scholarly journals Analytical stability conditions for elastic composite materials with a non-positive-definite phase

2012 ◽  
Vol 468 (2144) ◽  
pp. 2230-2254 ◽  
Author(s):  
D. M. Kochmann ◽  
W. J. Drugan
Keyword(s):  
Stability Analysis ◽  
Composite Materials ◽  
Closed Form ◽  
Positive Definite ◽  
Stability Conditions ◽  
Two Phase ◽  
Analytical Stability ◽  
Sufficient Stability ◽  
Multi Phase ◽  
Sufficient Stability Conditions

Elastic multi-phase materials with a phase having appropriately tuned non-positive-definite elastic moduli have been shown theoretically to permit extreme increases in multiple desirable material properties. Stability analyses of such composites were only recently initiated. Here, we provide a thorough stability analysis for general composites when one phase violates positive-definiteness. We first investigate the dynamic deformation modes leading to instability in the fundamental two-phase solids of a coated cylinder (two dimensions) and a coated sphere (three dimensions), from which we derive closed-form analytical sufficient stability conditions for the full range of coating thicknesses. Next, we apply the energy method to derive a general correlation between composite stability limit and composite bulk modulus that enables determination of closed-form analytical sufficient stability conditions for arbitrary multi-phase materials by employing effective modulus formulas coupled with a numerical finite-element stability analysis. We demonstrate and confirm this new approach by applying it to (i) the two basic two-phase solids already analysed dynamically; and (ii) a more geometrically complex matrix/distributed-inclusions composite. The specific new analytical stability results, and new methods presented, provide a basis for creation of novel, stable composite materials.

scholarly journals Stability Analysis of a Fractional-Order Linear System Described by the Caputo–Fabrizio Derivative

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 200 ◽  
Author(s):  
Hong Li ◽  
Jun Cheng ◽  
Hou-Biao Li ◽  
Shou-Ming Zhong
Keyword(s):  
Stability Analysis ◽  
Linear System ◽  
Fractional Order ◽  
Stability Domain ◽  
Order System ◽  
Stability Conditions ◽  
Order Linear ◽  
Dynamic Matrix ◽  
Sufficient Stability ◽  
Sufficient Stability Conditions

In this paper, stability analysis of a fractional-order linear system described by the Caputo–Fabrizio (CF) derivative is studied. In order to solve the problem, character equation of the system is defined at first by using the Laplace transform. Then, some simple necessary and sufficient stability conditions and sufficient stability conditions are given which will be the basis of doing research of a fractional-order system with a CF derivative. In addition, the difference of stability domain between two linear systems described by two different fractional derivatives is also studied. Our results permit researchers to check the stability by judging the locations in the complex plane of the dynamic matrix eigenvalues of the state space.

Volterra funktional equations in the stability problem for the existence of global solutions of distributed controlled systems

2020 ◽  
pp. 422-440
Author(s):  
Vladimir I. Sumin
Keyword(s):  
Lipschitz Condition ◽  
Functional Equations ◽  
Volterra Operator ◽  
Global Solutions ◽  
Equivalent Norm ◽  
Stability Conditions ◽  
Sufficient Stability ◽  
Definition Of ◽  
The Right ◽  
Sufficient Stability Conditions

Earlier the author proposed a rather general form of describing controlled initial–boundary value problems (CIBVPs) by means of Volterra functional equations (VFE) z(t)=f(t,A[z](t),v(t) ), t≡{t^1,⋯,t^n }∈Π⊂R^n, z∈L_p^m≡(L_p (Π) )^m, where f(.,.,.):Π×R^l×R^s→R^m; v(.)∈D⊂L_k^s – control function; A:L_p^m (Π)→L_q^l (Π)- linear operator; the operator A is a Volterra operator for some system T of subsets of the set Π in the following sense: for any H∈T, the restriction A├ [z]┤|_H does not depend on the values of ├ z┤|_(Π\H); (this definition of the Volterra operator is a direct multidimensional generalization of the well-known Tikhonov definition of a functional Volterra type operator). Various CIBVP (for nonlinear hyperbolic and parabolic equations, integro-differential equations, equations with delay, etc.) are reduced by the method of conversion the main part to such functional equations. The transition to equivalent VFE-description of CIBVP is adequate to many problems of distributed optimization. In particular, the author proposed (using such description) a scheme for obtaining sufficient stability conditions (under perturbations of control) of the existence of global solutions for CIBVP. The scheme uses continuation local solutions of functional equation (that is, solutions on the sets H∈T). This continuation is realized with the help of the chain {H_1⊂H_2⊂⋯⊂H_(k-1)⊂H_k≡Π}, where H_i∈T, i=¯(1,k.) A special local existence theorem is applied. This theorem is based on the principle of contraction mappings. In the case p=q=k=∞ under natural assumptions, the possibility of applying this principle is provided by the following: the right-hand side operator F_v [z(.) ](t)≡f(t,A[z](t),v(t)) satisfies the Lipschitz condition in the operator form with the quasi-nilpotent «Lipschitz operator». This allows (using well-known results of functional analysis) to introduce in the space L_∞^m (H) such an equivalent norm in which the operator of the right-hand side will be contractive. In the general case 1≤p,q,k ≤∞, (this case covers a much wider class of CIBVP), the operator F_v; as a rule, does not satisfy such Lipschitz condition. From the results obtained by the author earlier, it follows that in this case there also exists an equivalent norm of the space L_p^m (H), for which the operator F_v is a contraction operator. The corresponding basic theorem (equivalent norm theorem) is based on the notion of equipotential quasi-nilpotency of a family of linear operators, acting in a Banach space. This article shows how this theorem can be applied to obtain sufficient stability conditions (under perturbations of control) of the existence of global solutions of VFE.

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