One-dimensional elasticity

To make the treatment given above more than just an abstraction, we now consider the case of one-dimensional elasticity. We imagine our one-dimensional elastic medium to have length a and to be characterized by an elastic stiffness k. The traction vector, t x), is directly related to the gradients in displacements by t = kdu/dx. Our weak statement of the equilibrium equations may be written as... 

First, we consider a simple one-dimensional elastic problem. 

The system for the one-dimensional elastic problem can be given by a governing equation (4.1a), the Dirichlet (i.e., displacement) boundary condition (4.1b) and the Neumann (i.e., traction) boundary condition (4.1c). This system is referred to as the strong form [SF]. 

We outline the essential features of a multiscale homogenization analysis. A problem of a one-dimensional elastic bar is given as an example. 

Charles L. Mader, One-Dimensional Elastic-Plastic Calculations for Aluminum , Los Alamos Scientific Laboratory report LA-3678 (1967). 

The smectic A phase is a liquid in two dimensions, i.e. in tire layer planes, but behaves elastically as a solid in the remaining direction. However, tme long-range order in tliis one-dimensional solid is suppressed by logaritlimic growth of tliennal layer fluctuations, an effect known as tire Landau-Peierls instability [H, 12 and 13]... 

As for crystals, tire elasticity of smectic and columnar phases is analysed in tenns of displacements of tire lattice witli respect to the undistorted state, described by tire field u(r). This represents tire distortion of tire layers in a smectic phase and, tluis, u(r) is a one-dimensional vector (conventionally defined along z), whereas tire columnar phase is two dimensional, so tliat u(r) is also. The symmetry of a smectic A phase leads to an elastic free energy density of tire fonn [86]... 

Substituting (1.22), (1.23) into (1.21), one can see that the differential equations (1.21) of second order with respect to U have the same structure as those of the three-dimensional elasticity equations (1.1)- (1.3). The system (1.24)-(1.25) contains the fourth derivatives of w. 

In the case of most nonporous minerals at sufficiently low-shock stresses, two shock fronts form. The first wave is the elastic shock, a finite-amplitude essentially elastic wave as indicated in Fig. 4.11. The amplitude of this shock is often called the Hugoniot elastic limit Phel- This would correspond to state 1 of Fig. 4.10(a). The Hugoniot elastic limit is defined as the maximum stress sustainable by a solid in one-dimensional shock compression without irreversible deformation taking place at the shock front. The particle velocity associated with a Hugoniot elastic limit shock is often measured by observing the free-surface velocity profile as, for example, in Fig. 4.16. In the case of a polycrystalline and/or isotropic material at shock stresses at or below HEL> the lateral compressive stress in a plane perpendicular to the shock front... 

A strength value associated with a Hugoniot elastic limit can be compared to quasi-static strengths or dynamic strengths observed values at various loading strain rates by the relation of the longitudinal stress component under the shock compression uniaxial strain tensor to the one-dimensional stress tensor. As shown in Sec. 2.3, the longitudinal components of a stress measured in the uniaxial strain condition of shock compression can be expressed in terms of a combination of an isotropic (hydrostatic) component of pressure and its deviatoric or shear stress component. 

Classical lamination theory consists of a coiiection of mechanics-of-materials type of stress and deformation hypotheses that are described in this section. By use of this theory, we can consistentiy proceed directiy from the basic building block, the lamina, to the end result, a structural laminate. The whole process is one of finding effective and reasonably accurate simplifying assumptions that enable us to reduce our attention from a complicated three-dimensional elasticity problem to a SQlvable two-dimensinnal merbanics of deformable bodies problem. 

To represent the elasticity and dispersion forces of the surface, an approach similar to that of Eqs. (3) and (4) can be taken. The waU molecules can be assumed to be smeared out. And after performing the necessary integration over the surface and over layers of molecules within the surface, a 10-4 or 9-3 version of the potential can be obtained [54,55], Discrete representation of a hexagonal lattice of wall molecules is also possible by the Steele potential [56], The potential is essentially one dimensional, depending on the distance from the wall, but with periodic variations according to lateral displacement from the lattice molecules. Such a representation, however, has not been developed in the cylindrical pore... 

