Research Papers

Cohesive Zone-Based Damage Evolution in Periodic Materials Via Finite-Volume Homogenization

[+] Author and Article Information
Wenqiong Tu

Civil Engineering Department,
University of Virginia,
Charlottesville, VA 22904-4742

Marek-Jerzy Pindera

Civil Engineering Department,
University of Virginia,
Charlottesville, VA 22904-4742

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received March 29, 2014; final manuscript received July 26, 2014; accepted manuscript posted August 13, 2014; published online August 13, 2014. Assoc. Editor: Nick Aravas.

J. Appl. Mech 81(10), 101005 (Aug 13, 2014) (16 pages) Paper No: JAM-14-1141; doi: 10.1115/1.4028103 History: Received March 29, 2014; Revised July 26, 2014; Accepted July 30, 2014

The zeroth-order parametric finite-volume direct averaging micromechanics (FVDAM) theory is further extended in order to model the evolution of damage in periodic heterogeneous materials. Toward this end, displacement discontinuity functions are introduced into the formulation, which may represent cracks or traction-interfacial separation laws within a unified framework. The cohesive zone model (CZM) is then implemented to simulate progressive separation of adjacent phases or subdomains. The new capability is verified in the linear region upon comparison with an exact elasticity solution for an inclusion surrounded by a linear interface of zero thickness in an infinite matrix that obeys the same law as CZM before the onset of degradation. The extended theory's utility is then demonstrated by revisiting the classical fiber/matrix debonding phenomenon observed in SiC/Ti composites, illustrating its ability to accurately capture the mechanics of progressive interfacial degradation.

