Inverse deformation tensors

Evaluate the inverse deformation tensors for the deformations given in Example 1.4.1. 

Inverse Deformation Tensors (a) Show that the inverse of the Finger tensor operates on unit vectors in the deformed state to give inverse square of length change ... 

Consequently, E has components relative to the reference configuration, and is a referential strain tensor. A complementary strain tensor may be defined from the inverse deformation gradient F ... 

The Cauchy tensor is a device for describing the change in length at any point of the material body. Equation (5.29) indicates that in performing such a function the Cauchy tensor operates on the unit vector u at some time t in the past. However, the stress tensor is always measured with respect to the present form of the material body or the deformed state. Thus, there is the need to find a deformation tensor that operates on the unit vector at the present time. This can be done by using the inverse of E, E to express dX in terms of dX, namely. 

Using the inverse of the deformation gradient we can also define inverses of the deformation tensors B and C... 

In accordance with the experiment, the transverse compression strains have an inverse sign to the longitudinal extensions, and their contribution, v, is on the order of 0.2-0.3. These two parameters, that is, the modulus with the units of stress (Pa), [E] = [o], and dimensionless Poisson ratio v are sufficient for describing arbitrary deformations of an isotropic solid. It is worth pointing out here that two parameters (E and v, G and v, or G and E) are the smallest pieces of fourth order tensors that are sufficient for the description of strains. It is not possible for one to get away with using a single parameter. Eor an arbitrary principal stress tensor. 

In this case inversion of the Cauchy tensor, which is necessary when deriving the Finger tensor, is trivial. For non-orthogonal deformations this is more complicated, and here one can make use of a direct expression for the components of B which reads... 

We need to note that mathematically the splay and bending deformations are vectors, i.e., the flexoelectric coupling constant is a second-rank tensor. Accordingly, the flexoelectricity should not be confused with piezoelectricity, which is described by a third-rank tensor coupling constant. Piezoelectricity requires lack of inversion s)mtimetry of the phase, whereas flexoelectricity can exist in materials with inversion symmetry, such as in nematics with macroscopic symmetry. 

Long-period moment-tensor inversion is a robust method to characterize the geometry and size of earthquakes worldwide. The Unear relationship between the amplitude of long-period seismograms and the seismic moment leads to precise and accurate measurements of earthquake size. Catalogs of moment tensors provide primary data on active faults and the geometry of regional seismotectonic deformation. 

Full moment tensors describing the inelastic deformation in the source region can be obtained by waveform inversion, but data from VT earthquakes are often too sparse for such methods, so... 

It is seen in Eq. (2.53) that the velocity gradient tensor L(t) can be determined from the rate of deformation gradient tensor t(t) and the inverse of the deformation gradient tensor that is,... 

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