Shear deformation theory

Ouyang, Z., Li, G., Ibekwe, S.I., Stubblefield, M.A., and Pang, S.S. (2010) Crack initiation process of DCB specimens based on first order shear deformation theory. Journal of Reinforced Plastics and Composites, 29, 651-663. 

More-over the vibration analysis of MWCNTs were implemented by Aydogdu [73] using generalized shear deformation beam theory (GSD-BT). Parabolic shear deformation theory (PSDT) was used in the specific solutions and the results showed remarkable difference between PSDT and Euler beam theory and also the importance of vdW force presence for small inner radius. Lei et al. [74] have presented a theoretical vibration analysis of the radial breathing mode (RBM) of DWCNTs subjected to pressure based on elastic continuum model. It was shown that the frequency of RBM increases perspicuously as the pressure increases under different conditions. 

Natsuki Toshiaki, Ni Qing-Qing, Endo Morinobu. (2008). Analysis of the Vibration Characteristics of Double-Walled Carbon Nanotubes. Carbon, 46, 1570-1573. Aydogdu Metin. (2008). Vibration of Multi-Walled Carbon Nanotubes by Generalized Shear Deformation Theory. Int. J. Mech. Sci., 50, 837-844. 

Obviously, the classical lamination theory stresses in Pagano s example converge to the exact solution much more rapidly than do the displacements as the span-to-thickness ratio increases. The stress errors are on the order of 10% or less for S as low as 20. The displacements are severely underestimated for S between 4 and 30, which are common values for laboratory characterization specimens. Thus, a practical means of accounting for transverse shearing deformations is required. That objective is attacked in the next section. 

The results shown in Figure 6-21 for the present shear-deformation approach versus classical lamination theory are quite similar qualitatively to the comparison between the exact cylindrical bending solution and classical lamination theory in Figure 6-17. 

Whitney and Pagano [6-32] extended Yang, Norris, and Stavsky s work [6-33] to the treatment of coupling between bending and extension. Whitney uses a higher order stress theory to obtain improved predictions of a, and and displacements at low width-to-thickness ratios [6-34], Meissner used his variational theorem to derive a consistent set of equations for inclusion of transverse shearing deformation effects in symmetrically laminated plates [6-35]. Finally, Ambartsumyan extended his treatment of transverse shearing deformation effects from plates to shells [6-36]. 

The Maxwell construction would determine the condition of two phase coexistence or the points on the curves where the first-order phase change occurs [6,7]. It is the condition that the two phases have the same value of g or j d II = 0 from Eq. (2.6) at zero osmotic pressure, v2 and vx being the values of v in the two phases. However, this criterion is questionable in the case Kcritical point). This is because the shear deformation energy has not been taken into account in the above theory. See Sect. 8 for further comments on this aspect. 

As in the simple 3- or 4-point bending of a beam the vibrating reed device assumes the validity of the differential Eq. (2.2) which is due to Euler. Timoshenko25 included both rotary inertia and shear deformation deriving a more exact differential equation which reduces to the Euler equation as a special case. Use of the Timoshenko beam theory for anisotropic materials has been made by Ritchie et al.26 who derive a pair of equations for torsion-flexure coupling (which will always occur unless the axis of the beam coincides with the symmetry axis of the anisotropic material). 

A shear deformation, of course, takes place also when the coils are not sorbed on the gel in this case, ASdef cannot be calculated by a simple theory, but the results would hardly be very different from that obtained from Eq. (36) with n = 1... 

It should be mentioned that according to some theories the elastic shear deformation should be defined as yc (Tn T22)/(2T2i). 

A comparison between the theory and experiment [33] for PVAc under shear deformation is made out in Figs. 9 and 10 where the master creep curve and its... 

As mentioned above, when the transverse dimensions of the beam are of the same order of magnitude as the length, the simple beam theory must be corrected to introduce the effects of the shear stresses, deformations, and rotary inertia. The theory becomes inadequate for the high frequency modes and for highly anisotropic materials, where large errors can be produced by neglecting shear deformations. This problem was addressed by Timoshenko et al. (7) for the elastic case starting from the balance equations of the respective moments and transverse forces on a beam element. Here the main lines of Timoshenko et al. s approach are followed to solve the viscoelastic counterpart problem. 

To satisfy the theory s assumption, it was postulated that the material be isotropic, frictional, cohesive, and compressible. Fig. 2 depicts material in a press that undergoes shear deformation into a solid mass. ... 