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Barenblatt, G. I., 1959, “The Formation of Equilibrium Cracks During Brittle Fracture. General Ideas and Hypothesis. Axially-Symmetric Cracks,” Prikl. Mat. Mekh., 23(3), pp. 434–444. [CrossRef]
Barenblatt, G. I., 1962, Mathematical Theory of Equilibrium Cracks in Brittle Fracture (Advances in Applied Mechanics, Vol. VII), H. L.Dryden, and T.von Karman, eds., Academic, New York, pp. 55–125.
Dugdale, D. S., 1960, “Yielding of Steel Sheets Containing Slits,” J. Mech. Phys. Solids, 8(2), pp. 100–104. [CrossRef]
Needleman, A., 1987, “A Continuum Model for Void Nucleation by Inclusion Debonding,” ASME J. Appl. Mech., 54(3), pp. 525–531. [CrossRef]
Xu, X. P., and Needleman, A., 1994, “Numerical Simulation of Fast Crack Growth in Brittle Solids,” J. Mech. Phys. Solids, 42(9), pp. 1397–1434. [CrossRef]
Ortiz, M., and Suresh, S., 1993, “Statistical Properties of Residual Stresses and Intergranular Fracture in Ceramic Materials,” ASME J. Appl. Mech., 60(1), pp. 77–84. [CrossRef]
Camacho, G. T., and Ortiz, M., 1996, “Computational Modeling of Impact Damage in Brittle Materials,” Int. J. Solids Struct., 33(20–22), pp. 2899–2938. [CrossRef]
Elices, M., Guinea, G. V., Gomerz, J., and Planas, J., 2002, “The Cohesive Zone Model: Advantages, Limitations and Challenges,” Eng. Fract. Mech., 69(2), pp. 137–163. [CrossRef]
Jiang, L. Y., Tan, H. L., Wu, J., Huang, Y. G., and Hwang, K. C., 2007, “Continuum Modeling of Interfaces in Polymer Matrix Composites Reinforced by Carbon Nanotubes,” Nano, 2(3), pp. 139–149. [CrossRef]
Banea, M. D., and da Silva, L. F. M., 2009, “Adhesively Bonded Joints in Composite Materials: An Overview,” Proc. Inst. Mech. Eng., Part L, 223, pp. 1–18. [CrossRef]
Kim, Y. P., 2011, “Cohesive Zone Model to Predict Fracture in Bituminous Materials and Asphaltic Pavements: State-of-the-Art Review,” Int. J. Pavement Eng., 12(4), pp. 343–356. [CrossRef]
Park, K., and Paulino, G. H., 2011, “Cohesive Zone Models: A Critical Review of Traction-Separation Relationships Across Fracture Surfaces,” ASME Appl. Mech. Rev., 64(6), p. 060802. [CrossRef]
Berezovski, A., Engelbrecht, J., and Maugin, G. A., 2008, Numerical Simulation of Waves and Fronts in Inhomogeneous Solids, Series A, Vol. 62, World Scientific, Hackensack, NJ.
Versteeg, H. K., and Malalasekera, W., 2007, An Introduction to Computational Fluid Dynamics: The Finite Volume Method, Pearson Education, Ltd., Prentice-Hall, New York.
Ivankovic, A., Demirdzic, I., Williams, J. G., and Leevers, P. S., 1994, “Application of the Finite Volume Method to the Analysis of Dynamic Fracture Problems,” Int. J. Fract., 66(4), pp. 357–371. [CrossRef]
Ivankovic, A., 1999, “Finite Volume Modelling of Dynamic Fracture Problems,” Comput. Model. Simul. Eng., 4, pp. 227–235.
Stylianou, V., and Ivankovic, A., 2002, “Finite Volume Analysis of Dynamic Fracture Phenomena, Part II: A Cohesive Zone Type Modelling,” Int. J. Fract., 113(2), pp. 107–123. [CrossRef]
Karac, A., Blackman, B. R. K., Cooper, V., Kinloch, A. J., Sanchez, S. R., Teo, W. S., and Ivankovic, A., 2011, “Modelling the Fracture Behaviour of Adhesively-Bonded Joints as a Function of Test Rate,” Eng. Fract. Mech., 78(6), pp. 973–989. [CrossRef]
Carolan, D., Tukovic, Z., Murphy, N., and Ivankovic, A., 2013, “Arbitrary Crack Propagation in Multi-Phase Materials Using the Finite Volume Method,” J. Comput. Mater. Sci., 69, pp. 153–159. [CrossRef]
Cavalcante, M. A. A., Pindera, M.-J., and Khatam, H., 2012, “Finite-Volume Micromechanics of Periodic Materials: Past, Present and Future,” Composites, Part B, 43(6), pp. 2521–2543. [CrossRef]
Pindera, M.-J., Khatam, H., Drago, A. S., and Bansal, Y., 2009, “Micromechanics of Spatially Uniform Heterogeneous Media: A Critical Review and Emerging Approaches,” Composites, Part B, 40(5), pp. 349–378. [CrossRef]
Tukovic, Z., Ivankovic, A., and Karac, A., 2013, “Finite-Volume Stress Analysis in Multi-Material Linear Elastic Body,” Int. J. Numer. Meth. Eng., 93(4), pp. 400–419. [CrossRef]
Alveen, P., McNamara, D., Carolan, D., Murphy, N., and Ivankovic, A., 2014, “Analysis of Two-Phase Ceramic Composites Using Micromechanical Models,” Comput. Mater. Sci., 92, pp. 318–324. [CrossRef]
Chen, L., and Pindera, M.-J., 2007, “Plane Analysis of Finite Multilayered Media With Multiple Aligned Cracks. Part I: Theory,” ASME J. Appl. Mech., 74(1), pp. 128–143. [CrossRef]
