Continuum Dynamics on Manifolds: Application to Elasticity of Residually-Stressed Bodies

This paper is concerned with the dynamics of continua on differentiable manifolds. We present a covariant derivation of the equations of motion, viewing motion as a curve in an infinite-dimensional Banach manifold of embeddings of a body manifold in a space manifold. Our main application is the motion of residually-stressed elastic bodies, where the residual stresses result from a geometric incompatibility between body and space manifolds. We then study a particular example of elastic vibrations of a two-dimensional curved annulus embedded in a sphere.     

The Matching Property of Infinitesimal Isometries on Elliptic Surfaces and Elasticity of Thin Shells

Using the notion of Γ-convergence, we discuss the limiting behavior of the three-dimensional nonlinear elastic energy for thin elliptic shells, as their thickness h converges to zero, under the assumption that the elastic energy of deformations scales like h ^β with 2 < β < 4. We establish that, for the given scaling regime, the limiting theory reduces to linear pure bending. Two major ingredients of the proofs are the density of smooth infinitesimal isometries in the space of W ^2,2 first order infinitesimal isometries, and a result on matching smooth infinitesimal isometries with exact isometric immersions on smooth elliptic surfaces.     

Interaction of a One-Dimensional Continuum with an Inertial Object Moving over the Continuum

Free transverse oscillations in a system consisting of an infinite moment continuum, such as the Euler-Bernoulli beam lying on the Winkler foundation, and a rigid body moving along the beam with a constant velocity and having a point contact with the guide are studied. The range of the considered velocities of the concentrated inertial object along the continuum is limited by the requirement of a finite energy of elastic deformation of the infinite continuum, corresponding to cojoint free osillations of an unbounded system. An analytical solution of the corresponding spectral problem in a system with a mixed spectrum is constructed. Limiting situations are analyzed, where the inertial rigid object moving along the beam is devoid of one “oscillatory” degree of freedom for some reasons. In particular, an inertial object devoid of mass but having a nonzero tensor of inertia is considered. Dependences of all characteristics of the discrete spectrum of oscillations and their shapes on the magnitude of object velocity along the moment elastoinertial guide are given.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 4, pp. 88–97, July– August, 2005.     

A Molecular-Statistical Basis for the Gent Constitutive Model of Rubber Elasticity

Molecular constitutive models for rubber based on non-Gaussian statistics generally involve the inverse Langevin function. Such models are widely used since they successfully capture the typical strain-hardening at large strains. Limiting chain extensibility constitutive models have also been developed on using phenomenological continuum mechanics approaches. One such model, the Gent model for incompressible isotropic hyperelastic materials, is particularly simple. The strain-energy density in the Gent model depends only on the first invariant I ₁ of the Cauchy–Green strain tensor, is a simple logarithmic function of I ₁ and involves just two material parameters, the shear modulus μ and a parameter J _m which measures a limiting value for I ₁−3 reflecting limiting chain extensibility. In this note, we show that the Gent phenomenological model is a very accurate approximation to a molecular based stretch averaged full-network model involving the inverse Langevin function. It is shown that the Gent model is closely related to that obtained by using a Padè approximant for this function. The constants μ and J _m in the Gent model are given in terms of microscopic properties. Since the Gent model is remarkably simple, and since analytic closed-form solutions to several benchmark boundary-value problems have been obtained recently on using this model, it is thus an attractive alternative to the comparatively complicated molecular models for incompressible rubber involving the inverse Langevin function. This revised version was published online in June 2006 with corrections to the Cover Date.     

Analytical Solutions for a 1D Scale Inhibitor Transport Model with Coupled Adsorption and Precipitation

Transport in Porous Media - In a previous publication (Sorbie and Stamatiou in Transp Porous Media 123:271–287, 2018), we presented a one-dimensional analytical solution for scale inhibitor...     

Vibrations of L-Shaped Beam Structures With a Crack: Analytical Approach and Experimental Validation

Experimental Techniques - A crack on a structural member introduces a local flexibility which is function of the crack depth. This new flexibility condition changes the dynamic behavior of the...     

