Archivio della ricerca dell'Università di Modena e Reggio Emiliahttps://iris.unimore.itIl sistema di repository digitale IRIS acquisisce, archivia, indicizza, conserva e rende accessibili prodotti digitali della ricerca.Tue, 24 May 2022 02:26:07 GMT2022-05-24T02:26:07Z10131Overall elastic properties of a plate containing inhomogeneities of irregular shapehttp://hdl.handle.net/11380/1163838Titolo: Overall elastic properties of a plate containing inhomogeneities of irregular shape
Abstract: The present work deals with the stiffness properties of an infinite 2D isotropic elastic system containing inhomogeneities having a circular contour. Starting from this general layout, the cases of a matrix with lenticular, perfectly circular, semi-circular, “C-shaped” and thin straight inclusions can be obtained as limit cases. Owing to the geometry of the system, reference is made to bipolar cylindrical coordinates ( ), which are linked to the Cartesian ones (x1, x2) through the conformal map [2]. The effective elastic properties of the system is analytically investigated by introducing a fourth-order compliance contribution tensor H, which represents the effect induced by the inhomogeneity on the compliance of the system according to [1], being S the compliance tensor for the homogeneous elastic matrix and e the stress field. It is remarked that the last term in eq (1) denotes the correction acting on the strain field owing to the presence of the inclusions. The system without inhomogeneities and subjected to a remote stress field is considered first. The corresponding fundamental stress field (0) within the matrix does not accomplish the BCs at the contour of the inhomogeneities. Thus, following the Jeffery approach, an auxiliary stress field deduced by a biharmonic stress function in bipolar coordinates is introduced and tensor H is then evaluated by performing proper contour integrals involving the total stress distribution along the contours of the inclusions. The study allows evaluating the effective elastic properties of a wide class of inhomogeneous materials, with particular reference to composites reinforced with natural or synthetic fibres having optimized cross sections.
References
[1] Sevostianov, I., and Kachanov, M., “Explicit cross-property correlations for anisotropic
two-phase composite materials” Journal of the Mechanics and Physics of Solids, 50, 253-282 (2002).
[2] Korn, G.A. and Korn, T.M., Mathematical handbook for scientists and engineers. Definitions, Theorems and Formulas for Reference and Review, Dover, New York (1968).
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/11380/11638382018-01-01T00:00:00ZEffect of pore coalescence on the effective conductivity of an isotropic materialhttp://hdl.handle.net/11380/1245195Titolo: Effect of pore coalescence on the effective conductivity of an isotropic material
Abstract: The purpose of this work is to evaluate effect of two coalesced pores or insulating inhomo-geneities on the overall conductive properties of an isotropic material. Analytical modeling of the effective properties of materials with microstructures formed by inhomogeneities of non-ellipsoidal shape has not been well developed. The inhomogeneities are typically assumed to be ellipsoids of identical aspect ratios. This unrealistic assumption is largely responsible for insufficient linkage between methods of micromechanics and material science applications. The resistivity contribution tensor gives the extra temperature gradient produced by introduction of the inhomogeneity into a material subjected to otherwise uniform heat flux. The main goal of this work is to obtain an analytical solution for the components of the resistivity contribution tensor of two overlapping pores, in the 2D and 3D frameworks [1, 2].
Fri, 01 Jan 2021 00:00:00 GMThttp://hdl.handle.net/11380/12451952021-01-01T00:00:00ZEffective elastic properties of media containing coalescing holeshttp://hdl.handle.net/11380/1181117Titolo: Effective elastic properties of media containing coalescing holes
Abstract: A recent study about the temperature and heat flux distributions around two nonconductive (separate or intersecting) circular holes in a plane system recently appeared in Literature [1]. These results have been used to construct the second-rank resistivity contribution tensor which allows assessing the effective thermal properties of a composite including circular inhomogeneities.
