Micromechanics Of Defects Research Articles

Micromechanics of defects in functional materials

We report on the unified approach to the classification of defects in continuum-based analysis of the defect self-distortions. Defects of various dimensionality including point, linear, surface, and volume defects are considered depending on the domain of self-distortion definition. The results of analytical calculations of elastic fields and the energies of dislocations, disclinations, and inclusions in bodies of various geometries with internal and external boundaries are considered. Examples of defects in modern functional materials, e.g., graphene, are given. It is discussed how the presence of defects and associated elastic fields affects the properties of modern functional materials and the characteristics of devices based on them.

Displacement field due to glide and climb of rectilinear dislocations in gradient elasticity

ABSTRACT The aim of this article is to provide the displacement field of a straight dislocation in gradient elasticity that can be implemented in dislocation dynamics simulations. The displacement and plastic strain fields of a dislocation depend on the history of the dislocation motion. The cut surface (or the branch cut) represents this history. When an edge dislocation glides, the branch cut must be parallel to the Burgers vector, and when a dislocation moves by a combination of glide and climb, extra terms need to be added to the displacement field to capture the correct discontinuity in the displacement field. The existing formulas for the dislocation displacement in gradient elasticity lack such considerations, which make them unsuitable for simulations. In addition, the previous formulas both in classical and gradient elasticity have contained some inaccuracies and miscalculations. In this paper, the displacement field of a dislocation and the corresponding plastic strain are derived in detail and a distinction between the dislocation due to glide and that due to climb has been made. Moreover, we discuss the shortcomings of Mura's [Micromechanics of Defects in Solids, 2nd ed., Martinus Nijhoff Publishers, 1987.] and Lazar and Maugin's [Dislocations in gradient elasticity revisited, Proc. R. Soc. Math. Phys. Engin. Sci. 462(2075) (2006), pp. 3465–3480.] calculations. We illustrate that Orowan's law is a direct result of the correct displacement and plastic strain fields.

Average and local strain fields in nanocrystals

This article presents a rigorous and self-consistent comparison of lattice distortion and deformation fields existing in energy-optimized pseudo-spherical gold nanoparticles obtained from real-space and powder diffraction strain analysis techniques. The changes in atomic positions resulting from energy optimization (relaxation) of ideally perfect gold nanoparticles were obtained using molecular dynamics modeling. The relaxed atomic coordinates were then used to compute the displacement, rotation and strain components in all unit cells within the energy-optimized (relaxed) particles. It was seen that all of these terms were distributed heterogeneously along the radial and tangential directions within the nanospheroids. The heterogeneity was largest in the first few atomic shells adjacent to the nanoparticle surface, where the continuity of crystal lattice vectors originating from the interior layers was broken because of local lattice rotations. These layers also exhibited maximum shear and normal strains. These (real-space) strain values were then compared with the average lattice strains obtained by refining the computed diffraction patterns of such particles. The results show that (i) relying solely on full-pattern refinement techniques for lattice strain analysis might lead to erroneous conclusions about the dimensionality and symmetry of deformation within relaxed nanoparticles; (ii) the lattice strains within such relaxed particles should be considered `eigenstrains' (`inherent strains') as defined by Mura [Micromechanics of Defects in Solids, (1991), 2nd ed., Springer]; and (iii) the stress/strain state within relaxed nanoparticles cannot be analyzed rigorously using the constitutive equations of linear elasticity.

Closed Form Solution of the Exterior-Point Eshelby Tensor for an Elliptic Cylindrical Inclusion

With the help of the I-integrals expressed by Mura (1987, Micromechanics of Defects in Solids, 2nd ed., Martinus Nijhoff, Dordrecht) and the outward unit normal vector introduced by Ju and Sun (1999, “A Novel Formulation for the Exterior-Point Eshelby’s Tensor of an Ellipsoidal Inclusion,” ASME Trans. J. Appl. Mech., 66, pp. 570–574), the closed form solution of the exterior-point Eshelby tensor for an elliptic cylindrical inclusion is derived in this work. The proposed closed form of the Eshelby tensor for an elliptic cylindrical inclusion is more explicit than that given by Mura, which is rough and unfinished. The Eshelby tensor for an elliptic cylindrical inclusion can be reduced to the Eshelby tensor for a circular cylindrical inclusion by letting the aspect ratio of the inclusion α=1. The closed form Eshelby tensor presented in this study can contribute to micromechanics-based analysis of composites with elliptic cylindrical inclusions.

