Maxwell’s stress

FIGURE 10.5 Dielectric elastomer actuator (DBA) demonstrated to expand and relax in a circular strain test. The Maxwell s stress associated with the force is perpendicular to the apphed force and is given by... 

We can now identify the first term in (A.4) with Maxwell s stress tensor, which acts on any dielectric in an electric field. The magnitude of this force Pe is given by... 

Now, find the force acting on the conducting drop. The momentum fiux density in an electric field is defined by Maxwell s stress tensor [77]... 

The retarded dispersion energy between macroscopic particles was treated by Liftshitz [28]. He considered half-spaces. Going half the way from the microscopic to the macroscopic approach, Lifshitz expanded the local fluctuations within the half-spaces in terms of plane waves and coupled them to the outgoing (reflected) radiation field. Then, satisfying the boundary conditions for the radiation field across the surfaces of the half-spaces under consideration, he found their force of attraction from Maxwell s stress tensor in the interspace. 

The calculation of the force of attraction from Maxwell s stress tensor, i.e. the omission of the Poynting vector in the total energy balance, is equivalent to calculating the dispersion energy merely from the real part of the free energy gain according to Eq. (3.72). 

We have just described the linearized theory of capillarity. In the electrostatic analogy the field M(r ) is identified with a 2D electrostatic potential ( capillary potential ) and Il(r ) with a charge density ( capillary charge ) Equation 2.8 reduces to the Poisson equation of electrostatics and Equation 2.9 relates the tensor Tn, which has the form of Maxwell s stress tensor, with the electric force exerted on the capillary charge n(rn) (also the usual boundary conditions imposed on the interface have a close electrostatic analogy [34,35]). 

The basis for the above-mentioned model47) was provided by Maxwell s nonlinear model obtained in general form in Ref. 48). Flere the total strain was divided into irreversible strain and elastic strain X, Stress a and velocity of irreversible strain ep were determined from the elastic strain. In Ref. 48) a number of functions a (a.) and ep(X) were defined more specifically. Beside that, Maxwell s nonlinear models were connected in parallel. Note that in case of one Maxwell s element X = a23), but in case of several elements connected in parallel this is not true and a is determined from the solution of the respective problem. In case of the uniaxial extension the model of Ref.47) takes the following form ... 

Note that the model given in Ref. 47) takes into account the distortion of potential barriers in activation flow under the action of stress. Relaxation time is 0k = 0k exp (—mWk) (here mk = yk/RT where yk is constant, R is gas constant, T is temperature). Normally, two Maxwell s elements (N = 2) and one viscous element with vis-... 

Note that the property 2 is surprising and beautiful for the Maxwell equations to hold, it is not necessary to consider any variational principle whatsoever. Given a scalar held that can be interpreted as a map cjj N3 — N2, the mere existence of a dual map 0 guarantees that the two pull-backs of the area 2-form in S2 obey Maxwell s equations in empty space. This fact must be stressed—the duality condition on the two scalars implies the Maxwell equations by itself. 

As noted elsewhere [67], Eq. (14) means that the continuity condition does not prohibit the existence of an electromagnetic current density J in free space. It is stressed that Eq. (14) is a mathematical prediction of Maxwell s equations, completely independent of any interpretation. 

For the Maxwell field, the energy-momentum tensor Tfi(A) derived from Noether s theorem is unsymmetric, and not gauge invariant, in contrast to the symmetric stress tensor derived directly from Maxwell s equations [318], Consider the symmetric tensor 0 = T + AT, where... 

If we now perform a creep experiment, applying a constant stress, a0 at time t = 0 and removing it after a time f, then the strain/ time plot shown at the top of Figure 13-89 is obtained. First, the elastic component of the model (spring) deforms instantaneously a certain amount, then the viscous component (dashpot) deforms linearly with time. When the stress is removed only the elastic part of the deformation is regained. Mathematically, we can take Maxwell s equation (Equation 13-85) and impose the creep experiment condition of constant stress da/dt = 0, which gives us Equation 13-84. In other words, the Maxwell model predicts that creep should be constant with time, which it isn t Creep is characterized by a retarded elastic response. 

Adamson (51) proposed a model for W/0 microemulsion formation in terms of a balance between Laplace pressure associated with the interfacial tension at the oil/water interface and the Donnan Osmotic pressure due to the total higher ionic concentration in the interior of aqueous droplets in oil phase. The microemulsion phase can exist in equilibrium with an essentially non-colloidal aqueous second phase provided there is an added electrolyte distributed between droplet s aqueous interior and the external aqueous medium. Both aqueous media contain some alcohol and the total ionic concentration inside the aqueous droplet exceeds that in the external aqueous phase. This model was further modified (52) for W/0 microemulsions to allow for the diffuse double layer in the interior of aqueous droplets. Levine and Robinson (52) proposed a relation governing the equilibrium of the droplet for 1-1 electrolyte, which was based on a balance between the surface tension of the film at the boundary in its charged state and the Maxwell electrostatic stress associated with the electric field in the internal diffuse double layer. 

According to Maxwell s rheological model of long-term action of constant loading, the stress o0 and viscous deformations ev are characterized by the equations ... 

In the preceding chronicles, those developments that supported the atomic idea were stressed. On the other hand. Maxwell s development of electromagnetic theory, an un-dulatory theory, is an important link in the chain. Another fact of great consequence is that in the latter part of the 19th century a great amount of experimental work was devoted to the study of spectra. 

The behavior of a polymer system is so complicated that we cannot represent it with the response time of a single Maxwell element. In other words, the simple model described above cannot approach the behavior of a real system. In 1893, Weichert showed that stress-relaxation experiments could be represented as a generalization of Maxwell s equation. The mechanical model according to Weichert s formulation is shown in Figure 3.11 it consists of a large number of Maxwell elements coupled in parallel. 

With taking into accoimt of the elastic properties of conformation of polymeric chains the shear stress should be described by the Maxwell s equation... 

Maxwell s relation is based on the following argument. In a solid body a finite deformation of the type considered in Fig. 8 can only be maintained if a stress is applied (symbolised by the arrows F). The force per unit area needed to maintain unit shear is called shear modulus pf. In other words... 

To study the kinetics of temperature stresses in polymers, to analyse the influence of various factors on the flow of the examined processes, and to model the relaxation behaviour in polymers, a nonlinear constitutive differential equation is used in the paper. This equation was proposed by G.I. Gurevich [1], who called it the nonlinear generalized Maxwell equation out of respect for J. Maxwell s ideas [2] that served as a partial basis for deducing the equation. Total deformation is regarded as the sum of elastic, viscoelastic and temperature deformations ... 

In what follows, the functions negative Helmholtz energy. Therefo on purely dimensional grounds, a Sijj, must represent surface charge d coefficient of the displacement vec represent a stress tensor. In fact electrostatic field (e.g. for p = 0 Maxwell s electric stress tensor, the next Section that Maxwell s str a part of in our colloid model,... 

In one of the first papers dealing with dielectric elastomers, Pelrine et al. (1998) proposed a widely used model to analyze stresses and strains in a thin elastomeric membrane subjected to a transverse electric field. The notion of Maxwell s pressure Pel = eE is well known among the EAP research community. How does this concept match with the theory developed here This can be easily explained recalling that fire stress measure adopted by Pelrine et al. (1998) corresponds to our and their model is valid only for incompressible materials. 

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