Uniaxial stress conditions

Like other polymeric materials, rheological modeling was first attempted to predict the constitutive behavior of SMPs. Although earher efforts [5-8] using rheological models were able to describe the characteristic thermomechanical behavior of SMPs, loss of the strain storage and release mechanisms usually led to limited prediction accuracy. Also, the models were 1-D and could only predict the behavior under a uniaxial stress condition, such as 1-D tension. Furthermore, these models can oidy work for a small strain. They cannot predict the thermomechanical behavior of SMPs with finite strain, which is the case for most SMPs. Later, meso-scale models [9,10] were developed to predict the constitutive behavior of SMPs. However, one limitation is that a meso-scale model cannot understand the shape memory mechanisms in detail because the mechanisms controlling the shape memory are in a molecular... 

Prompt instrumentation is usually intended to measure quantities while uniaxial strain conditions still prevail, i.e., before the arrival of any lateral edge effects. The quantities of interest are nearly always the shock velocity or stress wave velocity, the material (particle) velocity behind the shock or throughout the wave, and the pressure behind the shock or throughout the wave. Knowledge of any two of these quantities allows one to calculate the pressure-volume-energy path followed by the specimen material during the experimental event, i.e., it provides basic information about the material s equation of state (EOS). Time-resolved temperature measurements can further define the equation-of-state characteristics. 

A strength value associated with a Hugoniot elastic limit can be compared to quasi-static strengths or dynamic strengths observed values at various loading strain rates by the relation of the longitudinal stress component under the shock compression uniaxial strain tensor to the one-dimensional stress tensor. As shown in Sec. 2.3, the longitudinal components of a stress measured in the uniaxial strain condition of shock compression can be expressed in terms of a combination of an isotropic (hydrostatic) component of pressure and its deviatoric or shear stress component. 

In the perfectly elastic, perfectly plastic models, the high pressure compressibility can be approximated from static high pressure experiments or from high-order elastic constant measurements. Based on an estimate of strength, the stress-volume relation under uniaxial strain conditions appropriate for shock compression can be constructed. Inversely, and more typically, strength corrections can be applied to shock data to remove the shear strength component. The stress-volume relation is composed of the isotropic (hydrostatic) stress to which a component of shear stress appropriate to the... 

The elastic energy of inhomogeneous, anisotropic, ellipsoidal inclusions can be studied using Eshelby s equivalent-inclusion method. Chang and Allen studied coherent ellipsoidal inclusions in cubic crystals and determined energyminimizing shapes under a variety of conditions, including the presence of applied uniaxial stresses [11]. 

Figure 45a-c shows an adaptation of the developed model to uniaxial stress-strain data of a pre-conditioned S-SBR-sample filled with 40 phr N220. The fits are obtained for the third stretching cycles at various prestrains by referring to Eqs. (38), (44), and (47) with different but constant strain amplification factors X=Xmax for every pre-strain. For illustrating the fitting procedure, the adaptation is performed in three steps. Since the evaluation of the nominal stress contribution of the strained filler clusters by the integral in Eq. (47) requires the nominal stress aR>1 of the rubber matrix, this quantity is developed in the first step shown in Fig. 45a. It is obtained by demanding an intersection of the simulated curves according to Eqs. (38) and (44) with the measured ones at maximum strain of each strain cycle, where all fragile filler clusters are broken and hence the stress contribution of the strained filler clusters vanishes. The adapted polymer parameters are Gc=0.176 MPa and neITe= 100, independent of pre-strain. According to the considerations at the end of Sect. 5.2.2, the tube constraint modulus is kept fixed at the value Ge=0.2 MPa, which is determined by the plateau modulus Gn° 0.4 MPa [174, 175] of the uncross-linked S-SBR-melt (Ge=l/2GN°). The adapted amplification factors Xmax for the different pre-strains ( max=l> 1-5, 2, 2.5, 3) are listed in the insert of Fig. 45a.

Figures 20A and B show the PL spectra, recorded at 290 K, at 600 nm, and as a function of pressure, for Cs9(SmW10O36) and SmWi0O36-LDH, respectively (Park et al., 2002). For the sake of comparison, the line shapes are normalized and displaced along the vertical axis. In both cases, the peak position is red-shifted by 4—5 nm when the hydrostatic pressure increases from 1 bar to 61 kbar. It was shown that the red-shift from A to A lies solely in the deformation of the samarium complexes by the uniaxial stress exerted by the host layers, whereas the shift from B to B is also influenced by the change in the cation environment. Under the same conditions, B is not at the same position for the non-intercalated (HN (n -b u t y 1) 3) 9 (SmW10O3e) and Cs9(SmWi0O36) compounds (Park et al., 2002). Thus only peak A is available to measure the unixial stress. This observation can be used to determine the uniaxial stress, when the external pressure is zero. For the SmW10O36—LDH system, the uniaxial stress varies significantly from 75 at 28 K to 140 kbar at 290 K (Park et al., 2002).

