Lagrangian strain measure

Following Hill (1978) the generalized Lagrangian strain measure is defined by... 

We can introduce the following family of n-th order Lagrangian strain measures ... 

The result (3.34) can be extended by introducing the rate of the generalized Lagrangian strain measure E ( ) ... 

For both mathematical and physical reasons, there are many instances in which the spatial variations in the field variables are sufficiently gentle to allow for an approximate treatment of the geometry of deformation in terms of linear strain measures as opposed to the description including geometric nonlinearities introduced above. In these cases, it suffices to build a kinematic description around a linearized version of the deformation measures discussed above. Note that in component form, the Lagrangian strain may be written as... 

In Eq. (46), the x(t) are the spatial (deformed) coordinates and the partial differentiation is performed with respect to the material (undeformed) coordinates. Expression (45) was introduced by Blatz et al. and Ogden, independently, who adopted the idea of a generalized strain measure to predict stress-strain relations of crosslinked samples of elastomers under various types of deformation. Representation in the principal axes system yields for the components of the Lagrangian stress tensor... 

Leaderman showed that if (A. - 1/X ) is used as a measure of the deformation, both creep and recovery, and creep curves at different load levels, can be described by a single time-dependent function. This is shown in Figures 11.5(a) and (b). The quantity (A. — 1 /X ) is the equivalent quantity to the Lagrangian strain measme in the theory of finite elasticity. 

The above description refers to a Lagrangian frame of reference in which the movement of the particle is followed along its trajectory. Instead of having a steady flow, it is possible to modulate the flow, for example sinusoidally as a function of time. At sufficiently high frequency, the molecular coil deformation will be dephased from the strain rate and the flow becomes transient even with a stagnant flow geometry. Oscillatory flow birefringence has been measured in simple shear and corresponds to some kind of frequency analysis of the flow... 

We can generalize this procedure If a function

Lagrangian description, and if

Eulerian description. The choice of the form is arbitrary but will be influenced by any advantage of a problem formulation in either description. For example, in solid mechanics, the Lagrangian description is commonly used, while in fluid mechanics the Eulerian description is popular. This is because in solid mechanics we can attach labels (e.g., visualize strain gauges at various points) on the surface of a solid body, and each material point can be easily traced from the reference state to the current state. On the other hand for a fluid we measure the velocity V or pressure p at the current position jc, therefore the Eulerian description better represents the fluid (note that for a fluid it is difficult to know the exact reference point X corresponding to all the current points jr). 

In this type of apparatus, the tensile stress is measured through the deflection of a spring associated with one of the rotating wheels. In determining with the methods described in this section, one may have to wait a long time for the stress to build up to the level where a steady-state (in both the Lagrangian and Eulerian senses) is attained because the specimen used is strain free before the test begins. 

The version of the apparatus used nowadays was introduced by Kolsky (1963), who added a second bar, from which the name Split Hopkinson Pressure Bar comes from the specimen of material to be tested is inserted between the two bars, as shown schematically in O Fig. 21.5a. The projectile, usually fired by means of a pneumatic gun, impacts the first bar (incident bar), generating the incident pulse which, at the bar/specimen interface, is partially reflected and partially propagates in the specimen. From the specimen, the pulse is transmitted to the second bar (transmitter bar). The situation is described graphically by the so-called Lagrangian diagram presented in O Fig. 21.5b. A concrete example of Split Hopkinson pressure bar is shown inO Fig. 21.6. The pulses are measured by means of strain gages placed on both incident and transmitter bar thus, their time history can be stored by means of a transient recorder, usually a digital oscilloscope or an acquisition board. From such measurements the stress (a), strain (e), and strain rate (s) in the specimen can be obtained as... 

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