The above problem has been addressed in (Li et al, 2003), where we have considered a quasi-one dimensional billiard model which consists of two parallel lines and a series of triangular scatterers (see Fig.3). In this geometry, no particle can move between the two reservoirs without suffering elastic collisions with the triangles. Therefore this model is... 

This model consists of a one-dimensional chain of elastically colliding particles with alternate masses m and M. In order to prevent total momentum conservation we confine the motion of particles of mass M (bars) inside separate cells. Schematically the model is shown in Fig.4 particles with mass m move horizontally and collide with bars of mass M which, besides suffering collisions with the particles, are elastically reflected back at the edges of their cells. In between collisions, particles and bars move freely. 

It is also possible that a membrane might have an even lower symmetry than a chiral smectic-C liquid crystal in particular, it might lose the twofold rotational symmetry. This would occur if the molecular tilt defines one orientation in the membrane plane and the direction of one-dimensional chains defines another orientation. In that case, the free energy would take a form similar to Eq. (5) but with additional elastic constants favoring curvature. The argument for tubule formation presented above would still apply, but it would become more mathematically complex because of the extra elastic constants. As an approximation, we can suppose that there is one principal direction of elastic anisotropy, with some slight perturbations about the ideal twofold symmetry. In that approximation, we can use the results presented above, with 4) representing the orientation of the principal elastic anisotropy. 

Fig. 14. One-dimensional cross section of an elliptic weighting filter. The characteristic length is defined as the section length when the relative weight has dropped to 2/a. The filter shape corresponds to the deformation profile of an elastic material under distributed load in a circle of radius Z./2.

In this overview we focus on the elastodynamical aspects of the transformation and intentionally exclude phase changes controlled by diffusion of heat or constituent. To emphasize ideas we use a one dimensional model which reduces to a nonlinear wave equation. Following Ericksen (1975) and James (1980), we interpret the behavior of transforming material as associated with the nonconvexity of elastic energy and demonstrate that a simplest initial value problem for the wave equation with a non-monotone stress-strain relation exhibits massive failure of uniqueness associated with the phenomena of nucleation and growth. 

Kunin, I.A., 1982, Elastic Media With Microstrucrure 1 (One Dimensional Model), Springer. 

DE - A Two-Dimensional Eulerian Hydro-dynamic Code for Computing One-Component Reactive Hydrodynamic Problems , Los Alamos Scientific Laboratory Report LA-3629-MS (1966) 23b) C.L. Mader, "FORTRAN-SIN - A One-Dimensional Hydrodynamic Code for Problems Which Include Chemical Reactions, Elastic-Plastic Flow, Spalling, and Phase Transitions , Los Alamos Scientific Laboratory Report LA-3720(1967) 24) R.C. Sprowls, "Com-... 

The trajectory of an ion moving in such a potential presents a sequence of rectilinear sections placed between the points of elastic reflections of an ion from the walls of the well. We consider two variants of such a model related to one-dimensional and spatial motion of ion, depicted, respectively, in Figs. 47a and 47b. In the first variant the ion s motion during its lifetime59 presents periodic oscillations on the rectilinear section 2 lc between two reflection points. In the second variant we consider a spherically symmetric potential well, to which a spherical hollow cavity corresponds with the radius lc. 

The interstitial fluid content of the skin is higher than in the subcutaneous fat layer and normal fluid movement is intrinsically finked to lymphatic drainage as governed by mechanical stresses of the tissue. A model of temporal profiles of pressure, stress, and convective ISF velocity has been developed based on hydraulic conductivity, overall fluid drainage (lymphatic function and capillary absorption), and elasticity of the tissue.34 Measurements on excised tissue and in vivo measurement on the one-dimensional rat tail have defined bulk average values for key parameters of the model and the hydration dependence of the hydraulic flow conductivity. Numerous in vivo characterization studies with nanoparticles and vaccines are currently underway, so a more detailed understanding of the interstitial/lymphatic system will likely be forthcoming. 

---