Kink Surfaces in a Directionally Reinforced neo-Hookean Material under Plane Deformation: I. Mechanical Equilibrium

Stationary kinks (elastostatic shocks) are examined in the context of a base neo-Hookean response augmented with unidirectional reinforcing that is characterized by a single additional constitutive parameter for the additional fiber reinforcing stiffness. Previous work has shown that such a transversely isotropic material can lose ellipticity in plane deformation if the reinforcing is sufficiently large and the fiber direction is sufficiently compressed. Here we show that the same reinforcing levels can give rise to piecewise smooth plane deformations separated by a plane stationary kink. Attention is restricted to deformations in which, on one side of the kink, the load axis is aligned with the fiber axis. Then the fiber stretch on this side of the kink is a natural load parameter. It is found that such a deformation can support a planar kink for a certain range of this load parameter. This range is dependent on the reinforcing parameter, and can even involve fiber extension if the reinforcing is sufficiently large. The set of all deformation states on the other side of the kink is precisely characterized in terms of a one-parameter family of (kink orientation, kink strength)-pairs. The results are interpreted in terms of the associated fiber alignment discontinuity and fiber stretch discontinuity.     

Kink Surfaces in a Directionally Reinforced neo-Hookean Material under Plane Deformation: II. Kink Band Stability and Maximally Dissipative Band Broadening

Stationary kinks (elastostatic shocks) are examined in the context of a base neo-Hookean response augmented with unidirectional reinforcing that is characterized by a single additional constitutive parameter for the additional fiber reinforcing stiffness. Previous work has shown that such a transversely isotropic material can support stationary kinks in plane deformation if the reinforcing is sufficiently great. If the deformation on one side of the kink involves a load axis aligned with the fiber axis, then the more general plane deformation on the other side of the kink is characterized in terms of a one-parameter family of (kink orientation, kink strength)-pairs. Here, the ellipticity status of the two correlated deformation states is shown to span all four possible ellipticity/nonellipticity permutations. If both deformation states are elliptic, then a suitable intermediate deformation is shown to be nonelliptic. Maximally dissipative quasi-static kink motion is examined and interpreted in terms of kink band broadening in on-axis loading. Such maximally dissipative kinks nucleate only in compression as weak kinks, with subsequent motion converting nonelliptic deformation to elliptic deformation. The associated fiber rotation involves three periods: an initial period of slow rotation, a secondary period of rapid rotation, and a final period of essentially constant orienation.     

Remarks on cavity formation in fiber-reinforced incompressible non-linearly elastic solids

In this paper we focus on cavity formation in fiber-reinforced incompressible non-linearly elastic solids. In particular, the material under consideration is a base neo-Hookean augmented by a function that accounts for the existence of a unidirectional reinforcing. This function characterizes the anisotropy of the material and is referred to as reinforcing model. Previous works has dealt with the analysis of a specific reinforcing model, the so called standard reinforcing model, that is a quadratic function that depends only on the fiber reinforcement stretch. Here, two different reinforcing models are examined: a power law that depends only on the fiber stretch and a quadratic function that depends simultaneously on the fiber stretch and fiber shearing. Closed form analytic solutions are found for the classical problem of cavity formation in a sphere under uniform radial tensile dead-load with the fiber in the radial direction. It is shown that for some of the new reinforcing models under study the cavitation instabilities obtained in previous works for the standard reinforcing model are not possible.     

A mathematical model for cylindrical,fiber reinforced electro-pneumatic actuators

In recent years, dielectric elastomers have received increasing attention due to their unparalleled large strain actuation response (>100%). The force output, however, has remained a major limiting factor for many applications. To address this limitation, a model for a fiber reinforced dielectric elastomer actuator based on the deformation mechanism of McKibben actuators is presented. In this novel configuration, the outer cylindrical surface of a dielectric elastomer is enclosed by a network of helical fibers that are thin, flexible and inextensible. This configuration yields an axially contractile actuator, in contrast to unreinforced actuators which extend. The role of the fiber network is twofold: (i) to serve as reinforcement to improve the load-bearing capability of dielectric elastomers, and (ii) to render the actuator inextensible in the axial direction such that the only free deformation path is simultaneous radial expansion and axial contraction. In this paper, a mathematical model of the electromechanical response of fiber reinforced dielectric elastomers is derived. The model is developed within a continuum mechanics framework for large deformations. The cylindrical electro-pneumatic actuator is modeled by adapting Green and Adkins’ theory of reinforced cylinders to account for the applied electric field. Using this approach, numerical solutions are obtained assuming a Mooney–Rivlin material model. The results indicate that the relationship between the contractile force and axial shortening is bilinear within the voltage range considered. The characteristic response as a function of various system parameters such as the fiber angle, inflation pressure, and the applied voltage are reported. In this paper, the elastic portion of the modeling approach is validated using experimental data for McKibben actuators.     

