Deformation elastic media

Assuming all deformations to be small, we can employ Hooke s law (13.16) and the deformation tensor expressed through the displacement vector (13.8). Substitution of (13.16) into (13.21) yields the following form for the equation of motion of a homogeneous isotropic elastic medium ... 

It is quite clear from definition (13.142) of function T that it depends only on the deformation tensor components at any given time and at any given point in the elastic body. This means that function W obtained by the integration of T over the entire volume V, occupied by the elastic medium, depends only on its deformations at a given moment of time. 

There have been many efforts for combining the atomistic and continuum levels, as mentioned in Sect. 1. Recently, Santos et al. [11] proposed an atomistic-continuum model. In this model, the three-dimensional system is composed of a matrix, described as a continuum and an inclusion, embedded in the continuum, where the inclusion is described by an atomistic model. The model is validated for homogeneous materials (an fee argon crystal and an amorphous polymer). Yang et al. [96] have applied the atomistic-continuum model to the plastic deformation of Bisphenol-A polycarbonate where an inclusion deforms plastically in an elastic medium under uniaxial extension and pure shear. Here the atomistic-continuum model is validated for a heterogeneous material and elastic constant of semi crystalline poly( trimethylene terephthalate) (PTT) is predicted. 

Persson and co-workers [265 267] consider a rough, rigid surface with a height prohle h x ). where x is a two-dimensional vector in the x-y plane. In reaction to /z(x) and its externally imposed motion, the rubber will experience a (time-dependent) normal deformation 8z(x, f). If one assumes the rubber to be an elastic medium, then it is possible to relate 52(q, ), which is the Fourier transform (F.T.) of 8z(x, f), to the F.T. of the stress a(q, ). Within linear-response theory, one can express this in the rubber-hxed frame (indicated by a prime) via... 

What this equation tells us is that a particular state of stress is nothing more than a linear combination (albeit perhaps a tedious one) of the entirety of components of the strain tensor. The tensor Cijn is known as the elastic modulus tensor or stiffness and for a linear elastic material provides nearly a complete description of the material properties related to deformation under mechanical loads. Eqn (2.52) is our first example of a constitutive equation and, as claimed earlier, provides an explicit statement of material response that allows for the emergence of material specificity in the equations of continuum dynamics as embodied in eqn (2.32). In particular, if we substitute the constitutive statement of eqn (2.52) into eqn (2.32) for the equilibrium case in which there are no accelerations, the resulting equilibrium equations for a linear elastic medium are given by... 

Butyl rubber like Hypalon, Neoprene or nitrile rubber is a speciality polymer which can be compounded for a soft, deformable elastic vulcanisate similar to the other elastomers, but having certain distinctive characteristics, like low permeability to all gases and resistance to ageing and ozone cracking. Butyl has poor oil resistance and medium low temperature flexibility. 

Under very rapid mechanical actions or in observations with characteristic time t < to, the substance behaves as an ideal elastic medium. For t to the developing flow becomes stronger than the elastic deformation, and the substance can be treated as a simple Newtonian fluid. It is only if t is of the same order of magnitude as to that the elastic and viscous effects act simultaneously, and the complex nature of the deformation displays itself. 

Zhao (1994) presented a model of coupled coal deformation and methane migration based on a consolidation theory of elastic medium with Darcy fluid flow and the Terzaghi effective stress law, and its numerical solution technique and applications to practical problems. Works using similar approaches were also reported in Liang et al. (1995,1996), Sun and Xian (1999), Ding et al. 

Kj is a measure of the stress-field Intensity near the tip of an ideal crack in a linear elastic medium deformed such that the crack faces are displaced apart, normal to the crack plane (that means Mode 1). Kj is directly proportional to the applied load and depends on the ratio of the specimen dimensions. ... 

The mechanical component includes the transformation of all deformation, elastic, and plastic energies in thermal energy chemical solicitation is manifested by mechanical and thermal activation of chemical reactions, at the contact surface, with the external or internal medium. Therefore, the friction processes through the... 

Here, the atomistic-continuum model is applied to the study of plastic deformation of BPA-PC. First, the elastic constants of BPA-PC are calculated by atomistic simulation. These values are used as the elastic constants for the matrix throughout the simulated deformation. Then, the system, i.e., the continuum matrix with its atomistic inclusion, is deformed stepwise up to a strain of about 0.2. The overall system is constrained to exactly follow a predescribed deformation sequence, but the atomistic inclusion is free to follow any strain path consistent with the misfit stresses acting between it and the matrix. The result is a new look at the behavior of a glassy polymeric inclusion deformed plastically in a surrounding elastic medium. 

The elastic constants of BPA-PC, calculated by atomistic modeling, show relatively good agreement with experimental values. The atomistic-continuum model is successfully applied to the large deformation behavior of a BPA-PC system where an inclusion described in an atomistically detailed manner deforms plastically in an elastic medium. In the atomistic box, the stress-strain curves show typical microstructural plastic behavior with sharp drops in the stress. Because the matrix is characterized by elastic constants of BPA-PC calculated by atomistic modeling, the strain of the atomistic box is equal to the system strain until a certain point. At large deformations, the strain in the atomistic box becomes higher than the system strain. 

We expect that a rod-like object embedded in an elastic medium will rotate as the matrix is deformed. Anisotropic objects within an elastic medium which retain their integrity on deformation will follow the predictions of the pseudo-affine model with respect to the orientation of the objects as shown in Fig. 3.5. 

Fig. 3.5 A plot of the orientation parameter (F2 c against extension ratio X calculated on the basis of the pseudo-affine model for non-deformable rods embedded in an elastic medium...

