scholarly journals The dynamic generalization of the Eshelby inclusion problem and its static limit

2016 ◽  
Vol 472 (2191) ◽  
pp. 20160256 ◽  
Author(s):  
Luqun Ni ◽  
Xanthippi Markenscoff
Keyword(s):  
Integral Form ◽  
Transformation Strain ◽  
The Self ◽  
Eshelby Tensor ◽  
Inclusion Problem ◽  
Static Limit ◽  
Governing System ◽  
Eshelby Inclusion ◽  
Interior Domain ◽  
Eshelby Problem

The dynamic generalization of the celebrated Eshelby inclusion with transformation strain is the (subsonically) self-similarly expanding ellipsoidal inclusion starting from the zero dimension. The solution of the governing system of partial differential equations was obtained recently by Ni & Markenscoff (In press. J. Mech. Phys. Solids ( doi:10.1016/j.jmps.2016.02.025 )) on the basis of the Radon transformation, while here an alternative method is presented. In the self-similarly expanding motion, the Eshelby property of constant constrained strain is valid in the interior domain of the expanding ellipsoid where the particle velocity vanishes (lacuna). The dynamic Eshelby tensor is obtained in integral form. From it, the static Eshelby tensor is obtained by a limiting procedure, as the axes' expansion velocities tend to zero and time to infinity, while their product is equal to the length of the static axis. This makes the Eshelby problem the limit of its dynamic generalization.

Properties of the self-similarly expanding Eshelby inclusions and application to the dynamic “Hill Jump Conditions”

2016 ◽  
Vol 22 (3) ◽  
pp. 573-578 ◽  
Author(s):  
Xanthippi Markenscoff
Keyword(s):  
Driving Force ◽  
Transformation Strain ◽  
Early Response ◽  
Jump Conditions ◽  
Eshelby Inclusion ◽  
Self Similar ◽  
Interior Domain ◽  
The Hill ◽  
Eshelby Problem

For a self-similarly subsonically dynamically expanding Eshelby inclusion, we show by an analytic argument (based on the analyticity of the coefficients of the ensuing elliptic system and the Cauchy–Kowalevska theorem) that the particle velocity vanishes in the whole interior domain of the expanding inclusion. Since the acceleration term is thus zero in the interior domain in the Navier equations of elastodynamics, this reduces to an Eshelby problem. The classical Hill jump conditions across the interface of a region with transformation strain are expanded here to dynamics when the interface is moving with inertia satisfying the Hadamard jump conditions. The validity of the Eshelby property and the determination of the constrained strain from the dynamic Eshelby tensor in the interior domain allow one to fully determine from the Hill jump conditions the stress across the moving phase boundary of a self-similarly expanding ellipsoidal Eshelby inhomogeneous inclusion. The driving force can then be obtained. Self-similar motion grasps the early response of the system.

On the dynamic generalization of the anisotropic Eshelby ellipsoidal inclusion and the dynamically expanding inhomogeneities with transformation strain

2016 ◽  
Vol 01 (03n04) ◽  
pp. 1640001 ◽  
Author(s):  
Xanthippi Markenscoff
Keyword(s):  
Spherical Inclusion ◽  
Transformation Strain ◽  
Positive Definiteness ◽  
Equivalent Transformation ◽  
Equivalent Inclusion Method ◽  
Equivalent Inclusion ◽  
Chemical Phase ◽  
Eshelby Inclusion ◽  
Interior Domain

The self-similarly dynamically (subsonically) expanding anisotropic ellipsoidal Eshelby inclusion is shown to exhibit the constant stress “Eshelby property” in the interior domain of the expanding inclusion on the basis of dimensional analysis, analytic properties and the proof for the static inclusion alone. As an example of this property and the application of the dynamic Eshelby tensor (constant in the interior domain), it is shown that the Eshelby equivalent inclusion method always allows for the determination of the equivalent transformation strain for a self-similarly dynamically expanding inhomogeneous spherical inclusion when the Poisson's ratio is in the real range (positive definiteness of the strain energy). Thus, the solution of dynamically self-similarly expanding inhomogeneities (chemical phase change) with transformation strain can be obtained, as well as the driving force per unit area of the expanding inhomogeneity.