Until we discovered the constancy of the surface potential from the uniaxial stress results, like most other people, I had been more interested in constant surface charge models. If you do not know how the valency of a macroion varies with the external conditions, it is reasonable to assume it to be constant unless given evidence to the contrary. Given the evidence that y/0 70 mV is roughly constant for the n-butylammonium vermiculite system, what other consequences follow from this In particular, what happens if we apply the coulombic attraction theory with the constant surface potential boundary condition ... 

The effect of stress state on the maximum algebraic stress concentration for various orientations of the ellipse (ft) is shown in Figure 4 for an ellipticity R = 0.1. The uniaxial tension condition = 0) generates the highest stress concentration. Equal biaxial tension ( = 45°) generates the same stress concentration for any ellipse orientation p. This is because the stress state is planar isotropic, and any pair of orthogonal axes can be chosen as the external principal axes. Consequently the orientation p is without effect. 

The x-ray method is best approached by first considering the case of uniaxial stress, where the stress acts only in a single direction, even though this condition is rare in practice. The more general case of biaxial stress will be dealt with later. 

Under stress, a one-valley EM donor state follows the energy shift of the valley it belongs to. A study of the effect of a uniaxial stress on the donor spectra in a multi-valley semiconductor (silicon) has been undertaken by Tekippe et al. [140]. The treatment given below follows this presentation closely, with minor changes in the notations. Following the deformation potential analysis of [60], the shift in energy Aof valley j of the CB of silicon or germanium with respect to the zero-stress conditions is ... 

JCS) values were used to represent the range of mechanical properties observed in all three formations (Formation 1 see Blum et al. 2(X)3 Formation 2 JRC (/- ) = 4.28, 5.98, 4.18, 2.29 JCS (1-4) = 39.3. 31.9, 90.9, 43.1 in MPa and the unchanged uniaxial compressive strength for all 4 cases UCS = 120.0 MPa Fault Zone JRC = 4.22, JCS = 105.9 MPa and the UCS = 128.4 MPa). In case of Formation 2 only four pairs were available, thus the entire data set is provided here. Stress conditions corresponding to five depths were also applied to the DFN. Table 1 summarizes the hydro-mechanical modelling results in terms of the median hydraulic apertures. Values range between 0.3 pm and 180.7 pm. 

A ceramic body composed of layers aligned parallel to a uniaxial stress, in which the strain is shared equally by the two phases, the iso-strain condition, has an elastic modulus ... 

A value is considered effect factor of rock mass stability under high stress condition in Mathews diagrammatize method. A value is a ratio of uniaxial compressive strength and maximum principal stress of parallel working face with complete block of coal and rock. The relationship of A value and shows a linear, and its variation range is from 0.1 to 1.0. 

Uniaxial load Condition where a material is stressed in only one direction along the axis or centerline of a component part. 

The stress field in the vicinity of a crack tip for an infinite plate containing a sharp, through-thickness, internal crack under plane stress conditions and uniaxially-applied tension (a) is expressed as... 

However, a temperature variation of a material can also occur under static loading conditions [29]. In 1851 Thomson (Lord Kelvin) [30,31] showed the proportionality between the load change applied and the resulting temperature variation for an isotropic material under adiabatic elastic deformation and uniaxial stress... 

Two engineering parameters are commonly used to rank the behavior of materials under creep conditions (constant temperature and applied uniaxial stress) the minimum strain rate and the time to rupture. The first parameter is related to the useful life of components susceptible to shape-change in service (in heat engines, for example), while the second estimates the time-dependent failure probability. 

In order to create a 2-D stress condition in a uniaxial loading device (MTS Q-TEST 150 gear driven machine) special specimens were prepared. The specimens were machined into biaxial... 

Fig. 1.45 Explosive failure of the S1C-N-UC02 specimen (12.7 mm in diameter and 25.4 mm in length) subjected to the unconfined uniaxial compressive stress condition (iTi = 3988 MPa at failure and 0-2 = (73 = 0) [33]. With kind permission of Professor Brannon...

As noted, the field of molecular simulation is relatively new, and a detailed review of it is beyond the scope of this text and we introduce here a few of the more relevant references. One of the first applications of molecular mechanics to polymers was by Theodorou and Suter (93,94), who modeled atactic polypropylene as an amorphous cell subjected to a range of stress conditions (hydrostatic pressure, pure strain, and uniaxial strain). Such modeling generally gives reasonable estimates of the elastic constants of a material [within 15% (79)], providing the density of the glass is correctly modeled. 

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