Variational eigenstrain multiscale finite element method

A new subgrid (subscale) finite element method is proposed, which is termed as the variational eigenstrain multiscale method. It combines the essential ideas of Hughes' variational multiscale formulation [Comput. Methods Appl. Mech. Engrg. 127 (1995) 387] with Eshelby's inclusion theory [Proc. R. Soc. Lond. A 252 (1957) 561] and Mura's equivalent eigenstrain principle [Micromechanics of Defects in Solids, Martinus Nijhoff Publisher, 1987] to homogenize numerical error due to finite element discretization. By synthesizing variational multiscale method with the equivalent eigenstrain principle, we have developed a new finite element formulation that can automatically homogenize its own discretization errors so that it may attain better accuracy in a coarse scale finite element computation than that of the original coarse scale finite element computation. The paper provides the theoretical foundation of the method as well as two numerical examples that illustrate and validate the proposed method.

Visco-Elastic Waves in Rock-Like Composites

The subject of this paper is the treatment of rocks - and, especially, fluid-saturated and partially saturated reservoir rocks, as composite visco-elastic media. By this we mean to study and partially answer the question of how the effective material (frequency-dependent and complex-valued stiffness/density) parameters can be estimated from a knowledge of the constituents of the rocks, their volume fractions, the statistical distribution of sizes, shapes, orientations and positions of the individual particles (minerals of quartz, clay, etc.) and cavities (pores, cracks, etc.); in addition to parameters related to the fluid and its ability to flow, at the scale of the microstructure as well as that of the wavelength (assumed to be long compared to the scale-size of the microstructure). Our approach is to develop and combine a theory of stochastic waves with established results for the micromechanics of defects in solids, as well as state-of-the-art models of wave-induced fluid flow. Specifically, we first derive an exact formal expression for the effective material parameters in terms of a dynamic T-matrix for the material, which satisfies a single integral equation of the Lippmann-Schwinger type (known from quantum scattering theory), but formulated in an abstract vector space, associated with the combination of the strain and velocity fields into a more general state vector Ψ. Inclusions-based models are developed on the basis of standard many-body techniques, known from the static T-matrix approach as well as nuclear collision theory. The t-matrix of a low-aspect-ratio spheroidal crack is expressed in terms of the familiar displacement discontinuity parameters of Hudson, via the so-called K-tensor, which is of interest in itself, for example, when connecting cracks to pores (in the presence of multiple solid constituents) on the basis of an expression for the t-matrix of a communicating cavity. The present theory can in principle be used beyond the Rayleigh limit, but explicit estimates of the effective material parameters have so far been derived only under the assumption that (scattering attenuation can be ignored) the wavelength is large compared to the scale-size of a representative volume element. Starting with the dynamic equations of motion, we show that the behaviour of the mean wave in the Rayleigh limit is indeed determined by the effective stiffness tensor associated with a static theory of composites, in conjunction with the spatially averaged density for the heterogeneous material as a whole. Thus, we have provided justification to the procedure we used in a series of related papers, where we started out with the static equilibrium condition and employed the elastic/visco-elastic correspondence principle. Numerical examples (dealing with the effects of randomly oriented cracks on the isotropic velocity and attenuation spectra of a dual porosity model of clay-sand mixtures, and the effects of spatial distribution on the anisotropic attenuation spectra of fully aligned cracks that are partially saturated with two different fluids) will be provided in order to complement those in our earlier papers.

An approximate algorithmic solution for the elastic fields in bonded patched sheets

An approximate algorithmic solution is derived for the elastic fields in an infinite isotropic sheet rigidly bonded with an orthotropic, polygon-shaped patch. The approach employed here combines the equivalent inclusion method by Eshelby, see work by Mura (Micromechanics of Defects in Solids, 1987) and the extension of Rodin’s (J. Mech. Phys. Solids 44, 1996) algorithmic solution for a polygonal inclusion with constant eigenstrains to the case of polynomial eigenstrains. A numerical example of an octagonal shaped patch symmetric with respect to two coordinate axes is presented using zero- and second-ordered approximation for eigenstrains, and the results are compared with those from finite element method. The study was motivated by the lack of analytical capabilities for analyzing and designing bonded repairs with a general shaped patch in aging aircraft.