An asymptotic analysis of finite deformations near the tip of an interface-crack

This work investigates the behavior of a traction-free crack at the interface of two semi-infinite slabs bonded together under the conditions of plane strain. A determination of the mathematical form of the deformation and stresses near the crack-tip, consistent with the fully non-linear equilibrium theory of compressible elastic solids, is found by an asymptotic treatment of the deformation.Each slab is assumed to be hyperelastic, homogeneous, and isotropic with Knowles-Sternberg type asymptotic conditions on its strain-energy density. It is shown that under these conditions, the interface-crack problem admits solutions in which oscillatory singularities do not occur. This suggests that it is the approximations made by the linear theory which produce these singularities.     

Rate type constitutive equations for fiber reinforced nonlinearly vicoelastic solids using spectral invariants

In this paper we are interested in developing constitutive equations for fiber-reinforced nonlinearly viscoelastic solids. It has been shown that constitutive equations for such bodies can be expressed in terms of a complete minimal set of 18 classical invariants associated with deformation and fiber orientation. In this paper, we give an alternative formulation using a set of spectral invariants. It is shown via the use of spectral invariants that only 11 of the 18 classical invariants are independent. We analyze the spectral invariants for two illustrative deformation gradients: (i) simple tension, and (ii) simple shear.     

Nonlinear waves in electromigration dispersion in a capillary

We construct exact solutions to an unusual nonlinear advection–diffusion equation arising in the study of Taylor–Aris (also known as shear) dispersion due to electroosmotic flow during electromigration in a capillary. An exact reduction to a Darboux equation is found under a traveling-wave ansatz. The equilibria of this ordinary differential equation are analyzed, showing that their stability is determined solely by the (dimensionless) wave speed without regard to any (dimensionless) physical parameters. Integral curves, connecting the appropriate equilibria of the Darboux equation that governs traveling waves, are constructed, which in turn are shown to be asymmetric kink solutions (i.e., non-Taylor shocks). Furthermore, it is shown that the governing Darboux equation exhibits bistability, which leads to two coexisting non-negative kink solutions for (dimensionless) wave speeds greater than unity. Finally, we give some remarks on other types of traveling-wave solutions and a discussion of some approximations of the governing partial differential equation of electromigration dispersion.     

On the bifurcations of the Lamé solutions in plane-strain elasticity

We consider the in-plane bifurcations experienced by the Lamé solutions corresponding to an elastic annulus subjected to radial tension on the curved boundaries. Numerical investigations of the relevant incremental problem reveal two main bifurcation modes: a long-wave local deformation around the central hole of the domain, or a material wrinkling-type instability along the same boundary. Strictly speaking, the latter scenario is related to the violation of the Shapiro–Lopatinskij condition in an appropriate traction boundary-value problem. It is further shown that the main features of this material instability mode can be found by using a singular-perturbation strategy.     

Dissipative acoustic solitons under a weakly-nonlinear, Lagrangian-averaged Euler-α model of single-phase lossless fluids

A one-dimensional weakly-nonlinear model equation based on a Lagrangian-averaged Euler-α model of compressible flow in lossless fluids is presented. Traveling wave solutions (TWS)s, in the form of a topological soliton (or kink), admitted by this fourth-order partial differential equation are derived and analyzed. An implicit finite-difference scheme with internal iterations is constructed in order to study soliton collisions. It is shown that, for certain parameters, the TWSs interact as solitons, i.e., they retain their “identity” after a collision. Kink-like solutions with an oscillatory tail are found to emerge in a signaling-type initial-boundary-value problem for the linearized equation of motion. Additionally, connections are drawn to related weakly-nonlinear acoustic models and the Korteweg-de Vries equation from shallow-water wave theory.     