Determination of Stress Intensity. The accurate determination of the elevation of stress intensity due to the crack-path obstacles in the specific DCB specimens requires taking note of the special features of the specimen and its mode of separation. The simple beam theory solution does not account for shear deformations or the compliance of the built-in cantilever ends. A... 

The unique cellular morphologies of foams play a key role in determining their deformation mechanisms [51. They also allow the development of very simple alternative equations based on the mechanical models of beam theory (a branch of civil engineering) combined with scaling concepts, to estimate both the thermoelastic properties and the strengths of foams. Such simple relationships have been developed for foams manifesting elastomeric, elastic-plastic and elastic-brittle responses to mechanical defonnation. While much of this work has focused on the responses of foams to compressive defonnation because of the special importance of this deformation mode in many applications of foams, the responses of foams to tensile and shear deformation have also been considered within this theoretical framework. 

General theory of mixing usually considers a non-random or segregated mass of at least two components and their deformation by a laminar or shearing deformation process. The object of the shearing is to mix the mass in such a way that samples taken from the mass exhibit minimal variations, ultimately tending to be zero. 

GSDBT Generalized Shear Deformation Beam Theory... 

This derivation of the ideal gas law is very close in all its steps to the theory developed in Sections 3.3 and 3.4 for the elasticity of rubber. In the case of rubber, which is essentially inconq>ressible, the deforming forces cause the specimen to change shape the deformation is shear deformation with no change in volume. 

The linear elastic stress analysis of a short fibre composite by the shear lag theory is due originally to H. L. Cox. The real composite (see Figure 6.32(a)) is assumed to deform in the vicinity of any particular fibre as if it were the model tem shown in Figure 6.32(b). In the model, the fibre (of diameter d) is surrounded 1 a cylinder of matrix of radius R, embedded in a homogeneous block of material which deforms as the composite as a... 

Fibre stress a, Is evaluated vrith the shear lag theory, (a) The mean fibre separation is labelled 2R. (b) The composite in the vicinity of the fibre is modelled as a cylinder of matrix with radius R surrounding the fibre, embedded in a block which deforms as the composite as a whole, (c) Within the cylinder of matrix, a shear stress r acts at radius r. 

The K-BKZ Theory Comparison with Experiment. The first data required to test the K-BKZ model is single-step stress relaxation data to determine the material parameters of interest. This is best seen from the following example for a simple shearing history. From equation 49, the shear stress for a simple shear deformation can be expressed as (see Ref 72)... 

K. Osaki, S. Kimura, and M. Kurata, Relaxation of Shear and Normal Stresses in Step-Shear Deformation of a Polystso ene Solution. Comparison with the Predictions of the Doi-Edwards Theory J. Polym. ScL, Polym. Phys. Ed. 19, 517—527... 

During load transfer, large shear stresses are transmitted by the adhesive layer to the adherend surfaces adjacent to the adhesive layer, which entails that shear stress equilibrium at the interface is maintained. These shear stresses trigger adherend shear deformations. Shear stresses are especially significant for adherends with relatively low transverse shear modulus, such as in the case of laminated FRP composites. This theory assumes a linear shear stress distribution through the thickness of the adherend, whereas... 

Based on the same notations of Fig. 10.11, the general solution for the adhesive shear stress governing equation (i.e. equation [10.13] below) is identieal to that of Hart-Smith s general solution. However, the exponent of the elastie shear stress distribution (A) in Hart-Smith s (1973) theory is ealled the elongation parameter in Tsai et a/. s (1998) theory, although its mathematical formula is identical (refer to equation [10.7]). It appears in the general equation for Hart-Smith s solution instead of the current solution s P parameter of equation [10.15], whieh is the eore difference between both theories. The parameter P is redefined by two parameters X and o) (refer to equation [10.15]). The latter is the shear deformation parameter which accounts for the shear deformations of the adherend in its mathematical formula (i.e. equation [10.14]). 

It appears that the first three-point, beam bending tests on structural-grade pultruded GFRP profiles were reported by Sims et al. (1987) and Bank (1989). The purpose of these tests was not to demonstrate directly that linear, first-order shear deformation beam theory could be used to predict the flexural... 

There have been a number of other investigations concerned with testing pultruded GFRP beams with various end support conditions. The primary purpose of these tests was to demonstrate that first-order shear deformation beam theory may be used to predict their small deflection response and to quantify its accuracy relative to classical shear-rigid beam theory. 

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