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Interaction of Fractures in Tensile Bars with Non-Local Spatial Dependence

  • Camillo De‐Lellis
  • Gianni Royer‐Carfagni
Article

Abstract

We propose to determine the displacement field u: IRR of a 1-D bar extended in a hard device by minimizing a non-local energy functional of the type
$$[u]: = \int_\mathcal{I} {U\;} \left( {u'(x) + \frac{1}{K}\sum\limits_{x_i \in J_u } {[u](x_i )} \;\rho (x - x_i )} \right)\;{\text{d}}x + \sum\limits_{x_i \in J_u } {\varphi ([u](x_i )),} $$
where K is a material parameter, [u](x i ) denotes the jump of u at x i and J u I the set of all jump points. For appropriate choices of the bulk energy U(⋅), of the surface energy ϕ(⋅) and of the weight function ρ(⋅), we prove an existence theorem for minimizers in the space SBV(I) of special bounded variation functions and we qualitatively discuss their form by investigating the corresponding Euler–Lagrange equations. We show that for sufficiently large values of the bar elongation, minimizers of the energy are discontinuous and, most of all, the non-local term [u](x i )ρ(xx i ) influences the relative position among the jump points, a finding that is of crucial importance to reproduce the experimental evidence.
nonlinear elasticity variational model fracture mechanics non-local model damage 

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References

  1. 1.
    L. Ambrosio, A compactness theorem for a special class of functions of bounded variation. Boll. Un. Mat. Ital. B 30 (1989) 857–881.MathSciNetGoogle Scholar
  2. 2.
    L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variations and Free Discontinuity Problems. Oxford Univ. Press, Oxford (2000).Google Scholar
  3. 3.
    G.I. Barenblatt, The mathematical theory of equilibrium cracks in brittle fracture. Adv. in Appl. Mech. 7 (1962) 44–129.Google Scholar
  4. 4.
    Z. Bažant and J. Planas, Fracture and Size Effect in Concrete and Other Quasi-Brittle Materials. CRC Press, New York (1998).Google Scholar
  5. 5.
    D. Broek, The Practical Use of Fracture Mechanics. Kluwer, Dordrecht (1989).Google Scholar
  6. 6.
    S. Cedolin, S. Dei Poli and I. Iori, Tensile behavior of concrete. ASCE J. Engrg. Mech. 113 (1987) 431–449.CrossRefGoogle Scholar
  7. 7.
    E.A. Davis, The effect of the speed of stretching and the rate of loading on the yielding of mild steel. ASME J. Appl. Mech. (1938) A137–A140.Google Scholar
  8. 8.
    G. Del Piero, Towards an unified approach to fracture, yielding and damage. In: E. Inan and K.Z. Markov (eds), Proc. of the 9th Internat. Symp. on Continuum Models and Discrete Systems. World Scientific, Singapore (1998) pp. 679–692.Google Scholar
  9. 9.
    G. Del Piero, One-dimensional ductile-brittle transition, yielding and structured deformations. In: P. Argoul, M. Frèmond and Q.S. Nguyen (eds), Variations de Domaines et Frontières Libres en Mécanique des Solides. Kluwer, Dordrecht (1999).Google Scholar
  10. 10.
    G. Del Piero and L. Truskinovsky, Macro-and micro-cracking in one-dimensional elasticity. Internat. J. Solids Struct. 38 (2001) 1135–1148.zbMATHCrossRefGoogle Scholar
  11. 11.
    D.S. Dugdale, Yielding of steel sheets containing slits. J. Mech. Phys. Solids 8 (1960) 100–108.CrossRefADSGoogle Scholar
  12. 12.
    C.F. Elam, Distortion in Metal Crystals. Claredon Press, Oxford (1935).Google Scholar
  13. 13.
    R. Fosdick and D. Mason, Single phase energy minimizers for materials with nonlocal spatial dependence. Quart. Appl. Math. 54 (1996) 161–195.zbMATHMathSciNetGoogle Scholar
  14. 14.
    M. Froli and G. Royer-Carfagni, On discontinuous deformation of tensile steel bars: Experimental results. ASCE J. Engrg. Mech. 125 (1999) 1243–1250.CrossRefGoogle Scholar
  15. 15.
    M. Froli and G. Royer-Carfagni, A mechanical model for elastic-plastic behavior of metallic bars. Internat. J. Solids Struct. 37 (2000) 3901–3918.zbMATHCrossRefGoogle Scholar
  16. 16.
    A. Hillerborg, M. Modèer and P.E. Petersson, Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement Concrete Res. 6 (1976) 773–782.CrossRefGoogle Scholar
  17. 17.
    J. Planas, M. Elices and G.V. Guinea, The extended cohesive crack. In: G. Bakker and B.L. Karihaloo(eds), Fracture of Brittle Disordered Materials: Concrete, Rock and Ceramics. E. & FN Spon, London (1995) pp. 51–65.Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Camillo De‐Lellis
    • 1
  • Gianni Royer‐Carfagni
    • 2
  1. 1.Scuola Normale SuperiorePiazza dei CavalieriPisaItaly
  2. 2.Dipartimento di Ingegneria CivileUniversità di ParmaParmaItaly

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