NUMERICAL SIMULATION ANALYSIS TO ESTIMATE STRESS FIELDS ALONG THE RECTANGULAR CRACK TIP IN FOAMS

International Journal of Plastic and Polymer Technology (IJPPT) ISSN (P): 2249–6904; ISSN (E): 2249–801X Vol. 11, Issue 2, Dec 2021, 15–22 © TJPRC Pvt. Ltd.

NUMERICAL SIMULATION ANALYSIS TO ESTIMATE STRESS FIELDS ALONG THE RECTANGULAR CRACK TIP IN FOAMS PIYUSH AGRAWAL & DR. SHUSHANT SINGH* Department of Mechanical Engineering, Uttaranchal University, Dehradun ABSTRACT Dynamic - Single boundary portrayal of the break/indent tip field utilizing break mechanics boundaries like K, J or CTOD has been very incredible in progressing prescient advancements for basic or sub-basic break development. It has additionally become clear in the course of the most recent 40 years that solitary boundary approaches have restrictions, especially in managing break development peculiarities emerging from break tip protecting, regularly coming about because of the plastic area encompassing a break. Impacts of this territory on the break tip pressure field in front of the break are augmented during cyclic stacking. On account of a boundary like pressure force factor, K, which describes the break tip field through a versatile guess, it is business as usual that any arrangement of pliancy actuated conditions that bother the size of the plastic area and its related strain field lead to prescient challenges. Over the most recent 30 years, prominent spaces of action identified with such troubles incorporate short breaks, pliancy initiated conclusion, variable especially in Europe, are chipping away at the subject of portrayal of break tip stresses utilizing more than one crack mechanics boundary. Consideration has been coordinated, for instance, towards fusing the T-stress into life forecast strategies. The T-stress is the second term in a Williams-type extension of the break tip stresses and it influences the degree and state of break tip versatility. It would hence be relied upon to be compelling in pliancy related break development peculiarities and various distributions have shown this to be valid. The circumstance is additionally convoluted where a break encounters multiaxial stacking and Modes II and III crack mechanics boundaries are likewise

Original Article

and multiaxial stacking and score impacts. In this manner, an expanding number of creators and exploration gatherings,

important. Other examination bunches have zeroed in consideration on joining extra flexible break mechanics boundaries into break/indent tip portrayal, which depict the impacts of an Eshelby-type 'plastic incorporation' on a versatile pressure field.

KEYWORDS: Stress Fields, Rectangular Crack, Numerical Simulation Analysis

Received: Nov 13, 2021; Accepted: Dec 03, 2021; Published: Dec 15, 2021; Paper Id.: IJPPTDEC20212

INTRODUCTION The current paper portrays a Software Simulation procedure as of late introduced that permits one to concentrate on associations between the break and miniature underlying hindrances with an uncommon degree of straight forwardness and detail. This strategy gives a simple method for recording and breaking down the impact of the microstructure upon break development rate. It was seen that the space between progressive break tip captures connects well with the material grain size. Another fascinating perception is that in most of the cases concentrated on the breaks didn't start at the place of greatest pressure focus. This is astonishing since the traditional techniques for scored exhaustion limit examination plainly show the even balance hub as the inception and spread bearing for push-pull stacking. Crack conclusion impacts weakness break development rate and should be remembered for the plan of parts. Versatility initiated break conclusion is personally connected with the break tip plastic distortion,

www.tjprc.org

editor@tjprc.org

Piyush Agrawal & Dr. Shushant Singh*

16

which becomes lingering as the break engenders. The target here is to concentrate on mathematically the impact of break engendering on break tip fields. The transient impact seen toward the start of break engendering is connected to the solidifying conduct of material. The impact of cross section refinement is examined, and solitary conduct is apparent, which is clarified by the sharp break related with network geography, made out of a standard example of square components. The plastic zone size estimated oppositely to break flank in the lingering plastic wake is measured and contrasted and writing models. At last, the expulsion of material at the principal hub behind break tip with load cycling was noticed for plane strain state and some solidifying models in plane pressure state. The impact of ecological medias on break engendering of various materials at high and extremely high-cycle weariness (VHCF) systems is explored dependent on the weakness tests Crack spread instruments because of various break main thrusts are researched as far as crack mechanics. A model is proposed to concentrate on the connection between exhaustion life, applied pressure and material property, which mirrors the variety of weariness existence with the applied pressure, grain size, consideration size and material yield pressure in fluctuating cycle and VHCF systems. The model forecast is in great concurrence with exploratory perceptions.

