Uniform Eigenstrain Research Articles

Elastic Fields in a Polygon-Shaped Inclusion With Uniform Eigenstrains

In this paper the elastic fields in an arbitrary, convex polygon-shaped inclusion with uniform eigenstrains are investigated under the condition of plane strain. Closed-form solutions are obtained for the elastic fields in a polygon-shaped inclusion. The applications to the evaluation of the effective elastic properties of composite materials with polygon-shaped reinforcements are also investigated for both dilute and dense systems. Numerical examples are presented for the strain field, strain energy, and stiffness of the composites with polygon shaped fibers. The results are also compared with some existing solutions.

The determination of the elastic field of a polygonal star shaped inclusion

Recently we found that the elastic field is uniform in a pentagonal star (five-pointed star inclusion) [1], and in a triangular inclusion [2], when an eigenstrain is distributed uniformly in these inclusions. This result is similar to the famous result of Eshelby (1957) that the elastic field is uniform in an ellipsoidal inclusion in an infinitely body when an eigenstrain is distributed uniformly in the ellipsoidal inclusion. We also found that for a Jewish star (Star of David or six points star) or a rectangular inclusion subjected to a uniform eigenstrain, the stress field is not uniform in these inclusions. These results also hold for two dimensional plane strain cases. Furthermore these analytical results are confirmed experimentally by photoelasticity method. In this paper, we investigate a more general inclusion of an m-pointed polygonal inclusion subjected to the uniform eigenstrain. We conclude that the stress field is uniform when m is odd number. This conclusion agrees with the speculation made by B. Boley after the author's talk at Shizuoka [2].

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

The calculation of the elastic strain energy due to a uniform eigenstrain in an inclusion continues to be of high concern for various problems in material science such as nucleation conditions or transformation conditions for the inclusion. It is the main goal of this note to show that very easily programmable equations can be formulated for a general uniform eigenstrain tensor consisting of three different normal and three different shear strains. Although the authors appreciate the recently published results very much, they do not see any necessity to demonstrate in detail specific results since the following derivation presents a consistent way to calculate the specific strain energy in very few steps. Specifically, a modified notation helps to split the usually lengthy expression into a group of easily expressible terms multiplied by the mixed product terms of the normal eigenstrains and the squares of the shear eigenstrains. Since all entities are expressed with respect to a coordinate frame attached to the inclusion any coordinate transformation can be avoided.

The Elastic Field in a Half-Space With a Circular Cylindrical Inclusion

The problem of a circular cylindrical inclusion with uniform eigenstrain in an elastic half-space is studied by using the Green’s function technique. Explicit solutions are obtained for the displacement and stress fields. It is shown that the present elastic fields can be expressed as functions of the complete elliptic integrals of the first, second, and third kind. Finally, numerical results are shown for the displacement and stress fields.

Thick-walled composite cylinders with optimal fiber prestress

A thick-walled composite cylinder consisting of many different cylindrically orthotropic layers is loaded by uniform, axisymmetric surface tractions and by piecewise uniform eigenstrains in the layers. A theoretical framework is established for evaluation of internal stress fields. An optimization procedure is implemented to find eigenstrain distributions that adjust the stresses in the layers to selected levels, while allowing the application of certain ranges of fiber prestrain magnitude to reduce fiber waviness. A particular fabrication process that utilizes fiber prestress as a source of the layer eigenstrains is analyzed, and a distribution of the fiber prestress forces is found that produces a desired distribution of both hoop and axial stresses in the cylinder wall. Both mandrel stiffness and the chosen prestress magnitude in the first layer influence the distribution. It is shown that application of a high fiber prestress that is constant through the wall thickness can be harmful as it may generate very high hoop stress gradients.

The Elastic Field Caused by a Circular Cylindrical Inclusion—Part I: Inside the Region x12+x22<a2,−∞<x3<∞ Where the Circular Cylindrical Inclusion is Expressed by x12+x22≤a2,−h≤x3≤h

The displacement and stress fields caused by uniform eigenstrains in a circular cylindrical inclusion are analyzed inside the region x12+x22<a2,−∞<x3<∞ and are given in terms of nonsingular surface integrals. Analytical solutions can be expressed as functions of the complete elliptic integrals of the first, second and third kind. The corresponding elastic fields in the region x12+x22>a2,−∞<x3<∞ are solved by using the same technique (by Green’s functions) in the companion paper (Part II).

