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Satya N. Atluri - One of the best experts on this subject based on the ideXlab platform.

  • trefftz Lekhnitskii grains tlgs for efficient direct numerical simulation dns of the micro meso mechanics of porous piezoelectric materials
    Computational Materials Science, 2014
    Co-Authors: Peter L. Bishay, Satya N. Atluri
    Abstract:

    Abstract We consider a class of piezoelectric materials with defects, voids, and/or elastic dielectric or piezoelectric inclusions. We develop computationally highly efficient as well as mathematically highly accurate methods for the Direct Numerical Simulation (DNS) of micro/meso mechanics of such materials, for the purposes of: 1. determining the meso/macro physical properties of such materials, and 2. studying the mechanics of damage initiation at the micro-level in such materials. In this paper, we develop what we label as “Trefftz-Lekhnitskii Grains (TLGs)”, each of which can model a single grain of piezoelectric materials with voids. These TLGs are of arbitrary geometrical shapes, to mimic the natural shape of each micro-grain of the material. The TLGs are based on expressing the mechanical and electrical fields in the interior of each grain in terms of the Trefftz solution functions derived from Lekhnitskii formulation for piezoelectric materials. The potential functions are written in terms of Laurent series which can describe interior or exterior domains where negative exponents are used only in the latter case. The boundary conditions at the outer boundaries of each TLG can be enforced using a boundary variational principle, collocation or least squares method, while the boundary conditions at the inner (void/inclusion) boundary can be enforced using collocation/least squares, or by using the special solution set which satisfy the traction-free, charge-free boundary conditions at the void periphery. These various methods of enforcing the boundary conditions generate different grains which are denoted as TLG-BVPs, TLG-C, TLG-Cs, TLG-LS, TLG-LSs (where BVP refers to “boundary variational principle”, C refers to “collocation”, LS refers to “Least Squares”, and s refers to “special solution set”). Several examples of the DNS of micro/meso mechanics of porous piezoelectric materials are presented, not only to determine the macro physical properties of such materials, but also to study the mechanisms for damage precursors in such intelligent materials.

  • Trefftz-Lekhnitskii Grains (TLGs) for efficient Direct Numerical Simulation (DNS) of the micro/meso mechanics of porous piezoelectric materials
    Computational Materials Science, 2014
    Co-Authors: Peter L. Bishay, Satya N. Atluri
    Abstract:

    Abstract We consider a class of piezoelectric materials with defects, voids, and/or elastic dielectric or piezoelectric inclusions. We develop computationally highly efficient as well as mathematically highly accurate methods for the Direct Numerical Simulation (DNS) of micro/meso mechanics of such materials, for the purposes of: 1. determining the meso/macro physical properties of such materials, and 2. studying the mechanics of damage initiation at the micro-level in such materials. In this paper, we develop what we label as “Trefftz-Lekhnitskii Grains (TLGs)”, each of which can model a single grain of piezoelectric materials with voids. These TLGs are of arbitrary geometrical shapes, to mimic the natural shape of each micro-grain of the material. The TLGs are based on expressing the mechanical and electrical fields in the interior of each grain in terms of the Trefftz solution functions derived from Lekhnitskii formulation for piezoelectric materials. The potential functions are written in terms of Laurent series which can describe interior or exterior domains where negative exponents are used only in the latter case. The boundary conditions at the outer boundaries of each TLG can be enforced using a boundary variational principle, collocation or least squares method, while the boundary conditions at the inner (void/inclusion) boundary can be enforced using collocation/least squares, or by using the special solution set which satisfy the traction-free, charge-free boundary conditions at the void periphery. These various methods of enforcing the boundary conditions generate different grains which are denoted as TLG-BVPs, TLG-C, TLG-Cs, TLG-LS, TLG-LSs (where BVP refers to “boundary variational principle”, C refers to “collocation”, LS refers to “Least Squares”, and s refers to “special solution set”). Several examples of the DNS of micro/meso mechanics of porous piezoelectric materials are presented, not only to determine the macro physical properties of such materials, but also to study the mechanisms for damage precursors in such intelligent materials.

W Wang - One of the best experts on this subject based on the ideXlab platform.

T C T Ting - One of the best experts on this subject based on the ideXlab platform.

