Inclusion Geometry Research Articles

The debonding of elastic inclusions and inhomogeneities

T his paper presents an anlytical study of the debonding of planar elastic inclusions and inhomogeneities. The standard inhomogeneity and inclusion (transformation) problems of Eshelby, which require a coherent interface, are reconsidered to allow for debonding. It is assumed that the inclusion or inhomogeneity interacts with the matrix through a prescribed interface force law which depends on the displacement jump at the interface. The solution procedure, which is not dependent on a particular inhomogeneity or inclusion geometry, force law or even planar elasticity, ultimately leads to a pair of integral equations governing the displacement jump at the interface. Detailed analytical solutions to the circular inclusion and inhomogeneity problems are obtained for the case where the interfacial traction varies linearly with displacement jump across the interface.

An arbitrarily oriented inclusion in a half-space subjected to an inclined surface load

The plane elastostatic problem for a half-space which contains an arbitrarily oriented flat inclusion is considered. The medium is loaded from its surface by an inclined concentrated force. It is assumed that the inclusion may be rigid or elastic with negligible bending rigidity. Fourier integral transforms are applied and the problem is formulated in terms of the unknown interface stresses using concentrated force solutions as Green's functions. The unknowns are then solved from a system of simultaneous singular integral equations. Stress intensity factors are presented for various inclusion and load geometries.

Photoelastic investigation of crack-inclusion interaction

An experimental procedure is presented for determining the mode I stress-intensity factor of an edge crack with a nearby rigid elliptical inclusion in a finite plate loaded in uniform tension. The rigid inclusion was modeled by bonding two identical steel inclusions on to the faces of a thin plate of polycarbonate. Models were constructed with edge cracks and various inclusion geometries so that the effect of parameters such as inclusion shape, orientation, and cracktip position on the stress-intensity factors of the crack could be determined. Photoelasticity experiments were used for this investigation and the results were compared to the results of a similar theoretical analysis of the interaction between a crack and an inclusion in an infinite plate. A good correlation was found between the experimental and theoretical models indicating that the results may help provide a better understanding of the toughening mechanisms in materials such as short-fiber-reinforced composites and ceramics. This experimental method is relatively easy to use making it an attractive candidate to be applied to similar problems involving cracks and inhomogeneities.

Inclusion complexes of purine nucleosides with cyclodextrins: I. Complexation and stabilization of a dideoxypurine nucleoside with 2-hydroxypropyl-β-cyclodextrin

2-Hydroxypropyl-β-cyclodextrin (HP-β-CD) markedly inhibits the acid-catalyzed hydrolysis of 2',3'-dideoxyadenosine (DDA), a compound under development for the treatment of HIV infections. The kinetic data conform to a model which assumes 1:1 complex formation between HP-β-CD and both the neutral and protonated forms of DDA. UV difference spectra were generated as a function of both HP-β-CD concentration and temperature to obtain independent estimates of formation constants and enthalpies and entropies for the binding of the neutral species of DDA to HP-β-CD. The acid lability of DDA at the low pH necessary to form the protonated species precluded the use of difference spectra to obtain complexation data for the protonated species. Binding constants for the protonated species were therefore obtained by combining apparent p K a values as a function of HP-β-CD concentration and temperature with the spectral data. Binding constants for the neutral species ranged from 28 M −1 at 50° C to 56 M −1 at 4° C and were at least 3-times larger than those of the protonated species over this temperature range. A negligible difference in the enthalpy of complexation was observed between the protonated and neutral forms of DDA, while the entropy of formation appeared to favor the neutral complex. Although hydrolysis is 100% suppressed in both the protonated and neutral complexes, due to the small binding constants, the maximum stabilization attainable in a 0.1 M solution of HP-gb-CD at 25° C was approx. 5-fold at pH 5 and 2-fold at pH 2. Possible inclusion geometries are considered in an attempt to account for the kinetic data.