Here, that study is extended to assess the overall elastic properties of an isotropic elastic matrix with two separate circular cavities or a cavity obtained by the union of two circles of generally different diameters (Figure 1). The problem is formulated in terms of stress functions expressed in Fourier series or Fourier transforms. Reference is made to bipolar cylindrical coordinates [2]. Once the displacement field u has been calculated, the extra strain due to the inhomogeneity is assessed according to (1), being n the normal vector and V the volume reference. Finally, the extra strain is used to assess the fourth-rank compliance contribution tensor varying the size of the circular arcs.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/11380/11811172019-01-01T00:00:00ZEffective properties of composites containing toroidal inhomogeneitieshttp://hdl.handle.net/11380/1165692Titolo: Effective properties of composites containing toroidal inhomogeneities
Abstract: The present work focuses on the problem of a rigid inhomogeneity of toroidal shape embedded in an elastic matrix. Inhomogeneities of this kind occur in both natural and man-made materials. Barium titanate nanotori are used as nonvolatile memory devices, transducers, optical modulators, sensors and possible energy storage in supercapacitors. Toroidal particles represent preferred morphology of Li2O2 deposition on porous carbon electrode in lithium-oxygen batteries. Polymeric “microdonuts” are used in bioengineering; toroidal shape of nanoparticles is preferred for microwave absorption properties of BaTiO3. Toroidal particles of SiO2 may form in a Cu matrix due to internal oxidation of a Cu-Si solid-solution polycrystal. Analytical modeling of materials with such microstructure has not been well developed. In the homogenization schemes, the inhomogeneities are usually assumed to be of ellipsoidal shape. This unrealistic assumption is responsible for insufficient linkage between micromechanics and materials science applications. While for 2D non-elliptical inhomogeneities many analytical and numerical results have been obtained, only a limited number of approximate estimates are available for non-ellipsoidal 3D shapes. Asymptotic methods have been used in [1] to evaluate the contribution of a thin rigid toroidal inhomogeneity into overall stiffness. Eshelby tensor for a toroidal inclusion has been also derived by Onata. However, Eshelby tensor for non-ellipsoidal inhomogeneities is irrelevant to the problem of effective properties of a heterogeneous material. The effective conductivity of a material containing toroidal insulating inhomogeneities has been addressed in [2].
We first consider a homogeneous elastic material, with isotropic stiffness tensor C0, containing a rigid inhomogeneity of volume V(1). The contribution of the inhomogeneity to the overall stress per representative volume V (the extra stress Δσ, as compared to the homogeneous matrix) is given by the fourth-rank stiffness contribution tensor N, defined by the following relation
where ε ∞ is the remotely applied strain, n is the outward unit normal to the inhomogeneity surface S. To calculate the components of N, a displacement boundary value problem has been solved for 3D elastic space containing a rigid toroidal inhomogeneity.
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/11380/11656922018-01-01T00:00:00ZEffect of cylindrical fibers, with cross-sections formed by two circular arcs, on the overall conductivity of a compositehttp://hdl.handle.net/11380/1150968Titolo: Effect of cylindrical fibers, with cross-sections formed by two circular arcs, on the overall conductivity of a composite
Abstract: An analytic solution for the steady-state temperature distribution in an infinite conductive medium, containing non-conductive fiber with the cross-section of irregular shape formed by two circles, and subjected to remotely applied uniform heat flux is obtained. The temperature flux on the surface of the inhomogeneity is then determined as a function of the geometrical parameters. This result is used to calculate resistivity contribution tensor for the fiber and to evaluate effective conductive properties of a material containing multiple inhomogeneities of this shape.
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/11380/11509682018-01-01T00:00:00ZEffective properties of materials containing multiple toroidal inhomogeneitieshttp://hdl.handle.net/11380/1103066Titolo: Effective properties of materials containing multiple toroidal inhomogeneities
Abstract: We calulate effective conductive properties of composite materials containing toroidal inclusion
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/11380/11030662016-01-01T00:00:00ZEffects of toroidal inhomogeneities on the effective properties of a compositehttp://hdl.handle.net/11380/1151698Titolo: Effects of toroidal inhomogeneities on the effective properties of a composite
Abstract: The present work focuses on the problem of rigid inhomogeneity of toroidal shape embedded in an elastic matrix. Inhomogeneities of this kind occur both in natural and man-made materials. Analytical modeling of materials with such microstructure has not been well developed. In the homogenization schemes, the inhomogeneities are usually assumed to be of ellipsoidal shape. This unrealistic assumption is largely responsible for insuffcient linkage between methods of micromechanics and materials science applications. While for 2-D non-elliptical inhomogeneities many analytical and numerical results have been obtained, only a limited number of numerical results and approximate estimates are available for non-ellipsoidal 3-D shapes. Most of them are related to pores and cracks. The problem of the effective conductivity (thermal or electric) of a material containing toroidal insulating inhomogeneities has been addressed in a pèrevious work, where an analytic solution is presented for the steady-state temperature distribution in an infinite conductive medium containing an insulated toroidal inclusion, under uniform heat flux in an arbitrary direction. The temperature flux on the torus surface is then determined as a function of torus parameters. This result is then used to determine resistivity contribution tensor for the toroidal inhomogeneity and for calculation of effective conductive
properties of a material containing multiple inhomogeneities of this shape.
