Abstract
It is well known that the application of the conventional \(J\)-integral is connected with severe restrictions when it is applied for elastic–plastic materials. The first restriction is that the \(J\)-integral can be used only, if the conditions of proportional loading are fulfilled, e.g. no unloading processes should occur in the material. The second restriction is that, even if this condition is fulfilled, the \(J\)-integral does not describe the crack driving force, but only the intensity of the crack tip field. Using the configurational force concept, Simha et al. (J Mech Phys Solids 56:2876–2895, 2008), have derived a \(J\)-integral, \(J^{\mathrm{ep}} \), which overcomes these restrictions: \(J^{\mathrm{ep}} \) is able to quantify the crack driving force in elastic–plastic materials in accordance with incremental theory of plasticity and it can be applied also in cases of non-proportionality, e.g. for a growing crack. The current paper deals with the characteristic properties of this new \(J\)-integral, \(J^{\mathrm{ep}}\), and works out the main differences to the conventional \(J\)-integral. In order to do this, numerical studies are performed to calculate the distribution of the configurational forces in a cyclically loaded tensile specimen and in fracture mechanics specimens. For the latter case contained, uncontained, and general yielding conditions are considered. The path dependence of \(J^{\mathrm{ep}} \) is determined for both a stationary and a growing crack. Much effort is spent in the investigation of the path dependence of \(J^{\mathrm{ep}} \) very close to the crack tip. Several numerical parameters are varied in order to separate numerical and physical effects and to deduce the magnitudes of the crack driving force for stationary and growing cracks. Interpretation of the numerical results leads to a new, completed picture of the \(J\)-integral in elastic–plastic materials where \(J^{\mathrm{ep}} \) and the conventional \(J\)-integral complement each other. This new view allows us also to shed new light on a long-term problem, which has been called the “paradox of elastic–plastic fracture mechanics”.
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Notes
It might be helpful to note that during the translation of the defects, the material points of the specimen remain (almost) at the same positions. (They do remain at the same positions in the load-free reference configuration.) If the crack tip moves, the bonding between the material points in front of the current crack tip, above and below the crack plane, is cut. If the plastic zone translates with the crack tip, the region, within which the material is deformed plastically, is shifted.
References
Anderson TL (1995) Fracture mechanics. CRC Press, Boca Raton, FL
ASTM E1820–05 (2005) Standard test method for measurement of fracture toughness. In: Annual Book of ASTM Standards, vol 03.01. ASTM International, West Conshohocken, PA, USA
Atkins AG (1995) Toughness given by accumulated work in LEFM: possible implications for elastoplastic \(J_{R}\) “resistance curves”. Fatigue Fract Eng Mater Struct 18:1007–1017
Brocks W, Cornec A, Scheider I (2003) Computational aspects of nonlinear fracture mechanics. In: de Borst R, Mang HA (eds) Comprehensive structural integrity, numerical and computational methods, vol 3. Elsevier, New York, pp 127–209
Bui HD (1987) Recent developments in fracture mechanics. In: Herrmann KP, Larsson LH (eds) Fracture of non-metallic materials. Kluwer, Dordrecht, pp 21–32
Chen JH, Wang GZ (2001) Study on cleavage fracture criteria of the quasi-brittle and micro-inhomogeneous materials. Int J Fract 108:143–164
Chen JH, Wang Q, Wang GZ, Li Z (2003) Fracture behavior at crack tip—a new framework for cleavage mechanism of steel. Acta Mater 51:1841–1855
Denzer R, Barth FJ, Steinmann P (2003) Studies in elastic fracture mechanics based on the material force method. Int J Numer Methods Eng 58:1817–1835
Eshelby JD (1951) The force on an elastic singularity. Philos Trans R Soc A 244:87–112
Eshelby JD (1970) Energy relations and the energy-momentum tensor in continuum mechanics. In: Kanninen M, Adler W, Rosenfield A, Jaffee R (eds) Inelastic behavior of solids. McGraw-Hill, New York, pp 77–115
ESIS P2-92 (1992) ESIS procedure for determining the fracture behaviour of materials. European Structural Integrity Society, Delft, The Netherlands
Fischer FD, Predan J, Fratzl P, Kolednik O (2012a) Semi-analytical approaches to assess the crack driving force in periodically heterogeneous elastic materials. Int J Fract 173:57–70
Fischer FD, Simha NK, Predan J, Schöngrundner R, Kolednik O (2012b) On configurational forces at boundaries in fracture mechanics. Int J Fract 174:61–74
Gurtin ME (1995) The nature of configurational forces. Arch Ration Mech Anal 131:67–100
