Abstract
In complex bodies, actions due to substructural changes alter (in some cases drastically) the force driving the tip of macroscopic cracks in quasi-static and dynamic growth, and must be represented directly. Here it is proven that tip balances of standard and substructural interactions are covariant. In fact, the former balance follows from the Lagrangian density’s requirement of invariance with respect to the action of the group of diffeomorphisms of the ambient space to itself, the latter balance accrues from an analogous invariance with respect to the action of a Lie group over the manifold of substructural shapes. The evolution equation of the crack tip can be obtained by exploiting invariance with respect to relabeling the material elements in the reference place. The analysis is developed by first focusing on general complex bodies that admit metastable states with substructural dissipation of viscous-like type inside each material element. Then we account for gradient dissipative effects that induce nonconservative stresses; the covariance of tip balances in simple bodies follows as a corollary. When body actions and boundary data of Dirichlet type are absent, the standard variational description of quasi-static crack growth is simply extended to the case of complex materials.
Similar content being viewed by others
References
Ambrosio, L., Tortorelli, V.M.: Approximation of functionals depending on jumps via Γ-convergence. Commun. Pure Appl. Math. 43, 999–1036 (1990)
Atkinson, C., Leppington, F.G.: The effect of couple stresses on the tip of a crack. Int. J. Solids Struct. 13, 1103–1122 (1977)
Bagchi, A., Evans, A.G.: The mechanics and the physics of thin film decohesion and its measurement. Interface Sci. 3, 169–193 (1996)
Beom, H.G., Atluri, S.N.: Effect of electric fields on fracture behavior of ferroelectric ceramics. J. Mech. Phys. Solids 51, 1107–1125 (2003)
Bloch, A., Krishnaprasad, P.S., Marsden, J.E., Ratiu, T.: The Euler–Poincaré equations and double bracket dissipation. Commun. Math. Phys. 174, 1–42 (1996)
Capriz, G.: Continua with Microstructure. Springer, Berlin (1989)
Capriz, G.: Continua with substructure (Part 1). Phys. Mesomech. 3, 5–14 (2000a)
Capriz, G.: Continua with substructure (Part 2). Phys. Mesomech. 6, 35–70 (2000b)
Capriz, G.: Elementary preamble to a theory of granular gases. Rend. Semin. Mat. Univ. Padova 110, 179–198 (2003)
Capriz, G., Pseudofluids. In: Capriz, G., Mariano, P.M. (eds.) Material Substructures in Complex Bodies: From Atomic Level to Continuum, pp. 238–261. Elsevier, Amsterdam (2007)
Capriz, G., Giovine, P.: On microstructural inertia. Mat. Models Method. Appl. Sci. 7, 211–216 (1997)
Capriz, G., Mariano, P.M.: Symmetries and Hamiltonian formalism for complex materials. J. Elast. 72, 57–70 (2003)
Dal Maso, G., Toader, R.: A model for the quasi-static growth of brittle fracture: existence and approximation results. Arch. Ration. Mech. Anal. 162, 101–135 (2004)
Dal Maso, G., Francfort, G.A., Toader, R.: Quasistatic crack growth in nonlinear elasticity. Arch. Ration. Mech. Anal. 176, 165–225 (2005)
Ding, D.-H., Yang, W., Hu, C., Wang, R.: Generalized theory of quasicrystals. Phys. Rev. B 48, 7003–7010 (1993)
de Fabritiis, C., Mariano, P.M.: Geometry of interactions in complex bodies. J. Geom. Phys. 54, 301–323 (2005)
Elssner, G., Korn, D., Rühle, M.: The influence of interface impurities on fracture energy of UHV diffusion bonded metal–ceramic bicrystals. Scr. Metall. Mat. 31, 1037–1042 (1994)
Francfort, G., Marigo, J.-J.: Revisiting brittle fracture as an energy minimizing problem. J. Mech. Phys. Solids 46, 1319–1342 (1998)
Freund, L.B.: Dynamic Fracture Mechanics. Cambridge University Press, Cambridge (1990)
Fu, R.K., Zhang, T.-Y.: Effects of an electric field on the fracture toughness of poled lead zirconate titanate ceramics. J. Am. Ceram. Soc. 83, 1215–1218 (2000)
Fulton, C.C., Gao, H.: Microstructural modeling of ferroelectric fracture. Acta Mater. 49, 2039–2054 (2001)
Gao, H., Zhang, T.-Y., Pin, T.: Local and global energy release rates for an electrically yelded crack in a piezoelectric ceramic. J. Mech. Phys. Solids 45, 491–510 (1997)
Giaquinta, M., Modica, G., Souček, J.: Cartesian Currents in the Calculus of Variations, vols. I and II. Springer, Berlin (1998)
Giusti, E.: Direct Methods in the Calculus of Variations. World Scientific, Singapore (2003)
