Finite deformation

Deformation and Stress A fluid is a substance which undergoes continuous deformation when subjected to a shear stress. Figure 6-1 illustrates this concept. A fluid is bounded by two large paraU plates, of area A, separated by a small distance H. The bottom plate is held fixed. Application of a force F to the upper plate causes it to move at a velocity U. The fluid continues to deform as long as the force is applied, unlike a sohd, which would undergo only a finite deformation. 

Carroll [4] developed their theories in the context of finite deformations using these measures. 

The referential constitutive equations for an inelastic material may be set into spatial terms. Casey and Naghdi [14] did so for their special case of finite deformation rigid plasticity discussed by Casey [15], Using the spatial (Almansi) strain tensor e and the relationships of the Appendix, it is possible to do so for the full inelastic referential constitutive equations of Section 5.4.2. 

Casey, J. and Naghdi, P.M., Constitutive Results for Finitely Deforming Elastic-Plastic Materials, in Constitutive Equations Macro and Computational Aspects (edited by K.J. Wiliam), ASME, 1984, pp. 53-71. 

Naghdi, P.M. and Trapp, J.A., Restrictions on Constitutive Equations of Finitely Deformed Elastic-Plastic Materials, Quart. J. Mech. Appl. Math. 28, Part 1,25-46(1975). 

Casey, J., On Finitely Deforming Rigid-Plastic Materials, Internat. J. Plasticity 2,247-277 (1986). 

The stored strain energy can also be determined for the general case of multiaxial stresses [1] and lattices of varying crystal structure and anisotropy. The latter could be important at interfaces where mode mixing can occur, or for fracture of rubber, where f/ is a function of the three stretch rations 1], A2 and A3, for example, via the Mooney-Rivlin equation, or suitable finite deformation strain energy functional. 

Murnaghan, F. D. "Finite Deformation of an Elastic Solid" Dover Publications, Inc. New York, 1967. 

As an indenter creates an indentation it causes at least three types of finite deformation. It punches material downwards creating approximately circular prismatic dislocation loops. At the surface of the material it pushes material sideways. It causes shear on the planes of maximum shear stress under itself. Therefore, the overall pattern of deformation is very complex, and is reflected... 

The effective potential governing torsional deformations could conceivably be quite anharmonic, so that overwinding is much more strongly resisted than underwinding for finite deformations. This question is addressed by examining the dependence of the torsion constant on temperature(40) and on superhelix density. 

Note 2 The Xi are effectively deformation gradients, or, for finite deformations, the deformation ratios characterizing the deformation. 

B. Stress-Strain Relation Under Finite Deformation.93... 

Murnaghan, F. D. Finite deformation of an elastic solid. New York John Wiley 1951. 

A finite element method is employed to study the nonlinear dynamic effect of a strong wind gust on a cooling tower. Geometric nonlinearities associated with finite deformations of the structure are considered but the material is assumed to remain elastic. Load is applied in small increments and the equation of motion is solved by a step-by-step integration technique. It has been found that the cooling tower will collapse under a wind gust of maximum pressure 1.2 psi. 13 refs, cited. 

Jiang HQ, Khang DY, Song JZ, Sun YG, Huang YG, Rogers JA (2007) Finite deformation mechanics in buckled thin films on compliant supports. Proc Nat Acad Sci USA 104 15607-15612... 

Levenston, M.E., Frank, E.H., and Grodzinsky, A.J. (1999) Electrokinetic and poroelastic coupling during finite deformations of charged porous media. Journal of Applied Mechanics 66, 323-333... 

For other models of flow of electrolytes through porous media the reader is referred to [2], [5], [6]. To take into account FCD (fixed charge density) one has to impose additional condition on the interface T (w) and the electroneutrality condition. A challenging problem is to use homogenisation methods for the case of finitely deformable skeleton, even hyperelastic. The permeability would then necessarily depend on strains. Such a dependence (nonlinear) is important even for small strain, cf. [7]. It is also important to include ion channels [8]. 

The proposed model consists of a biphasic mechanical description of the tissue engineered construct. The resulting fluid velocity and displacement fields are used for evaluating solute transport. Solute concentrations determine biosynthetic behavior. A finite deformation biphasic displacement-velocity-pressure (u-v-p) formulation is implemented [12, 7], Compared to the more standard u-p element the mixed treatment of the Darcy problem enables an increased accuracy for the fluid velocity field which is of primary interest here. The system to be solved increases however considerably and for multidimensional flow the use of either stabilized methods or Raviart-Thomas type elements is required [15, 10]. To model solute transport the input features of a standard convection-diffusion element for compressible flows are employed [20], For flexibility (non-linear) solute uptake is included using Strang operator splitting, decoupling the transport equations [9],... 

