Shear deformation kinematics

In addition to the importance that attaches to rigid body motions, shearing deformations occupy a central position in the mechanics of solids. In particular, permanent deformation by either dislocation motion or twinning can be thought of as a shearing motion that can be captured kinematically in terms of a shear in a direction s on a plane with normal n. 

Rudd et al. developed a kinematic drape model. Its numerical solution with constrained fiber paths predicts fiber shear deformations and their effect on the fiber volume fraction. Bickerton et al modeled the draping of a compound curved preform and its effect on the resin flow. They validated the model results with mold filling experiments. 

Consider a slender in-plane curved beam as shown in Fig. 4. The circumferential reference frame is placed such that the s-axis is along the neutral axis of the beam, here supposed to be equidistant of the upper and lower surfaces of the beam. The beam thickness h varies in the range —h/2 < r < +/r/2 along the radial r-axis and its width lies along the z-axis. Based on the Euler-Bemoulli beam theory, which neglects shear deformation, the kinematics of a curved beam can be written as ... 

While we have considered only shearing deformations, the superposition principle applies to deformations having any kinematics. To generalize Eq. 4.3 to account for this, we need only replace the shear stress, shear strain, and shear rate by the corresponding tensorial quantities. The tensorial form of the Boltzmann superposition principle can then be used to determine... 

A fluid packet, like a solid, can experience motion in the form of translation and rotation, and strain in the form of dilatation and shear. Unlike a solid, which achieves a certain finite strain for a given stress, a fluid continues to deform. Therefore we will work in terms of a strain rate rather than a strain. We will soon derive the relationships between how forces act to move and strain a fluid. First, however, we must establish some definitions and kinematic relationships. 

Equivalent considerations for nonstatic, sheared systems demonstrate the kinematical possibility of such shearing motions. This requires, inter alia, that the distance between any two sphere centers remains larger than 2a. The static viewpoint can be generalized to such circumstances as follows Rather than considering the lattice deformation, it suffices to examine the deformed collision sphere. The latter body 3 is defined as the set of points... 

The data in this work were obtained by quenching either uniaxially stretched samples for which the low thickness of the specimens allows rapid cooling, or samples deformed in simple shear where higher thickness of the specimens was required in order to perform the scattering experiments along the three principal shear directions. For both types of flow, special devices were developed to control the flow kinematics in the molten state as well as the quenching process which freezes the molecular orientation. These devices are briefly described in the next paragraphs. 

Kinematics of Mixing Spencer and Wiley [1957] have found that the deformation of an interface, subject to large unidirectional shear, is proportional to the imposed shear, and that the proportionality factor depends on the orientation of the surface prior to deformation. Erwin [1978] developed an expression, which described the stretch of area under deformation. The stretch ratio (i.e., deformed area to initial area) is a function of the principal values of the strain tensor and the orientation of the fluid. Deformation of a plane in a fluid is a transient phenomenon. So, the Eulerian frame of deformation that is traditionally used in fluid mechanical analysis is not suitable for the general analysis of deformation of a plane, and a local Lagrangian frame is more convenient [Chella, 1994]. 

In the case of oscillatory shear experiments, for example, the strain amplitude must usually be low. For large and more rapid deformations, the linear theory has not been validated. The response to an imposed deformation depends on (1) the size of the deformation, (2) the rate of deformation, and (3) the kinematics of the deformation. 

The preliminary reduction (reconsolidation) of the samples before recovery of its natural density, was made by recovery phase composition method. Vertical load was put with earlier reached pressure in a chamber and blocked drainage under kinematic mode of loading, with a speed of 0.5% in a minute. Defended deformation characteristics were modulus of deformation and Poisson s ratio. The durability of samples was estimated by the meaning of the resistance of undrainaged shear. For their definition the soil samples were tested by putting vertical load under the set comprehensive pressure... 

Consider a steady simple shear flow with the kinematics given by u = yi3JC3, U2 = 0, Ms = 0 where the shear rate yj3 is a constant. This flow has the rate of deformation... 

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