Solid deformation equations

A solid, by definition, is a portion of matter that is rigid and resists stress. Although the surface of a solid must, in principle, be characterized by surface free energy, it is evident that the usual methods of capillarity are not very useful since they depend on measurements of equilibrium surface properties given by Laplace s equation (Eq. II-7). Since a solid deforms in an elastic manner, its shape will be determined more by its past history than by surface tension forces. 

Shock-compressed solids and shock-compression processes have been described in this book from a perspective of solid state physics and solid state chemistry. This viewpoint has been developed independently from the traditional emphasis on mechanical deformation as determined from measurements of shock and particle velocities, or from time-resolved wave profiles. The physical and chemical studies show that the mechanical descriptions provide an overly restrictive basis for identifying and quantifying shock processes in solids. These equations of state or strength investigations are certainly necessary to the description of shock-compressed matter, and are of great value, but they are not sufficient to develop a fundamental understanding of the processes. 

Note 3 For a Voigt-Kelvin solid, the equation describing the deformation becomes HJ/L )ay + (HJ/L )f dy/dt) =f cos cot with solution... 

Let s look at this in a little more detail, to make sure that you understand what we mean by strain rate in a shear experiment First, let s go back to Newton s good friend, Hooke. (If you ve read the introduction to this chapter, you know we re being facetious ) We have seen above (Figure 13-11) that for shear the most convenient way to describe the deformation of a solid is in terms of the angle 6 through which a block of the material is deformed (Equation 13-61) ... 

The SIP methods by Stone (1986) for the gas leak flow equations and the rock mass deformation equations for double parallel coal seams have been developed by Sun (1998,2002) with Microsoft Visual C+-I- 6.0 under Windows2000 on a Pentium rv rc. The program is suitable for isotropic heterogeneous coal seams with irregular shapes as well as anisotropic heterogeneous coal seams. The systems with the first or the second boundary conditions as well as the hybrid boundary conditions can all be solved. The overview of the numerical implementations of the SIP methods for the solid-gas coupled mathematical model for double parallel coal seams is described as follows. 

Supposing that the solid matrix has an elasto-plastic behavior, the theory of plasticity can then be used to describe the matrix deformation in oil reservoir. Equations of the solid deformation include the following three groups equilibrium equations, geometry equations and constitutive equations. 

The numerical solution method for the above fluid-solid coupling model is an iterative computation process. To reduce the computational complexity, the solid deformation and fluid flow are regarded as two coupled equation systems, solved by FEM. The equilibrium in solid matrix is solved using Eq.(6) with an added coupling item apS j and the pore pressure is treated as an equivalent initial stress term. The flow equation (5) is solved with an added term of volume strain, reflecting the effect of solid deformation on fluid flow. It can be treated as a source or converge. In each iterative loop, the solid matrix deformation is solved firstly. The stress and strain results are then taken as inputs for the flow calculation with modified hydraulic parameters. After flow model is solved, the pore pressure values are transferred into solid matrix deformation model and begins next iterative loop. In this way, the flow and deformation of oil reservoir can be simulated. 

The Navier-Stokes equations are the differential momentum balances for a three-dimensional flow, subject to the assumptions that the flow is laminar and of a constant-density newtonian fluid and that the stress deformation behavior of such a fluid is analogous to the stress deformation behavior of a perfectly elastic isotropic solid. These equations are useful in setting up momentum balances for three-dimensional flows, particularly in cylindrical or spherical geometries. 

Viscoeiasticity. As already noted, the time-dependent properties of polymer-based materials are due to the phenomenon of viscoelasticity (qv), a combination of solid-like elastic behavior with liquid-like flow behavior. During deformation, equations 3 and 6 above applied to an isotropic, perfectly elastic solid. The work done on such a solid is stored as the energy of deformation that energy is released completely when the stresses are removed and the original shape is restored. A metal spring approximates this behavior. 

The simplest is that proposed by Wall and developed by him and Flory using only the distribution function equation (83). Despite Brownian motion. Wall supposed that the crosslinks did not really move at all, and thus the problem reduces to that of a single chain between two fixed ends, and when the solid deforms all chains deform affinely, i.e. all microscopic dimensions deform the same way as the macroscopic dimensions. Under these circumstances the number of configurations Q is given by A Cl R)... 

The viscosity given by Eq. (3.98) not only follows from a different model than the Debye viscosity equation, but it also describes a totally different experimental situation. Viscoelastic studies are done on solid samples for which flow is not measurable. A viscous deformation is present, however, and this result shows that it is equivalent to what would be measured directly, if such a measurement were possible. 

Let a solid body occupy a domain fl c with the smooth boundary L. The deformation of the solid inside fl is described by equilibrium, constitutive and geometrical equations discussed in Sections 1.1.1-1.1.5. To formulate the boundary value problem we need boundary conditions at T. The principal types of boundary conditions are considered in this subsection. 

In plate theory, the problem is reduced from the deformation of a solid body to the deformation of a surface by use of the Kirchhoff hypothesis (normals to the undeformed middle surface remain straight and normal after deformation, etc., as discussed in Chapter 4). Then, we attempt to apply boundary conditions to that surface which is usually the middle surface of the plate. There should be no surprise that the boundary conditions for the unapproximated solid body are not the same as those for the solid approximated with a surface. The problem arises when these boundary conditions are applied to an approximate set of equilibrium equations that result when force-strain and moment-curvature... 

There is greater similarity in the behavior of stretched melts and solid samples prepared by, e.g. pressure molding, probably, for the reason of parallelism in structure formation and destruction caused by deformation in melts and the amorphous regions of solid matrices. It is also possible to use identical equations for longitudinal viscosity and strength which present them as functions of the filler concentration [34]. 

Because of the difficulty in explaining the observed U-series excesses by time-independent models, interpretations of how disequilibria are created have evolved into models based on residence times. In these models, a melt phase coexists with the solid mantle but moves relative to it due to a driving force, most typically buoyancy. The physical situation under ridges can be referred to as two-phase flow because both the solid and the liquid flow. McKenzie (1984) and Scott and Stevenson (1984, 1986) derived the equations describing flow in a viscously deforming porous media. McKenzie... 

Here we describe the strain history with the Finger strain tensor C 1(t t ) as proposed by Lodge [55] in his rubber-like liquid theory. This equation was found to describe the stress in deforming polymer melts as long as the strains are small (second strain invariant below about 3 [56] ). The permanent contribution GcC 1 (r t0) has to be added for a linear viscoelastic solid only. C 1(t t0) is the strain between the stress free state t0 and the instantaneous state t. Other strain measures or a combination of strain tensors, as discussed in detail by Larson [57], might also be appropriate and will be considered in future studies. A combination of Finger C 1(t t ) and Cauchy C(t /. ) strain tensors is known to express the finite second normal stress difference in shear, for instance. 

Note 1 In the equation for g, the term go is usually equal to zero because the undistorted state of nematics is the state of uniform alignment. However, for chiral nematics, a nonzero value of go allows for the intrinsic twist in the structure. In order to describe g for smectic phases, an additional term must be added, due to the partially solid-like character of the smectic state and arising from positional molecular deformations. 

Note 5 For small deformations of an incompressible, inelastic solid, the constitutive equation may be written... 

Equation relating stress and deformation in an incompressible viscoelastic liquid or solid. Note 1 A possible general form of constitutive equation when there is no dependence of stress on amount of strain is... 

Note 5 For a Voigt-Kelvin solid (cf note 2), the equation in note 4 describing the deformation becomes... 

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