Elastic Field Due to a Dynamically Transforming Spherical Inclusion

1990 ◽  
Vol 57 (4) ◽  
pp. 845-849 ◽  
Author(s):  
Y. Mikata ◽  
S. Nemat-Nasser
Keyword(s):  
Phase Transformation ◽  
Systematic Study ◽  
Spherical Inclusion ◽  
Closed Form Solution ◽  
Elastic Field ◽  
Form Solution ◽  
Eshelby Tensor ◽  
Inclusion Problem ◽  
Static Limit ◽  
Stress Components

As a first step towards a systematic study of the interaction between a stress-pulse traveling in transformation toughened ceramics and possible phase transformation of zirconia particles, a dynamic inclusion problem is investigated. An exact closed-form solution is obtained for the case of a spherical inclusion. With this result, the dynamic Eshelby tensors for the inside and outside fields of the spherical inclusion are defined and determined. It is confirmed that the static Eshelby tensor is obtained as a static limit of the dynamic Eshelby tensor. It is found in the numerical results that the frequency of the dynamic inclusion has a relatively large influence on the amplitudes of the stress components inside and outside the inclusion.

Development of Generalized Plane-Strain Tensors for the Concentric Cylinder

1995 ◽  
Vol 62 (3) ◽  
pp. 590-594
Author(s):  
N. Chandra ◽  
Zhiyum Xie
Keyword(s):  
Fiber Volume Fraction ◽  
Volume Fraction ◽  
Transformation Strain ◽  
Inclusion Problem ◽  
Infinite Domain ◽  
Concentric Cylinder ◽  
Finite Radius ◽  
Fiber Volume ◽  
Thermal Residual Stresses ◽  
The Matrix

A pair of two new tensors called GPS tensors S and D is proposed for the concentric cylindrical inclusion problem. GPS tensor S relates the strain in the inclusion constrained by the matrix of finite radius to the uniform transformation strain (eigenstrain), whereas tensor D relates the strain in the matrix to the same eigenstrain. When the cylindrical matrix is of infinite radius, tensor S reduces to the appropriate Eshelby’s tensor. Explicit expressions to evaluate thermal residual stresses σr, σθ and σz in the matrix and the fiber using tensor D and tensor S, respectively, are developed. Since the geometry of the present problem is of finite radius, the effect of fiber volume fraction on the stress distribution can be easily studied. Results for the thermal residual stress distributions are compared with Eshelby’s infinite domain solution and finite element results for a specified fiber volume fraction.

Implementation of the self-consistent Kröner–Eshelby model for the calculation of X-ray elastic constants for any crystal symmetry

2019 ◽  
Vol 34 (2) ◽  
pp. 103-109
Author(s):  
Arnold C. Vermeulen ◽  
Christopher M. Kube ◽  
Nicholas Norberg
Keyword(s):  
Elastic Constants ◽  
Ultrasonic Waves ◽  
The Self ◽  
Crystal Symmetry ◽  
Intermediate Step ◽  
X Rays ◽  
X Ray ◽  
Eshelby Inclusion ◽  
Eshelby Model ◽  
Self Consistent