Micromechanics of defects in transversely isotropic elastic solids with interfaces

The response of microstructural defects to internal and external fields is an important subject in materials science and engineering. The knowledge of elastic deformations due to defects is the first step towards understanding the fundamental effects of defects on the mechanical and physical properties of materials. A recently developed method for obtaining the explicit three- dimensional closed form elastic solution in solids with defects in the presence of an interface is reviewed. The interface is a planar boundary between two semi-infinite solids. The solids are either perfectly bonded together or in frictionless contact with each other at the planar interface. The defects are point defects, dislocations, disclinations, inclusions, elastic inhomogeneities, and thermal inhomogeneities. The method is a modified point force method. The elastic solution is expressed in terms of force double Green's functions in a simple form. The force double Green's functions are the partial derivatives of point force solutions. The only task in finding the solution is to find the appropriate surface or volume integral of functions of the distance from the defects to the point of interest. The integrals are the Newtonian potential of the defects with appropriate mass density and other biharmonic and harmonic potentials that are associated with the Newtonian potentials. The same method applies to both isotropic and transversely isotropic solids.

Some new problems in the micromechanics

Main fundamental subjects in the micromechanics are mathematical and mechanical formulations for inclusions and dislocations as shown in Mura [T. Mura, Micromechanics of Defects in Solids, Martinus Nijhoff, Dordrecht, 1987]. Literature surveying inclusion problems, further mathematical development and applications of the inclusion theory to material transformation and composite materials were reported in [T. Mura, Appl. Mech. Rev. 41 (1988) 15–20] and [T. Mura, H.M. Shodja, Y. Hirose, Appl. Mech. Rev. 49 (1996) 118–127]. This paper proposes some new problems associated with the fundamental mathematical mechanical formulations of the inclusion. If an important old problem has not been solved yet, the old problem is a new frontier. Eshelby conjecture states that the ellipsoidal inclusions in an isotropic elastic medium are the only kind inclusions with the remarkable property that the stress field inside the inclusion is uniform if the prescribed eigenstrain in uniform [J.D. Eshelby, Proc. R. Soc. London, Ser. A. 46 (1957) 376–396; J.D. Eshelby, Elastic inclusions and inhomogeneities, in: I.N. Snedon, R. Hill (Eds.), Progress in Solid Mechanics, vol. 2, North Holland, Amsterdam, 1961, pp. 89–140]. Recently, Mura [T. Mura, Mech. Res. Commun. 24 (1997) 473–482] found that a m-pointed polygonal inclusion subjected to the uniform eigenstrain would also produce the uniform stress field inside the inclusion, if m is an odd number. Contradictory to this result, several papers have been reported as well, for instance, Rodin [G.J. Rodin, J. Mech. Phys. Solids 44 (1996) 473–482], Nozaki and Taya [H. Nozaki, M. Taya, J. Appl. Mech. (1997) 495–502], Markenscoff [X. Markenscoff, J. Mech. Phys. Solids (1998)], and Shozu et al. [W. Shozu, K. Hirashima, Y. Hirose, Jpn. Soc. Mech. Eng. (1996) 1643–1648]. The so-called mechanics of materials should study physical properties of materials more than the elastic moduli. A new method is proposed to satisfy the need mentioned above. The Schrödinger equation is used in determination of eigenstrain [T. Mura, The eigenstrain method applied to fracture and fatigue mechanics, in: G.C. Sih, H. Nishitani, T. Ishibara (Eds.), Role of Fracture Mechanics in Modern Technology, 1987, pp. 145–152]. The semi-infinite chain of point atoms bonded longitudinally by linear bendable elements and linear stretchable elements with spacing, a 0, zero-stress separation, b 0, and spring constants, α and β, employed by Kitamura et al. [K. Kitamura, I.L. Marksimov, K. Nishioka, Phil. Mag. Lett. 75 (6) (1997) 343–350], can be considered in terms of the eigendistortion to explain the effect of temperature on the crack-lattice trapping (CLT) phenomenon.

Modelling the effect of fluid communication on velocities in anisotropic porous rocks

An inclusion-based anisotropic poroelasticity model (IBAPM) has been developed from the J. D. Eshelby (1957, Proceedings of the Royal Society of London Series A, 241, 376–396) and T. Mura (1987, Micromechanics of Defects in Solids, Martinus Nijhoff, Dordrecht) theories. Eshelby’s interaction energy approach is employed to calculate the effective elastic constants of a solid containing dilute pores, since it offers two approximations : one corresponding to the condition of constant load and the other to the condition of constant displacement. The effect of pore fluid communication on the poroelasticity of a porous rock is modelled via pore connectivity. A totally isolated pore system implies that pore pressure is equilibrated within each individual pore only and may vary from pore to pore, depending on the shape and orientation of each pore. In terms of the local flow mechanism, this gives us a high-frequency estimate of the effective elastic moduli. The low-frequency estimate, on the other hand, is modelled by a communicating pore system. In this case all pores are assumed to be connected and, consequently, pore pressure is equilibrated within the whole pore system. Other related problems discussed in the paper include : (1) extension of the model to the case of pores having arbitrary orientations, (2) extension of the non-interacting model to higher pore concentrations and (3) consistency checks of IBAPM with previous models. In particular the low-frequency version of the model is compared with the anisotropic version of the Gassmann model (R. J. S. Brown and J. Korringa, 1975, Geophysics, 40, 608–616) . Finally the model is applied to the laboratory measurements reported by J. S. Rathore, E. Ejaer, R. M. Holt and L. Renlie (1995, Geophys. Prosp., 43, 711–728) .