Computational technique for assurance of the particular and the non-solitary pressure terms along the front of the 3D surface break is proposed. Assessment of the terms depends on complete examination between disfigurement reactions (for estimation focuses on a superficial level) acquired tentatively and from mathematical arrangements of the relating limit issue of strong mechanics. The proposed approach permits doing a satisfactory and a far-reaching evaluation of stress fields nearby the surface break front.

Contacts with sharp edges subject to oscillatory stacking are probably going to nucleate breaks from the corners, assuming that the stacking is adequately extreme. To a first estimate, the corners act like indents, where the neighborhood versatile conduct is soothed by pliancy, and which thus causes irreversibility's that lead to break nucleation, yet additionally by frictional slip.

The paper manages the three-dimensional nature and the multi-parametric portrayal of the pressure field in front of breaks around a v indent break tip math of various shapes. Limited thickness plates are thought of, under various stacking conditions. In the current review, break tip fields are researched for plastically compressible, solidifying relaxing solidifying solids. Here, plane strain and limited scope yielding conditions have been expected. Mathematical outcomes unmistakably demonstrate that close to tip field amounts of a mode I break rely extensively upon the joined impact of plastic compressibility and solidifying relaxing solidifying sort of hardness work. As the compressibility and relaxing because of the hardness work increment, the dulling at the tip takes various shapes and the zone of most extreme plastic deformity moves some distance in front of the break tip. In this manner the plastic zone in commonplace plastically compressible, solidifying mellowing solidifying solids is viewed as very unique when contrasted with yielding in solidifying metals.

While the impact of plastic dilatancy is disregarded in the traditional versatility hypothesis, numerous materials display plastic volume changes. Now and again, for instance, some plastic materials likewise show a solidifying mellowing solidifying reaction. In a new report, it has been seen that the disfigurement of the in an upward direction adjusted carbon nanotube (VACNT) columns follow flexible viscoplastic constitutive connection which joins plastic compressibility, plastic non-ordinariness and a solidifying mellowing solidifying type hardness work It is viewed as that pressure yielding and plastic volumetric strains of these materials come from an assortment of variables like essential stream instrument, Impact Factor (JCC): 2.5466

IBI Factor: 3.2

Numerical Simulation Analysis to Estimate Stress Fields Along the Rectangular Crack Tip in Foams

17

cavitation, enraging in lustrous polymers and a few other associating miniature system.

Concentrates on identified with fixed and developing breaks in plastically compressible as well as tension delicate solids have been done by numerous analysts, [3-7]. The consequences of Spitzig and Richmond [3] exhibit that for polyethylene and polycarbonate sort of polymeric materials, the stream pressure is altogether subject to hydrostatic pressure. Hwang and Luo [4] gave the answer for close to tip fields for consistently developing breaks in versatile impeccably plastic compressible materials. In view of the Drucker-Prager yield basis, Li and Pan [5], concentrated on the impact of strain touchy yielding on break tip fields for power law solidifying materials with KI field limit conditions.

Asymptotic break tip fields dependent on Drucker-Prager yield measures have likewise been introduced by Yuan and Lin [6]. Lai and Giessen [7] detailed the limited component examination on the break tip plastic zone and close to tip fields in viscoplastic lustrous nebulous polymers. Their outcomes show something else altogether of the plastic zone in front of the break tip. In this review, the mode I break tip fields are explored utilizing an isotropic, plastically compressible Finite Element Model The constitutive model with solidifying mellowing solidifying type of the hardness work is executed in limited component code. As the tip of a sharp break is dulled in the beginning phases of pliancy after a specific measure of stacking is applied, in the current review, we think about an at first dulled break tip with a root span. A round locale encompassing the break tip is thought of, notwithstanding, due to balance, limited component computations are done uniquely for a semiroundabout area as portrayed in Fig. 1. Uprooting stacking is forced on the external semi-roundabout limit as per the Kfield around the underlying break tip. The upsides of KI are slowly expanded on the example from the underlying worth.

Square components; each containing four "crossed" three-sided components, have been utilized for discritization of the semi-round district. These components with an appropriate perspective proportion and direction, are broadly used to replicate confined deformity designs at limited strains. A cross section affectability concentrate on has been done with various lattice densities (16x44, 18x54, 20x58, 22x64, 28x74 square components). However, it was viewed as that 20x58, 22x64, and 28x74 square components yielded practically indistinguishable outcomes with great refinement close to the tip, in the current review we have utilized 22x64 square components.

Square elements; each being comprised of four “crossed” triangular elements have been used for discritization of the semi-circular region. These elements with a proper aspect ratio and orientation, are extensively used to reproduce localized deformation patterns at finite strains. A mesh sensitivity study has been carried out with different mesh densities (16x44, 18x54, 20x58, 22x64, 28x74 square elements). Though, it was found that 20x58, 22x64, and 28x74 square elements yielded almost identical results with good refinement near the tip, in the present study we have used 22x64 square elements.