Gotay, M.J., Isemberg, J., Marsden, J.E., Montgomery, R., with the collaboration of Śniatycki, J. and Yasskin, P.B.: Momentum maps and classical fields. Part I: Covariant field theory. arXiv. physics/9801019v2 (2003)
Green, A.E., Rivlin, R.S.: On Cauchy’s equation of motion. Z. Angew. Math. Phys. 15, 290–293 (1964)
Gurtin, M.E., Podio Guidugli, P.: Configurational forces and the basic laws of crack propagation. J. Mech. Phys. Solids 44, 905–927 (1996)
Gurtin, M.E., Shvartsman, M.M.: Configurational forces and the dynamics of planar cracks in three-dimensional bodies. J. Elast. 48, 167–191 (1997)
Jaric, J.: The energy release rate in quasistatic crack propagation and J-integral. Int. J. Solids Struct. 22, 767–778 (1986)
Khludnev, A.M.: Invariant integrals in problems of a crack at the locus of inhomogeneity and in contact problems. Dokl. Phys. 49, 603–607 (2004)
Kreher, W.S.: Influence of domain switching zones on the fracture thoughness of ferroelectrics. J. Mech. Phys. Solids 50, 1029–1050 (2002)
Landau, L.D., Lifshitz, E.M.: Theory of Elasticity. Butterworth-Heinmann, Oxford (1986)
Landis, C.M.: On the fracture toughness of ferroelectric materials. J. Mech. Phys. Solids 51, 1347–1369 (2003)
Lubarda, V.A., Markescoff, X.: Conservation integrals in couple stress elasticity. J. Mech. Phys. Solids 48, 553–564 (2000)
Mariano, P.M.: Multifield theories in mechanics of solids. Adv. Appl. Mech. 38, 1–93 (2002)
Mariano, P.M.: Influence of the material substructure on crack propagation: a unified treatment. Proc. R. Soc. Lond. A 461, 371–395 (2005)
Mariano, P.M.: Mechanics of quasi-periodic alloys. J. Nonlinear Sci. 16, 45–77 (2006)
Mariano, P.M., Modica, G.: Ground states in complex bodies (2007, submitted)
Mariano, P.M., Stazi, F.L.: Computational aspects of the mechanics of complex materials. Arch. Comput. Method. Eng. 12, 391–478 (2006)
Mariano, P.M., Stazi, F.L., Gioffrè, M.: Stochastic clustering and self-organization of gross deformation and phason activity in quasicrystals: modeling and simulations. J. Comput. Theor. Nanosci. 3, 478–486 (2006)
Marsden, J.E., Hughes, T.R.J.: Mathematical Foundations of Elasticity. Dover, New York (1994)
Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry. Springer, Berlin (1999)
Mikulla, R., Stadler, J., Krul, F., Trebin, H.-R., Gumbsch, P.: Crack propagation in quasicrystals. Phys. Rev. Lett. 81, 3163–3166 (1998)
Obrezanova, O., Movchan, A.B., Willis, J.R.: Dynamic stability of a propagating crack. J. Mech. Phys. Solids 50, 2637–2668 (2003)
Oleaga, G.: On the path of a quasi-static crack in mode III. J. Elast. 76, 163–189 (2004)
Shen, S., Nishioka, T.: Finite element simulation of impact interfacial crack problem in piezoelectric bimaterials. In: Electromagnetic Mechanics of Solids, pp. 211–227 (2003)
Schmicker, D., van Smaalen, S.: Dynamical behavior of aperiodic intergrowth crystals. Int. J. Mod. Phys. B 10, 2049–2080 (1996)
Simo, J.C., Marsden, J.E., Krishnaprasad, P.S.: The Hamiltonian structure of nonlinear elasticity: the material and convective representations of solids, rods and plates. Arch. Ration. Mech. Anal. 104, 125–183 (1988)
Stazi, F.L., Budyn, E., Chessa, J., Belytschko, T.: An extended finite element method with higher-order element for crack problems with curvature. Comput. Mech. 31, 38–48 (2002)
Stolarska, M., Chopp, D.L., Moës, N., Belytschko, T.: Modelling crack growth by level sets and the extended finite element method. Int. J. Numer. Method. Eng. 51, 943–960 (2001)
Vukobrat, M.D.: Conservation laws in micropolar elastodynamics and path-independent integrals. Int. J. Eng. Sci. 27, 1093–1106 (1989)
Wei, Y., Hutchinson, J.W.: Steady-state crack growth and work of fracture for solids characterized by strain gradient plasticity. J. Mech. Phys. Solids 45, 1253–1273 (1997)
Willis, J.R., Movchan, A.B.: Three-dimensional dynamic perturbation of a propagating crack. J. Mech. Phys. Solids 45, 591–610 (1997)
Yavari, A., Marsden, J.E., Ortiz, M.: On spatial and material covariant balance laws in elasticity. J. Math. Phys. 47, 042903 (2006)
Author information
Authors and Affiliations
Corresponding author
Additional information
To the memory of my friend Dusan Krajcinovic, an elegant scholar, a true gentleman.
Rights and permissions
About this article
Cite this article
Mariano, P.M. Cracks in Complex Bodies: Covariance of Tip Balances. J Nonlinear Sci 18, 99–141 (2008). https://doi.org/10.1007/s00332-007-9008-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00332-007-9008-4