Deformed configuration

Here r(t) is the stress at a fluid particle given by an integral of deformation history along the fluid particle trajectory between a deformed configuration at time f and the current reference time t. 

It is important to understand what gives rise to the established coexisting intruder band in doubly closed subshell 9(>Zr. The slight predominance of the intruder deformed configuration in the ground state wavefunction of lO Mo shown in Fig. 1(b) cannot explain the tremendous difference between the alpha-pickup strengths to the two 0+ states of 9 >Zr clearly demonstrated by Fig. 1(a). This is not unexpected, since the simultaneous occurrence of proton and neutron subshell closures should produce a marked difference with respect to other nuclei near the middle of the 50 to 82 neutron shell where two-proton, two-hole excitations can account entirely for the observed shape coexistence phenomena. 

In this paper a static linear elastic deformation problem for a fluid saturated solid is formulated in which the behavior of the solid matrix is described by a second gradient model. The non-deformed configuration, chosen as a reference configuration, for the considered mixture can not be stress-free indeed the saturating fluid must exhibit internal stresses acting both on the solid constituent and on its sub-bodies. 

The possibility that reacting species prefer to react along those paths in which they undergo the least modification has always been intuitively attractive. At one time or another, so-called principles of minimum structural change or deformation, configurational change, and minimum atomic and electronic motion have been invoked (Wheland, 1960 Hine, 1966). To account for Michael s rule of favored anti 1,2-addition, Pfeiffer formulated acetylenes as tram-heat structures in 1904 Frankland (1912) suggested that anti elimination is favored by an inherent tendency to centric symmetry. The more conscious applications of PLM by Muller after 1886, are probably misapplications of the principle, since they were usually concerned with complex pyrolytic reactions above 1000° (Muller and Peytral, 1924). 

Here, we assume a dipole fluid flow initialized by the infiltration of cold water in a hot rock and its extraction after the heating-up, see Fig. 4. A fracture with a stress dependent transmissivity is located in the front part of the domain. Due to the thermal contraction of the rock the transmissivity of the fracture increases. Fig. 5 shows the different velocity curves in the fracture and in the non-affected matrix. Fig. 6 shows the final temperature field in the deformed configuration. 

Figure 6. Temperature field in the deformed configuration (t=IO days).

If another, deformed configuration x is defined in the same way as Equation 54.4, the fiber-segment-extension ratio ds/ds associated with a change in the elHpsoid geometry [16] can be derived from... 

It can be proved that U is a symmetric and positive definite tensor, which is a measure of the local stretching (or contraction) of material at X. V is also is a symmetric and positive definite second-order tensor called the left stretch tensor, which is a measure of the local stretching (or contraction) of the material in the deformed configuration at x. R is a proper orthogonal tensor, that is, R R = I or detR = 1, where T means transpose, I is the identity tensor, and det is the determinant. 

The velocity gradient, L, an Eulerian quantity rather than a Lagrangian quantity, is defined in the deformed configuration as... 

In strength of materials text, it is weU known that Cauchy stress, small strain because the area in the undeformed body and deformed body is almost the same. For a large strain, however, we generally do not know the area of the deformed configuration. Thus we need to define a stress measure that we can use in the reference configuration. However, it is noted that Cauchy stress is still the most used stress definition or tme stress because the equilibrium is about the deformed body but not the undeformed body. Therefore, other definitions of stress are for convenience of mathematical operation. Stress is generally reported as the Cauchy stress. This shows a distinct departure from the strain definitions. 

This is the first Piola-Kirchoff stress tensor. It means the force in the deformed configuration but in the undeformed area. It is noted that T is not symmetric so it is not easy to use. Similarly, we can obtain the second Piola-Kirchoff stress tensor, which is... 

Fig. 5.15 The reference configuration in Cartesian coordinates and the deformed configuration in cylindrical coordinates for a trilayer bender. Reprinted from [Fang et al.

Suppose that the deformation takes a particle at location X with Cartesian coordinates (x, y, z) in the reference configuration to the location with cylindrical coordinates in the deformed configuration. The... 

The energy of a two-particle excitation is higher at the saddle point than that in the stably deformed configuration. Thus, it is sufficient to consider only collective excited states as transition states. 

In the second case (Fig. 26.10), strain time histories on the opposite sides show an offset, indicating a residual deformed configuration of the sectimi, with a residual bending moment. This moment is slightly lower than the maximum moment induced during excitation. 

Figure I Three-dimensional elastoplastic solid in free flight. Sequence of deformed configurations in the early stages of the motion with the distribution of the equivalent plastic strain a, obtained with the EDMC scheme.

Figure 2 Deformed configurations of BIRF of ground-tie system at selected time steps (a) vertical excitation, (b) horizontal excitation.

If the stressed polymer has deformed viscoelastically by relaxation, the deformed configuration has a higher energy than the initial one. Upon unloading, the molecules return to their initial positions. This process again requires thermal activation and is therefore time-dependent as well. 

The local change in area from the undeformed configuration to the strained configuration represented by efj is the trace of the surface strain tensor e j,. It follows that the surface energy per unit area in the deformed configuration is C s(l + e j.) = Ug. In terms of this measure of surface energy density, (1.6) becomes... 

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