Cavalcante, M. A. A., Marques, S. P. C., and Pindera, M.-J., 2007, “Parametric Formulation of the Finite-Volume Theory for Functionally Graded Materials. Part I: Analysis,” ASME J. Appl. Mech., 74(5), pp. 935–945. [CrossRef]
Bansal, Y., and Pindera, M.-J., 2003, “Efficient Reformulation of the Thermoelastic Higher-Order Theory for Functionally Graded Materials,” J. Therm. Stresses, 26(11/12), pp. 1055–1092. [CrossRef]
Gattu, M., Khatam, H., Drago, A. S., and Pindera, M.-J., 2008, “Parametric Finite-Volume Micromechanics of Uniaxial, Continuously-Reinforced Periodic Materials With Elastic Phases,” ASME J. Eng. Mater. Technol., 130(3), p. 031015. [CrossRef]
Khatam, H., and Pindera, M.-J., 2009, “Thermo-Elastic Moduli of Lamellar Composites With Wavy Architectures,” Composites, Part B, 40(1), pp. 50–64. [CrossRef]
Khatam, H., and Pindera, M.-J., 2009, “Parametric Finite-Volume Micromechanics of Periodic Materials With Elastoplastic Phases,” Int. J. Plast., 25(7), pp. 1386–1411. [CrossRef]
Bansal, Y., and Pindera, M.-J., 2005, “A Second Look at the Higher-Order Theory for Periodic Multiphase Materials,” ASME J. Appl. Mech., 72(2), pp. 177–195. [CrossRef]
Bansal, Y., and Pindera, M.-J., 2006, “Finite-Volume Direct Averaging Micromechanics of Heterogeneous Materials With Elastic-Plastic Phases,” Int. J. Plast., 22(5), pp. 775–825. [CrossRef]
Bensoussan, A., Lions, J.-L., and Papanicolaou, G., 1978, Asymptotic Analysis for Periodic Structures, North Holland, Amsterdam, Netherlands.
Suquet, P. M., 1987, Elements of Homogenization for Inelastic Solid Mechanics (Lecture Notes in Physics, Vol. 272), Springer-Verlag, Berlin, pp. 193–278.
Charalambakis, N., 2010, “Homogenization Techniques and Micromechanics. A Survey and Perspectives,” ASME Appl. Mech. Rev., 63(3), p. 0308031. [CrossRef]
Achenbach, J. D., 1975, A Theory of Elasticity With Microstructure for Directionally Reinforced Composites, Springer-Verlag, New York.
Geubelle, P. H., and Baylor, J. S., 1998, “Impact-Induced Delamination of Composites: A 2D Simulation,” Composites, Part B, 29(5), pp. 589–602. [CrossRef]
Chandra, N., Li, H., Shet, C., and Ghonem, H., 2002, “Some Issues in the Application of Cohesive Zone Models for Metal–Ceramic Interfaces,” Int. J. Solids Struct., 39(10), pp. 2827–2855. [CrossRef]
Matous, K., and Geubelle, P. H., 2006, “Multiscale Modelling of Particle Debonding in Reinforced Elastomers Subjected to Finite Deformations,” Int. J. Numer. Meth. Eng., 65(2), pp. 190–223. [CrossRef]
Song, S. H., Paulino, G. H., and Buttlar, W. G., 2006, “A Bilinear Cohesive Zone Model Tailored for Fracture of Asphalt Concrete Considering Viscoelastic Bulk Material,” Eng. Fract. Mech., 73(18), pp. 2829–2848. [CrossRef]
Hill, R., 1963, “Elastic Properties of Reinforced Solids: Some Theoretical Principles,” J. Mech. Phys. Solids, 11(5), pp. 357–372. [CrossRef]
Eshelby, J. D., 1957, “The Determination of the Elastic Field of an Ellipsoidal Inclusion and Related Problems,” Proc. R. Soc., London, Ser. A, 241(1226), pp. 376–396. [CrossRef]
Drago, A. S., and Pindera, M.-J., 2008, “A Locally Exact Homogenization Theory for Periodic Microstructures With Isotropic Phases,” ASME J. Appl. Mech., 75(5), p. 051010. [CrossRef]
Johnson, W. S., Lubowinski, S. J., and Highsmith, A. L., 1990, “Mechanical Characterization of Unnotched SCS6/Ti-15-3 Metal Matrix Composites at Room Temperature, Thermal and Mechanical Behavior of Metal Matrix and Ceramic Matrix Composites,” J. M.Kennedy, H. H.Moeller, W. S.Johnson, eds., American Society for Testing and Materials, Philadelphia, PA, pp. 193–218, Paper No. ASTM STP 1080.
Dieter, G. E., 1976, Mechanical Metallurgy, 2nd ed., McGraw-Hill, New York, pp. 451–489.
Raghavan, P., and Ghosh, S., 2005, “A Continuum Damage Mechanics Model for Unidirectional Composites Undergoing Interfacial Debonding,” Mech. Mater., 37(9), pp. 955–979. [CrossRef]
Cavalcante, M. A. A., and Pindera, M.-J., 2014, “Generalized FVDAM Theory for Periodic Materials Undergoing Finite Deformation. Part I: Framework,” ASME J. Appl. Mech., 81(2), p. 021005. [CrossRef]
Tu, W., and Pindera, M.-J., 2013, “Targeting the Finite-Deformation Response of Wavy Biological Tissues With Bio-Inspired Material Architectures,” J. Mech. Behav. Biomed. Mater., 28, pp. 291–308. [CrossRef] [PubMed]
Kennedy, J., and Eberhart, R., 1995, “Particle Swarm Optimization,” IEEE International Conference on Neural Networks, Perth, WA, November 27–December 1, pp. 1942–1948. [CrossRef]