Gurtin ME (2000) Configurational forces as basic concepts of continuum physics. Springer, New York
Gurtin ME, Podio-Guidugli P (1996) Configurational forces and the basic laws for crack propagation. J Mech Phys Solids 44:905–927
Gürses E, Miehe C (2009) A computational framework of three-dimensional configurational-force-driven brittle crack propagation. Comput Methods Appl Mech Eng 198:1413–1428
Häusler SM, Lindhorst K, Horst P (2011) Combination of the material force concept and the extended finite element method for mixed mode crack growth simulation. Int J Numer Methods Eng 85:1522–1542
Honein T, Herrmann G (1997) Conservation laws in nonhomogeneous plane elastostatics. J Mech Phys Solids 45:789–805
Hutchinson JW (1968) Singular behavior at the end of a tensile crack tip in a hardening material. J Mech Phys Solids 16:13–31
Hutchinson JW, Paris PC (1979) Stability analysis of \(J\)-controlled crack growth. ASTM STP 668:37–64
Kfouri AP, Miller KJ (1976) Crack separation energy rate for crack advance in finite growth steps. Proc Inst Mech Eng 190:571–584
Kfouri AP, Rice JR (1977) Elastic/plastic separation energy rate for crack advance in finite growth steps. In: Taplin DMR (ed) Fracture 1977, vol 1. University of Waterloo Press, Waterloo, pp 43–59
Kienzler R, Herrmann G (2000) Mechanics in material space. Springer, Berlin
Kishimoto K, Aoki S, Sakata M (1980) On the path independent integral \(\hat{{J}}\). Eng Fract Mech 13:841–850
Kolednik O (1991) On the physical meaning of the \(J-\Delta \) a-curves. Eng Fract Mech 38:403–412
Kolednik O (1992) Loading conditions may influence the shape of \(J-\Delta \) a-curves. Eng Fract Mech 41:251–255
Kolednik O (1993) A simple model to explain the geometry dependence of the \(J-\Delta \) a-curves. Int J Fract 63:263–274
Kolednik O (2000) The yield-stress gradient effect in inhomogeneous materials. Int J Solids Struct 37:781–808
Kolednik O, Stüwe HP (1985) The stereophotogrammetric determination of the critical crack tip opening displacement. Eng Fract Mech 21:145–155
Kolednik O, Stüwe HP (1986) An extensive analysis of a \(J_{{\rm IC}}\)-test. Eng Fract Mech 24:277–290
Kolednik O, Turner CE (1994) Application of energy dissipation rate arguments to ductile instability. Fatigue Fract Eng Mater Struct 17:1129–1145
Kolednik O, Albrecht M, Berchthaler M, Germ H, Pippan R, Riemelmoser F, Stampfl J, Wei J (1996) The fracture resistance of a ferritic-austenitic duplex steel. Acta Mater 44:3307–3319
Kolednik O, Predan J, Shan GX, Simha NK, Fischer FD (2005) On the fracture behavior of inhomogeneous materials–a case study for elastically inhomogeneous bimaterials. Int J Solids Struct 42:605–620
Kolednik O, Predan J, Fischer FD (2010) Reprint of “Cracks in inhomogeneous materials: comprehensive assessment using the configurational forces concept”. Eng Fract Mech 77:3611–3624
Kolling S, Baaser H, Gross D (2002) Material forces due to crack-inclusion interaction. Int J Fract 118:229–238
Kuhn C, Müller R (2010) A continuum phase field model for fracture. Eng Fract Mech 77:3625–3634
Li Q, Kuna M (2012) Evaluation of electromechanical fracture behavior by configurational forces in cracked ferroelectric polycrystals. Comput Mater Sci 57:94–101
Maugin GA (1995) Material forces: concepts and applications. ASME J Appl Mech Rev 48:213–245
Maugin GA (2011) Configurational forces: thermomechanics, physics, mathematics and numerics. CRC Press, Boca Raton
Maugin GA, Trimarco C (1992) Pseudo-momentum and material forces in nonlinear elasticity: variational formulation and application to brittle fracture. Acta Mech 94:1–28
McMeeking RM (1977) Path dependence of the \(J\)-integral and the role of \(J\) as a parameter characterizing the near tip field. ASTM STP 631:28–41
McMeeking RM, Parks DM (1979) On criteria for \(J\)-dominance of crack-tip fields in large-scale yielding. ASTM STP 668:175–194
Miyazaki N, Nakagaki M (1995) Two-dimensional finite element analysis of stably growing cracks in inhomogeneous materials. Int J Press Vessel Pip 63:249–260
Mueller R, Kolling S, Gross D (2002) On configurational forces in the context of the finite element method. Int J Numer Methods Eng 53:1557–1574
Mueller R, Gross D, Maugin GA (2004) Use of material forces in adaptive finite element methods. Comput Mech 33:421–434
Nguyen TD, Govindjee S, Klein PA, Gao H (2005) A material force method for inelastic fracture mechanics. J Mech Phys Solids 53:91–121
Ochensberger W, Kolednik O (2014) \(J\)-integral and crack driving force in elastic–plastic materials under cyclic loading, to be published
Parks DM (1977) The virtual crack extension method for nonlinear material behavior. Comput. Methods Appl Mech Eng 12:353–364
Predan J, Gubeljak N, Kolednik O (2007) On the local variation of the crack driving force in a mismatched weld. Eng Fract Mech 74:1739–1757
