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Liqun Qi - One of the best experts on this subject based on the ideXlab platform.
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elasticity m tensors and the strong Ellipticity Condition
Applied Mathematics and Computation, 2020Co-Authors: Weiyang Ding, Liqun QiAbstract:Abstract In this paper, we establish two sufficient Conditions for the strong Ellipticity of any fourth-order elasticity tensor and investigate a class of tensors satisfying the strong Ellipticity Condition, the elasticity M -tensor. The first sufficient Condition is that the strong Ellipticity holds if the unfolding matrix of this fourth-order elasticity tensor can be modified into a positive definite one by preserving the summations of some corresponding entries. Second, an alternating projection algorithm is proposed to verify whether an elasticity tensor satisfies the first Condition or not. Besides, the elasticity M -tensor is defined with respect to the M-eigenvalues of elasticity tensors. We prove that any nonsingular elasticity M -tensor satisfies the strong Ellipticity Condition by employing a Perron-Frobenius-type theorem for M-spectral radii of nonnegative elasticity tensors. Other equivalent definitions of nonsingular elasticity M -tensors are also established.
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on the m eigenvalues of elasticity tensor and the strong Ellipticity Condition
arXiv: Rings and Algebras, 2017Co-Authors: Hua Xiang, Liqun QiAbstract:Strong Ellipticity is an important property in the elasticity theory. In 2009, M-eigenvalues were introduced for the elasticity tensor. It was shown that M-eigenvalues are invariant under coordinate system choices, and the strong Ellipticity Condition holds if and only if all the M-eigenvalues of the elasticity tensor are positive. Thus, M-eigenvalues are some intrinsic parameters of the elasticity tensor. In this paper, we show that the M-eigenvalues of the elasticity tensor are closely related with some elastic moduli, such as the bulk modulus, the shear modulus, Lam\'e's first parameter, the P-wave modulus, etc, and the positiveness of the M-eigenvalues are corresponding to some existing Conditions for strong Ellipticity in some special cases, such as the isotropic case, the cubic case, the polar anisotropic case and the tetragonal case. We also present new sufficient Conditions for the strong Ellipticity of the orthotropic case. These, in a certain sense, further reveal the physical meanings of M-eigenvalues.
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elasticity mathscr m tensors and the strong Ellipticity Condition
arXiv: Mathematical Physics, 2017Co-Authors: Weiyang Ding, Liqun QiAbstract:In this paper, we establish two sufficient Conditions for the strong Ellipticity of any fourth-order elasticity tensor and investigate a class of tensors satisfying the strong Ellipticity Condition, the elasticity $\mathscr{M}$-tensor. The first sufficient Condition is that the strong Ellipticity holds if the unfolding matrix of this fourth-order elasticity tensor can be modified into a positive definite one by preserving the summations of some corresponding entries. Second, an alternating projection algorithm is proposed to verify whether an elasticity tensor satisfies the first Condition or not. Besides, the elasticity $\mathscr{M}$-tensor is defined with respect to the M-eigenvalues of elasticity tensors. We prove that any nonsingular elasticity $\mathscr{M}$-tensor satisfies the strong Ellipticity Condition by employing a Perron-Frobenius-type theorem for M-spectral radii of nonnegative elasticity tensors. Other equivalent definitions of nonsingular elasticity $\mathscr{M}$-tensors are also established.
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Singular values of a real rectangular tensor
Journal of Mathematical Analysis and Applications, 2010Co-Authors: Kung-ching Chang, Liqun Qi, Guanglu ZhouAbstract:Real rectangular tensors arise from the strong Ellipticity Condition problem in solid mechanics and the entanglement problem in quantum physics. In this paper, we systematically study properties of singular values of a real rectangular tensor, and give an algorithm to find the largest singular value of a nonnegative rectangular tensor. Numerical results show that the algorithm is efficient.