The Elastic Field Caused by a Circular Cylindrical Inclusion—Part II: Inside the Region x12+x22>a2,−∞<x3<∞ Where the Circular Cylindrical Inclusion is Expressed by x12+x22≤a2,−h≤x3≤h

Analytical solutions are presented for the displacement and stress fields caused by a circular cylindrical inclusion with arbitrary uniform eigenstrains in an infinite elastic medium. The expressions obtained and those presented in Part I constitute the solutions of the whole elastic field, −∞<x1,x2,x3<∞. In the present paper, it is found that the analytical solutions within the region x12+x22>a2,−∞<x3<∞ can also be expressed as functions of the complete elliptic integrals of the first, second, and third kind. When the length of inclusion tends towards the limit (infinity), the present solutions agree with Eshelby’s results. Finally, numerical results are shown for the stress field.

A unified treatment of the elastic elliptical inclusion under antiplane shear

A generalized and unified treatment is presented for the antiplane problem of an elastic elliptical inclusion undergoing uniform eigenstrains and subjected to arbitrary loading in the surrounding matrix. The general solution to the problem is obtained through the use of conformal mapping technique and Laurent series expansion of the associated complex potentials. The resulting elastic fields are derived explicitly in both transformed and physical planes for the inclusion and the surrounding matrix. These relations are universal in the sense of being independent of any particular loading as well as the geometry of the matrix. The complete field solutions are provided for an elliptical inclusion under uniform loading at inifinity, and for a screw dislocation interacting with the elastic elliptical inclusion.

Interaction of two elliptic inclusions

The paper presents the solution of a boundary value problem involving two interacting elliptical inhomogeneities in an infinite elastic body. The loading cases considered include a far-field biaxial tension and a thermally induced residual field that is modeled by uniform eigenstrains sustained by the inhomogeneities. The mathematical formulation is based upon the Papkovich-Neuber displacement approach. Certain interesting aspects of the solution and the effects of perfectly bonded and slipping interfaces are discussed in some detail.

Load Diffusion and Absorption Problems From a Finite Fiber to Elastic Infinite Matrix

The load transfer behavior of a finite fiber perfectly bonded to an infinite matrix of distinct elastic moduli is investigated in this paper. The fiber is subjected to the uniformly distributed loading applied at infinity or on one cross-section of the fiber. The stress disturbance due to the existing fiber is simulated by the equivalent inclusion method, which formulates the inhomogeneity problem to a system of integral equations. By dividing the fiber into finite numbers of ring elements with uniform distributed eigenstrains, the integral equations can be further reduced to a system of algebraic equations with coefficients expressed in terms of the integrals of Lipschitz-Hankel type. Numerical results are presented for resultant axial force for various fiber length and material properties. The limiting cases of the infinite and semi-infinite fibers are also compared with the exact and approximate solutions.

Composite homogenization via the equivalent poly-inclusion approach

For the efficient use and development of advanced materials it is necessary to develop simple yet accurate analytic relationships linking their average physical response to as large a set as possible of microstructural quantifiers. With this objective in mind, the theory of the effective elastic response of biphase composite materials, with arbitrary second-phase geometry, concentration, and orientation distribution, is examined here. A formulation is presented, which employs an equivalent poly-inclusion methodology, in the sense that its governing variable is the average inclusion strain concentrator for an entirely uniform body undergoing a uniform eigenstrain in a non-dilute family of morphologically identical internal regions (inclusions). Admissibility criteria for approximate inclusion strain concentrators are formulated. The relation of the present approach with the traditional one, based on the inhomogenity average strain concentrator, is investigated, and some deficiencies of current methods based on the latter approach are discussed. A family of fully admissible poly-inclusion concentrators is proposed, which differs by a single scalar-valued function of the volume fraction—the so-called “interaction function”. A specific interaction function is finally identified, which satisfies all admissibility criteria, is consistent with the known analytic solutions for biphase composites, and reproduces literature data to within experimental error.

Hollow Circular Cylindrical Inclusion at the Surface of a Half-Space

Exact solutions are presented in closed form for the axisymmetric stresses and displacement fields caused by a solid or hollow circular cylindrical inclusion in the present of uniform eigenstrain in a half space. The elastic fields for interior and exterior points are expressed by one analytical form. The strain energy is also obtained in closed forms.

Frictional sliding inclusions

S olutions are presented in closed form by using an averaging method for inclusions sliding along an interface due to uniform eigenstrains precribed in the inclusions. The associated stress fields are also analytically determined. A parameter s is introduced to indicate the relative magnitude of sliding compared with the extreme cases of perfect bonding and perfect sliding. When the parameter s becomes zero, the present solution coincides with Eshelby's solution which is the perfectly bonded case. In contrast, when the parameter s is unity, the solution agrees with Volterra's solution (M ura and F uruhashi, 1984, J. appl. Mech. 51, 308] for the perfect sliding case. Because of non-uniform elastic fields caused by sliding along the interface, the well-known Eshelby tensor is modified for the sliding inclusions. Moreover, based on the Mori-Tanaka theory (M ori and T anaka, 1973, Acta Metall. 21, 571), an overall stress-strain relation is established to characterize the sliding effect on the overall elastic moduli.