Rakin M, Kolednik O, Medjo B, Simha NK, Fischer FD (2009) A case study on the effect of thermal residual stresses on the crack-driving force in linear-elastic bimaterials. Int J Mech Sci 51:531–540
Rice JR (1966) An examination of the fracture mechanics energy balance from the point of view of continuum mechanics. In: Yokobori T (ed) Proceedings 1st international conference fracture. Japanese Society for Strength and Fracture, Tokyo, pp 309–340
Rice JR (1968a) A path independent integral and the approximate analysis of strain concentration by notches and cracks. ASME J Appl Mech 35:379–386
Rice JR (1968b) Mathematical analysis in the mechanics of fracture. In: Liebowitz H (ed) Fracture—an advanced treatise, vol 2. Academic Press, New York, pp 191–311
Rice JR (1979) The mechanics of quasi-static crack growth. In: Kelly RE (ed) Proceedings of the eighth U.S. National congress of applied mechanics. ASME, New York, pp 191–216
Rice JR, Johnson MA (1970) The role of large crack tip geometry changes in plane strain fracture. In: Kanninen MF (ed) Inelastic behavior of solids. McGraw-Hill, New York, pp 641–672
Rice JR, Rosengren GF (1968) Plane strain deformation near a crack tip in a power-law hardening material. J Mech Phys Solids 16:1–12
Rice JR, Sorensen EP (1978) Continuing crack-tip deformation and fracture for plane-strain crack growth in elastic–plastic solids. J Mech Phys Solids 26:163–186
Rice JR, Paris PC, Merkle JG (1973) Some further results of \(J\)-integral analysis and estimates. ASTM STP 536:231– 245
Rice JR, Drugan WJ, Sham TL (1980) Elastic–plastic analysis of growing cracks. ASTM STP 700:189–221
Schöngrundner R (2011) Numerische Studien zur Ermittlung der risstreibenden Kraft in elastisch-plastischen Materialien bei unterschiedlichen Belastungsbedingungen. Fortschritt-Berichte VDI, Reihe 18 Mechanik/Bruchmechanik, Nr. 329. VDI-Verlag, Düsseldorf
Schöngrundner R, Kolednik O, Fischer FD (2010) The configurational force concept in elastic–plastic fracture mechanics—instructive examples. Key Eng Mater 417–418:297–300
Simha NK, Fischer FD, Kolednik O, Chen CR (2003) Inhomogeneity effects on the crack driving force in elastic and elastic–plastic materials. J Mech Phys Solids 51:209–240
Simha NK, Fischer FD, Kolednik O, Predan J, Shan GX (2005a) Crack tip shielding due to smooth and discontinuous material inhomogeneities. Int J Fract 135:73–93
Simha NK, Kolednik O, Fischer FD (2005b) Material force models for cracks–influences of eigenstrains, thermal strains & residual stresses. In: Carpinteri A (ed) 11th International conference fracture. Torino, Italy, Paper 5329
Simha NK, Fischer FD, Shan GX, Chen CR, Kolednik O (2008) \(J\)-integral and crack driving force in elastic–plastic materials. J Mech Phys Solids 56:2876–2895
Steinmann P, Ackermann D, Barth FJ (2001) Application of material forces to hyperelastostatic fracture mechanics. Part II: computational Setting. Int J Solids Struct 38:5509– 5526
Stumpf H, Makowski J, Hackl K (2010) Configurational forces and couples in fracture mechanics accounting for microstructures and dissipation. Int J Solids Struct 47:2380–2389
Tillberg J, Larsson F, Runesson K (2010) On the role of material dissipation for the crack-driving force. Int J Plasticity 26:992–1012
Turner CE, Kolednik O (1994a) A micro and macro approach to the energy dissipation rate model of stable ductile crack growth. Fatigue Fract Eng Mater Struct 17:1089–1107
Turner CE, Kolednik O (1994b) Application of energy dissipation rate arguments to stable crack growth. Fatigue Fract Eng Mater Struct 17:1109–1127
Wadier Y (2004) Reconsidering the paradox of rice for a linear strain hardening material. Int J Fract 127:L125–L132
Wadier Y, Le HN, Bargellini R (2013) An energy approach to predict cleavage fracture under non-proportional loading. Eng Fract Mech 97:30–51
van der Meer FP, Moës N, Sluys LJ (2012) A level set model for delamination—modeling crack growth without cohesive zone or stress singularity. Eng Fract Mech 79:191–212
Acknowledgments
O.K. and F.D.F. gratefully acknowledge valuable discussions with Dr. N.K. Simha. Financial support by the Austrian Federal Government and the Styrian Provincial Government within the research activities of the K2 Competence Center on “Integrated Research in Materials, Processing and Product Engineering”, under the frame of the Austrian COMET Competence Center Programme, is gratefully acknowledged (strategic Projects A4.11-WP2 and A4.20-WP1).
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Kolednik, O., Schöngrundner, R. & Fischer, F.D. A new view on J-integrals in elastic–plastic materials. Int J Fract 187, 77–107 (2014). https://doi.org/10.1007/s10704-013-9920-6
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DOI: https://doi.org/10.1007/s10704-013-9920-6