A Rigidity Result for a Reduced Model of a Cubic-to-Orthorhombic Phase Transition in the Geometrically Linear Theory of Elasticity

In this article we study a simplified two-dimensional model for a cubic-to-orthorhombic phase transition occurring in certain shape-memory-alloys. In the low temperature regime the linear theory of elasticity predicts various possible patterns of martensite arrangements: Apart from the well known laminates this phase transition displays additional structures involving four martensitic variants—so called crossing twins.Introducing a variational model including surface energy, we show that these structures are rigid under small energy perturbations. Combined with an upper bound construction this gives the optimal scaling behavior of incompatible microstructures. These results are related to papers by Capella and Otto (Commun. Pure Appl. Math. 62(12):1632–1669, 2009; Proc. R. Soc. Edinb., Sect. A, Math. 142:273–327, 2012) as well as to a paper by Dolzmann and Müller (Meccanica 30:527–539, 1995).     

On a Nonlinear Model for Tumor Growth in a Cellular Medium

We investigate the dynamics of a nonlinear model for tumor growth within a cellular medium. In this setting the “tumor” is viewed as a multiphase flow consisting of cancerous cells in either proliferating phase or quiescent phase and a collection of cells accounting for the “waste” and/or dead cells in the presence of a nutrient. Here, the tumor is thought of as a growing continuum \(\Omega \) with boundary \(\partial \Omega \) both of which evolve in time. In particular, the evolution of the boundary \(\partial \Omega \) is prescibed by a given velocity \({{{\varvec{V}}}.}\) The key characteristic of the present model is that the total density of cancerous cells is allowed to vary, which is often the case within cellular media. We refer the reader to the articles (Enault in Mathematical study of models of tumor growth, 2010; Li and Lowengrub in J Theor Biol, 343:79–91, 2014) where compressible type tumor growth models are investigated. Global-in-time weak solutions are obtained using an approach based on penalization of the boundary behavior, diffusion, viscosity and pressure in the weak formulation, as well as convergence and compactness arguments in the spirit of Lions (Mathematical topics in fluid dynamics. Compressible models, 1998) [see also Donatelli and Trivisa (J Math Fluid Mech 16: 787–803, 2004), Feireisl (Dynamics of viscous compressible fluids, 2014)].     

Construction and Analysis of an Analytical Solution for the Surface Rayleigh Wave within the Framework of the Cosserat Continuum

The problem of propagation of an acoustic surface Rayleigh wave in an infinite half-space is considered within the framework of the asymmetric theory of elasticity (Cosserat medium). It is assumed that material deformation is described not only by the displacement vector but also by an independent rotation vector. A global analytical solution of the problem in displacements is obtained. A comparative analysis of the solution obtained and the corresponding solution for the classical elastic medium is performed. Macroparameters characterizing the difference of the stress-strain state from that predicted by the classical theory of elasticity are introduced.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 4, pp. 116–124, July– August, 2005.     

Nonlinear Vibrations of a Viscoelastic Rod with Quadratic Elasticity

A study is made of the effect of the constant component and the second harmonic on the vibrations of a Voigt rod with quadratic elasticity and linear viscosity in the case of monoharmonic excitation at the frequency of the principal linear resonance. A numerical algorithm is proposed to solve one-dimensional problems on the vibrations of viscoelastic bodies taking into account higher harmonics.     

Continuum modeling of a porous solid with pressure-sensitive dilatant matrix

The pressure-sensitive plastic response of a material has been studied in terms of the intrinsic sensitivity of its yield stress to pressure and the presence and growth of cavities. This work focuses on the interplay between these two distinctly different mechanisms and the attendant material behavior. To this end, a constitutive model is proposed taking both mechanisms into account. Using Gurson's homogenization, an upper bound model is developed for a voided solid with a plastically dilatant matrix material. This model is built around a three-parameter axisymmetric velocity field for a unit sphere containing a spherical void. The void is also subjected to internal pressure; this can be relevant for polymeric adhesives permeated by moisture that vaporizes at elevated temperatures. The plastic response of the matrix material is described by Drucker–Prager's yield criterion and an associated flow rule. The resulting yield surface and porosity evolution law of the homogenized constitutive model are presented in parametric form. Using the solutions to special cases as building blocks, approximate models with explicit forms are proposed. The parametric form and an approximate explicit form are compared against full-field solutions obtained from finite element analysis. They are also studied for loading under generalized tension conditions. These computational simulations shed light on the interplay between the two mechanisms and its enhanced effect on yield strength and plastic flow. Among other things, the tensile yield strength of the porous solid is greatly reduced by the internal void pressure, particularly when a liquid/vapor phase is the source of the internal pressure.     

