Research Papers

Rice’s Internal Variables Formalism and Its Implications for the Elastic and Conductive Properties of Cracked Materials, and for the Attempts to Relate Strength to Stiffness

[+] Author and Article Information
Mark Kachanov

Department of Mechanical Engineering  Tufts University Medford, MA 02155 mark.kachanov@tufts.edu

Igor Sevostianov

Department of Mechanical and Aerospace Engineering  New Mexico State University Las Cruces, NM 88003-8001 igor@nmsu.edu

J. Appl. Mech 79(3), 031002 (Apr 05, 2012) (10 pages) doi:10.1115/1.4005957 History: Received June 15, 2011; Revised January 11, 2012; Posted February 13, 2012; Published April 04, 2012; Online April 05, 2012

Rice’s internal variables formalism [1975, “Continuum Mechanics and Thermodynamics of Plasticity in Relation to Microscale Deformation Mechanisms,” in Constitutive Equations in Plasticity, edited by A. Argon, MIT Press, Cambridge, MA, pp. 23–75] is one of the basic tools in the micromechanics of materials. One of its implications is the possibility to relate the compliance/resistivity contributions of cracks—the key quantities in the problem of effective elastic/conductive properties—to the stress intensity factors (SIFs) and thus to utilize a large library of available solutions for SIFs. Examples include configurations that are common in materials science applications: branched and intersecting cracks, cracks with partial contact between crack faces, and cracks emanating from pores. The formalism also yields valuable physical insights of a qualitative character, such as the impossibility to correlate, in a quantitative way, the strength of microcracking materials and their stiffness reduction.

Copyright © 2012 by American Society of Mechanical Engineers
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Figure 1

Stress-strain curves for materials undergoing microstructural changes: (a) Total strain as a sum of accumulated plastic strain ɛp and elastic strain S(W)σ. The difference between elastic slopes in loading and unloading is due to changes in microstructure. (b) Elastic material subjected to microstructural changes (for example, microcracking brittle material); the slope at unloading is smaller due to the mentioned changes. (c) Slopes at loading and unloading are the same in the case of metal plasticity. (d) In the general case, microstructural changes lead to both accumulation of plastic strain and changes in elastic compliance.

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Figure 2

(a) Circular arc crack and (b) components of the compliance contribution tensor normalized to d=π(1-ν2)AER2(1-cosα)

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Figure 3

Branched crack: (a) Geometry of the configuration and (b)–(d) Components of the compliance contribution tensor, as functions of the branch length λ=b/baa normalized to d=1A1-v2Eπa2

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Figure 4

Crack emanating from a circular hole under biaxial tension: (a) Geometry of the configuration. (b) Function F(a/aRR) entering expressions (3.7) for the SIFs. The effect of the hole decreases as the crack length increases. (c) and (d) Components of the compliance contribution tensor normalized to d=π(1-ν02)/d=π(1-ν02)AE0AE0.

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Figure 5

Annular crack emanating from a spherical pore (“Saturn ring”): (a) Geometry of the configuration. (b) As the crack size increases, the SIF quickly approaches the one of a circular crack (the effect of the pore becomes negligible). (c) Component of the compliance contribution tensor normalized to d=1V4π2(1-ν02)E0a3 as a function of dimensionless crack length λ=c/caa. Note quick convergence to H1111 for a circular crack. (d) Accuracy of the approximation by a circular crack of radius a+c.

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Figure 6

Annular crack (circular crack with partial contact in the middle): (a) Geometry of the configuration. (b) Functions entering the numerical solution for the SIF at the inner edge (solid line) and outer edge (dashed line) entering (3.18). (c) Compliance contribution of the crack {normalized to d=[π2(1-ν02)a3]/d=[π2(1-ν02)a3]VE0VE0 as a function of the relative area of the contact}. (d) Radius of the circular crack (without contact) that has the same compliance contribution, as a function of the width of the ring.

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Figure 7

Resistivity contribution of a circular arc crack. (a) Geometry of the configuration and (b) components of the resistivity contribution tensor normalized to d=1Vπ2k0R2(1-cosα).

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Figure 8

Infinite array of parallel cracks: paradoxical strength-stiffness correlation. (a) Geometry of the configuration and (b) as the spacing between cracks decreases, the SIFs decrease (due to increased shielding), whereas the compliance increases.




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