Related papers: On Some Sufficient Conditions for Strong Elliptici…
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…
In this paper, we consider higher order paired symmetric tensors and strongly paired symmetric tensors. Elasticity tensors and higher order elasticity tensors in solid mechanics are strongly paired symmetric tensors. A (strongly) paired…
For a 4th order 3-dimensional cyclic symmetric tensor, a sufficient and necessary condition is bulit for its positive semi-definiteness. A sufficient and necessary condition of positive definiteness is showed for a 4th order $n$-dimensional…
For a 4th order 3-dimensional symmetric tensor with its some entries $1$ or $-1$, we show the analytic sufficient and necessary conditions of its positive definiteness. By applying these conclusions, several strict inequalities is bulit for…
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…
In particle physics, scalar potentials have to be bounded from below in order for the physics to make sense. The precise expressions of checking lower bound of scalar potentials are essential, which is an analytical expression of checking…
We consider uniformly strongly elliptic systems of the second order with bounded coefficients. First, sufficient conditions for the invariance of convex bodies obtained for linear systems without zero order term in bounded domains and…
This paper focuses on the strict copositivity analysis of 4th-order 3-dimensional symmetric tensors. A necessary and sufficient condition is provided for the strict copositivity of a fourth-order symmetric tensor. Subsequently, building…
We introduce a novel technique for proving global strong discrete maximum principles for finite element discretizations of linear and semilinear elliptic equations for cases when the common, matrix-based sufficient conditions are not…
Fiber-orientation tensors describe the relevant features of the fiber-orientation distribution compactly and are thus ubiquitous in injection-molding simulations and subsequent mechanical analyses. In engineering applications to date, the…
We study strong hyperbolicity of first order partial differential equations for systems with differential constraints. In these cases, the number of equations is larger than the unknown fields, therefore, the standard Kreiss necessary and…
We establish an explicit criterion for determining whether a $4 \times 4$ upper-triangular matrix is a contraction with respect to the Euclidean operator norm.
We consider second-order partial differential operators $H$ in divergence form on $\Ri^d$ with a positive-semidefinite, symmetric, matrix $C$ of real $L_\infty$-coefficients and establish that $H$ is strongly elliptic if and only if the…
We formulate effective necessary and sufficient conditions to identify the symmetry class of an elasticity tensor, a fourth-order tensor which is the cornerstone of the theory of elasticity and a toy model for linear constitutive laws in…
The paper addresses the problem of finding the necessary and sufficient conditions to be satisfied by the engineering moduli of an anisotropic material for the elastic energy to be positive for each state of strain or stress. The problem is…
We exploit the so called atomic condition, recently defined by De Philippis, De Rosa, and Ghiraldin in [Comm. Pure Appl. Math.] and proved to be necessary and sufficient for the validity of the anisotropic counterpart of the Allard…
We address the problem about under what conditions an endomorphism having a dense orbit, verifies that a sufficiently close perturbed map also exhibits a dense orbit. In this direction, we give sufficient conditions, that cover a large…
We make progress towards proving the strong Eshelby's conjecture in three dimensions. We prove that if for a single nonzero uniform loading the strain inside inclusion is constant and further the eigenvalues of this strain are either all…
The strict opositivity of 4th order symmetric tensor may apply to detect vacuum stability of general scalar potential. For finding analytical expressions of (strict) opositivity of 4th order symmetric tensor, we may reduce its order to 3rd…
We derive fundamentally new equations that are satisfied by first-order flexes of a flexible polyhedron. Moreover, we indicate two sources of such new equations. These sources are the Dehn invariants and rigidity matrix. The equations…