NUMERICAL RESULTS AND DISCUSSION We demonstrate here the effect of hardening-softening-hardening type of hardness function in the finite strain deformation at the crack tip of pressure sensitive dilatants material. It should be mentioned here that the two different materials E and G used in this study are taken from paper [2]. The constant parameters considered for the present analyses are 𝐸⁄𝜎0=100, reference strain rate C0̇ =1 and rate hardening exponent m = 0.02. Initially, in order to look at how the hardening-softening-hardening type hardness function will affect the crack tip opening displacement in plastically compressible solid, we have plotted the cracktip opening displacement of both material E and G for incompressible and www.tjprc.org

editor@tjprc.org

Piyush Agrawal & Dr. Shushant Singh*

18

compressible (𝛼𝑝 = 0.28 and 𝛼𝑝= 0.20) conditions. It appears from the plots of Figure. 2 that for very small values of Jintegral, the curves for both materials merge onto a single parabolic curve which indicates that crack opening is dominated by elastic deformations Thereafter, we can observe that the slope of the curve increases with the increase in compressibility, as expected. For 𝛼𝑝=0.20, the curves turn with a sharper rise in crack opening after some point of application of load. This change in crack tip opening displacement indicates that the opening is now controlled jointly by the action of the hardening- softening-hardening behavior and the compressibility effects.

For the material E and G, the crack tip deformation along with the accumulated plastic strain εp for incompressible and compressible condition (αp=0.28) are shown in Figure. 3 and 4, respectively. It is clear from Figure. 3(a) and 4(a) (i. e. for incompressible condition) that maximum plastic strain occurs at the crack tip and extends along the crack surface for both materials. On the other hand, it is interesting to note that for compressible for both the materials.

The crack tip blunting for the compressible condition is also entirely different as compared to the corresponding incompressible situation for both the materials E and G. For the material G and for the present compressible situation (αp=0.28), due to the more softening the zone of maximum plastic strain gets completely separated and moves away from the rectangular crack tip geometry.

As plastic deformation takes place by the initiation and subsequent propagation of bands of localized plastic shearing hence, we present the numerical results of the crack tip by contours of the accumulated plastic strain. All the x and y-coordinate are normalized by the initial half crack widthb0.

Figure 1: Distribution of plastic Strain εp at the Crack Tip in Material E; a) Compressible with J/s0b0=1.20, b) αp=0.28.

Impact Factor (JCC): 2.5466

IBI Factor: 3.2

Numerical Simulation Analysis to Estimate Stress Fields Along the Rectangular Crack Tip in Foams

19

Figure 2: Distribution of Plastic Strain εp at the Crack tip in Material E; Incompressible with J/s0b0=1.20, and αp=0.33.

For the material E and G, the crack tip deformation along with the accumulated plastic strain εp for incompressible and compressible condition (αp=0.28) are shown in Figure. 3 and 4, respectively. It is clear from Figure.

3(a) and 4(a) (i. e. for incompressible condition) that maximum plastic strain occurs at the crack tip and extends along the crack surface for both materials. On the other hand, it is interesting to note that for compressible condition and for both the materials, the tip of the plastic zone is moving in the positive x-direction in a self-similar manner. The crack tip blunting for the compressible condition is also entirely different as compared to the corresponding incompressible situation for both the materials E and G. For the material G and for the present compressible situation (αp=0.28), due to the more softening the zone of maximum plastic strain gets completely separated and moves away from the crack tip.

REFERENCES 1.

Hutchens, S. B., Needleman, A., Greer, J. R., 2012. A microstructurally motivated description of the deformation of vertically aligned carbon nanotube structures. Appl. Phys. Letter, 100, 121910-1.

2.

Mohan, N., Cheng, J., Greer, J. R., Needleman, A., 2013.Uniaxial tension of a class of compressible solids with plastic nonnormality. J. Appl. Mech., 80, 040912-1-8.

3.

Spitizig, W.A., Richmond, O., 1979. Effect of hydrostatic pressure on the deformation behavior of polyethylene and polycarbonate in tension and compression. Polymer. Eng. Sci., 19, 1129-1139.

4.

Hwang K.C., Luo X.F., 1988. Near-tip fields for cracks growing steadly in elastic-perfectly- plastic compressible material.

IUTAM Symposium on recent advances in nonlinear fracture mechanics, Caltech, Pasadena, CA, USA.

5.

