This work reported a novel magnetorheological plastomer doped with hard magnetic NdFeB particles (named as H-MRP). The dynamic properties of the H-MRP were systematically tested, and the mechanism of its unusual response to magnetic field was discussed. Unlike the MRP filled with soft magnetic particles (S-MRP), the storage modulus (G′) of H-MRPs kept growing by 20% when the magnetic field decreased, while that of S-MRPs decreased with decreasing of the magnetic field immediately without an obvious change. Under the magnetic field ranging from 0 to 1 T, there was a peak value in the rising stage of G′ at about 400 mT magnetic field, while that of S-MRPs changed monotonically. After an off−on progress of the magnetic field, the final G′ of H-MRPs increased by over 10%. A possible mechanism was proposed to study the microstructure dependent mechanical properties. It was found that the complex viscoelastic behavior originated from the large hysteresis characteristics of the H-MRPs.
In this article, we study the dynamical behaviour of solutions of the non-autonomous stochastic Fitzhugh-Nagumo system on R N with both multiplicative noises and non-autonomous forces, where the nonlinearity is a polynomial-like growth function of arbitrary order. An asymptotic smoothing effect of this system is demonstrated, namely, that the random pullback attractor in the initial space L 2 (R N) × L 2 (R N) is actually a compact, measurable and attracting set in H 1 (R N) × L 2 (R N). A difference estimates method, rather than the usual truncation estimate and spectrum decomposition technique, is employed to overcome the lack of Sobolev compact embedding in H 1 (R N) × L 2 (R N), despite of the loss of the high-order integrability of the difference of solutions for this system.
In this paper, we investigate the approximations of stochastic p-Laplacian equation with additive white noise by a family of piecewise deterministic partial differential equations driven by a stationary stochastic process. We firstly obtain the tempered pullback attractors for the random p-Laplacian equation with a general diffusion. We secondly prove the convergence of solutions and the upper semi-continuity of pullback attractors of the Wong-Zakai approximation equations in a Hilbert space for the additive case. Thirdly, by a truncation technique, the uniform compactness of pullback attractor with respect to the quantity of approximations is derived in the space of q-times integrable functions, where the upper semi-continuity of the attractors of the approximation equations is well established.
In this article, a notion of bi-spatial continuous random dynamical system is introduced between two completely separable metric spaces. It is show that roughly speaking, if such a random dynamical system is asymptotically compact and random absorbing in the initial space, then it admits a bispatial pullback attractor which is measurable in two spaces. The measurability of pullback attractor in the regular spaces is completely solved theoretically. As applications, we study the dynamical behaviour of solutions of the nonautonomous stochastic fractional power dissipative equation on R N with additive white noise and a polynomial-like growth nonlinearity of order p, p ≥ 2. We prove that this equation generates a bi-spatial (L 2 (R N ), H s (R N ) ∩ L p (R N ))continuous random dynamical system, and the random dynamics for this system is captured by a bi-spatial pullback attractor which is compact and attracting in H s (R N ) ∩ L p (R N ), where H s (R N ) is a fractional Sobolev space with s ∈ (0, 1). Especially, the measurability of pullback attractor is individually derived by proving the the continuity of solutions in H s (R N ) and L p (R N ) with respect to the sample. A difference estimates approach, rather than the usual truncation estimate and spectral decomposition technique, is employed to overcome the loss of Sobolev compact embedding in H s (R N ) ∩ L p (R N ), s ∈ (0, 1), N ≥ 1.2010 Mathematics Subject Classification. Primary: 35R60, 35B40, 35B41; Secondary: 35B65. Key words and phrases. Bi-spatial continuous random dynamical system, fractional power dissipative equation, fractional Sobolev space, pullback attractor, measurability.The operator (−∆) s with s ∈ (0, 1) is called a fractional power Laplacian, whose limit as s ↑ 1 is the classic Laplacian ∆, see [17]. It is a nonlocal generalization of the classic Laplacian that is often used to model the diffusive processes, see [17,27,40] and the references therein. As for the long-time dynamics, Hu et al [32] discussed the existence of a random attractor in L 2 (R N ) for the fractional power dissipative stochastic equation with additive noises. Wang [46] discussed the existence and uniqueness of random attractor for the same equation with multiplicative noise on bounded domain. The dynamics of 3D Ginzburg-Landau equation involving fractional power Laplacian ware studied in [33,34]. Most recently, Gu et al [21] obtained the regularity of the random attractor for problem (1) in H s (R N ) in the case of multiplicative noise, where the authors employed the tail estimate and spectral decomposition technique to overcome the loss of compactness on unbounded domains. But little is known about the continuity and measurability of its solutions in H s (R N ) and L p (R N ) for s ∈ (0, 1) and p ≥ 2. In this paper, we prove the existence, regularity and measurability of pullback attractors for the stochastic fractional power dissipative equation (1) defined on the whole space R N .Most of the stochastic partial differential equations (SPDEs) present a regular solut...
As a kind of novel magneto-sensitive smart soft material, magnetorheological plastomers (MPPs) have great application potential in the areas of absorber, damper and sensor. In this work, the mechanical behaviors of MRPs under several different varying magnetic fields were studied. The viscoelastic characteristic of the MRPs was seriously depended on the type of external magnetic field. A linearly changed magnetic field would cause the increase of zero field storage modulus ΔG0′ of MPRs for about 1 MPa, which would severely reduce the magnetorheological effect of MRPs (from 500% to 50%). The direction of the external magnetic field had no influence on the variation of G0′. While under the square wave magnetic field, no variation of the magnetic storage modulus was found when the external magnetic field was removed. A particle-level dynamics simulation was performed to observe the microstructure evolution of the MRP sample under different magnetic loading model. And a possible mechanism was proposed to explain the different residual storage modulus of the MRPs sample under different magnetic loading mode. The discovery would contribute to enhance the regulatory capacity of the MRP material in the application of the smart device.
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