Lack of porphyroblast rotation in the Otago schists, New Zealand: implications for crenulation cleavage development, folding and deformation partitioning

Porphyroblasts of garnet and plagioclase in the Otago schists have not rotated relative to geographic coordinates during non‐coaxial deformation that post‐dates their growth. Inclusion trails in most of the porphyroblasts are oriented near‐vertical and near‐horizontal, and the strike of near‐vertical inclusion trails is consistent over 3000 km2. Microstructural relationships indicate that the porphyroblasts grew in zones of progressive shortening strain, and that the sense of shear affecting the geometry of porphyroblast inclusion trails on the long limbs of folds is the same as the bulk sense of displacement of fold closures. This is contrary to the sense of shear inferred when porphyroblasts are interpreted as having rotated during folding.Several crenulation cleavage/fold models have previously been developed to accommodate the apparent sense of rotation of porphyroblasts that grew during folding. In the light of accumulating evidence that porphyroblasts do not generally rotate, the applicability of these models to deformed rocks is questionable.Whether or not porphyroblasts rotate depends on how deformation is partitioned. Lack of rotation requires that progressive shearing strain (rotational deformation) be partitioned around rigid heterogeneities, such as porphyroblasts, which occupy zones of progressive shortening or no strain (non‐rotational deformation). Therefore, processes operating at the porphyroblast/matrix boundary are important considerations. Five qualitative models are presented that accommodate stress and strain energy at the boundary without rotating the porphyroblast: (a) a thin layer of fluid at the porphyroblast boundary; (2) grain‐boundary sliding; (3) a locked porphyroblast/matrix boundary; (4) dissolution at the porphyroblast/matrix boundary, and (5) an ellipsoidal porphyroblast/shadow unit.

Micromechanical Stresses in SiC‐Reinforced Al2O3 Composites

Applying an Eshelby approach, the internal micromechanical stresses within an SiC‐inclusion‐reinforced (platelet to whisker geometries) polycrystalline alumina matrix composite were calculated. The results are compared to the experimental residual stress measurements of a SiC‐whisker‐reinforced Al2O3 by Predecki, Abuhasan, and Barrett and found to be in excellent agreement. The calculations are then extended to SiC‐reinforced composites with polycrystalline mullite, silicon nitride, and cordierite matrices. It is concluded that the internal stresses are significantly influenced by the inclusion geometry as well as the thermoelastic differences between the inclusion and the matrix and also the volume fraction.

Model for prediction of the elastic response of reinforced materials over wide ranges of concentration

AbstractA model for the prediction of the elastic response of reinforced materials over wide ranges of concentration is presented. The method is based on the mathematical analogy between the motion of particles suspended in viscous media and the elastic deformation of solids. The system is locally described by linearized forms of the elastic moduli. The validity of these relations is then extended to all concetrations under the assumption that any new portion of filler “sees” the existing structure as a noniteracting homogeneous matrix. The method predicts the behavior of reinforced materials with solid spherical inclusions and foams over wide ranges of concentration. The model, free from adjustable parameters, shows excellent agreement with existing experimental data. The extension of the method to other inclusion geometries is straightforward.

Selection of Guest Inclusion Geometry by a Series of Water-soluble Paracyclophanes Having Well-defined Hydrophobic Cavities

Selection of Guest Inclusion Geometry by a Series of Water-soluble Paracyclophanes Having Well-defined Hydrophobic Cavities

Internal stresses due to an oblate spheroidal inclusion: Misfit, inhomogeneity and plastic deformation effects

Internal stress in and around an oblate spheroidal inclusion is evaluated by using Eshelby theory. We consider three different sources of internal stress as a function of the aspect ratio and elastic moduli ratio; these are 1. (1) misfit effect, 2. (2) inhomogeneity effect 3. (3) plastic deformation effect. The misfit effect arises from the difference of thermal expansion coefficients of the matrix and inclusion. Normal stresses within the inclusion (or the principal stresses), maximum shear stress inside the inclusion, total strain energy and the normal stress at the matrix-inclusion boundary are determined. The inhomogeneity effect arises from the difference of the elastic moduli of the matrix and inclusion, which perturbs otherwise uniform applied stress. The direction of applied stress is parallel to the axis of revolution of the inclusion. Normal stresses within the inclusion and at the matrix-inclusion boundary are obtained. Plastic deformation of the matrix in the presence of a plastically non-deformable inclusion produces internal stresses, as shown by Ashby and by Tanaka and Mori. The approach developed by Tanaka and Mori is extended here to evaluate the internal stresses for the inclusion geometries appropriate to the analysis of technically important cases. The results of the present calculations in limiting cases agree with previously published calculations. Implications of the results are discussed in connection with several experimental studies.

