Finite strain

Bernstein, B., Kearslcy, E.A. and Zapaa, L., 1963. A study of stress relaxation with finite strain. Trans. Soc. Rheol. 7, 391-410. 

In order to relate the parameters of (4.5), the shock-wave equation of state, to the isentropie and isothermal finite strain equations of state (discussed in Section 4.3), it is useful to expand the shock velocity normalized by Cq into a series expansion (e.g., Ruoff, 1967 Jeanloz and Grover, 1988 Jeanloz, 1989). 

The relationship of the above parameters to finite strain equations of state is given in the next section. 

In the following treatment we show how the usual Birch-Murnaghan finite strain equation of state is derived and is related to the Hugoniot parameters. Using the Eulerian definition of finite strain. 

Birch, F. (1978), Finite Strain Isotherm and Velocities for Single-Crystal and Polycrystalline NaCl at High Pressures and 300 K, J. Geophys. Res. 83, 1257-1268. 

Jeanloz, R. (1989), Shock Wave Equation of State and Finite Strain Theory, J. Geophys. Res 94, 5873-5886. 

Atluri, S.N., On Constitutive Relations at Finite Strain Hypo-Elasticity and Elasto-Plasticity with Isotropic or Kinematic Hardening, Comput. Methods Appl. Mech. Engrg. 43, 137-171 (1984). 

The elastic-shock region is characterized by a single, narrow shock front that carries the material from an initial state to a stress less than the elastic limit. After a quiescent period controlled by the loading and material properties, the unloading wave smoothly reduces the stress to atmospheric pressure over a time controlled by the speeds of release waves at the finite strains of the loading. Even though experiments in shock-compression science are typically... 

The relative motion of materials points in a solid body in finite strain is best represented by a deformation gradient having components... 

To describe properties of solids in the nonlinear elastic strain state, a set of higher-order constitutive relations must be employed. In continuum elasticity theory, the notation typically employed differs from typical high pressure science notations. In the present section it is more appropriate to use conventional elasticity notation as far as possible. Accordingly, the following notation is employed for studies within the elastic range t = stress, t] = finite strain, with both taken positive in tension. 

The stress relation obtained from an expansion of the internal energy function to fourth order in the finite strain t] takes the following form [79D01] ... 

In this chapter physical properties of solids at finite strain within their purely elastic ranges will be investigated. Although the strain levels of a few percent are small relative to the total compressions of typical shock-compression studies, they are large compared to those typically encountered in higher-order elastic property investigations. 

Fig. 4.3. Typical normalized piezoelectric current-versus-time responses are compared for x-cut quartz, z-cut lithium niobate, and y-cut lithium niobate. The y-cut response is distorted in time due to propagation of both longitudinal and shear components. In the other crystals, the increases of current in time can be described with finite strain, dielectric constant change, and electromechanical coupling as predicted by theory (after Davison and Graham [79D01]).

Fig. 5.21. The shock-induced polarization of polymers as studied under impact loading is shown. For the current pulse shown, time increases from left to right. The increase of current in time is due to finite strain and dielectric constant change. (See Graham [79G01]).

Mars, W.V., Heuristic approach for approximating energy release rates of small cracks under finite strain, multiaxial loading, in Elastomers and Components—Service Life Prediction Progress and Challenges, Coveney, V., Ed., OCT Science, Philadelphia, 2006, 89. 

Fig. 13. Shear stress t12 and first normal stress difference N1 during start-up of shear flow at constant rate, y0 = 0.5 s 1, for PDMS near the gel point [71]. The broken line with a slope of one is predicted by the gel equation for finite strain. The critical strain for network rupture is reached at the point at which the shear stress attains its maximum value...

Lodge AS (1964) Elastic liquids an introductory vector treatment of finite-strain polymer rheology. Academic, London... 

The two most usual equations of state for representation of experimental data at high pressure are the Murnaghan and Birch-Murnaghan equations of state. Both models are based on finite strain theory, the Birch-Murnaghan or Eulerian strain [26], The main assumption in finite strain theory is the formal relationship between compression and strain [27] ... 