  • Anisotropic Elastic Materials With a Parabolic or Hyperbolic Boundary: A Classical Problem Revisited
    Journal of Applied Mechanics, 2001
    Co-Authors: T C T Ting, H O K Kirchner
    Abstract:

    When an anisotropic elastic material is under a two-dimensional deformation that has a hole of given geometry Γ subjected to a prescribed boundary condition, the problem can be solved by mapping Γ to a circle of unit radius. It is important that (i) each point on Γ is mapped to the same point for the three Stroh eigenvalues p 1 , p 2 , p 3 and (ii) the mapping is one-to-one for the region outside Γ. In an earlier paper it was shown that conditions (i) and (ii) are satisfied when Γ is an ellipse. The paper did not address to the case when Γ is an open boundary, such as a parabola or hyperbola that was studied by Lekhnitskii. We examine the mappings employed by Lekhnitskii for a parabola and hyperbola, and show that while the mapping for a parabola satisfies conditions (i) and (ii), the mapping for a hyperbola does not satisfy condition (i). Nevertheless, a valid solution can be obtained for the problem with a hyperbolic boundary, although the prescription of the boundary condition is restricted. We generalize Lekhnitskii's solutions for general anisotropic elastic materials and for more general boundary conditions. Using known identities and new identities presented here, real form expressions are given for the displacement and hoop stress vector at the parabolic and hyperbolic boundary.

  • A new modified Lekhnitskii formalism à la Stroh for steady-state waves in anisotropic elastic materials
    Wave Motion, 2000
    Co-Authors: T C T Ting
    Abstract:

    Abstract For the analysis of a two-dimensional steady-state motion such as the surface wave in an anisotropic elastic half-space, the Stroh formalism has always been employed. The solutions are in terms of the elastic stiffnesses Cαβ. The Lekhnitskii formalism for elastostatics that provides the solutions in terms of the reduced elastic compliances sαβ′ is not applicable for two-dimensional steady-state motion. We present a new modified Lekhnitskii formalism in the style of Stroh that can be employed for analyzing two-dimensional steady-state motion. In contrast to the Stroh formalism for which one computes the eigenvector b in terms of the eigenvector a, the new modified Lekhnitskii formalism can compute the eigenvector b without computing the vector a. This feature is attractive in the study of surface waves because the vector b is related to the surface traction. The vanishing of the surface traction at the boundary of the half-space is the key in the surface wave theory. Application to one-component surface waves shows that the conditions for such waves are easily deduced. Motivated by the new modified Lekhnitskii formalism we show that an eigenrelation for the vector b can also be derived for the Stroh formalism.

  • Recent developments in anisotropic elasticity
    International Journal of Solids and Structures, 1999
    Co-Authors: T C T Ting
    Abstract:

    Anisotropic elasticity has been an active research subject for the last thirty years due to its applications to composite materials. There are essentially two formalisms for two-dimensional deformations of a general anisotropic elastic material. The Lekhnitskii formalism [Lekhnitskii, S.G., 1950. Theory of Elasticity of an Anisotropic Elastic Body. Gostekhizdat, Moscow (in Russian)] has been the favorite among the engineering community, while the newer Stroh formalism is well-known in the material sciences, applied mathematics and physics community. The Stroh formalism (Stroh, A.N., 1958. Dislocations and cracks in anisotropic elasticity. Phil. Mag. 3, 625–646.) is mathematically elegant and technically powerful. It began to be noticed by the engineering community in recent years, specially among the younger researchers. A comprehensive treatment of both formalisms and applications of the theory have been presented in a book by Ting. Since the appearance of the book in 1996, there have been several new developments in the theory and applications of anisotropic elasticity. We present here new results that have appeared since 1996. Only linear anisotropic elasticity is considered here; for nonlinear elasticity, the reader is referred to the book by Antman (Antman, S.S., 1995. Nonlinear Problems in Elasticity. Springer–Verlag, New York).

