi tudy we adopt von Mie tre (Timohenko and Goodier, 98) a the criterion (tructural repone) for analying the tre field and elatic wave propagation in the pecimen. The von Mie tre combine normal and hear component of the deviatoric tre tenor at a point, and i a commonly ued criterion to ae failure (deign) trength of material. (a) t =. m Uniform Velocity Loading In thi example the piton wa moved vertically downward at a contant velocity of. mm/, while the bottom end of the pecimen wa kept fixed by placing it on a rigid plate, ee Figure. In the initial tage of loading, we oberve the tranient phenomenon of elatic tre wave propagation within the pecimen. Thi tate will be termed a the tranient tate in thi paper. Figure demontrate the nature of wave propagation through the pecimen. A a conequence of the current boundary condition, the wave initiate from the top of the pecimen and propagate downward. The rapid variation in the tre pattern in the pecimen i hown in Figure. On reaching the bottom urface, the elatic wave reflect from the rigid plate at the bottom of the pecimen. The reflected wave then propagate back upward and interfere with the (newly generated) incident wave from the top (Figure ). Thi create a wave pattern by uperpoition of the incident wave and the reflected wave (from the bottom plate). The uperpoed wave propagate further up through the pecimen and are then reflected form the moving top piton. Thee reflected wave again interact with the wave reflected from the bottom plate. The uperpoition of the wave alternately reflected from the top piton and bottom plate continue. Thi phenomenon gradually lead to a complex interacting tre wave pattern in the pecimen, a een in Figure. The variation in the amplitude of the uperpoed elatic wave lead to patial fluctuation in the von Mie tre ditribution. The reulting tre variation i hown in Figure. After ometime the interacting wave reach a teady tate with little change in the tre wave pattern with time. However, the magnitude of the tree at all point in the pecimen continue to increae uniformly. A the top piton i being puhed vertically down, the pecimen i compreed axially, which teadily increae the magnitude of the tre. (b) t =.8 m (c) t =. m Figure : Stre wave propagation from the top piton toward the bottom plate for uniform velocity loading cae.

4 (a) t =. m (a) t = m (b) t =. m Figure : Reflected wave propagation from the bottom plate and their interaction with the wave originating from the top piton for uniform velocity loading cae. The teady tate tre ditribution i hown in Figure and i extremely even. It may be noted that the corner of the pecimen, due to the preence of harp geometry change, create region of (theoretical) tre ingularitie. Thi induce and maintain localied high tree at the corner throughout the imulation, which i a phyically intuitive phenomenon. To monitor the tre variation, we elect three repreentative point in the pecimen, a hown in Figure. The rationale behind chooing thee three pecific point are: Point A ( mm, mm) lie on the vertical and horizontal plane of ymmetry (plane v-v and h-h ). Point B (. mm,. mm) lie on neither of the ymmetry plane. Point C (. mm, mm) lie on a ingle ymmetry plane (plane h-h ). Thee repreentative location characterie the tre variation taking into account the problem ymmetry. (b) t = m Figure : Steady tate repone of the pecimen with little change in tre pattern for uniform velocity loading cae. The variation of the von Mie tre a the pecimen i loaded i hown in Figure -8 for both the tranient and teady repone. The intantaneou repone of the tructure when the load i jut applied i hown in Figure for a very hort period of. m. The tre at point B i raied firt by the load (blue line in Figure ). Thi i followed by rie in tre level at point A and C, ee Figure, a the initial tre wave reache thee location. Thi i phyically intuitive a point B i near the loading (top) edge. So the initial tre wave firt reach point B and then point A and C. A point A and C are at ame ditance from the top edge, the rie in tre level at thee location i oberved almot imultaneouly. After the intantaneou harp rie and the initial ocillation in the tre level, the ubequent tranient tre pattern (up to ~. m) exhibit reduced wavine in tre variation. Thi i becaue the amplitude of the elatic wave diminihe rapidly and the repone become approximately linear, reaching a teady tate, ee Figure.