An elastic-plastic micromechanical analysis of a cross-ply composite

Multi-directional fiber-reinforced composites are often used in aerospace structures. The design of such structures needs a realistic macroscopic stress/strain relationship for the composite material. In this paper an analytical approach for determining this stress/strain relationship is developed which is based on detailed analysis of the microscopic stress/strain fields, which govern the spread of plastic strain and the amount of material damage. In this approach, the heterogeneous composite is transformed into a homogeneous solid by Eshelby's equivalent-inclusion method (Mura, T., Micromechanics of Defects in Solids. Martinus Nijhoff, The Hague, 1982). 1 The microstresses and strains are then analyzed by using Lin's method of analyzing the microstress fields of elastically homogeneous polycrystals of periodic crystal orientations based on Green's functions of an infinite solid (Lin, T. H., Adv. Appl. Mech., 1971, 11, 255–311). 2 The major advantage of the present approach is that it does not need to isolate a representative volume (or ‘unit cell’) and specify the unknown boundary conditions. A numerical example for a cross-plied boron aluminum composite is given.

Micromechanics determinations of thermoelectroelastic fields and effective thermoelectroelastic moduli of piezoelectric composites

By modelling thermal expansion strain and electric field induced by a uniform temperature change as eigenfields, the eigenstrain formulation ( T. Mura, Micromechanics of Defects in Solids, 2nd revised edn., Martinus Nijhoff, 1987 ) for anisotropic inclusion problems is extended to consider the inherently anisotropic coupled behaviour of a piezoelectric composite. Utilizing the extended eigenstrain formulation, a unified explicit expression for the electroelastic tensors analogous to Eshelby tensors for elastic ellipsoidal inclusions is obtained. Particularly, closed forms for the electroelastic Eshelby tensors are presented when both the piezoelectric matrix and piezoelectric inclusions are transversely isotropic, and the shapes of the inclusions are spheroidal, elliptic cylindrical, circular cylindrical, penny-shaped, and ribbon-like. With the resulting tensors, analytical expressions for the thermoelectroelastic fields of the piezoelectric composite can then be obtained. Moreover based on the Mori-Tanaka theory and the equivalent inclusion method to account for interaction at finite concentrations of inhomogeneities, the effective thermal expansion coefficients, pyroelectric coefficients, elastic moduli, piezoelectric coefficients, and dielectric constants of the piezoelectric composite material in terms of phase properties, volume fraction, and the shape of the inhomogeneities are presented in closed forms.

Experimental study of the dynamic deformation of elastic cylindrical shells in water with evacuation of the internal cavity

ties in elastic solids," J. Elasticity, 16, No. 2, 201-211 (1986). 12. R. Hill, "An invariant treatment of interfacial discontinuities in elastic composites," in: Mechanics of a Continuous Medium and Kindred Analysis Problems [in Russian], (1972), pp. 596-604. 13. N. Laws, "On interfacial discontinuities in elastic composites," J. Elasticity, ~, No. 2, 227-235 (1975). 14. T. Mura, Micromechanics of Defects in Solids, Martinus Nijhoff Publ., The Hague (1982). 15. K. Yamazaki and J. Oda, "Optimum shape of elastic inclusion in infinite elastic solid under triaxial stress field," Comput. Mech. 86: Theory and Appl. Proc. Int. Conf., Tokyo, May 25-29, 1986, Vol. 2, X/83-X/89 (1986).

Micromechanics of Defects in Solids (2nd rev. ed.)

Micromechanics of Defects in Solids (2nd rev. ed.)

Micromechanics of Defects in Solids, by Toshio Mura

<i>Micromechanics of Defects in Solids</i>, by Toshio Mura

Micromechanics of Defects in Solids

Micromechanics of Defects in Solids