Research Papers

Application of the Shakedown Theory to Brittle-Matrix Fiber-Reinforced Cracked Composite Beams Under Combined Traction and Flexure

[+] Author and Article Information
Andrea Spagnoli

e-mai: spagnoli@unipr.it

Lorenzo Montanari

Department of Civil-Environmental
Engineering and Architecture,
University of Parma,
Parco Area delle Scienze 181/A,
Parma 43124, Italy

Manuscript received April 3, 2013; final manuscript received August 29, 2013; accepted manuscript posted September 3, 2013; published online October 16, 2013. Assoc. Editor: Daining Fang.

J. Appl. Mech 81(3), 031012 (Oct 16, 2013) (8 pages) Paper No: JAM-13-1147; doi: 10.1115/1.4025313 History: Received April 03, 2013; Revised August 29, 2013; Accepted September 03, 2013

The cracking behavior of a composite beam with multiple reinforcing fibers under periodic traction-flexure is analyzed through a fracture mechanics-based model, where the edge-cracked beam section is exposed to external loads and crack bridging reactions due to the fibers. Assuming a rigid-perfectly plastic bridging law for the fibers and a linear-elastic law for the matrix, the statically indeterminate bridging forces are obtained from compatibility conditions. Under general load paths, shakedown conditions are explored by making use of the Melan's theorem, here reformulated for the discrete problem under consideration, where crack opening displacement at the fiber level plays the role of plastic strain in the counterpart problem of an elastic-plastic solid. The limit of shakedown is determined through an optimization procedure based on a linear programming technique.

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Marshall, D. B., Cox, B. N., and Evans, A. G., 1985, “The Mechanics of Matrix Cracking in Brittle-Matrix Fiber Composites,” Acta Metall., 33(11), pp. 2013–2021. [CrossRef]
McCartney, L. N., 1987, “Mechanics of Matrix Cracking in Brittle-Matrix Fibre-Reinforced Composites,” Proc. R. Soc. London, Ser. A, 409(1837), pp. 329–350. [CrossRef]
Budiansky, B., Hutchinson, J. W., and Evans, A. G., 1986, “Matrix Fracture in Fiber-Reinforced Ceramics,” J. Mech. Phys. Solids, 34(2), pp. 167–189. [CrossRef]
Li, V. C., 1992, “Postcrack Scaling Relations for Fiber Reinforced Cementitious Composites,” J. Mater. Civ. Eng., 4(1), pp. 41–57. [CrossRef]
Carpinteri, A., and Carpinteri, A., 1984, “Hysteretic Behavior of RC Beams,” J. Struct. Eng., 110(9), pp. 2073–2084. [CrossRef]
Carpinteri, A., 1991, “Energy Dissipation in R.C. Beams Under Cyclic Loadings,” Eng. Fract. Mech., 39(2), pp. 177–184. [CrossRef]
Begley, M. R., and McMeeking, R. M., 1995, “Fatigue Crack Growth With Fiber Failure in Metal–Matrix Composites,” Compos. Sci. Tech., 53(4), pp. 365–382. [CrossRef]
McMeeking, R. M., and Evans, A. G., 1990, “Matrix Fatigue Cracking in Fiber Composites,” Mech. Mater., 9(3), pp. 217–227. [CrossRef]
Matsumoto, T., and Li, V. C., 1999, “Fatigue Life of Fiber Reinforced Concrete With a Fracture Mechanics Based Model,” Cement Concrete Compos., 21(4), pp. 249–261. [CrossRef]
Carpinteri, A., Spagnoli, A., and Vantadori, S., 2004, “A Fracture Mechanics Model for a Composite Beam With Multiple Reinforcements Under Cyclic Bending,” Int. J. Solids Struct., 41(20), pp. 5499–5515. [CrossRef]
Carpinteri, A., Spagnoli, A., and Vantadori, S., 2006, “An Elastic–Plastic Crack Bridging Model for Brittle-Matrix Fibrous Composite Beams Under Cyclic Loading,” Int. J. Solids Struct., 43(16), pp. 4917–4936. [CrossRef]
Carpinteri, A., and Puzzi, S., 2007, “The Bridged Crack Model for the Analysis of Brittle Matrix Fibrous Composited Under Repeated Bending Loading,” ASME J. Appl. Mech., 74(6), pp. 1239–1246. [CrossRef]
Melan, E., 1936, “Theorie Statisch Unbestimmter Systeme aus Ideal-Plastischem Baustoff,” Sitzungsber. Akad. Wiss. Wien, 2A(145), pp. 195–218 (in German).
Carpinteri, A., and Massabò, R., 1997, “Continuous vs Discontinuous Bridged-Crack Model for Fiber-Reinforced Materials in Flexure,” Int. J. Solids Struct., 34(18), pp. 2321–2338. [CrossRef]
Tada, H., Paris, P. C., and Irwin, G. R., 1985, The Stress Analysis of Crack Handbook, Del Research, St. Louis, MO.
Drucker, D. C., 1960, “Plasticity,” Structural Mechanics, J. N.Goodier and N. J.Hoff, eds., Pergamon, Oxford, pp. 407–455.
Maier, G., 1969, “Shakedown Theory in Perfect Elastoplasticity With Associated and Nonassociated Flow-Laws: A Finite Element, Linear Programming Approach,” Meccanica, 4(3), pp. 250–260. [CrossRef]
Polizzotto, C., Borino, G., and Fuschi, P., 1996, “An Extended Shakedown Theory for Elastic-Plastic-Damage Material Models,” Eur. J. Mech. A/Solids, 15(5), pp. 825–858.
Churchman, C. M., and Hills, D. A., 2006, “General Results for Complete Contacts Subject to Oscillatory Shear,” J. Mech. Phys. Solids, 54(6), pp. 1186–1205. [CrossRef]
Churchman, C. M., Korsunsky, A. M., and Hills, D. A., 2006, “The Application of Plasticity Principles to Friction,” J. Strain Anal. Eng. Des., 41(4), pp. 323–328. [CrossRef]
Klarbring, A., Ciavarella, M., and Barber, J. R., 2007, “Shakedown in Elastic Contact Problems With Coulomb Friction,” Int. J. Solids Struct., 44(25–26), pp. 8355–8365. [CrossRef]


