Uniaxial stressed solid

K. Kassner, C. Misbah. Nonlinear evolution of a uniaxially stressed solid a route to fracture Europhys Lett 28 145, 1994. 

W. Araki, Y. Arai, Oxygen diffusion in yttria-stabilized zirconia subjected to uniaxial stress. Solid State Ionics 181, 441-446 (2010)... 

The material properties used in the simulations pertain to a new X70/X80 steel with an acicular ferrite microstructure and a uniaxial stress-strain curve described by er, =tr0(l + / )", where ep is the plastic strain, tr0 = 595 MPa is the yield stress, e0=ff0l E the yield strain, and n = 0.059 the work hardening coefficient. The Poisson s ratio is 0.3 and Young s modulus 201.88 OPa. The system s temperature is 0 = 300 K. We assume the hydrogen lattice diffusion coefficient at this temperature to be D = 1.271x10 m2/s. The partial molar volume of hydrogen in solid solution is... 

We are now able to analyze the demixing of a solid solution (A, B) O ( = MeO) exposed to a non-hydrostatic, uniaxial stress as shown in Figure 8-9. If transport occurs... 

Figure 8-9. Demixing of a solid solution (A,B)0 exposed to uniaxial stress. Cross hatched solid electrolytes.

Fig. 45 a Uniaxial stress-strain data (symbols) of S-SBR samples filled with 40 phr N220 at various pre-strains emax and simulation curves (lines) of the polymer contribution according to Eqs. (38) and (44). The set of polymer parameters is found as Gc=0.176 MPa, Ge=0.2 MPa, and njTe= 100. b Stress contributions of the strained filler clusters for the different pre-strains (upper part), obtained by subtracting the polymer contributions from the experimental stress-strain data of a. The solid lines are adaptations with the integral term of Eq. (47) and the cluster size distribution Eq. (37), shown... 

Fig. 47 a Uniaxial stress-strain data in stretching direction (symbols) of S-SBR samples filled with 60 phr N 220 at various pre-strains smax and simulations (solid lines) of the third up- and down-cycles with the cluster size distribution Eq. (55). Fit parameters are listed in the insert and Table 4, sample type C60. b Simulation of uniaxial stress-strain cycles for various pre-strains between 10 and 50% (solid lines) with material parameters from the adaptation in a. The dashed lines represent the polymer contributions according to Eqs. (38) and (44) with different strain amplification factors... 

The concept of stress-induced dilatation affecting the relaxation time or rate has been suggested by others (5, 6, 7, 8). The density of most solids decreases under uniaxial stress because the lateral contraction of the solid body does not quite compensate for the longitudinal extension in the direction of the stress, and the body expands. The Poisson ratio, the ratio of such contraction to the extension, is about 0.35 for many polymeric solids it would be 0.5 if no change in density occurred, as in an ideal rubber. The volume increase, AV, accompanying the tensile strain of c, can be described by the following equation ... 

For measurements at 77 K we used a simple optical cryostat with an inserted uniaxial stress device that transfers the applied tensile stress to an elastic ring with the firmly fixed sample inside [2]. The ring converts the stretching force in one direction into a compressive one along the sample. The axial distribution of the stress in the ring and a solid junction secured the sample of a premature... 

When a solid is subjected to a tensile stress, it extends in the direction of the stress but contracts in the perpendicular direction. This is quantified by the Poissson s ratio which is the ratio of transverse strain to the longitudinal strain. This can be understood with reference to the application of a uniaxial stress as shown in Figure 10.01 (a) where the elongation in x-direction is associated with a shrinkage in y and z directions. Thus v is defined by the transverse, -e, (e, has a negative sign) and the longitudinal strain, e/, as. 

Figure 18.7. Calculated shear yield stress under uniaxial tension (solid line) and brittle fracture stress (dashed line) of polystyrene below the glass transition temperature. Note that the shear yield stress has a much stronger temperature dependence than the brittle fracture stress. 