Deflections of uniformly stressed reinforced plates with account for their weakened resistance to the transverse shear

The hypotheses of the Reissner theory are used to formulate the problem of uniformly stressed reinforcement (USR) of transversely bent elastic plates by fibers with a constant cross section. The analysis of the corresponding system of governing equations and boundary conditions is performed. The model problem of USR of rectangular elongated plates subjected to cylindrical bending with various types of loading of one of the longitudinal edges and rigid clamping of the other one. It is shown that there could exist two solutions of the USR problem, one of which is regular and the other one singular. The edge effects occurring in the presence of the torque applied to the edge, which significantly affects both the stress–strain state of the binder material and the reinforcement structure, are revealed.     

Continuum-mechanical modelling of kink-band formation in fibre-reinforced composites

A kink is a singular surface across which the displacement is continuous but the deformation gradient and the fibre direction suffer a discontinuity. A kink band is a highly deformed or even damaged region bounded by two kinks. The objective of modelling kink-band formation, within the framework of finite elasticity theory, is to find a suitable strain–energy function, guided by results from a finite number of simple experiments, that can be used to predict what have been observed and what might be possible under other loading conditions. In this paper, we explain a theoretical basis for choosing such strain–energy functions. More precisely, for a given strain–energy function that allows formation of kinks and a given deformation field, we characterize all possible deformation fields that can join the given deformation field through a kink and explain a procedure that can be used to assess the stability properties of any kink solution that is mathematically possible. In contrast with most previous studies in the engineering community where, for instance, the kink orientation angle is undetermined, the present theory completely determines the kink propagation stress, the kink orientation angle and the fibre direction within the kink band.     

An exact solution of crack problems in piezoelectric materials

IntroductionInrecentyearscrackproblemsinpiezoelectricmaterialhavereceivedmuchattention.Manytheoreticalanalyseshavebeengivenby[1～16].Itshouldbe,however,notedthatalltheaboveanalysesarebasedonaso-calledimpermeablecrackassumphon,i.e.thecrackfacesareassumedtobeimpermeabletoelectricfield,sotheelectricdisplacementvanishesinsidethecrack.Usingthisassumption,onewillobtainthefollowingresultS[2'3'5,6'9'16]=whentheelectricloadsaresolelyaPPliedatinLfinity,theelectricdisplacementissquare-rootsingularatthe…     

Kink-fold onset and development based on the maximum strength theorem

The structure of interest occupies initially a rectangular-shaped domain and is composed of a laminated material with weak interfaces defining the horizontal bedding. It is subjected to a compressive lateral force parallel to the bedding. The postulated fold kinematics relies on the existence of a kink band with two parallel sharp boundaries, or hinges, separating two rigid blocks. One block moves upward, the other sideways, by distances defining the fold amplitude and the horizontal shortening, respectively. Internal work is assumed to be done only along the hinges and the weak interfaces. The orientations of the hinges (hinge dip) and of the beds within the kink band (kink dip) are optimised for every fold amplitude to provide the least upper bound on the compressive force according to the maximum strength theorem.It is shown that the fold onset requires the introduction of a compaction mechanism for the hinges. The kink initiates as a sub-vertical compaction band of finite thickness equal to the bed thickness times the sine of the friction angle over the bedding. During the first phase of the fold development, the kink band rotates with the property that the sum of the kink dip and the hinge dip is always complementary to the friction angle over the bedding. Each bed along the hinges sees the activation of two deformation mechanisms: compaction and opening. The boundary between the regions over each bed where they are activated migrates from the bottom of the bed—pure compaction at the onset—towards a position at 90% of the bed thickness—development dominantly controlled by opening. The second phase of the fold development is marked by a thickening of the kink band with minor evolution of the dips. This two-phase development leads to a sharp decrease of the compressive load from the onset, a minimum as the two dips are approximately equal and then a moderate increase in applied load. In conclusion, it is noted that such combination of postulated fold kinematics and the application of the maximum strength theorem to optimise the structure could be generalised to folding in the presence of ramps, providing a useful tool to comprehend the mechanics of fold-and-thrust belts and of accretionary wedges.     