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Fig. 1

A reference square subvolume in the η–ξ plane (left) mapped onto a quadrilateral subvolume in the y2y3 plane (right) of the actual microstructure

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Fig. 2

Interfacial discontinuity between two adjacent subvolumes

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Fig. 3

Traction-interfacial separation relations for the CZM in normal (left column) and tangential (right column) directions to the interface: graphical representations of (a) coupled and (b) uncoupled relations

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Fig. 11

Interfacial displacement discontinuity and traction distributions around the fiber/matrix interface with progressively greater applied load after fabrication cooldown

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Fig. 12

Full-field stress distributions at progressively greater applied load after fabrication cooldown

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Fig. 4

Unit cell geometry containing 0.05 fiber volume fraction (left) and a detailed close-up (right) used for comparison with the Eshelby solution for an inclusion with a linear interface in an infinite matrix

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Fig. 5

Comparison of radial and tangential displacement discontinuities around the fiber/matrix interface obtained from modified Eshelby and dilute FVDAM solutions

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Fig. 6

Comparison of normal and tangential stress fields in the region occupied by the unit cell obtained from modified Eshelby and dilute FVDAM solutions based on the interfacial properties in Table 1

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Fig. 7

Comparison of normal and tangential stress fields in the region occupied by the unit cell obtained from modified Eshelby and dilute FVDAM solutions with the interfacial properties set to simulate the Kirsch solution

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Fig. 8

Initial transverse response of the unidirectional SiC/Ti composite with different interfacial strengths, illustrating the effect of fabrication-induced residual stresses

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Fig. 9

Comparison of the applied (vertical axis) and calculated (horizontal axis) homogenized strains based on Eq. (33), demonstrating consistency and accuracy of the implemented solution technique for the nonlinear response of the unit cell based on the implemented CZM, and importance of the contributions of the interfacial displacement discontinuities toward total homogenized strains

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Fig. 10

Initial transverse response of the unidirectional SiC/Ti composite immediately after fabrication cooldown, illustrating the effect of (a) uncoupled and (b) coupled interfacial separation laws

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Fig. 14

Interfacial displacement discontinuity and traction distributions around the fiber/matrix interface with progressively greater applied load after initial preloading

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Fig. 15

Full-field stress distributions at progressively greater applied load after fiber/matrix degradation by initial preloading

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Fig. 13

Transverse response of the representative unit cell of unidirectional SiC/Ti composite with different interfacial debonding lengths after initial preloading




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