5 v method uing fundamentally different numerical concept (ued in FE and SPH). The effect of different particle reolution on the SPH olution and their comparion with the FE olution i reported in the later ection. h A y v B C x h Figure : Location of the repreentative point in the pecimen for monitoring von Mie tre variation. In a laboratory cale uniaxial tet, our primary focu i the teady tate repone of the tructure. Ideally, for a linear elatic tructure the teady tate repone hould conit of a uniform tre ditribution at variou location (apart from the loading region where contact mechanic play a dominant role and affect the local tre field). Once the ytem ha reached teady tate (t >. m), the tree at point A, B, and C are found to be the ame at any given time and vary linearly with load (and therefore time). Thi demontrate that the SPH method i producing the expected uniform (patial) tre ditribution at teady tate and linear elatic tructural behaviour (Figure 8). von Mie Stre (Pa) 8 Point A Point B Point C Time (m) Figure : Very early tre variation in repone to loading at the repreentative point. FINITE ELEMENT ANALYSIS AND VALIDATION The Finite Element Method wa alo ued to model the uniaxial tet o a to provide a high quality olution with which to compare the SPH olution. The domain of the pecimen wa mehed with quadratic (8-node) reducedintegration quadrilateral element. The finite element olution wa obtained uing an implicit dynamic analyi. The reult preented in thi ection ue a SPH particle reolution of mm, which i the ame a the element ize ued with the Finite Element Method. The meh reolution with FE and the particle reolution with SPH were kept ame o a to enable a comparative tudy of the two von Mie Stre (MPa) Point A Point B Point C Time (m) Figure : Stre variation at the three repreentative point in the pecimen a a function of time (load) in the initial tage. The ame uniform velocity boundary condition of. mm/ (vertically downward) wa impoed on the top urface of the pecimen. The bottom urface wa again contrained to be fixed (a in Figure ). von Mie Stre (MPa) 9 8 Point A Point B Point C 8 Time (m) Figure 8: Stre variation at the three repreentative point in the pecimen over longer time when the ytem ha reached teady tate. The finite element olution wa taken a the bai for evaluating the accuracy of the SPH method for tre analyi. Figure 9 how the comparion between the SPH and the FEM prediction of the von Mie tre over time at point A (from Figure ) in the pecimen. The olution agree very well for both the tranient and teady tate tage of compreion. The maximum relative (%) deviation from the FE olution were.9%,.%, and.% for point A, B, and C (in Figure ) repectively. For the tranient region the FE olution have coniderable ocillation (a een in Figure 9(a)), wherea the SPH olution are mooth and non-ocillatory, ee Figure 9(a). Thi indicate that SPH olution how more tability in capturing the initial tranient repone (elatic tre wave propagation) in the pecimen. Effect of Particle Reolution on the Stre Field The application of Smoothed Particle Hydrodynamic for tructural tre analyi i relatively new. Hence, it i intructive to perform a convergence tudy to ae the effect of particle reolution on the SPH olution. The

6 uniaxial tet wa modelled with two more different particle reolution (. mm and mm) in addition to the mm reolution cae reported o far. The tre field obtained for each particle reolution cae wa compared with the FE olution at each of the repreentative location (point A, B and C in Figure ). The reult are preented in Figure for point B. It can be een that all the particle reolution produce deired linear tre variation and how no intability. von Mie Stre (MPa) SPH FE Time (m) (a) Tranient tate at point A von Mie Stre (MPa) 9 8 SPH FE 8 Time (m) (b) Steady tate at point A Figure 9: Comparion of von Mie tre at point A in the pecimen uing SPH and FEM. The relative difference between the SPH and the FE reult are found to be SPH reolution dependent (a one hould expect). A fine particle reolution lead to a olution cloer to the correponding finite element olution. For example, the deviation from the FE olution i illutrated in Figure for point B in the teady tate region. It can be oberved that the difference grow gradually, reache a limiting value, and then doe not vary ignificantly. A expected, the coarer the particle reolution, the higher the difference between the SPH and FE olution. Furthermore, the relative difference between the SPH and FE olution i found to be approximately proportional to the quare of the particle reolution (Figure ). Thi i conitent with the econd order accuracy expected in a two dimenional problem domain. A detailed reolution tudy againt an analytical olution or a numerical olution with a coniderably fine meh will appear in a later paper. Gravity Loading One load cae wa invetigated to tudy the effect of gravity on the pecimen. The purpoe of thi tudy i to ae the tability of the inherently tranient SPH methodology to predict a teady tate olution. The pecimen wa placed on a rigid platform under gravity only, and the von Mie tre field in the pecimen wa monitored. Thi pecific example wa choen a it i a imple problem with a tatic olution. The variation in the von Mie tre in the pecimen i hown in Figure. Initially the tre level reach to the tatic tre value (a determined by the gravity), and thereafter the tre level remain eentially the ame. The magnitude of the tree at variou point depend on their ditance from the reference horizontal plane (i.e. the bottom urface of the pecimen). Thi i evident from the graded tre pattern in Figure. Furthermore, the olution i found to tabilie rapidly. Thi demontrate the tability of the current SPH implementation in predicting teady tate olution. von Mie Stre (MPa) 9 8 mm reolution. mm reolution mm reolution Finite Element 8 Time (m) Figure : Steady tate tre variation at point B for different SPH particle reolution and their comparion with the finite element olution. Difference in VM Stre (%) mm reolution. mm reolution mm reolution 8 9 Time (m) Figure : Deviation of von Mie tre from the finite element olution at point B for different SPH particle reolution. Figure : von Mie tre ditribution in the pecimen at t = m ubjected to gravity loading.