Biot, M.A. (1972) Theory of finite deformations of porous solids. Indiana University... 

McKenzie, D. (1979) Finite deformation during fluid flow. Geophys. J. Roy. Astr. Soc., 58, 689-715. 

H.D. Espinosa et al A 3-D finite deformation anisotropic visco-plasticity model for fiber composites. J. Compos. Mater. 35(5) 369-410 (2001)... 

M.F. Horstemeyer, M.I. Baskes Atomistic finite deformation simulations a discussion on length scale effects in relation to mechanical stresses. J. Eng. Matls. Techn. Trans. ASME 121, 114-119 (1999)... 

D.J. Bammann, E.C. Aifantis A model for finite-deformation plasticity. Acta Mech. 69, 97-117 (1987)... 

Though a simple Maxwell model in the form of equations (1) and (2) is powerful to describe the linear viscoelastic behaviour of polymer melts, it can do nothing more than what it is made for, that is to describe mechanical deformations involving only infinitesimal deformations or small perturbations of molecules towards their equilibrium state. But, as soon as finite deformations are concerned, which are typically those encountered in processing operations on pol rmers, these equations fail. For example, the steady state shear and elongational viscosities remain constant throughout the entire rate of strain range, normal stresses are not predicted. 

A great number of experiments in soft cross-linked rubbers are made at substantial finite deformations. There are experimental grounds suggesting that the relaxation stress can be factored into a function of time and a function of strain (39,40),... 

Mumaghan FD (1937) Finite deformations of an elastic solid. Am J Math 49 235-260 Niesler H, Jackson I (1989) Pressure derivatives of elastic wave velocities from ultrasonic interferometric measurements on jacketed poly crystals. J Acoust Soc America 86 1573-1585 Nomura M, Nishizaka T, Hirata Y, Nakagiri N, Fujiwara H (1982) Measurement of the resistance of Manganin under liquid pressure to 100 kbar and its application to the measurement of the transition pressures of Bi and Sn. Jap J Appl Phys 21 936-939 Nye JF (1957) Physical Properties of Crystals. Oxford University Press, Oxford... 

It is worth noting that this expression is generic as a means of extracting the elastic moduli from microscopic calculations. We have not, as yet, specialized to any particular choice of microscopic energy functional. The physical interpretation of this result is recalled in fig. 5.2. The basic observations may be summarized as follows. First, we note that the actual energy landscape is nonlinear. One next adopts the view that even in the presence of finite deformations, there is a... 

One point that has not been emphasized is that all of the preceding analysis and discussion pertains only to the steady-state problem. From this type of analysis, we cannot deduce anything about the stability of the spherical (Hadamard Rybczynski) shape. In particular, if a drop or bubble is initially nonspherical or is perturbed to a nonspherical shape, we cannot ascertain whether the drop will evolve toward a steady, spherical shape. The answer to this question requires additional analysis that is not given here. The result of this analysis26 is that the spherical shape is stable to infinitesimal perturbations of shape for all finite capillary numbers but is unstable in the limit Ca = oo (y = 0). In the latter case, a drop that is initially elongated in the direction of motion is predicted to develop a tail. A drop that is initially flattened in the direction of motion, on the other hand, is predicted to develop an indentation at the rear. Further analysis is required to determine whether the magnitude of the shape perturbation is a factor in the stability of the spherical shape for arbitrary, finite Ca.21 Again, the details are not presented here. The result is that finite deformation can lead to instability even for finite Ca. Once unstable, the drop behavior for finite Ca is qualitatively similar to that predicted for infinitesimal perturbations of shape at Ca = oo that is, oblate drops form an indentation at the rear, and prolate drops form a tail. 

The problem on finite deformations arising in the motion of a solid sphere toward a free interface and in the motion of a deformed drop to the solid plane wall, which is vital for chemical industry, was studied numerically in [17, 517],... 

Fig. 22.18 Schematic of a crack tip the broken iine represents the ciassicai shape obtained from iinear eiasticity and the fuii iine represents a possibie evoiution of the crack shape when finite deformations and non-...

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