Relative deformation

As it was determined by the test, the stretch diagram at the uniaxial load carrying ability of the carbon plastic UKN-5000 is almost linear until the destruction point. The samples are breaking brittle, and the relative deformation is small (E < 2%). 

Figure 7. A projection of the Fermi surfaces on a plane parallel to the axis of the symmetry breaking. The concentric circles correspond to the two populations of spin/isospin-up and down fermions in spherically symmetric state (Se = 0), while the deformed figures correspond to the state with relative deformation Se = 0.64. The density asymmetry is a = 0.35.

Figure 8. The dependence of the free energy of the combined DFS and LOFF phases on the center-of-mass momentum of the pairs Q (in units of Fermi momentum pf) and the relative deformations Se for a fixed density asymmetry [18].

Figure 9. The color superconducting gap (left panel) and the free energy (right panel) as a function of relative deformation parameter a for the values of the flavor asymmetry a = 0 (solid lines), 0.1 (dashed lines), 0.2 (short dashed lines) and 0.3 (dashed dotted lines) at density pB = 0.31 I m 3 and temperature T = 2 MeV [25].

When a stress is applied to a material, a deformation will occur. In order to make calculations tractable, we define the strain as the relative deformation, i.e. the deformation per unit length. The length that we use is the one over which the deformation occurs. This is illustrated in Figures 1.1 and 1.2. 

These two equations show that the pad deformation is proportional to the applied load and inversely proportional to the pad material stiffness (represented by the Young s modulus). The equations involve complete elliptic integrals that can be readily evaluated numerically. The relative deformation represented by Eqs. (24) and (25) is plotted in Fig. 13, where the inner region corresponds to Eq. (24) and the outer corresponds to Eq. (25). 

The ratio between these relative deformations is Ifn and can be used to define the deformation profile or length scale. Due to the presence of a softer back pad, more deformation is expected for the stacked pad but the shape, which is the main concern, will be approximately similar [45,46], The deformation is relatively small compared to the region of apphcation of the force. Using approximate material properties for the ICIOOO pad (Young s modulus of 2.9 x 10 Pa [41] and approximate Poisson ratio of 1/3) with force applied in a circular region of radius 2 mm, and a local pressure of 7 psi, the maximum deflection is about 6 fira. This deformation is referenced to the origin as illustrated in Fig. 13. It is also important to note that the transition shape is very gradual and this sets the polish limit for the down areas. 

Let us suppose that we now apply uniaxial force to the network sample along the axis oZ. Let be the force per cross sectional area of the sample in the reference state (i.e. is the stress normalized in a special way). Applied force leads to some relative deformation along oZ az = p. The network dimensions along axes x and y, ax = cty = a are varied freely. It has been shown [20] that in the case of a homogeneous solvent containing no salt (ns = 0) the equilibrium dimensions of the network are described by the following system of equations ... 

As shown in Figure 10.25, there is an /-dependent barrier to fusion that is the sum of the nuclear, Coulomb, and centrifugal potentials. This barrier is also a sensitive function of the relative deformation and orientation of the colliding ions. In Figure 10.27, we show the excitation function for fusion of 160 with various isotopes of Sm that span a wide range of deformations. 

Fig. 6. Dependence of strain, r, on relative deformation, X, in Mooney-Rivlin coordinates (f = x/r0, X )...

The problem of determination of the partition function Z(k, N) for the iV-link chain having the fc-step primitive path was at first solved in Ref. [17] for the case a = c by application of rather complicated combinatorial methods. The generalization of the method proposed in Ref. [17] for the case c> a was performed in Refs. [19,23] by means of matrix methods which allow one to determine the value Z(k,N) numerically for the isotropic lattice of obstacles. The basic ideas of the paper [17] were used in Ref. [19] for investigation of the influence of topological effects in the problem of rubber elasticity of polymer networks. The dependence of the strain x on the relative deformation A for the uniaxial tension Ax = Xy = 1/Va, kz = A calculated in this paper is presented in Fig. 6 in Moon-ey-Rivlin coordinates (t/t0, A ), where r0 = vT/V0(k — 1/A2) represents the classical elasticity law [13]. (The direct Edwards approach to this problem was used in Ref. [26].) Within the framework of the theory proposed, the swelling properties of polymer networks were investigated in Refs. [19, 23] and the t(A)-dependence for the partially swollen gels was obtained [23]. In these papers, it was shown that the theory presented can be applied to a quantitative description of the experimental data. 

Dimensional stability is one of the most important properties of solid materials, but few materials are perfect in this respect. Creep is the time-dependent relative deformation under a constant force (tension, shear or compression). Hence, creep is a function of time and stress. For small stresses the strain is linear, which means that the strain increases linearly with the applied stress. For higher stresses creep becomes non-linear. In Fig. 13.44 typical creep behaviour of a glassy amorphous polymer is shown for low stresses creep seems to be linear. As long as creep is linear, time-dependence and stress-dependence are separable this is not possible at higher stresses. The two possibilities are expressed as (Haward, 1973)... 

More recently, Yoon and Chen68 developed a theory to predict the deformation behavior of particulate composites. Their theory treats the case of rigid particles embedded in a nonNewtonian matrix. The relative deformation rate, e/k0, is related to the volume fraction of particles, , the creep stress exponent of the matrix, n, and the stress concentration factor, k, of the inclusion in the matrix ... 