The eigenstrain formulation for classical plates

In the present paper, the eigenstrain formulation is developed for classical plates. By employing this approach. Green's function of an isotropic plate is obtained in closed form. The solution for a plate with semi-infinite hinge is derived, which shows a l/r singularity near the tip of the hinge. The Eshelby-type problem, an elliptic inclusion with uniform eigenstrain. is studied and the result has the same features as that of the Eshelby solution for ellipsoidal inclusion in threedimensional elasticity, i.e., the general stress Mαβ and strain Wαβ are uniform inside the inclusion. By using the equivalent inclusion method outlined in this paper, an elliptic inhomogeneity which has different material properties from the matrix can be made equivalent to an elliptic inclusion with appropriate uniform eigenstrain. and can therefore be solved readily.

Surface Displacements and Stress Field Generated by a Semi-Ellipsoidal Surface Inclusion

This paper presents calculations of the displacement and stress fields generated by semi-ellipsoidal surface inclusions containing uniform transformation strains or eigenstrains. The inclusion is assumed to have the same elastic constants as the rest of the material. This is a reasonable assumption for modeling transformed zones in transformation toughened ceramics and localized plasticity in individual surface grains in alloys. Analytical results are obtained for special cases and numerical results for general cases. The approach is particularly useful for accurately calculating the anomalous fields at the intersection of the boundary of the inclusion and the free surface. It is shown that, in many physically important cases, all components of the stress tensor are zero or constant on the free surface within the inclusion. For shallow inclusions or inclusions of general geometry suffering volume conserving transformation strains, the stress fields are also approximately uniform throughout the inclusion. This result greatly simplifies modeling of localized deformation in certain materials under complex external loads.

The elastic field of a hemispherical inhomogeneity at the free surface of an elastic half space

This paper analyses the elastic fields around a hemispherical inhomogeneity at the free surface of a semiinfinite elastic body. The loading is either all-around tension at infinity, perpendicular to the axis of symmetry of the inhomogeneity or uniform eigenstrains introduced in the inhomogeneity. The interface between the inclusion and the matrix assumes perfect bonding, shear-traction-free sliding or intermediate between them. A spring-type resistance mode) is used to investigate this intermediate stage. The method of Boussinesq's displacement potentials is used in the analysis and the solution is expressed in terms of five sets of spherical harmonics. Numerical calculations are performed to illustrate the results. It was found that the non-vanishing stress components at the free surface of the inclusion are constant for equal Poisson's ratios.

A note on eigenstrains

Abstract Problems of uniform eigenstrains prescribed within regions that are bounded by two ellipsoidal surfaces are considered. The uniformity of the constrained strain in the inner ellipsoid is observed. An application to polycrystal plasticity is also discussed.

The Elastic Inclusion With a Sliding Interface

It is found that when an ellipsoidal inclusion undergoes a shear eigenstrain and the inclusion is free to slip along the interface, the stress field vanishes everywhere in the inclusion and the matrix. It is assumed in the analysis that the inclusion interface cannot sustain any shear traction. There exists a shear deformation that transforms an ellipsoid into the identical ellipsoid without changing its orientation (ellipsoid invariant transformation). This is not true, however, for a spheroidal inclusion. The amount of slip and the associated stress field are calculated for a spherical inclusion for a given uniform eigenstrain εij*.

On eigenstresses in a semi-infinite solid

AbstractA uniform eigenstrain is prescribed within a spherical subregion of an isotropic linearly elastic half-space. Combining an application of potential theory with the stress-function approach of Papkovitch and Neuber, and starting with Eshelby's well-known solution for the homogeneous infinite solid, it is shown that the residual problem of potential to be solved is the determination of Boussinesq's first and second three-dimensional logarithmic potentials of volume distributions. Although explicit results are supplied only for the case of a spherical inclusion, the dependence of the solution on the infinite solid solution holds good for an arbitrarily shaped transforming inclusion. This can be established on the basis of the principle of superposition, considering that an arbitrary volume is essentially made up from an infinite spectrum of spherical regions.

The elastic field caused by a general ellipsoidal inclusion and the application to martensite formation †

The elastic field throughout an ellipsoidal inclusion in an indefinitely-extended anisotropic material is investigated when an eigenstrain (a stress-free transformation strain) is periodically distributed throughout the inclusion. This is an extention of the results obtained by J.D. Eshelby (1961) for uniform eigenstrains and by R.J. Asaro and D.M. Barnett (1975) for polynomial eigenstrains. The solution is applied to the evaluation of elastic strain energies of a disc-shaped martensite with alternating twins and of a spherical precipitate with a banded structure. The significant amount of the elastic strain energies explains the necessity of the supercooling of austenite steel for the martensitic transformation to occur.