Effect of rigid inclusions on the sintering of mullite synthesized by sol-gel processing

The effect of crystalline particulate inclusions of mullite or zirconia on the sintering and crystallization of a mullite powder matrix was investigated as a function of the inclusion volume fraction and size. The mullite powder was synthesized by sol-gel processing and, within the limits of X-ray diffraction, was amorphous. Composites containing up to 22.5 vol % zirconia reached almost full density after sintering at 1500 °C for 1 h. Under identical conditions, the sintered density of the composites containing crystalline mullite inclusions was considerably lower. The zirconia inclusions were inert but the mullite inclusions enhanced the independent nucleation and growth rate of the mullite crystals in the matrix. The lower sintering rate of the matrix reinforced with crystalline mullite is attributed to the enhanced matrix crystallization.     

On the extension of a crack due to rigid inclusions

Dislocation layers have been utilized to study the effect of rigid inclusion on the crack extension in an infinite body and thereby to derive the crack extension condition from Irwin's criterion. Some particular cases have also been considered.

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Résumé On a eu recours aux couches de dislocation pour étudier l'effet d'une inclusion rigide sur l'extension d'une fissure dans un corps infini et déterminer en conséquence les conditions d'extension d'une fissure en utilisant le critère d'Irwin. Certains cas particuliers ont également été considérés.

     

Interaction of cracks with rigid inclusions in longitudinal shear deformation

The interaction of a crack with rigid circular cylindrical inclusions is considered for the case of longitudinal shear deformation. General representations of the solutions for a radial crack near a single and midway between two inclusions are given. The particular case of uniform shearing stress applied at infinity is discussed in detail.Expressions for the crack tip stress intensity factorK ₃ are derived and it is shown thatK ₃=0 for a crack tip at the inclusion boundary, provided that the crack is radial. Generalization of the results to any number of inclusions with common line of centers is indicated.

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Zusammenfassung Man untersucht die Zusammenwirkung eines Risses mit steifen kreisförmigen und zylindrischen Inklusionen im Falle von Längsquerverformungen. Allgemeine Ausdrücke der Lösungen für einen Radialriss in der Nähe einer Einzelinklusion und halbwegs zwischen zwei Inklusionen werden gegeben.Der Sonderfall von gleichmässigen Querspannungen die im Unendlichen angebracht werden, wird im einzeln besprochen. Formeln für den SpannungsintensitätsfaktorK ₃ an der Risspitze werden abgeleitet und man zeigt dassK ₃=0 für eine Risspitze an der Grenze der Inklusion, unter der Bedingung dass es sich um einen Radialriss handelt. Eine Verallgemeinerung der Ergebnisse für jegliche Anzahl von Inklusion mit einer gemeinsamen Mittelpunktslinie wird angegeben.

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Résumé On considère l'interaction d'une fissure avec des inclusions rigides circulaires et cylindriques dans le cas d'une déformation par cisaillement longitudinal. On donne des représentations générales des solutions pour une fissure radiale au voisinage d'une simple inclusion ou à mi-chemin entre deux inclusions.Le cas particulier de la contrainte uniforme de cisaillement appliquée à l'infini est discuté dans le détail. On en tire des expressions pour le facteur d'intensité des contraintesK ₃, et on montre queK ₃=0 lorsque la fissure est radiale et que son extrémité se situe à la frontière de l'inclusion.Une généralisation des résultats au cas d'un nombre quelconque d'inclusions possédant une ligne de centres commune est possible.

     

Geometrically nonlinear analysis of elastic membranes with embedded rigid inclusions

In this paper the deformation of membranes containing rigid inclusions is analyzed. These rigid inclusions can significantly change the entire stress distribution in the membrane and therefore create major difficulties for the design. The initially flat membrane, which may be prestretched by boundary in-plane tractions or displacements, is subjected to externally applied loads and to the weight of the rigid inclusions. The composite system is examined in cases where its deformation reaches a state for which the undeformed and deformed shapes are substantially different. In such cases large deflections of membranes are considered, which result from nonlinear kinematic relations. The three coupled nonlinear equations in terms of the displacements governing the response of the membrane are solved using the analog equation method, which reduces the problem to the solution of three uncoupled Poisson's equations with fictitious domain source densities. The problem is strongly nonlinear [Katsikadelis JT, Nerantzaki MS. The boundary element method for nonlinear problems. Eng Anal Boundary Elements 1999;23:365–73]. In addition to the geometrical nonlinearity, the problem is itself nonlinear, since the membrane's reactions on the boundary of the rigid inclusions are not a priori known as they depend on the produced deflection surface. Iterative schemes are developed for calculation of deformed membrane's configuration, which converge to the final equilibrium state of the membrane with the given external applied loads. Several example problems are presented, which illustrate the method and demonstrate its accuracy and efficiency. The method employed for the solution is boundary only with all the advantages of the pure BEM.     

Longitudinal shear of a body with mutually immobile rigid collinear inclusions

We study the problem of antiplane deformation of an elastic body with collinear perfectly rigid inclusions connected to form a single skeleton. The problem is reduced to a system of singular integral equations with an additional condition guaranteeing the absence of mutual displacements of the inclusions. The influence of mutual immobility and arrangement of the inclusions on the distributions of stresses and displacements in the body is analyzed.Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 40, No. 3, pp. 69–73, May–June, 2004.     