The application of finite strains and stresses leads to a very wide range of responses. We have seen in Chapters 4 and 5 well-developed linear viscoelastic models, which were particularly important in the area of colloids and polymers, where unifying features are readily achievable in a manner not available to atomic fluids or solids. In Chapter 1 we introduced the Peclet number ... 

The non-linear response of plastic materials is more challenging in many respects than pseudoplastic materials. While some yield phenomena, such as that seen in clay dispersions of montmorillonite, can be catastrophic in nature and recover very rapidly, others such as polymer particle blends can yield slowly. Not all clay structures catastrophically thin. Clay platelets forming an elastic structure can be deformed by a finite strain such that they align with the deforming field. When the strain... 

In a later study [56], the effect of gas volume fraction (foam rheology was investigated. Two models were considered one in which the liquid was confined to the Plateau borders, with thin films of negligible thickness and the second, which involves a finite (strain-dependent) film thickness. For small deformations, no differences were observed in the stress/strain results for the two cases. This was attributed to the film thickness being very much smaller than the cell size. Thus, it was possible to neglect the effect of finite film thickness on stress/strain behaviour, for small strains. 

The notion of fluid strains and stresses and how they relate to the velocity field is one of the fundamental underpinnings of the fluid equations of motion—the Navier-Stokes equations. While there is some overlap with solid mechanics, the fact that fluids deform continuously under even the smallest stress also leads to some fundamental differences. Unlike solid mechanics, where strain (displacement per unit length) is a fundamental concept, strain itself makes little practical sense in fluid mechanics. This is because fluids can strain indefinitely under the smallest of stresses—they do not come to a finite-strain equilibrium under the influence of a particular stress. However, there can be an equilibrium relationship between stress and strain rate. Therefore, in fluid mechanics, it is appropriate to use the concept of strain rate rather than strain. It is the relationship between stress and strain rate that serves as the backbone principle in viscous fluid mechanics. 

A fluid packet, like a solid, can experience motion in the form of translation and rotation, and strain in the form of dilatation and shear. Unlike a solid, which achieves a certain finite strain for a given stress, a fluid continues to deform. Therefore we will work in terms of a strain rate rather than a strain. We will soon derive the relationships between how forces act to move and strain a fluid. First, however, we must establish some definitions and kinematic relationships. 

England PC, Houseman G A (1986) Finite strain calculations of continental deformation, 2, Comparison with the India-Asia collision. J Geophys Res 91 3664-3676... 

Ehlers, W. and Markert, B. (2003). A macroscopic finite strain model for cellular polymers. International Journal of Plasticity, 19 961-976. 

Soft biological structures exhibit finite strains and nonlinear anisotropic material response. The hydrated tissue can be viewed as a fluid-saturated porous medium or a continuum mixture of incompressible solid (s), mobile incompressible fluid (f), and three (or an arbitrary number) mobile charged species a, (3 = p,m, b). A mixed Electro-Mechano-Chemical-Porous-Media-Transport or EMCPMT theory (previously denoted as the LMPHETS theory) is presented with (a) primary fields (continuous at material interfaces) displacements, Ui and generalized potentials, ifi ( , r/ = /, e, to, b) and (b) secondary fields (discontinuous) pore fluid pressure, pf electrical potential, /7e and species concentration (molarity), ca = dna/dVf or apparent concentration, ca = nca and c = Jnca = dna/dVo. The porosity, n = 1 — J-1(l — no) and no = no(Xi) = dVj/dVo for a fluid-saturated solid. Fixed charge density (FCD) in the solid is defined as cF = dnF/dV , cF = ncF, and cF = cF (Xf = JncF = dnF/d o. 

J. T. Oden and T. Sato, Finite Strains and Displacements of Elastic Membranes by the Finite Element Method, Int. J. Solids and Struct., 3, 471 -88 (1967). 

We present a finite element study which includes both shear yielding and crazing within a finite strain description. This provides a way of putting together all aspects of glassy polymer fracture crazing and shear yielding but also thermal effects. 

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