In this paper, we will report about the implementation of the self-consistent Kröner–Eshelby model for the calculation of X-ray elastic constants for general, triclinic crystal symmetry. With applying appropriate symmetry relations, the point groups of higher crystal symmetries are covered as well. This simplifies the implementation effort to cover the calculations for any crystal symmetry. In the literature, several models can be found to estimate the polycrystalline elastic properties from single crystal elastic constants. In general, this is an intermediate step toward the calculation of the polycrystalline response to different techniques using X-rays, neutrons, or ultrasonic waves. In the case of X-ray residual stress analysis, the final goal is the calculation of X-ray Elastic constants. Contrary to the models of Reuss, Voigt, and Hill, the Kröner–Eshelby model has the benefit that, because of the implementation of the Eshelby inclusion model, it can be expanded to cover more complicated systems that exhibit multiple phases, inclusions or pores and that these can be optionally combined with a polycrystalline matrix that is anisotropic, i.e., contains texture. We will discuss a recent theoretical development where the approaches of calculating bounds of Reuss and Voigt, the tighter bounds of Hashin–Shtrikman and Dederichs–Zeller are brought together in one unifying model that converges to the self-consistent solution of Kröner–Eshelby. For the implementation of the Kröner–Eshelby model the well-known Voigt notation is adopted. The 4-rank tensor operations have been rewritten into 2-rank matrix operations. The practical difficulties of the Voigt notation, as usually concealed in the scientific literature, will be discussed. Last, we will show a practical X-ray example in which the various models are applied and compared.

Self-consistent estimates for the viscoelastic creep compliances of composite materials

1978 ◽  
Vol 359 (1697) ◽  
pp. 251-273 ◽  
Keyword(s):  
Composite Materials ◽  
Numerical Solutions ◽  
The Self ◽  
Inclusion Problem ◽  
Inclusion Problems ◽  
Viscoelastic Creep ◽  
Self Consistent ◽  
Title Problem ◽  
Consistent Method ◽  
Selection Of

In this paper the viscoelastic creep compliances of various composites are estimated by the self-consistent method. The phases may be arbitrarily anisotropic and in any concentrations but we demand that one of the phases be a matrix and the remaining phases consist of ellipsoidal inclusions. The theory is succinctly formulated with the help of Stieltjes convolutions. In order to solve the title problem, we first solve the misfitting viscoelastic inclusion problem. Numerical solutions are given for a selection of inclusion problems and for two common composite materials, namely an isotropic dispersion of spheres, and a uni-directional fibre reinforced material.

Eshelby's tensor for embedded inclusions and the Elasto-Capillary phenomenon

2016 ◽  
Vol 01 (03n04) ◽  
pp. 1630002 ◽  
Author(s):  
Shengyou Yang ◽  
Pradeep Sharma
Keyword(s):  
Closed Form ◽  
Solid Mechanics ◽  
Transformation Strain ◽  
Approximate Solutions ◽  
Inclusion Problem ◽  
Physical Sciences ◽  
Classical Paper ◽  
Eshelby’S Tensor ◽  
The Subject ◽  
Curvature Sphere

The elastic state of an embedded inclusion undergoing a stress-free transformation strain was the subject of John Douglas Eshelby's now classical paper in 1957. This paper, the subject of which is now widely known as “Eshelby's inclusion problem”, is arguably one of the most cited papers in solid mechanics and several other branches of physical sciences. Applications have ranged from geophysics, quantum dots to composites. Over the past two decades, due to an interest in all things “small”, attempts have been made to extend Eshelby's elastic analysis to the nanoscale by incorporating capillary or surface energy effects. In this note, we revisit a particular formulation that derives a very general expression for the elasto-capillary state of an embedded inclusion. This approach, that closely mimics that of Eshelby's original paper, appears to have the advantage that it can be readily used for inclusions of arbitrary shape (for numerical calculations) and provides a facile route for approximate solutions when closed-form expressions are not possible. Specifically, in the case of inclusions of constant curvature (sphere, cylinder) subject to some simplifications, closed-form expressions are obtained.

scholarly journals The Eshelby inclusion problem in ageing linear viscoelasticity

2016 ◽  
Vol 97-98 ◽  
pp. 530-542 ◽  
Author(s):  
Jean-François Barthélémy ◽  
Albert Giraud ◽  
Francis Lavergne ◽  
Julien Sanahuja
Keyword(s):  
Linear Viscoelasticity ◽  
Inclusion Problem ◽  
Eshelby Inclusion
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