Effect of rotation on a fiber-reinforced thermo-elastic under Green-Naghdi theory and influence of gravity

A general model of the equation of Green-Naghdi theory (G-N), which is applied to study the influence of reinforcement on the total deformation for an infinite space weakened by a finite linear opening thermal shock, is solved. We study the influence of reinforcement on the total deformation of a rotating thermo-elastic half-space and the mutual interaction under the influence of gravity. The material is homogeneous, isotropic and elastic half-space. The methodology applied here consists of the use of a normal mode analysis to obtain the exact expressions for the temperature, the displacement components, and the stress components. Some particular cases are also discussed in this context. The deformation of a body depends on the nature of the force applied as well as the type of boundary conditions. The variations of the variables considered with the horizontal distance are illustrated graphically. Comparisons are made with the results predicted by the type II and type III cases in the presence and in the absence of the effect of fiber reinforcement. It is found that reinforcement, rotation, and gravity have great effect on the distribution of the field quantities.     

Construction of two-phase equilibria in a non-elliptic hyperelastic material

This work focuses on the construction of equilibrated two-phase antiplane shear deformations of a non-elliptic isotropic and incompressible hyperelastic material. It is shown that this material can sustain metastable, two-phase equilibria which are neither piecewise homogeneous nor axisymmetric, but, rather, involve non-planar interfaces which completely segregate inhomogeneously deformed material in distinct elliptic phases. These results are obtained by studying a constrained boundary value problem involving an interface across which the deformation gradient jumps. The boundary value problem is recast as an integral equation and conditions on the interface sufficient to guarantee the existence of a solution to this equation are obtained. The contraints, which enforce the segregation of material in the two elliptic phases, are then studied. Sufficient conditions for their satisfaction are also secured. These involve additional restrictions on the interface across which the deformation gradient jumps — which, with all restrictions satisfied, constitutes a phase boundary. An uncountably infinite number of such phase boundaries are shown to exist. It is demonstrated that, for each of these, there exists a solution — unique up to an additive constant — for the constrained boundary value problem. As an illustration, approximate solutions which correspond to a particular class of phase boundaries are then constructed. Finally, the kinetics and stability of an arbitrary element within this class of phase boundaries are analyzed in the context of a quasistatic motion.     

Bulk Cavitation and the Possibility of Localized Interface Deformation due to Surface Layer Swelling

We consider the uniform swelling of a compressible hyperelastic surface layer with finite thickness that is attached to an underlying bulk material composed of a non-swelling incompressible hyperelastic material. In addition to classically smooth solutions, two additional phenomena may occur for sufficiently large swelling. One is the formation of cavities in the interior of the underlying bulk material. The other is the disappearance of smooth solutions in the surface layer while the underlying bulk material remains intact. It is conjectured that the latter may be associated with the concentration of deformation at the swelling interface. Both phenomena are investigated by the consideration of solutions to a boundary value problem for a sphere involving radial deformation with a prescribed swelling field that acts as an effective loading device. Specific material models for both the compressible swollen surface layer and the non-swollen incompressible bulk are invoked so as to permit an analytical treatment. Swelling thresholds are obtained that depend on the thickness of the surface layer for the onset of these separate phenomena.     

Kink wave determined by parabola solution of a nonlinear ordinary differential equation

By finding a parabola solution connecting two equilibrium points of a planar dynamical system,the existence of the kink wave solution for 6 classes of nonlinear wave equations is shown.Some exact explicit parametric representations of kink wave solutions are given.Explicit parameter conditions to guarantee the existence of kink wave solutions are determined.     

聚丙烯纤维和水泥对粘性土强度的影响及机理研究

选取聚丙烯纤维和水泥作为加固材料,在室内试验的基础上,研究了单独的纤维和水泥对粘性土强度的影响及其的综合作用,分析了它们的作用机理。试验中,3种不同百分比(0.05%,0.15%和0.25%素土重)的纤维和两种不同百分比的(5%和8%素土重)水泥分别掺入到粘土试样中,配制了12组试样,进行了无侧限抗压试验。试验结果表明,纤维和水泥均能够提高土体的强度,纤维的加入改善了水泥土样的脆性破坏模式;纤维水泥土样的强度远远高于相同掺量下的纯纤维土和纯水泥的强度,甚至高于它们的强度之和。运用扫描电镜(SEM)从微观层次上分析了纤维和水泥加固粘性土的力学机理,发现纤维表面与土介质之间的粘结力和摩擦力对加固效果有直接影响。