Material models, including softening effects due to, for example, damage and localizations, share the problem of ill-posed boundary value problems that yield mesh-dependent finite element results. It is thus necessary to apply regularization techniques that couple local behavior described, for example, by internal variables, at a spatial level. This can take account of the gradient of the internal variable to yield mesh-independent finite element results. In this paper, we present a new approach to damage modeling that does not use common field functions, inclusion of gradients or complex integration techniques: Appropriate modifications of the relaxed (condensed) energy hold the same advantage as other methods, but with much less numerical effort. We start with the theoretical derivation and then discuss the numerical treatment. Finally, we present finite element results that prove empirically how the new approach works.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Abu Al-Rub, R.K., Voyiadjis, G.Z.: A direct finite element implementation of the gradient-dependent theory. Int. J. Numer. Methods Eng. 63(4), 603–629 (2005)
Aifantis, E.C.: On the role of gradients in the localization of deformation and fracture. Int. J. Eng. Sci. 30(10), 1279–1299 (1992)
Ball, J.M.: Constitutive inequalities and existence theorems in nonlinear elastostatics. In: Nonlinear Analysis and Mechanics: Heriot–Watt Symposium, vol. 1, pp. 187–241. Pitman Publishing Ltd., Boston (1977)
Bazant, Z.P., Jirásek, M.: Nonlocal integral formulations of plasticity and damage: survey of progress. J. Eng. Mech. 128(11), 1119–1149 (2002)
Carstensen, C., Hackl, K., Mielke, A.: Non-convex potentials and microstructures in finite-strain plasticity. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 458(2018), 299–317 (2002)
Comi, C.: A non-local model with tension and compression damage mechanisms. Euro. J. Mech. A/Solids 20(1), 1–22 (2001)
Dacorogna, B.: Direct Methods in the Calculus of Variations, vol. 78. Springer Science & Business Media, New York (2007)
Dimitrijevic, B., Hackl, K.: A method for gradient enhancement of continuum damage models. Tech. Mech. 28(1), 43–52 (2008)
Dimitrijevic, B., Hackl, K.: A regularization framework for damage-plasticity models via gradient enhancement of the free energy. Int. J. Numer. Methods Biomed. Eng. 27(8), 1199–1210 (2011)
Forest, S., Lorentz, E., et al.: Localization Phenomena and Regularization Methods. Local Approach to Fracture, pp. 311–371 (2004)
Francfort, G.A., Marigo, J.-J.: Stable damage evolution in a brittle continuous medium. Euro. J. Mech. Ser. A Solids 12, 149–189 (1993)
Francfort, G., Mielke, A.: Existence results for a class of rate-independent material models with nonconvex elastic energies. J. für die reine und angewandte Math. (Crelles J) 2006(595), 55–91 (2006)
Govindjee, S., Miehe, C.: A multi-variant martensitic phase transformation model: formulation and numerical implementation. Comput. Methods Appl. Mech. Eng. 191(3), 215–238 (2001)
Gurson, A.L.: Continuum theory of ductile rupture by void nucleation and growth: part I-yield criteria and flow rules for porous ductile media. J. Eng. Mater. Technol. 99(1), 2–15 (1977)
Junker, P., Hackl, K.: A variational growth approach to topology optimization. Struct. Multi. Optim. 52(2), 293–304 (2015)
Junker, P.: An accurate, fast and stable material model for shape memory alloys. Smart Mater. Struct. 23(11), 115010 (2014)
Junker, P.: A novel approach to representative orientation distribution functions for modeling and simulation of polycrystalline shape memory alloys. Int. J. Numer. Meth. Eng. 98(11), 799–818 (2014)
Junker, P., Hackl, K.: A thermo-mechanically coupled field model for shape memory alloys. In: Continuum Mechanics and Thermodynamics, pp. 1–19 (2014)
Junker, P., Hackl, K.: A discontinuous phase field approach to variational growth-based topology optimization. Struct. Multi. Optim. 54(1), 81–94 (2016)
Junker, P., Jaeger, S., Kastner, O., Eggeler, G., Hackl, K.: Variational prediction of the mechanical behavior of shape memory alloys based on thermal experiments. J. Mech. Phys. Solids 80, 86–102 (2015)
Junker, P., Jerzy, M., Hackl, K.: The principle of the minimum of the dissipation potential for non-isothermal processes. Continuum Mech. Thermodyn. 26(3), 259–268 (2014)
Lemaitre, J.: Coupled elasto-plasticity and damage constitutive equations. Comput. Methods Appl. Mech. Eng. 51(1–3), 31–49 (1985)
Liu, W.K., Hao, S., Belytschko, T., Li, S.F., Chang, C.T.: Multiple scale meshfree methods for damage fracture and localization. Comput. Mater. Sci. 16(1), 197–205 (1999)
Lorentz, E., Andrieux, S.: Analysis of non-local models through energetic formulations. Int. J. Solids Struct. 40(12), 2905–2936 (2003)
Lorentz, E., Benallal, A.: Gradient constitutive relations: numerical aspects and application to gradient damage. Comput. Methods Appl. Mech. Eng. 194(50), 5191–5220 (2005)
Mielke, A., Roubíček, T.: Rate-independent damage processes in nonlinear elasticity. Math. Models Methods Appl. Sci. 16(02), 177–209 (2006)
Peerlings, R.H.J., Brekelmans, W.A.M., de Vree, J.H.P.: Gradient enhanced damage for quasi-brittle materials. Int. J. Numer. Methods Eng. 39, 3391–3403 (1996)
Peerlings, R.H.J., Geers, M.G.D., de Borst, R., Brekelmans, W.A.M.: A critical comparison of nonlocal and gradient-enhanced softening continua. Int. J. Solids Struct. 38(44), 7723–7746 (2001)
Yan, Y., Wen, W.-D., Chang, F.-K., Shyprykevich, P.: Experimental study on clamping effects on the tensile strength of composite plates with a bolt-filled hole. Compos. Part A Appl. Sci. Manuf. 30(10), 1215–1229 (1999)
Yang, Y., Misra, A.: Micromechanics based second gradient continuum theory for shear band modeling in cohesive granular materials following damage elasticity. Int. J. Solids Struct. 49(18), 2500–2514 (2012)
Communicated by Andreas Öchsner.
About this article
Cite this article
Junker, P., Schwarz, S., Makowski, J. et al. A relaxation-based approach to damage modeling. Continuum Mech. Thermodyn. 29, 291–310 (2017). https://doi.org/10.1007/s00161-016-0528-8