  • a modified Lekhnitskii formalism a la stroh for anisotropic elasticity and classifications of the 6 6 matrix n
    Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 1999
    Co-Authors: T C T Ting
    Abstract:

    The Stroh formalism for two-dimensional deformations of anisotropic elastic materials computes the eigenvalue p and the eigenvector a from an eigenrelation. The vector b is then determined from a . Depending on the number of repeated eigenvalues p and the number of independent eigenvectors ( a , b ) the system has, it can be classified into six groups. The Lekhnitskii formalism has no eigenrelation to speak of. We present a modified Lekhnitskii formalism that computes the eigenvalue p and the eigenvector b from an eigenrelation. The vector a is then determined from b . Thus the modified Lekhnitskii formalism is a dual to the Stroh formalism. Not only does the modified formalism enable us to do the classifications, it is much simpler than using the Stroh formalism. The six groups are the SP, SS, D1, D2, ED and ES groups. The ES group does not exist for a real material. The SS group (that has p 1 = p 2 ≠ p 3) and the D2 group (that has p 1 = p 2 ≠ p 3) can be identified without computing the eigenvalues p and the eigenvectors ( a , b ). We show that the repeated eigenvalue p 1 = p 2 in the SS and D2 groups is simply a root of the quadratic equation l 2 = 0. We present an explicit expression of p 3 for the SS group. The ED group that has three identical p can also be identified without computing p and ( a , b ); however, we do present an explicit expression of p . We show that monoclinic materials with the symmetry plane at x 3 = 0 cannot belong to the ED group. The identification of the SP and D1 groups is the only one that requires computation of p but not ( a , b ). For special classes of materials, however, they can be identified without computing p . In all cases, the eigenvectors and the generalized eigenvectors are obtained explicitly.

  • A modified Lekhnitskii formalism à la Stroh for anisotropic elasticity and classifications of the 6 × 6 matrix N
    Proceedings of the Royal Society of London. Series A: Mathematical Physical and Engineering Sciences, 1999
    Co-Authors: T C T Ting
    Abstract:

    The Stroh formalism for two-dimensional deformations of anisotropic elastic materials computes the eigenvalue p and the eigenvector a from an eigenrelation. The vector b is then determined from a . Depending on the number of repeated eigenvalues p and the number of independent eigenvectors ( a , b ) the system has, it can be classified into six groups. The Lekhnitskii formalism has no eigenrelation to speak of. We present a modified Lekhnitskii formalism that computes the eigenvalue p and the eigenvector b from an eigenrelation. The vector a is then determined from b . Thus the modified Lekhnitskii formalism is a dual to the Stroh formalism. Not only does the modified formalism enable us to do the classifications, it is much simpler than using the Stroh formalism. The six groups are the SP, SS, D1, D2, ED and ES groups. The ES group does not exist for a real material. The SS group (that has p 1 = p 2 ≠ p 3) and the D2 group (that has p 1 = p 2 ≠ p 3) can be identified without computing the eigenvalues p and the eigenvectors ( a , b ). We show that the repeated eigenvalue p 1 = p 2 in the SS and D2 groups is simply a root of the quadratic equation l 2 = 0. We present an explicit expression of p 3 for the SS group. The ED group that has three identical p can also be identified without computing p and ( a , b ); however, we do present an explicit expression of p . We show that monoclinic materials with the symmetry plane at x 3 = 0 cannot belong to the ED group. The identification of the SP and D1 groups is the only one that requires computation of p but not ( a , b ). For special classes of materials, however, they can be identified without computing p . In all cases, the eigenvectors and the generalized eigenvectors are obtained explicitly.

Morteza Eskandari-ghadi - One of the best experts on this subject based on the ideXlab platform.

  • A Complete Solution of the Wave Equations for Transversely Isotropic Media
    Journal of Elasticity, 2005
    Co-Authors: Morteza Eskandari-ghadi
    Abstract:

    A transversely isotropic material in the sense of Green is considered. A complete solution in terms of retarded potential functions for the wave equations in transversely isotropic media is presented. In this paper we reduce the number of potential functions to only one, and we discuss the required conditions. As a special case, the torsionless and rotationally symmetric configuration with respect to the axis of symmetry of the material is discussed. The limiting case of elastostatics is cited, where the solution is reduced to the Lekhnitskii–Hu–Nowacki solution. The solution is simplified for the special case of isotropy. In this way, a new series of potential functions (to the best knowledge of the author) for the elastodynamics problem of isotropic materials is presented This solution is reduced to a special case of the Cauchy–Kovalevski–Somigliana solution, if the displacements satisfy specific conditions. Finally, Boggio's Theorem is generalized for transversely isotropic media which may be of interest to the reader beyond the present application.