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Conditions for Strong Ellipticity of Anisotropic Elastic Materials
Journal of Elasticity, 2009Co-Authors: Liqun QiAbstract:In this paper, we derive necessary and sufficient Conditions for the strong Ellipticity Condition of anisotropic elastic materials. We first observe that the strong Ellipticity Condition holds if and only if a second order tensor function is positive definite for any unit vectors. Then we further link this Condition to the rank-one positive definiteness of three second-order tensors, three fourth-order tensors and a sixth-order tensor. In particular, we consider Conditions of strong Ellipticity of the rhombic classes, for which we need to check the copositivity of three second-order tensors and the positive definiteness of a sixth-order tensor. A direct method is presented to verify our Conditions.
Francesco Nicolosi - One of the best experts on this subject based on the ideXlab platform.
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On the sets of boundedness of solutions for a class of degenerate nonlinear elliptic fourth-order equations with L 1 -data
Journal of Mathematical Sciences, 2008Co-Authors: Alexander Kovalevsky, Francesco NicolosiAbstract:In this article, we deal with a class of degenerate, nonlinear, elliptic fourth-order equations in divergence form with coefficients satisfying a strengthened Ellipticity Condition and right-hand sides of the class L1 depending on the unknown function. We consider the Dirichlet problem for equations of the given class and prove the existence of solutions of this problem bounded on the sets where the behavior of the data of the problem and the weighted functions involved is sufficiently regular.
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On the sets of L∞-regularity of solutions for a class of degenerate nonlinear problems with slightly regular data
Nonlinear Analysis-theory Methods & Applications, 2008Co-Authors: Alexander Kovalevsky, Francesco NicolosiAbstract:We consider the Dirichlet problem for a class of degenerate nonlinear elliptic fourth-order equations in divergence form. Coefficients of the equations satisfy a strengthened Ellipticity Condition involving two weighted functions. The right-hand sides F(x,u) of the equations depend on the unknown function u. Under some Conditions, including Lm-regularity of F(⋅,0) with m close to 1, we establish that the problem under consideration has a W-solution bounded on the sets where the behaviour of the data of the problem is regular enough.
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On the convergence of solutions of degenerate nonlinear elliptic high order equations
Nonlinear Analysis-theory Methods & Applications, 2002Co-Authors: Alexander Kovalevsky, Francesco NicolosiAbstract:Here is a bounded open set of Rn; m?2; q?p?2; and are positive functions in ; ◦ W 1; q m;p( ; ; ) is the Banach space of all functions u : → R with the properties |u|q; |D u|q; |D u|p ∈ L1( ); | |=1; | |=m, and “zero” boundary values. Symbols and all through the paper denote n-dimensional multiindices, | |; | | are their lengths, mu= {D u: | | 6 m}. The class of operators under consideration is characterized by the following degenerate Ellipticity Condition for their coe>cients: ∑
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Existence of Solutions of Some Degenerate Nonlinear Elliptic Fourth-order Equations with L 1 -data
Applicable Analysis, 2002Co-Authors: Alexander Kovalevsky, Francesco NicolosiAbstract:This article deals with a class of degenerate nonlinear elliptic fourth-order equations with L 1 -right-hand sides. Equations of the given class have divergence form and their coefficients satisfy a strengthened Ellipticity Condition with two different weights associated, respectively, to the first- and the second-order derivatives of unknown function. Under suitable hypotheses on the weighted functions involved we establish solvability in a Sobolev space of Dirichlet problem for equations under consideration.
Alexander Kovalevsky - One of the best experts on this subject based on the ideXlab platform.
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On the sets of boundedness of solutions for a class of degenerate nonlinear elliptic fourth-order equations with L 1 -data
Journal of Mathematical Sciences, 2008Co-Authors: Alexander Kovalevsky, Francesco NicolosiAbstract:In this article, we deal with a class of degenerate, nonlinear, elliptic fourth-order equations in divergence form with coefficients satisfying a strengthened Ellipticity Condition and right-hand sides of the class L1 depending on the unknown function. We consider the Dirichlet problem for equations of the given class and prove the existence of solutions of this problem bounded on the sets where the behavior of the data of the problem and the weighted functions involved is sufficiently regular.