Grahic Jump Location
Fig. 1

(a) Schematic of the model, (b) crack profile for fibers in their rigid stage, (c) crack profile for some fibers in their plastic stage

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Fig. 2

(a) Time histories of axial force and bending moment in the case of Mmax/Nmax = 0.25 and cross load path; (b) time histories of axial force and bending moment in the case of Mmax/Nmax = 0.25 and elliptic load path; (c) corresponding M-N load paths

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Fig. 3

Strength distribution of fibers along the height of the beam: (a) constant distribution; (b) linear distribution; (c) symmetrical and linear distribution

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Fig. 4

Bree-like diagram showing the elastic domain and the shakedown domain for the (a), (b), and (c) fiber distributions with ξ = 0.1

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Fig. 5

Bending moment versus rotation and axial force versus rotation for (a) cross load path and (b) elliptic load path in the case of Mmax/Nmax = 0.25 and symmetrical and linear fiber distribution

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Fig. 6

Bridging force versus crack-opening displacement in the case of Mmax/Nmax = 0.25: (a) μ = μSD and cross load path; (b) μ = 1.05μSD and cross load path; (c) μ = μSD and elliptic load path; (d) μ = 1.05μSD and elliptic load path. The symmetrical and linear fiber distribution is considered.

Grahic Jump Location
Fig. 7

Shakedown domain for different crack depths and fiber strength distributions: (a) constant distribution; (b) linear distribution; (c) symmetrical and linear distribution

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Fig. 8

Shakedown domain for intermediate crack depth (ξ = 0.3) and different distributions of fiber strength




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