Li F.Z., Pan j., 1990. Plane-strain crack-tip fields for pressure-sensitive dilatent materials. J.Appl. Mech., 57, 40-49.

6.

Yuan H., Lin G., 1993. Elastoplastic crack analysis for pressure sensitive dilatent materialsPart I: Higher-order solutions and two-parameter characterization. Int. J. Fract., 61, 295-330.

7.

Lai J., Van der Giessen E., 1997.A numerical study of crack tip plasticity in glassy polymers.Mech. Mater., 25, 183-197.

8.

Tovo, R. Livieri, P., An implicit gradient application to fatigue of sharp notches and weldments, Engineering Fracture Mechanics, 74 (2007) 515–526.

9.

Tovo, R., Livieri, P., An implicit gradient application to fatigue of complex structures, Engineering Fracture Mechanics, 75(7) (2008) 1804–1814.

www.tjprc.org

editor@tjprc.org

Piyush Agrawal & Dr. Shushant Singh*

20

10. Tanaka, K., Engineering formulae for fatigue strength reduction due to crack-like notches, International Journal of Fracture, 22 (1983) R39–R46.

11. Livieri, P. Tovo, R., Fatigue limit evaluation of notches, small cracks and defects: an engineering approach, Fatigue and Fracture of Engineering Materials and Structures, 27 (2004) 1037–1049.

12. Cristofori, A., Livieri, P., Tovo, R., An Application of the Implicit Gradient Method to welded structures under multiaxial fatigue loadings, International Journal of Fatigue, 31(1) (2009) 12–19.

13. Peerlings, R.H.J., de Borst, R., Brekelmans, W.A.M., de Vree, J.H.P., Gradient enhanced damage for quasi brittle material, International Journal of Numerical Methods in Engineering, 39 (1996) 3391–3403.

14. Dalquist, G., Björck, Å., Numerical Methods in Scientific Computing, Dover Publications, Inc., Mineola, New York, 1 (1974).

15. (8) Toribio, J., Lancha, A.M., Overload retardation effects on stress corrosion behaviour of prestressing steel, Constr.

Building Mater., 10 (1996) 501-505.

16. Ford, F.P., Stress corrosion cracking of iron-base alloys in aqueous environments, in: C.L. Briant, S.K. Banerji (Eds.), Treatise on Materials Science and Technology, Embrittlement of Engineering Alloys, Academic Press, New York, 25 (1983) 235-274.

17. Overview. Advances in fatigue crack closure measurement and analysis, ASTM STP 1343, R.C. McClung, J.C. Newman, Eds., ASTM International, West Gonshohocken, (1999) XI. 18. Louat, N., Sadananda, K., Duesbery, M., Vasudevan, A. K., A theoretical evaluation of crack closure, Met. Trans., A24 (1993) 2225-2232.

19. Vasudevan, A. K., Sadananda, K., Glinka, G., Critical parameters for fatigue damage, Int. J. of Fatigue, 23 (2001) S39-S53.

20. Elber, W., Fatigue crack growth under cyclic tension, Eng. Fract. Mech., 2 (1970) 37-45.

21. Macha, D. E., Corbly, D. M., Jones, J. W., On the variation of fatigue-crack-opening load with measurement location, Exp. Mech., 19 (1979) 207-213.

22. Xu Yigeng, Gregson, P. J., Sinclair, I., Systematic assessment of compliance-based crack closure measurements in fatigue, Mater. Sci. and Eng., A284 (2000) 114-125.

23. Deshpande, V. S., Needleman, A., van der Giessen, E., A discrete dislocation analysis of near-threshold fatigue crack growth, Acta Mater., 49 (2001) 3189-3203.

24. Pippan, R., Riemelmoser, F. O., Visualization of the plasticity-induced crack closure under plane strain conditions, Eng. Fract. Mech., 60 (1998) 315-322.

25. Bjerkén, C., Melin, S., Growth of a short fatigue crack – A long term simulation using a dislocation technique, Int. J. of Solids and Struct., 46 (2009) 1196-1204.

26. Pelloux, R. M. N., Crack extension by alternating shear, Eng. Fract. Mech., 1 (1970) 697-704.

27. Neumann, P., New experiments concerning the slip process at propagating fatigue cracks, Acta Met., 22 (1974) 1155-1165.

28. Suresh S., Fatigue of Materials, Cambridge University Press, Cambridge, (1991) 617.

29. Riemelmoser, F. O., Pippan, R., Stüwe, H. P., An argument for a cycle-by-cycle propagation of fatigue cracks at small stress intensity ranges, Acta Mater., 46 (1998) 1793-1799.

