Abstract
We introduce a regularized model for free fracture propagation based on nonlocal potentials. We work within the small deformation setting, and the model is developed within a state-based peridynamic formulation. At each instant of the evolution, we identify the softening zone where strains lie above the strength of the material. We show that deformation discontinuities associated with flaws larger than the length scale of nonlocality δ can become unstable and grow. An explicit inequality is found that shows that the volume of the softening zone goes to zero linearly with the length scale of nonlocal interaction. This scaling is consistent with the notion that a softening zone of width proportional to δ converges to a sharp fracture set as the length scale of nonlocal interaction goes to zero. Here the softening zone is interpreted as a regularization of the crack network. Inside quiescent regions with no cracks or softening, the nonlocal operator converges to the local elastic operator at a rate proportional to the radius of nonlocal interaction. This model is designed to be calibrated to measured values of critical energy release rate, shear modulus, and bulk modulus of material samples. For this model one is not restricted to Poisson ratios of 1/4 and can choose the potentials so that small strain behavior is specified by the isotropic elasticity tensor for any material with prescribed shear and Lamé moduli.
References
A. Agwai, I. Guven, E. Madenci, Predicting crack propagation with peridynamics: a comparative study. Int. J. Fract. 171, 65–78 (2011)
F. Bobaru, W. Hu, The meaning, selection, and use of the peridynamic horizon and its relation to crack branching in brittle materials. Int. J. Fract. 176, 215–222 (2012)
F. Bobaru, J.T. Foster, P.H. Geubelle, S.A. Silling, Handbook of Peridynamic Modeling (CRC Press, Boca Raton, 2016)
K. Dayal, K. Bhattacharya, Kinetics of phase transformations in the peridynamic formulation of continuum mechanics. J. Mech. Phys. Solids 54, 1811–1842 (2006)
P. Diehl, R. Lipton, M.A. Schweitzer, Numerical verification of a bond-based softening peridynamic model for small displacements: deducing material parameters from classical linear theory. Institut für Numerische Simulation Preprint No. 1630 (2016)
B.K. Driver, Analysis Tools with Applications. E-book (Springer, Berlin, 2003)
Q. Du, M. Gunzburger, R. Lehoucq, K. Zhou, Analysis of the volume-constrained peridynamic Navier equation of linear elasticity. J. Elast. 113, 193–217 (2013)
J.T. Foster, S.A. Silling, W. Chen, An energy based failure criterion for use with peridynamic states. Int. J. Multiscale Comput. Eng. 9, 675–688 (2011)
W. Gerstle, N. Sau, S. Silling, Peridynamic modeling of concrete structures. Nucl. Eng. Des. 237, 1250–1258 (2007)
Y.D. Ha, F. Bobaru, Studies of dynamic crack propagation and crack branching with peridynamics. Int. J. Fract. 162, 229–244 (2010)
I. Jasiuk, J. Chen, M.F. Thorpe, Elastic moduli of two dimensional materials with polygonal and elliptical holes. Appl. Mech. Rev. 47, S18–S28 (1994)
R. Lipton, Dynamic brittle fracture as a small horizon limit of peridynamics. J. Elast. 117, 21–50 (2014)
R. Lipton, Cohesive dynamics and brittle fracture. J. Elast. 124(2), 143–191 (2016)
R. Lipton, S. Silling, R. Lehoucq, Complex fracture nucleation and evolution with nonlocal elastodynamics. arXiv preprint arXiv:1602.00247 (2016)
G.W. Milton, The Theory of Composites (Cambridge University Press, Cambridge, 2002)
S.A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48, 175–209 (2000)
S.A. Silling, E. Askari, A meshfree method based on the peridynamic model of solid mechanics. Comput. Struct 83, 1526–1535 (2005)
S.A. Silling, F. Bobaru, Peridynamic modeling of membranes and fibers. Int. J. Nonlinear Mech. 40, 395–409 (2005)
S.A. Silling, R.B. Lehoucq, Convergence of peridynamics to classical elasticity theory. J. Elast. 93, 13–37 (2008)
S. A. Silling, M. Epton, O. Weckner, J. Xu, E. Askari, Peridynamic states and constitutive modeling. J. Elast. 88, 151–184 (2007)
S. Silling, O. Weckner, E. Askari, F. Bobaru, Crack nucleation in a peridynamic solid. Int. J. Fract. 162, 219–227 (2010)
O. Weckner, R. Abeyaratne, The effect of long-range forces on the dynamics of a bar. J. Mech. Phys. Solids 53, 705–728 (2005)
Acknowledgements
This material is based upon the work supported by the US Army Research Laboratory and the US Army Research Office under contract/grant number W911NF1610456.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this entry
Cite this entry
Lipton, R., Said, E., Jha, P.K. (2018). Dynamic Brittle Fracture from Nonlocal Double-Well Potentials: A State-Based Model. In: Voyiadjis, G. (eds) Handbook of Nonlocal Continuum Mechanics for Materials and Structures. Springer, Cham. https://doi.org/10.1007/978-3-319-22977-5_33-1
Download citation
DOI: https://doi.org/10.1007/978-3-319-22977-5_33-1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-22977-5
Online ISBN: 978-3-319-22977-5
eBook Packages: Springer Reference EngineeringReference Module Computer Science and Engineering