In-plane impact behavior of honeycomb structures randomly filled with rigid inclusions

The in-plane impact behavior of honeycomb structures randomly filled with rigid inclusions was studied by using the finite element method to clarify the effect of inclusions on the deformation process, mean stress, densification strain, and absorbed energy. The deformation processes of the models were disturbed by inclusions; shear bands were pinned, and the cell regions surrounded by inclusions were shielded. Mean stress, densification strain, and absorbed energy per unit volume normalized by the values of the model without inclusions were found to be only dependent on the fraction of inclusions. As the volume fraction of inclusions increased, the normalized mean stress linearly increased and the normalized densification strain linearly decreased. The normalized absorbed energy per unit volume could be approximated by an inverted parabolic equation. The energy absorption of models with inclusions having volume fractions from 0 to 0.25 was larger than that of the models without inclusions. In particular, honeycomb models with fractions of inclusion from 0.1 to 0.2 exhibited the maximum absorbed energy. The model with a volume fraction larger than 0.4 could not be compressed because the inclusions in the model had already percolated before deformation. The in-plane impact behavior of honeycomb structures as energy absorbing materials can be designed by using the approximate equation and selecting the volume fraction of inclusions.     

On the influence of the orientation of spheroidal hollows and rigid inclusions in orthotropic media on stress concentration

We consider the problem of stress concentration in an orthotropic elastic medium containing an arbitrarily oriented spheroidal hollow or an inclusion. For the solution of the problem, we use the method of equivalent inclusion, triple Fourier transformation over the space variables, and the Fourier transform of the Green function for an infinite anisotropic space. In computing some double integrals over a finite domain, we use the Gaussian quadrature formulas. In special cases, the results of our investigations are compared with the data of other authors. The influence of the orientation of the inhomogeneity on stress concentration is investigated. __________ Translated from Problemy Prochnosti, No. 1, pp. 58–68, January–February, 2006.     

Spread of plasticity from cuspidal points of rigid inclusions

Summary The plastic zones developed around cuspidal points of rigid inclusions embedded in an elastic plate under small-scale yielding were determined. For this reason the singular solution of the stress field in the local neighborhood of the cuspidal points in conjunction with the Mises yield criterion was used. Formulae given the radius of the plastic zone under conditions of plane strain and generalized plane stress were derived. The cases of the astroidal and hypocycloidal inclusions embedded in a plate which was subjected to a uniform uniaxial stress were examined in detail. The plastic zones at all cuspidal points of these inclusions for various values of the Poisson's ratio of the plate, the orientation of the inclusion and the state of stress (plane strain or generalized plane stress) were determined and discussed.With 15 Figures     

Debonding of rigid curvilinear inclusions in longitudinal shear deformation

The problem of two symmetrically placed interface cracks at rigid curvilinear inclusions under longitudinal shear deformation is considered. A solution valid for arbitrary inclusion shapes is found. It depends on a parameter β describing the cracks. For $\text{β = e}{}^{\mathrm{i\alpha }}$ where α is an angle, the cracks lie in the interface. For β real and greater than unity, we have two radial cracks emanating from a curvilinear cavity. The solution for $\text{β = 1}$ corresponds to a completely debonded inclusion.Examples of elliptic, square with rounded corners, and rectangular inclusions are worked out in detail. It is shown that the crack tip stress intensity factor becomes infinite for interface cracks terminating at cusps and corners. This phenomenon is attributed to the change in the nature of the singularity as the crack tip approaches a cusp or corner. The singularity is three-quarter power at a cusp and two-thirds power at a corner of a rectangular inclusion. Finally, the application of the results to composite materials is indicated.     

A hybrid finite element method for heterogeneous materials with randomly dispersed rigid inclusions

A hybrid finite element approach is proposed for the mechanical response of two-dimensional heterogeneous materials with linearly elastic matrix and randomly dispersed rigid circular inclusions of arbitrary sizes. In conventional finite element methods, many elements must be used to represent one inclusion. In this work, each inclusion is embedded inside a polygonal element and only one element is required to represent one inclusion. In numerically approximating stress and displacement distributions around the inclusion, classical elasticity solutions for a multiply-connected region are employed. A modified hybrid functional is used as the basis of the element formulation where the displacement boundary conditions of the element are automatically considered in a variational sense. The accuracy and efficiency of the proposed method are demonstrated by two boundary value problems. In one example, the results based on the proposed method with only 64 hybrid elements (450 degrees of freedom) are shown to be almost identical to those based on the traditional method with 2928 conventional elements (5526 degrees of freedom).     

Constitutive models for ductile solids reinforced by rigid spheroidal inclusions

We study the effective constitutive response of composite materials made of rigid spheroidal inclusions dispersed in a ductile matrix phase. Given a general convex potential characterizing the plastic “in the context of J₂-deformation theory” behavior of the isotropic matrix, we derive expressions for the corresponding effective potentials of the rigidly reinforced composites, under general loading conditions. The derivation of the effective potentials for the nonlinear composites is based on a variational procedure developed recently by Ponte Castaneda (1991a, J. Mech. Phys. Solids 39, 45–71). We consider two classes of composites. In the first class, the spheroidal inclusions are aligned, resulting in overall transversely isotropic symmetry for the composite. In the second class, the inclusions are randomly oriented, and thus the composite is macroscopically isotropic. The effective response of composites with aligned inclusions depends on both the orientation of the loading relative to the inclusions and on the inclusion concentration and shape. Comparing the strengthening effects of rigid oblate and prolate spheroids, we find that prolate spheroids give rise to stiffer effective response under axisymmetric “relative to the axis of transverse isotropy” loading, while oblate spheroids provide greater reinforcement for materials loaded in transverse shear. On the other hand, nearly spherical “slightly prolaterd spheroids are most effective in strengthening the composite under longitudinal shear. Thus, the optimal shape for strengthening composites with aligned inclusions depends strongly on the loading mode. Alternatively, the properties of composites with randomly oriented spheroidal inclusions, being isotropic, depend only on the concentration and shape of the inclusions. We find that both oblate and prolate inclusions lead to significant strengthening for this class of composites.