Virtual strain

The process is attractive for a number of reasons. Firstly, since it is a low pressure process the moulds are generally simple and relatively inexpensive. Also the moulded articles can have a very uniform thickness, can contain reinforcement, are virtually strain free and their surface can be textured if desired. The use of this moulding method is growing steadily because although the cycle times are slow compared with injection or blow moulding, it can produce very large, thick walled articles which could not be produced economically by any other technique. Wall thicknesses of 10 mm are not a problem for rotationally moulded articles. 

Only product 16 from intermediate 14 is formed.1 This is partly because 14 contains two fused virtually strain-free five-membered rings while 6 has strained bridged rings (these are by... 

Why are six-membered rings and large rings virtually strain-free ... 

The heats of combustion data show that cyclohexane is virtually strain-free. This must include strain from eclipsing interaction as well as angle strain. A model of the chair conformation of cyclohexane including all the hydrogen atoms looks like this. 

The C-0 and C-C bond lengths in tetrahydrofuran (oxolan) are 142.8 and 153.5 pm, respectively. These values are close to the corresponding bond lengths in dialkyl ethers. The ring is virtually strain-free, but not planar. There are 10 twist and 10 envelope conformations which interconvert rapidly through pseudorotation (activation energy 0.7 kJ mol ). This results in a molecule which has almost free conformational mobility (see p 68). 

We can regard the arbitrary function v as a virtual displacement 8u, then (4.4) corresponds to a null condition of the virtual displacement Su on x . We write a virtual strain as... 

This formulation of the principle of virtual work is the principle of virtual displacements, which appears in the hterature sometimes under the name of the preceding. Naturally, the virtual strain energy 6U exists only for mechanical systems with deformable parts. As the contained virtual strain tensor is assembled from derivatives of the virtual displacements, these have to be continuously differentiable. The virtual work of external impressed loads ymd includes the limiting cases of line or point loads. External reactive loads do not contribute when the virtual displacements are required to vanish at the points of action of these loads, and thus the virtual displacements have to comply with the actual geometric or displacement boimdary conditions of Eq. (3.16). With these presumptions, the initial axiom of Remark 3.1 may now be reformulated for the virtual displacements. 

Here JV is the complementary virtual work of external loads, and (JW the complementary virtual strain energy. The initial axiom of Remark 3.1 may now be reformulated for the virtual loads. 

The criteria of admissibility for the virtual displacements have been discussed in Section 3.4.2. As rigidity has been assumed in the case at hand, the occurring displacements do not cause strains. Therefore, virtual strains do not exist and, consequently, there are no contributions of internal loads to the virtual work. As expected, the virtual work of external impressed loads is identical to the term in the static principle of virtual displacements. The accelerated motion results in the additional term representing the virtual work of the loads of inertia. In general, the principle may be formulated as follows ... 

Since the forces fg and charges q A on the boundary are zero apart from their respective working surface, the surface integrals may be summarized. Then the integrands can be collated in vector form, as shown in the last line. Similarly, the virtual work of internal contributions can be formulated, where the vectors of virtual strains 6e and virtual electric field strength SE, as well as the vectors of actual stresses electric flux density D, be merged ... 

The virtual strain measures are related to the virtual displacements, just as it is given for the actual case in Eqs. (6.12), (6.13), and (6.14a). When these kinematic relations are substituted into the principle, different derivatives of the virtual displacements appear. These may be eliminated with the aid of integration by parts to summarize the contributions connected to every virtual displacement. In order to satisfy the principle, each of the resulting integrands needs to vanish. Not to be pursued here, the natural boundary conditions therewith can be determined. The sought-after equilibrium conditions in directions of the coordinates x, s, and n take the following form... 

The virtual work of internal contributions is assembled in Section 3.4.6 from the virtual strain energy and virtual work of internal charges, as supplied by the principle of virtual displacements and of virtual electric potential, respectively. In Ek[. (3.63), the virtual work of internal contributions is given for a volumetric object. The preceding analysis accomplished a reduction to two dimensions for the sheU-like wall and to one dimension for the beam. Consequently, the expression for the virtual work of internal contributions may be reformulated for the wall SLi t) and for the beam SLi t) as follows ... 

As shown here, the strains may be split into the linear part Se(x) and the non-linear part S (x). The prior corresponds to the variation of the linear strain measures as they are obtained for the thin-walled beam in Section 7.2 and given by Eq. (7.31). Further on, the internal loads vector N x,t) of the beam can be subdivided into the portion N x,t), associated with the initial configuration, and the portion N x,t) related to the superposed deformation. Then the virtual strain energy based on the general formulation of... 

In the last line of Eq. (8.40), Eq. (8.39) is introduced and expressions are multiplied out. The first term represents the linearized virtual strain energy (t). The initial internal loads vector N x,t) can be determined in advance, while the vector N x,t) of the other internal loads needs to be substituted with the aid of a constitutive relation. Therefore, the second term is free of non-linear products, while the third term contains such products and, consequently, will be neglected. Such a second-order theory corresponds to the equilibrium of the slightly deformed system and contributes the virtual work of initial stresses Thus, the virtual work of internal mechanical... 

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