This formula was given by Hunter (1961) for the special case of plane strain conditions and a cylindrical punch. For an elastic medium, this force is zero. However, for a viscoelastic medium, we shall see that this is not the case. The deformation caused by the moving load results in mechanical energy loss, which is manifested by the presence of a resisting force. This is the well-known force of hysteretic friction, first demonstrated experimentally by Tabor (1952). 

Figure 4.11. Diagrammatic sketches of atomic lattice rearrangements as a result of dynamic compression, which give rise to (a) elastic shock, (b) deformational shock, and (c) shock-induced phase change. In the case of an elastic shock in an isotropic medium, the lateral stress is a factor v/(l — v) less than the stress in the shock propagation direction. Here v is Poisson s ratio. In cases (b) and (c) stresses are assumed equal in all directions if the shock stress amplitude is much greater than the material strength.

The existence of yield stress Y at shear strains seems to be the most typical feature of rheological properties of highly filled polymers. A formal meaing of this term is quite obvious. It means that at stresses lower than Y the material behaves like a solid, i.e. it deforms only elastically, while at stresses higher than Y, like a liquid, i.e. it can flow. At a first approximation it may be assumed that the material is not deformed at all, if stresses are lower than Y. In this sense, filled polymers behave as visco-plastic media with a low-molecular and low-viscosity dispersion medium. This analogy is not random as will be stressed below when the values of the yield stress are compared for the systems with different dispersion media. The existence of yield stress in its physical meaning must be correlated with the strength of a structure formed by the interaction between the particles of a filler. 

In particular it can be shown that the dynamic flocculation model of stress softening and hysteresis fulfils a plausibility criterion, important, e.g., for finite element (FE) apphcations. Accordingly, any deformation mode can be predicted based solely on uniaxial stress-strain measurements, which can be carried out relatively easily. From the simulations of stress-strain cycles at medium and large strain it can be concluded that the model of cluster breakdown and reaggregation for prestrained samples represents a fundamental micromechanical basis for the description of nonlinear viscoelasticity of filler-reinforced rubbers. Thereby, the mechanisms of energy storage and dissipation are traced back to the elastic response of tender but fragile filler clusters [24]. 

In an elastic material medium a deformation (strain) caused by an external stress induces reactive forces that tend to recall the system to its initial state. When the medium is perturbed at a given time and place the perturbation propagates at a constant speed (or celerity) c that is characteristic of the medium. This propagating strain is called an elastic (or acoustic or mechanical) wave and corresponds to energy transport without matter transport. Under a periodic stress the particles of matter undergo a periodic motion around their equilibrium position and may be considered as harmonic oscillators. 

Propagation of small deformations in a linear elastic isotropic homogeneous medium... 

Model simulating the hydrodynamic properties of a chain macromolecule consisting of a sequence of beads, each of which offers hydrodynamic resistance to the flow of the surrounding medium and is connected to the next bead by a spring which does not contribute to the frictional interaction but which is responsible for the elastic and deformational properties of the chain. The mutual orientation of the springs is random. 

As the rubber hardness is a measurement of almost completely elastic deformation, it can be related to elastic modulus. Most rubber hardness tests measure the depth of penetration of an inden-tor under either a fixed weight or a spring load, and when rubber is assumed to be an elastic isotropic medium, the indentation obtained at small deformation depends on the elastic modulus, the... 

In this chapter, two simple cases of stereomechanical collision of spheres are analyzed. The fundamentals of contact mechanics of solids are introduced to illustrate the interrelationship between the collisional forces and deformations of solids. Specifically, the general theories of stresses and strains inside a solid medium under the application of an external force are described. The intrinsic relations between the contact force and the corresponding elastic deformations of contacting bodies are discussed. In this connection, it is assumed that the deformations are processed at an infinitely small impact velocity and for an infinitely long period of contact. The normal impact of elastic bodies is modeled by the Hertzian theory [Hertz, 1881], and the oblique impact is delineated by Mindlin s theory [Mindlin, 1949]. In order to link the contact theories to collisional mechanics, it is assumed that the process of a dynamic impact of two solids can be regarded as quasi-static. This quasi-static approach is valid when the impact velocity is small compared to the speed of the elastic... 

The dilational rheology behavior of polymer monolayers is a very interesting aspect. If a polymer film is viewed as a macroscopy continuum medium, several types of motion are possible [96], As it has been explained by Monroy et al. [59], it is possible to distinguish two main types capillary (or out of plane) and dilational (or in plane) [59,60,97], The first one is a shear deformation, while for the second one there are both a compression - dilatation motion and a shear motion. Since dissipative effects do exist within the film, each of the motions consists of elastic and viscous components. The elastic constant for the capillary motion is the surface tension y, while for the second it is the dilatation elasticity e. The latter modulus depends upon the stress applied to the monolayer. For a uniaxial stress (as it is the case for capillary waves or for compression in a single barrier Langmuir trough) the dilatational modulus is the sum of the compression and shear moduli [98]... 

Rejection of protein adsorption to the outermost grafted surface is attributed to a steric hinderance due to the tethered chains. A grafted surface in contact with an aqueous medium, a good solvent of the chains, has been identified to have a diffuse structure [67]. Reversible deformation of tethered chains due to invasion of mobile protein molecules into the layer would lead to a repulsive force which is governed by the balance of entropic elasticity of the chains and osmotic pressure owing to the rise in the segment concentration. The overlapped repulsive force would prevent the direct contact of protein molecules with the substrate surface. 

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