Inclusions in steel plates and mechanical anisotropy

AbstractThe work conducted at IRSI D on a large number of fully killed steels has confirmed the importance of the inclusion system with respect to short-transverse ductility and transverse impact properties. Concerning through-thickness ductility, as assessed by tensile testing, a very good correlation wasfound between reduction of area and the projected inclusion length. Transverse impact properties were alsofound to be dependent to some extent upon the same parameter but no satisfactory correlation wasfound. This is discussed in terms of matrix inclusion interactions. Practical methods of inclusion shape control are reviewed and illustrated. The temperature dependence of inclusion elongation is examined and rare-earth (cerium) effects, with respect to inclusion composition and geometry, are described.

A phenomenological theory of interface shear stresses generated by hydrostatic pressure in martensitic materials

The application of hydrostatic pressure to martensitic materials generates a shear stress at the martensite-austenite interface due to the different elastic properties of the martensite and austenite phases. A closed-form solution for the maximum interface shear stress is developed for a single martensite plate in an infinite austenite matrix. The magnitude of this interface shear stress is a function of the applied hydrostatic pressure, the geometry of the martensite plate inclusion, and the elastic properties of the phases. Interface shear stresses are calculated and compared to hydrostatic pressure data from 0 to 2.3 kbar. It is found that the interface shear stress is of the correct order of magnitude, when combined with the pressure-volume-change term, to account for the large depression of the austenite start temperature with pressure (−30°C/kbar). The shear modulus and Poisson's ratio are reported for the temperature range of 22°–350°C for both the austenite and a 95 per cent martensite specimens.

Elastic and thermoelastic properties of a statistically isotropic heterogeneous medium

The method proposed by the authors in their earlier paper on thermal conductivity has been utilized to estimate the effective elastic constants and the coefficient of thermal expansion of a statistically isotropic heterogeneous medium. The method considers the inclusion geometry in the lower-order correlation functions, that is, those of order two and less. The upper and lower bounds and the improvements on these bounds have been derived for the elastic constants. Expressions containing the two-point correlation functions are also derived for the coefficient of thermal expansion and the related stress tensor. The theoretical values are compared with the experimental results and the comparison can be considered satisfactory in view of the difficulty in obtaining perfectly isotropic distribution experimentally and the sensitivity of the final properties to mild anisotropy when the difference in the constituent properties is large as in the case of Araldite and copper. The theoretical results for the aluminium-copper system show that convergence of the bounds is extremely good when properties of the constituents are of the same order.

The thermal conductivity of a statistically isotropic heterogeneous medium

A method for estimating values of the effective constants of a heterogeneous medium, consisting of a continuous and a discontinuous phase, is proposed. This method incorporates a new physical model that reduces this problem of a two-phase system to an equivalent problem of a polycrystalline aggregate. It also takes the inclusion geometry into account in the lower order correlation functions - that is, those of order two and less. The upper and lower bounds and the improvements on them are derived for the thermal conductivity of a statistically isotropic heterogeneous system, and compared with the experimental results of other authors in order to check the effectiveness of this model. Good agreement between theoretical and experimental results is observed in spite of large differences between the conductivity of the two phases, that of the inclusions being about a thousand times that of the matrix. This method can therefore be expected to yield very good results when the differences are small.

Interaction Between An Elliptic Inclusion And ACrack

The interaction between an elliptic inclusion and a crack is analyzed by body force method. Based on numerical results, effects of the inclusion shape on the crack tip stress intensity factor are discussed. It is found that for small cracks emanating from a stress-higher point on the inclusion interface the stress intensity factors are mainly determined by the stresses, occurring at the crack starting point before the crack initiation, and the inclusion root radius, besides the crack length. However, for the cracks occurring in a stress-lower region around the inclusion, it is difficult to characterize the effect of the inclusion geometry on the stress intensity factors of small cracks by the inclusion root radius alone.

Some three-dimensional inclusion problems in elasticity

The theory of potential functions is applied to solve a number of three-dimensional problems involving sheet-like inclusions embedded in elastic solids. Two types of inclusions are considered; namely, that of a rigid elliptical disk and a rigid sheet containing an elliptical hole. By varying the ellipticity of the disk and hole, certain information on the general character of the stresses around a plane inclusion of arbitrary shape may be obtained. More precisely, if reference is made to a suitable coordinate system, the functional forms of the stresses in the close neighborhood of the inclusion border can be expressed independently of uncertainties of both the inclusion geometry and of the applied stresses or displacements. In general, the intensification of the local stresses can be described by three parameters which may be used to establish criteria for the failure of the solid containing the inclusions.