7 DISCUSSION Thi work demontrate that Smoothed Particle Hydrodynamic i able to accurately predict tre wave propagation and material deformation in uniaxial compreion tet on laboratory cale pecimen. The boundary condition on the pecimen can be impoed in a variety of way, uch a velocity baed loading, preure baed loading, and gravity loading. The effect of varying tet parameter on pecimen of variou dimenion and material can be rapidly aeed by the SPH technique a pre-proceing (mehing and boundary condition) requirement with SPH i coniderably le than thoe needed with conventional meh-baed technique, and SPH can produce good accuracy with a (relative to FE) coarer reolution. In the current SPH implementation, the loading piton and the fixed jaw (plate) are explicitly modelled, and the load and contraint are applied through them. Therefore in the preent application of SPH, the boundary condition are impoed by modelling the realitic loading or contraining agent explicitly, uch a the loading piton and the fixed plate here. Thi direct contact modelling imulate realitic tet condition better than imulated uing pecified boundary condition (uniform preure or velocity), which are commonly ued with the Finite Element Method. Indeed, when uing FEM, the imulation of contact between the urface (e.g. piton and pecimen here) involve detailed pre-proceing requirement for accurate contact definition and very fine meh to capture the interaction between the urface in contact. With SPH the modelling of contact for the purpoe of normal load tranmiion can be performed without reolving complex apect of detailed contact mechanim and without requiring a finely graded meh near the load boundarie (a needed in the Finite Element Method). Thi i particularly advantageou when imulating mechanical tet that involve urface tranmitting normal load. The typical uniaxial tet problem wa alo analyed uing the Finite Element Method. It wa found that the repone of the pecimen obtained uing SPH agreed very well with the finite element olution. CONCLUSIONS Thi tudy ha evaluated and etablihed Smoothed Particle Hydrodynamic a an effective and efficient numerical tool for tre analyi, modelling elatic tre wave propagation and imulating mechanical tet. The generation, reflection and uperpoition of the elatic wave are well captured uing SPH. The SPH olution can predict attainment of teady tate condition and how no intability. The table repone under gravity loading illutrate the ability of SPH in modelling tatic tre analyi problem. The proper load tranfer to the pecimen indicate that an SPH baed approach can be ued for imulating realitic tet condition involving a variety of load cae. Moreover, the SPH formulation ued here can provide accuracy comparable to that of FEM. Thi will enable rapid analyi of a wide range of tet cenario. eaily handle the dicontinuou large cale deformation involved in fracture problem due to it meh-le nature. Furthermore, the Lagrangian formulation make SPH well uited to imulating damage initiation and propagation by tracking the hitory of tre-train tate of the particle. Therefore, the underlying numerical concept and formulation embedded in Smoothed Particle Hydrodynamic provide an effective framework for modelling fracture problem. REFERENCES ALIABADI, M. H. and ROOKE, D. P., (99), "Numerical Fracture Mechanic": Computational Mechanic Publication and Kluwer Academic Publiher. BRADLEY, G. L., CHANG, P. C. and MCKENNA, G. B., (), "Rubber modeling uing uniaxial tet data", Journal of Applied Polymer Science, 8, (), CLEARY, P. W., PRAKASH, M. and HA, J., (), "Novel application of moothed particle hydrodynamic (SPH) in metal forming", Journal of Material Proceing Technology,, (-), -8. CLEARY, P. W., PRAKASH, M., HA, J., STOKES, N. and SCOTT, C., (), "Smooth Particle Hydrodynamic; Statu and future potential", In Proc. th International Conference on CFD in the Oil and Ga, Metallurgical & Proce Indutrie, Norway. ed. S. T. Johanen. FERNANDEZ-MENDEZ, S., BONET, J. and HUERTA, A., (), "Continuou blending of SPH with finite element", Computer and Structure, 8, 8-8. GRAY, J. P., MONAGHAN, J. J. and SWIFT, R. P., (), "SPH elatic dynamic", Computer Method in Applied Mechanic and Engineering, 9, (9-), -. LI, S.-H. and WANG, Y.-N., (), "Stochatic model and numerical imulation of uniaxial loading tet for rock and oil blending by D-DEM", Yantu Gongcheng Xuebao/Chinee Journal of Geotechnical Engineering,, (),. LIBERSKY, L. D. and PETSCHEK, A. G., (99), "Smooth particle hydrodynamic with trength of material", In Advance in the Free-Lagrange Method, ed. T. a. Crowley, Berlin: Springer. MONAGHAN, J. J., (99), "Smoothed particle hydrodynamic", Ann. Rev. Atron. Atrophy.,, -. NAWROCKI, P. A., DUSSEAULT, M. B. and BRATLI, R. K., (998), "Ue of uniaxial compreion tet reult in tre modelling around opening in nonlinear geomaterial", Journal of Petroleum Science & Engineering,, (-), 9-9. TIMOSHENKO, S. and GOODIER, J. N., (98), "Theory of elaticity", rd ed. McGraw-Hill. WINGATE, C. A. and FISHER, H. N., 99. Strength Modeling in SPHC. Lo Alamo National Laboratory, Report No. LA-UR-9-9. Thi tudy ha alo etablihed the potential trength of SPH a a numerical tool that can be ued for modelling fracture in compreion tet. The SPH olution matche the FE reult for non-fracturing cae in accuracy and can