Shear elastic stress

Therefore, the rate at which chemical bonds break increases with elastic shear stressing of the material. The rupture of chemical bonds, hence fracture of material, leads to its fragmentation into particles. This reduces the average particle size in powder as fractured particles multiply into even smaller particles. Equation (1.24) points to the importance of elastic shear strains in mechanical activation of chemical bonds for particle size refinement and production of nanoparticles. 

Wake(37) and by Adams and Wake(5), and Kinloch(4) summarises the evolution of the approach of the many stress analysts. The most common shear test comprises the single lap shear joint embodied in BS 5350(10) and ASTM 01002-72(11) (Fig. 4.7(a)). With reference to Figs. 4.1(a) and 4.8 it can be seen that the resulting stress concentrations can be extremely large at the joint ends (points X and Y in Fig. 4.8(b)). The analysis of Volkersen(15) predicts that for identical adherends, the elastic shear stress concentration factor, for the adhesive due to adherend tensile strain is given by... 

Shape factor of adhesive elastic shear stress distributiou (1/m)... 

Favis [7] has summarized in a well-structured chapter the effect of the viscosity ratio, blend composition, elasticity, shear stress, and interfacial modification in immiscible blends on the final phase morphology. 

Rheological properties of foams (elasticity, plasticity, and viscosity) play an important role in foam production, transportation, and applications. In the absence of external stress, the bubbles in foams are symmetrical and the tensions of the formed foam films are balanced inside the foam and close to the walls of the vessel [929], At low external shear stresses, the bubbles deform and the deformations of the thin liquid films between them create elastic shear stresses. At a sufficiently large applied shear stress, the foam begins to flow. This stress is called the yield stress, Tq- Then, Equation 4.326 has to be replaced with the Bingham plastic model [930] ... 

If the bar is simply twisted, there is no volume change ( = 0), only distortion. (Lines scratched along the sides of the bar would be converted to helices by the strain.) This type of deformation is called shear, and the constant of proportionality between the instantaneous elastic shear stress and strain is the shear modulus, G, The shear modulus is related to Young s modulus by... 

Fig. 28.8 a Comparison of the measured size dependence of elasticity, shear stress and hardness of Cu nanostructures [77], and the nanoindentation depth dependence of b the hardness and c the correlation between the hardness and elastic modulus of Ni films [111], indicating the significance of extrinsic factors on the hardness measurement of nanocrystals but contribute little to films (Reprinted with permission from [15])... 

Figure 6.10 Schematic representation of single lap joint, (a) Unloaded, (b) loaded in tension, inextensible substrates, (c) loaded in tension, elastic substrates, and (d) distribution of elastic shear stress in the adhesive layer.

Although failure is caused by tensile stresses near the adherend corner, most of the load is transferred by the adhesive in the overlap region. It is instructive, therefore, to examine the effect of non-linear behaviour on the shear stress distributions in the overlap region. The predicted shear stress distribution for an AY103 lap joint just before failure is compared with the elastic shear stress distribution in Fig. 44. The highest shear stress near the end of the outer adherend has been reduced to 83% of the value predicted by the elastic analysis. 

Although the adhesive fails in tension at the end of the joint, most of the load is transferred in the overlap region. It is, therefore, instructive to examine the effect of non-linear behaviour on the adhesive shear stress distribution. The shear stress distributions corresponding to maximum adhesive strains of 10% and 20% in a double-lap joint are shown in Fig. 56(a). The shear stress distributions are only modified significantly at high values of maximum adhesive strain, by which time the joint is likely to have failed. The adherend cTx stress concentrations are reduced, but not significantly, by plastic flow in the adhesive. For a double-scarf joint, the elastic shear stress distributions are similarly modified by yielding of the adhesive (see Fig. 56(b)). 

The elastic shear stress distribution for a double-lap joint between 0/90/90/0 crossply adherends is compared with the shear stress distribution for unidirectional adherends in Fig. 57. The higher shear stresses at each end are caused by the lower tensile stiffness of the crossply adherends. This also produces a higher stress concentration of lOT compared with 7 3 for a similar joint between unidirectional type II CFRP adherends. 

As would be expected, the imposition of compressive, shear, or torsional stresses also evokes elastic behavior. The stress-strain characteristics at low stress levels are virtually the same for both tensile and compressive situations, to include the magnitude of the modulus of elasticity. Shear stress and strain are proportional to each other through... 

Based on these assumptions, Cox [2] derived the following equations for the tensile stress, a(x), in the fibre, and the elastic shear stress at the interface, t(x),... 

The efficiency of fibre reinforcement depends to a large extent on the maximum tensile stress that can be transferred to the fibre. The maximum value would, of course, be the yield strength or tensile strength of the fibre. The shear lag theory (Eqs 3.1-3.3) provides an analytical tool to predict the shear stresses that will develop at the interface in order to achieve this maximum tensile stress. An estimate of the maximum elastic shear stress developed for different levels of... 

The stress distribution predicted by Eg. (3.8) is essentially similar to that of the shear lag theory, with a maximum elastic shear stress at the point atwhich the fibre enters the matrix (Figure 3.5(b)). The value of this stress is ... 

Once the elastic shear strength has been exceeded (at the point of entry of the fibre into the matrix) debonding will occur. Assuming that the debonding is limited to the zone in which the elastic shear stress exceeds the adhesional shear bond strength, then the load transfer process will be made up of frictional slip at the debonded end and elastic shear transfer in the rest of the fibre. This implies that when the elastic shear bond strength is exceeded, catastrophic failure will not necessarily occur. This state of combined stress-transfer mechanisms was treated analytically by Lawrence [8], and was later reviewed and extended by others... 

Beyond this point, the stress distribution in the bonded zones is governed by elastic considerations, and can be calculated by equations such as Eqs 3.8 and 3.9, assuming a fibre with a length of I - b) and a pull-out load of P (Eq. 3.12). Thus, the interfacial elastic shear stress at the end of the debonded zone is ... 

Elastic shear stress without any debonding (i.e. tfu/tau = 0), Figure 3.16(a). This elastic curve is shown also as dotted lines in Figures 3.16(c) and (d), and is independent of the fibre strength. 

For elastic shear stress transfer (Figure 3.16(a)), and I < l., the mode of failure will be instantaneous debonding and fibre extraction. A longer embedded... 

T (max) maximum elastic shear stress at the interface t (max) interfacial elastic shear stress at the end of the debonded zone along a fibre... 

An insight into the performance of real bonded structures may only be obtained from knowledge of the engineering characteristics of the adhesive concerned and their interaction with a mathematical model of the structure itself The appropriate characteristics — shear modulus, elastic shear stress limit, asymptotic shear stress — are probably best obtained from the thick adherend shear test . Typical figures for two contrasting types are given in Table 7.3. 

Fatigue resistance is maximised by ensuring that tensile shear loads do not oscillate between compression and tension, and that the elastic shear stress limit of the adhesive is not exceeded... 

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