A general analytical solution is developed here for the problem of an infinite elastic medium containing a rigid toroidal inhomogeneity, under remotely applied uniform strain. The traction vector on the torus surface is determined as a function of torus parameters and remote strain components. The results are utilized to calculate the components of the fourth-rank stiffness contribution tensor of the rigid toroidal inhomogeneity that are required for calculation of the overall elastic properties of a material containing multiple toroidal inhomogeneities. The analytical results are verified by comparison with FEM calculations.
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/11380/11516982018-01-01T00:00:00ZEffect Of Pair Coalescence Of Circular Pores On The Overall Elastic Propertieshttp://hdl.handle.net/11380/1177188Titolo: Effect Of Pair Coalescence Of Circular Pores On The Overall Elastic Properties
Abstract: The paper focuses on the effect of the pair coalescence of circular pores on the overall elastic properties. An analytic solution for the stress and displacement fields in an infinite elastic medium, containing cylindrical pore with the cross-section formed by two circles, and subjected to remotely applied uniform stresses is obtained. The displacement field on the surface of the pore is then determined as a function of the geometrical parameters. This result is used to calculate compliance contribution tensor for the pore and to evaluate effective elastic properties of a material containing multiple pores of such a shape.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/11380/11771882019-01-01T00:00:00ZOverall thermal conductivity of fibre reinforced materialshttp://hdl.handle.net/11380/1166896Titolo: Overall thermal conductivity of fibre reinforced materials
Abstract: The overall thermal conductivity of composites involving cylindrical fibres of irregular shape is investigated in the present work. Isotropic and homogeneous thermal conductivity is assumed for both the matrix and fibre. The system consists of an infinite plate with an embedded fibre subjected to a remotely applied steady state heat flux q acting along a given
direction. Once the alteration of the heat flux and temperature field T due to the presence of the inclusion is assessed, the homogeneized thermal properties of the composite material can be computed following the procedure reported in [1].
As an example, the dimensionless temperature distribution Tk/Rq and heat flow q/q in an infinite with a non-conductive
circula fiber is sketched in Figure 1, being k the thermal conductivity of the matrix and R denotes the radius of the fiber.
The study extends the results reported in [2] performed for non-conductive inclusions accounting for the real thermal conductivity of the fibres. The analysis allows assessing the effective thermal properties of a fibre reinforced material based on fibres with cross section formed by circular arcs, as polystyrene, polyacrylonitrile and sisal fibres.
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/11380/11668962018-01-01T00:00:00ZEffect of spherical pores coalescence on the overall conductivity of a materialhttp://hdl.handle.net/11380/1202189Titolo: Effect of spherical pores coalescence on the overall conductivity of a material
Abstract: The problem about steady-state temperature distribution in a homogeneous isotropic medium containing a pore or an insulating inhomogeneity formed by two coalesced spheres of the same radius, under arbitrarily oriented uniform heat flux, is solved analytically. The limiting case of two touching spheres is analyzed separately. The solution is obtained in the form of converged integrals that can be calculated using Gauss-Laguerre quadrature rule. The temperature on the inhomogeneity’s surface is used to determine components of the resistivity contribution tensor for the insulating inhomogeneity of the mentioned shape. An interesting observation is that the extreme values of these components are achieved when the spheres are already slightly coalesced
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/11380/12021892020-01-01T00:00:00Z