30. Kanninen, M.F., Popelar, C.H., Advanced fracture mechanics, Oxford University Press, New York, (1985) 562.

Impact Factor (JCC): 2.5466

IBI Factor: 3.2

Numerical Simulation Analysis to Estimate Stress Fields Along the Rectangular Crack Tip in Foams

21

31. McMeeking, R. M., Finite deformation analysis of crack tip opening in elastic-plastic materials and implications for fracture, J. Mech. and Phys. of Solids, 25 (1977) 357-381.

32. Needleman, A., Tvergaard, V., Crack-tip stress and deformation fields in a solid with vertex on its yield surface, Elastic-Plastic Fracture: Second Symposium - Inelastic Crack Analysis, ASTM STP 803, C.F. Shih, J.P. Gudas, Eds., ASTM International, 1 (1983) 80.

33. Rice, J. R., McMeeking, R. M, Parks, D. M., Sorensen, E. P., Recent finite element studies in plasticity and fracture mechanics, Comput. Meth. Applied Mech. and Eng., 17/18 (1979) 411-442.

34. Handerhan, K. J., Garrison, W. M., Jr., A study of crack tip blunting and the influence of blunting behavior on the fracture toughness of ultra high strength steels, Acta Met. et Mater., 40 (1992) 1337-1355.

35. Savruk, M. P., Stress intensity factors in solids with cracks, Naukova Dumka, Kiev, (1988) 619 36. Toribio, J., Kharin, V., Comments on simulations of fatigue crack propagation by blunting and re-sharpening: the mesh sensitivity, Int. J. of Fract., 140 (2006) 285-292.

37. Hill, R., Acceleration waves in solids, J. Mech. and Phys. of Solids, 10 (1962) 1-16.

38. Rice, J. R., The localization of plastic deformation, Theoretical and Applied Mechanics, North-Holland, Amsterdam, (1977) 207-220.

39. Croft, M., Zhong, Z., Jisrawi, N., Zakharchenko, I., Holtz, R.L., Skaritka, J., Fast, T., Sadananda, K., Lakshmipathy, M., Tsakalakos, T., Strain profiling of fatigue crack overload effects using energy dispersive X-ray diffraction, Int. J. of Fatigue, 27 (2005) 1408-1419.

40. Lorenzino, P., Navarro, A., Krupp, U., Naked eye observations of microstructurally short fatigue cracks. Submitted for publication to Int. J. of Fatigue (2013).

41. Lorenzino, P., Fatiga en componentes con concentradores de tención bajo carga en modo I, Ph.D. Thesis, University of Seville, Spain, (2012).

42. Taylor, D., Geometrical effects in fatigue: a unifying theoretical approach, Int. J. of Fatigue, 21 (1999) 413–20.

43. Taylor, D., The theory of critical distances: a new perspective in fracture mechanics, Elsevier, (2007).

44. Peterson, RE, Notch sensitivity. In: Sines G, Waisman JL, Eds. Metal Fatigue, McGraw-Hill, (1959) 293–307.

45. Neuber, H., Kerbspannungslehre. Springer Verlag; (1937). Translated into English, Theory of Notches, Edwards, J. W., Ann Arbor, MI, (1946).

46. Herbig, M., King, A., Reischig, P., Proudhon, H., Lauridsen, E. M., Marrow, J., Buffière, J-Y, 3-D growth of a short fatigue crack within a polycrystalline microstructure studied using combined diffraction and phase-contrast X-ray tomography, Acta Materialia, 59 (2011) 590–601.

47. Senthiil, P. V., V. S. Aakash Sirusshti, and T. Sathish. "Equivalent stress prediction of Automobile structural member using FEA-ANN Technique." International Journal of Mechanical and Production Engineering Research and Development 9.2 (2019): 757-768.

48. Senthiil, P. V., V. S. Aakash Sirusshti, and T. Sathish. "Equivalent stress prediction of Automobile structural member using FEA-ANN Technique." International Journal of Mechanical and Production Engineering Research and Development 9.2 (2019): 757-768.

49. Reddy, A. Chennakesava. "Simulation analysis of four-pass shape roll forming of I-sections." International Journal of www.tjprc.org

editor@tjprc.org

Piyush Agrawal & Dr. Shushant Singh*

22

Mechanical and Production Engineering Research and Development 5.1 (2015): 35-44.

50. KRYUCHKOV, VASSILIY, et al. "Investigation of dynamic motion processes of modernized uav using mathematical model of numerical simulation." International Journal of Mechanical and Production Engineering Research and Development 10.2 (2020): 535-554.

Impact Factor (JCC): 2.5466

IBI Factor: 3.2