  • Wave Equations in Transversely Isotropic Media in Terms of Potential Functions (RESEARCH NOTE)
    International Journal of Engineering, 2003
    Co-Authors: A. Noorzad, Morteza Eskandari-ghadi
    Abstract:

    A complete series of potential functions for solving the wave equations in an almost transversely isotropic media is presented. The potential functions are reduced to only one potential function particularly for axisymmetric wave propagation problems. The potential functions presented in this paper can be reduced to Lekhnitskii-Hu-Nowacki solution for elastostatics problems.

Peter L. Bishay - One of the best experts on this subject based on the ideXlab platform.

  • trefftz Lekhnitskii grains tlgs for efficient direct numerical simulation dns of the micro meso mechanics of porous piezoelectric materials
    Computational Materials Science, 2014
    Co-Authors: Peter L. Bishay, Satya N. Atluri
    Abstract:

    Abstract We consider a class of piezoelectric materials with defects, voids, and/or elastic dielectric or piezoelectric inclusions. We develop computationally highly efficient as well as mathematically highly accurate methods for the Direct Numerical Simulation (DNS) of micro/meso mechanics of such materials, for the purposes of: 1. determining the meso/macro physical properties of such materials, and 2. studying the mechanics of damage initiation at the micro-level in such materials. In this paper, we develop what we label as “Trefftz-Lekhnitskii Grains (TLGs)”, each of which can model a single grain of piezoelectric materials with voids. These TLGs are of arbitrary geometrical shapes, to mimic the natural shape of each micro-grain of the material. The TLGs are based on expressing the mechanical and electrical fields in the interior of each grain in terms of the Trefftz solution functions derived from Lekhnitskii formulation for piezoelectric materials. The potential functions are written in terms of Laurent series which can describe interior or exterior domains where negative exponents are used only in the latter case. The boundary conditions at the outer boundaries of each TLG can be enforced using a boundary variational principle, collocation or least squares method, while the boundary conditions at the inner (void/inclusion) boundary can be enforced using collocation/least squares, or by using the special solution set which satisfy the traction-free, charge-free boundary conditions at the void periphery. These various methods of enforcing the boundary conditions generate different grains which are denoted as TLG-BVPs, TLG-C, TLG-Cs, TLG-LS, TLG-LSs (where BVP refers to “boundary variational principle”, C refers to “collocation”, LS refers to “Least Squares”, and s refers to “special solution set”). Several examples of the DNS of micro/meso mechanics of porous piezoelectric materials are presented, not only to determine the macro physical properties of such materials, but also to study the mechanisms for damage precursors in such intelligent materials.

  • Trefftz-Lekhnitskii Grains (TLGs) for efficient Direct Numerical Simulation (DNS) of the micro/meso mechanics of porous piezoelectric materials
    Computational Materials Science, 2014
    Co-Authors: Peter L. Bishay, Satya N. Atluri
    Abstract:

    Abstract We consider a class of piezoelectric materials with defects, voids, and/or elastic dielectric or piezoelectric inclusions. We develop computationally highly efficient as well as mathematically highly accurate methods for the Direct Numerical Simulation (DNS) of micro/meso mechanics of such materials, for the purposes of: 1. determining the meso/macro physical properties of such materials, and 2. studying the mechanics of damage initiation at the micro-level in such materials. In this paper, we develop what we label as “Trefftz-Lekhnitskii Grains (TLGs)”, each of which can model a single grain of piezoelectric materials with voids. These TLGs are of arbitrary geometrical shapes, to mimic the natural shape of each micro-grain of the material. The TLGs are based on expressing the mechanical and electrical fields in the interior of each grain in terms of the Trefftz solution functions derived from Lekhnitskii formulation for piezoelectric materials. The potential functions are written in terms of Laurent series which can describe interior or exterior domains where negative exponents are used only in the latter case. The boundary conditions at the outer boundaries of each TLG can be enforced using a boundary variational principle, collocation or least squares method, while the boundary conditions at the inner (void/inclusion) boundary can be enforced using collocation/least squares, or by using the special solution set which satisfy the traction-free, charge-free boundary conditions at the void periphery. These various methods of enforcing the boundary conditions generate different grains which are denoted as TLG-BVPs, TLG-C, TLG-Cs, TLG-LS, TLG-LSs (where BVP refers to “boundary variational principle”, C refers to “collocation”, LS refers to “Least Squares”, and s refers to “special solution set”). Several examples of the DNS of micro/meso mechanics of porous piezoelectric materials are presented, not only to determine the macro physical properties of such materials, but also to study the mechanisms for damage precursors in such intelligent materials.