Analytical solutions for long waves on a circular island with combined topographies

We derived analytical solutions for long waves on a circular island with combined topographies. The inner region is a circular island on a flat bottom and the outer region is a cylindrical island on a shoal. The solution for the circular island on a flat bottom was developed previously by Jung et al. (2010) [13]. The solution for the cylindrical island on a shoal was developed in the present study using the methods of separation of variables, Taylor series expansion, and Frobenius series. The present solution for the cylindrical island on a shoal was verified by comparing with previous solutions which was developed for limited cases. Also, some cases were investigated with different values of wave period, shoal vertex level, and power of the radius. The solutions for two different types of real islands were obtained using the solution for the combined topographies. The present solution for the combined topographies can be applied more accurately to the real island than the previous solution for the single bottom topography.     

Spectral Analysis of the Slip-Length Model for Turbulence over Textured Superhydrophobic Surfaces

We assess the applicability of slip-length models to represent textured superhydrophobic surfaces. From the results of direct numerical simulations, and by considering the slip length from a spectral perspective, we discriminate between the apparent boundary conditions experienced by different lengthscales in the overlying turbulent flow. In particular, we focus on the slip lengths experienced by lengthscales relevant to the near wall turbulent dynamics. Our results indicate that the apparent failure of homogeneous slip-length models is not the direct effect of the texture size becoming comparable to the size of eddies in the flow. The texture-induced signal scatters to the entire wavenumber space, affecting the perceived slip length across all lengthscales, even those much larger than the texture. We propose that the failure is caused by the intensity of the texture-induced flow, rather than its wavelength, becoming comparable to the background turbulence.     

On a Model of Nonlocal Continuum Mechanics Part II: Structure, Asymptotics, and Computations

We consider the asymptotic behavior and local structure of solutions to the nonlocal variational problem developed in the companion article to this work, On a Model of Nonlocal Continuum Mechanics Part I: Existence and Regularity. After a brief review of the basic setup and results of Part I, we conduct a thorough analysis of the phase plane related to an integro-differential Euler--Lagrange equation and classify all admissible structures that arise as energy minimizing strain states. We find that for highly elastic materials with relatively weak particle-particle interactions, the maximum number of internal phase boundaries is two. Moreover, we also develop explicit bounds for the number of internal phase boundaries supported by any material and show that this bound is essentially inversely related to the particle size. To understand the question of asymptotics, we utilize the Young measure and show that in the sense of energetics and averages, minimizers of the full nonlocal problem converge to minimizers of two limiting problems corresponding to both the large and small particle limits. In fact, in the small particle limit, we find that the minimizing fields converge, up to a subsequence in strong-L^p, for 1 ≤ p < ∞, to fields that support either a single internal phase boundary, or two internal phase boundaries that are distributed symmetrically about the body midpoint. We close this work with some computations that illustrate these asymptotic limits and provide insight into the notion of nonlocal metastability. This revised version was published online in August 2006 with corrections to the Cover Date.     

钢筋混凝土框架杆件弹塑性动力分析模型的改进

对Roufaiel杆件弹塑性动力分析模型进行修改,克服了原模型在杆件弹塑性区域分界面处刚度产生突变的缺陷,使弹塑性动力分析模型在理论上进一步完善。     

Development and Validation of a New Model for In Situ Foam Generation Using Foamer Droplets Injection

Transport in Porous Media - Foam generation and transport in porous media are a proven method to improve the sweep efficiency of a flooding fluid in enhanced oil recovery process and increase the...     

Analytical solution of attitude motion for spacecraft with a slewing appendage

This paper investigates pitch motion and in orbital plane elastic vibration of a spacecraft with a flexible beam type appendage undergoing prescribed slew maneuver. The governing equations are transformed into a standard quasi-linear form, and then solved by Butenin's variation of parameters approach. Validity of the analytical solutions is assessed over a range of system parameters and initial conditions by comparing them with the results of numerical integration. The results show that they are very good approximations and provide extensive insight into the dynamical response of the system.