A uniaxial stress is usually designated by the symbol o, and a shearing stress by X. For certain lipids that behave like solid food systems, the relationship between stress and strain is represented by a straight line through the origin, up to the so-called limit of elasticity. The proportionality factor E for uniaxial stress is called Young s modulus, or the modulus of elasticity. For a shear stress, the modulus is called Coulomb modulus, or the tensile modulus G. 

These are most easily represented by the equation E = E + iE". where E is the ratio between (the amplitude of the in-phase stress component strain, a/e) and E" is the loss modulus (the amplitude of the out-of-phase component. strain amplitude). Similarly for (7 and K and the ratio between the Young s modulus E and the shear modulus C includes Poisson s ratio u, for an isotropic linear elastic solid with a uniaxial stress. (Poisson s ratio is more correctly defined as minus the ratio of the perpendicular. strain to the plane strain, or for one orthogonal direction 22 which equals the. 3.3 strain if the sample is... 

It is shown below that this equation describes a neo-Hookeian sohd as defined earlier if C = E/6, where E is the modulus for vanishingly low uniaxial stress. A neo-Hookeian solid can thus more generally be defined as one that obeys equation (6.26). 

Fig. 5.23 a A single crystal subjected to a uniaxial stress, showing the direction of atomic flux. A representative grain in polycrystalline solids, showing the expected atomic flux by lattice difiusion (b) and grain boundary diffusion (c). a, c Reproduced from [1]. Copyright 2003, CRC Press, b Reproduced with permission from [54]. Copyright 1950, American Institute of Physics... 

AnastassaJds, E., Knczuk, A., Burstein, E., PoUack, RH., Cardona, M. Effect of static uniaxial stress on the Raman spectrum of silicon. Solid State Commun. 8, 133-138 (1970)... 

Chung S, Wang Y, Greenbaum S, Golodnitsky D, Peled E (1999) Uniaxial stress effects in poly(ethylene oxide)-LiI polymer electrolyte film - a Li-7 nuclear magnetic resonance study. Electrochem Solid State Lett 2(11) 553-555... 

The procedure used in the derivation of Eq. (8.116) can also be used to analyze diffusional creep of porous solids under an applied uniaxial stress p. In this case, the mean stress on the grain boundary is 4>p, and the general creep equation has the form... 

As indicated earlier the different stress-strain curves are not characteristic for particular, chemically defined species of polymers but for the physical state of a polymeric solid. If the environmental parameters are chosen accordingly transitions from one type of behavior (e.g. brittle, curve a) to another (e.g. ductile, curve c) will be observed. These phenomenological aspects of polymer deformation are discussed in detail in [14], [52—53], [55—57], and in the general references of Chapter 1. A decrease of rate of strain or an increase of temperature generally tend to increase the ductility and to shift the type of response from that of curve a) towards that of curves c) and d). At small strains (between zero and about one per cent) the uniaxial stress o and the strain e are linearly related (Hooke s law) ... 

Stress in crystalline solids produces small shifts, typically a few wavenumbers, in the Raman lines that sometimes are accompanied by a small amount of line broadening. Measurement of a series of Raman spectra in high-pressure equipment under static or uniaxial pressure allows the line shifts to be calibrated in terms of stress level. This information can be used to characterize built-in stress in thin films, along grain boundaries, and in thermally stressed materials. Microfocus spectra can be obtained from crack tips in ceramic material and by a careful spatial mapping along and across the crack estimates can be obtained of the stress fields around the crack. ... 

In solids of cubic symmetry or in isotropic, homogeneous polycrystalline solids, the lateral component of stress is related to the longitudinal component of stress through appropriate elastic constants. A representation of these uniaxial strain, hydrostatic (isotropic) and shear stress states is depicted in Fig. 2.4. Such relationships are thought to apply to many solids, but exceptions are certainly possible as in the case of vitreous silica [88C02]. 

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