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On the sets of L∞-regularity of solutions for a class of degenerate nonlinear problems with slightly regular data
Nonlinear Analysis-theory Methods & Applications, 2008Co-Authors: Alexander Kovalevsky, Francesco NicolosiAbstract:We consider the Dirichlet problem for a class of degenerate nonlinear elliptic fourth-order equations in divergence form. Coefficients of the equations satisfy a strengthened Ellipticity Condition involving two weighted functions. The right-hand sides F(x,u) of the equations depend on the unknown function u. Under some Conditions, including Lm-regularity of F(⋅,0) with m close to 1, we establish that the problem under consideration has a W-solution bounded on the sets where the behaviour of the data of the problem is regular enough.
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On the convergence of solutions of degenerate nonlinear elliptic high order equations
Nonlinear Analysis-theory Methods & Applications, 2002Co-Authors: Alexander Kovalevsky, Francesco NicolosiAbstract:Here is a bounded open set of Rn; m?2; q?p?2; and are positive functions in ; ◦ W 1; q m;p( ; ; ) is the Banach space of all functions u : → R with the properties |u|q; |D u|q; |D u|p ∈ L1( ); | |=1; | |=m, and “zero” boundary values. Symbols and all through the paper denote n-dimensional multiindices, | |; | | are their lengths, mu= {D u: | | 6 m}. The class of operators under consideration is characterized by the following degenerate Ellipticity Condition for their coe>cients: ∑
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Existence of Solutions of Some Degenerate Nonlinear Elliptic Fourth-order Equations with L 1 -data
Applicable Analysis, 2002Co-Authors: Alexander Kovalevsky, Francesco NicolosiAbstract:This article deals with a class of degenerate nonlinear elliptic fourth-order equations with L 1 -right-hand sides. Equations of the given class have divergence form and their coefficients satisfy a strengthened Ellipticity Condition with two different weights associated, respectively, to the first- and the second-order derivatives of unknown function. Under suitable hypotheses on the weighted functions involved we establish solvability in a Sobolev space of Dirichlet problem for equations under consideration.
Klausjurgen Bathe - One of the best experts on this subject based on the ideXlab platform.
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On the Ellipticity Condition for model-parameter dependent mixed formulations
Computers & Structures, 2020Co-Authors: Dominique Chapelle, Klausjurgen BatheAbstract:International audienceWhen establishing and analyzing model-parameter dependent mixed formulations, it is common to consider required Ellipticity and inf-sup Conditions for the continuous and discrete problems. However, in the modeling of some important categories of problems, like in the analysis of plates and shells, the Ellipticity Condition usually considered does not naturally hold, and the inf-sup Condition can only be stated in an abstract form and can hardly be evaluated analytically. In this paper we present a new and practical Ellipticity Condition which together with the inf-sup Condition guarantees that (i) when the model parameter goes to zero, the limit problem solution is uniformly approached, and (ii) an optimal finite element discretization has been established (for the interpolations used). In practice, a numerical test might be performed to see whether the proposed Ellipticity Condition is satisfied
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on the Ellipticity Condition for model parameter dependent mixed formulations
Computers & Structures, 2010Co-Authors: Dominique Chapelle, Klausjurgen BatheAbstract:When establishing and analyzing model-parameter dependent mixed formulations, it is common to consider required Ellipticity and inf-sup Conditions for the continuous and discrete problems. However, in the modeling of some important categories of problems, like in the analysis of plates and shells, the Ellipticity Condition usually considered does not naturally hold, and the inf-sup Condition can only be stated in an abstract form and can hardly be evaluated analytically. In this paper we present a new and practical Ellipticity Condition which together with the inf-sup Condition guarantees that (i) when the model parameter goes to zero, the limit problem solution is uniformly approached, and (ii) an optimal finite element discretization has been established (for the interpolations used). In practice, a numerical test might be performed to see whether the proposed Ellipticity Condition is satisfied.
Yichao Chen - One of the best experts on this subject based on the ideXlab platform.
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on strong Ellipticity and the legendre hadamard Condition
Archive for Rational Mechanics and Analysis, 1991Co-Authors: Yichao ChenAbstract:First, we derive Conditions characterizing all fourth order tensors that satisfy the strong Ellipticity Condition or the Legendre-Hadamard Condition. Secondly, we establish the sufficiency of the Legendre-Hadamard Condition for the non negativiy of the second variation in non linear elasticity