We present an efficient and reliable approach for the numerical modelling of failure with nonlocal damage models. The two major numerical challenges––the strongly nonlinear, highly localized and parameter-dependent structural response of quasi-brittle materials, and the interaction between nonadjacent finite elements associated to nonlocality––are addressed in detail. Reliability of the numerical results is ensured by an h-adaptive strategy based on error estimation. We use a residual-type error estimator for nonlinear FE analysis based on local computations, which, at the same time, accounts for the nonlocality of the damage model. Efficiency is achieved by a proper combination of load-stepping control technique and iterative solver for the nonlinear equilibrium equations. A major issue is the computation of the consistent tangent matrix, which is nontrivial due to nonlocal interaction between Gauss points. With computational efficiency in mind, we also present a new nonlocal damage model based on the nonlocal average of displacements. For this new model, the consistent tangent matrix is considerably simpler to compute than for current models. The various ideas discussed in the paper are illustrated by means of three application examples: the uniaxial tension test, the three-point bending test and the single-edge notched beam test.

J. Bobiński, J. Tejchman. Comparison of continuous and discontinuous constitutive models to simulate concrete behaviour under mixed-mode failure conditions. Int. J. Numer. Anal. Meth. Geomech. 40(3) (2015) DOI 10.1002/nag.2411

A. Rodríguez-Ferran, I. Morata, A. Huerta. A new damage model based on non-local displacements. Int. J. Numer. Anal. Meth. Geomech. 29(5) (2005) DOI 10.1002/nag.422

L. Chen, G. Duveau, J. Shao. Modelling of plastic deformation and damage in cement-based material subjected to desiccation. Int. J. Numer. Anal. Meth. Geomech. 35(17) (2010) DOI 10.1002/nag.985

L. Pérez Pozo, A. Campos, S. Lascano, S. Oller, A. Rodríguez-Ferran. A Finite Points Method Approach for Strain Localization Using the Gradient Plasticity Formulation. Mathematical Problems in Engineering 2014 DOI 10.1155/2014/782079

E. Tamayo‐Mas, J. Feliu‐Fabà, M. Casado‐Antolin, A. Rodríguez‐Ferran. A continuous‐discontinuous model for crack branching. Int J Numer Methods Eng 120(1) (2019) DOI 10.1002/nme.6125

F. Bobaru, M. Yang, L. Alves, S. Silling, E. Askari, J. Xu. Convergence, adaptive refinement, and scaling in 1D peridynamics. Int. J. Numer. Meth. Engng 77(6) DOI 10.1002/nme.2439

G. Nguyen, I. Einav. A stress-return algorithm for nonlocal constitutive models of softening materials. Int. J. Numer. Meth. Engng DOI 10.1002/nme.2790

M. Cervera, M. Chiumenti, R. Codina. Mesh objective modeling of cracks using continuous linear strain and displacement interpolations. Int. J. Numer. Meth. Engng. 87(10) (2011) DOI 10.1002/nme.3148

A. Martowicz, J. Roemer, W. Staszewski, M. Ruzzene, T. Uhl. Solving partial differential equations in computational mechanics via nonlocal numerical approaches. Z. Angew. Math. Mech. 99(4) (2019) DOI 10.1002/zamm.201800342

A. Mohamad-Hussein, J. Shao. Mechanical modeling of rock using a non-local elastoplastic damage model. (2010) DOI 10.1201/9781439833469.ch102

E. Benvenuti, A. Tralli. The fast Gauss transform for non-local integral FE models. Commun. Numer. Meth. Engng. 22(6) (2005) DOI 10.1002/cnm.827

H. Askes, T. Bennett, S. Kulasegaram. Meshless discretisation of nonlocal damage theories. DOI 10.1007/978-1-4020-6530-9_1

A. Martowicz, J. Bryła, W. Staszewski, M. Ruzzene, T. Uhl. Nonlocal elasticity in shape memory alloys modeled using peridynamics for solving dynamic problems. Nonlinear Dyn 97(3) (2019) DOI 10.1007/s11071-019-04943-5

L. Driemeier, C. Comi, S. Proença. On nonlocal regularization in one dimensional finite strain elasticity and plasticity. Comput Mech 36(1) (2005) DOI 10.1007/s00466-004-0640-7

M. Huang, X. Qu, X. Lü. Regularized finite element modeling of progressive failure in soils within nonlocal softening plasticity. Comput Mech 62(3) (2017) DOI 10.1007/s00466-017-1500-6