The border cross-section shape (border profile) was determined photographically. The size of the balloon was chosen so that at the instant of contact with dodecahedron faces a sufficiently high inflection (> 300%) was ensured. This was done to diminish the possible anisotropy of rubber balloon elements. Thus, further deformation of the balloon during its transformation into a polyhedron would not be more than 10 - 30%. To check whether the balloon regions deformed, differently the position of control points situated in the centre of films and in the middle of borders was monitored. When the dodecahedron faces were wetted, the relative deformation of the balloon regions was practically identical and the shape of the border cross-section surface was spherical. The thickness of liquid films between the dodecahedron faces and the balloon surface was determined conductometrically. 

It is different from the value obtained by the model of truncated octahedron with planar faces in which // = 0.672 or 0.516 (depending on the calculation technique used). There is also a difference from the analogous values, calculated by the formulae of Derjaguin (1.1), Stamenovich (0.75), and Budansky and Kimmel (0.794). Maximum stress was reached at relative deformation of 0.213 and was equal to... 

Equations 1.2 to 1.4 represent material functions under large deformations (e.g., continuous shear of a fluid). One may recall a simple experiment in an introductory physics course where a stress (a) is applied to a rod of length Z, in a tension mode and that results in a small deformation AL. The linear relationship between stress (ct) and strain (j/) (also relative deformation, y = AL/L) is used to define the Young s modulus of elasticity E (Pa) ... 

Oakenflill et al. (1989) presented a method for determining the absolute shear modulus (E) of gels from compression tests in which the force, F, the strain or relative deformation (S/L) are measured with a cylindrical plunger with radius r, on samples in cylindrical containers of radius R, as illustrated in Figure 3-47. Assuming that the gel is an incompressible elastic solid, the following relationships were derived ... 

Teflon. This synthetic material is chemically inert, and has a safe working temperature of 120°C. It has a relatively high permeability to gases and is suitable for dynamic vacuum work, but is not recommended for static systems. Teflon (or P.T.F.E.) is a relatively deformable material, has a low coefficient of. friction and is eminently suitable for making seals in stopcocks and taps ... 

Obviously, when 0 = t, no deformation with respect to the reference configurations has taken place, and Ffxk, t t) = bg, where 6, is the Kronecker delta. The actual strain of the relative deformation tensor is better expressed by the symmetrical tensor... 

The components of the relative deformation gradient Fy and the symmetrical tensor Jy can be calculated, respectively, from Eqs. (13.7) and (13.8), giving... 

ME deformation along axis of orientation ME deformation perpendicular to axis of orientation Fig. 3. Dependence of stress in a sample on its relative deformation. 

H measurements of PBT under strains of up to 20% relative deformation (e) were performed using a stretching device. The strain s is defined ass = ( — o) /f o where q and are the starting and stretched lengths of the sample, respectively. The indentation anisotropy AH = 1 — (d /dj ) was also derived (see eq. (2.6)). In order to evaluate the contribution of each polymorphic phase to the total H it is necessary to know their mass fractions at the different deformation stages as required by the additivity law (eq. (4.3)). For this purpose the data of Tashiro et al. (1980) obtained by the infrared study of the a-fi transition have been used. The same authors have shown that just in the transition deformation interval (e = 4-16%) the relationship between s and the amount of the phase is linear. 

Figure 6.8. Variation of microhardness H with increasing relative deformation e of the blend PBT/PEE = 49/51 wt% (with PEE of PBT/PEG-1000 = 51 /49 wt%). (From Boneva et al., 1998.)...

Figure 6.9. Variation of microhardness H with increasing relative deformation e for (a) homo-PBT (from Fig. 6.2), (b) the blend PBT/PEE = 49/51 wt% (with PEE of PBT/PEG-1000 = 51/49 wt% (from Fig. 6.8), and (c) the multiblock copolymer PEE with PBT/PEG = 57/43 wt% (from Fig. 6.5). For a better visual presentation a different H scale for stunples (b) and (c) has been used. (From Boneva et al., 1998.)...

The same PEE material was used for this study as in the previous sections. Using a stretching device it was possible to perform measurements up to 70% overall relative deformation s at which point the sample broke. Again the deformation was increased in steps of 5%. The main difference was that in previous studies (Sections 6.2.1-6.2.3) the deformation was increased continuously without relaxation, whereas in this case the sample was unloaded and allowed to relax after each H measurement under stress before the next H measurement without stress (a = 0) was performed. The sample was then stretched to the next deformation and H was measured again. It should be noted that beyond some overall deformation (typically s > 20%) the unstressed sample shows some residual (plastic) deformation amounting about 50% of the strain s under stress. The same deformation cycle has been used for SAXS measurements aimed at the morphological characterization of these PEE samples. 

This standard specifies a method for determining a) the compressive strength and corresponding relative deformation or b) the compression stress at 10% relative deformation, of rigid cellular plastics. 

Consider a rubber network with undeformed dimensions Lyo, and L o (Fig- 7.2). If the network experiences relative deformations in the X, y, and z directions by the factors X, Xy, and A, then the dimensions of the deformed network are... 

A constant stress a is applied on the system (that may be placed in the gap between two concentric cylinders or a cone and plate geometry) and the strain (relative deformation) y or compliance J =Y/ff, Pa" ) is followed as a function of time for a period of t. At t = t, the stress is removed and the strain y or compliance J is followed for another period t [1]. 

Suppose that the lowest cavity mode is resonant. Then one can evaluate the dimensionless coupling constant asK (e2/In mc2L) (here we return to the dimensional variables). The maximum value of parameter is (see the discussion in Section X) max dmaxvs/2nc, where 5max 0.01 is the maximal possible relative deformation in the material of the wall, and vs 5 103 m/s is the sound velocity inside the wall. Then the ratio e/k cannot exceed the value 5max (mirL/Hne2) 0.05 for L 1 cm and m the mass of electron (for these parameters k 2 10 7). Consequently, one may believe that in the real conditions e/k [Pg.369]

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