Evaluation of elastic strain energy of spheroidal inclusions with uniform volumetric and shear eigenstrains

Böhm, H.J.; Fischer, Franz Dieter; Reisner, G.

doi:10.1016/s1359-6462(96)00476-9

Cited by 19 publications

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“…According to Eshelby's seminal work on inclusions, see the references in [2]; the eigenstress state is spatially constant in the cylindrical inclusion. The inside eigenstress state can immediately be calculated in Cartesian coordinates (x, y, r = x 2 + y 2 , z axis of rotation), according to [3,4], aspect ratio α → ∞, Eqs. (4.1-2) and (A4) 4 there, with the shear modulus…”

Section: Solution Conceptmentioning

confidence: 99%

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Stress–strain analysis of the antiplane shear problem for an infinite cylindrical inclusion with eigenstrain: an addendum to Arch. Appl. Mech. 2018

2018

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As outlined in the previous paper, an eigenstrain state due to an interstitial atom, acting as misfitting inclusion in the crystal lattice, may interact with a load stress state and/or a stress state due to a defect. Since the interstitial atoms can be assembled in a cylinder, as in the established case of a so-called Cottrell cloud, the according stress state can be investigated by a cylindrical inclusion with a general eigenstrain state. Both eigenstrain components belonging to the antiplane shear, however, were not dealt with in the past. According stress and deformation equations are now offered in analytical form in this addendum.

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Section: Solution Conceptmentioning

confidence: 99%

“…The inside eigenstress state can immediately be calculated in Cartesian coordinates (x, y, r = x 2 + y 2 , z axis of rotation), according to [3,4], aspect ratio α → ∞, Eqs. (4.1-2) and (A4) 4 there, with the shear modulus…”

Section: Solution Conceptmentioning

confidence: 99%

Stress–strain analysis of the antiplane shear problem for an infinite cylindrical inclusion with eigenstrain: an addendum to Arch. Appl. Mech. 2018

2018

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“…Expressions given by Böhm, Fischer and Reisner (1997) for the non-zero components of the Eshelby tensor for ellipsoids of revolution in an isotropic system are listed below using the contracted matrix notation rules given by Eq. (2.88).…”

Section: Components Of the Eshelby Tensor In Isotropic Systemmentioning

confidence: 99%

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2016

Introduction to Elasticity Theory for Crystal Defects

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A summary is presented of a number of relationships involving the del operator, ∇, in the cylindrical and spherical orthogonal curvilinear coordinate systems shown in Fig. A.1. A.1 Cylindrical Orthogonal Curvilinear CoordinatesIf f = f(r, θ, z), and f =ê r f r +ê θ f θ +ê z f z , where the unit base vectors (ê r ,ê θ ,ê z ) are given in terms of Cartesian base vectors by Eq. (G. 2 Introduction to Elasticity Theory for Crystal Defects 2nd Ed A.2 Spherical Orthogonal Curvilinear CoordinatesIf f = f (r, θ, φ), and f =ê r f r +ê θ f θ +ê φ f φ , where the unit base vectors (ê r ,ê θ ,ê φ ) are given in terms of Cartesian base vectors by Eq. (G. Appendix B9 Integral Relationships B.1 Divergence (Gauss's) TheoremIf A(x) is a vector field, and V is a region of volume enclosed by the surface S, andn is the positive unit vector normal to S (and therefore in the outward direction), as in Fig. B B.2 Stokes' TheoremIf M(x) is a vector field, Stokes' theorem is usually expressed aswhere the line integral is over the closed curve, C, in Fig. B.2 on which the surface, S, terminates. The surface integral is over S, andn is the positive unit vector normal to S, which is defined for this unclosed surface by the requirement that if C were shrunk down on S until it just traversed a circuit aroundn in the direction of the tangent vector,t, the circuit would be clockwise when sighting alongn. is a scalar function of x in a body,as can be verified by applying Eq. (B.5) to the case where G has the form G lj = e lij g i . Then, , into Eq. (B.10), then yields the further relationship Appendix C The Tensor Product of Two VectorsThe tensor product of two vectors can be used to represent tensors as products of vectors. In many cases equations involving tensors can then be written in more compact vector forms. The tensor product, P, of two vectors a and b is written as P = a ⊗ b (C.1) and possesses components given by Appendix D Properties of the Delta FunctionThe one-dimensional delta function, δ(x − x • ) vanishes everywhere except at x = x • , and has the property, e.g., Hassani (2000), thatThen, in three dimensions with vector arguments, the delta function appears asThe N th derivative of the delta function with respect to its argument, indicated here by the superscript (N), obeys the ruleFurther properties, as given by Bacon, Barnett and Scattergood (1979b), are listed in Still further useful relationships can be obtained from potential theory. The electrostatic potential, v(x), due to a distribution of electrical charge density, ρ(x), must satisfy Poisson's equationTherefore, inserting Eqs. (3.14) andIntroduction to Elasticity Theory for Crystal Defects Downloaded from www.worldscientific.com by 54.245.55.244 on 05/11/18. For personal use only. Appendix E The Alternator OperatorThe alternator operator, e ijk , is conveniently expressed in the formwhere, as usual, theê i are the base unit vectors of a Cartesian, righthanded, orthogonal coordinate system. Therefore, for example, it has the properties e 123 = e 231 = e 312 = 1, e 13...

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Thermodynamics and Kinetics of Phase and Twin Boundaries

Fischer

Simha

CISM International Centre for Mechanical Sciences

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Evaluation of elastic strain energy of spheroidal inclusions with uniform volumetric and shear eigenstrains

Cited by 19 publications

References 14 publications

Stress–strain analysis of the antiplane shear problem for an infinite cylindrical inclusion with eigenstrain: an addendum to Arch. Appl. Mech. 2018

Stress–strain analysis of the antiplane shear problem for an infinite cylindrical inclusion with eigenstrain: an addendum to Arch. Appl. Mech. 2018

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Thermodynamics and Kinetics of Phase and Twin Boundaries

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