Theoretical shear strength

At small displacement values, the material will be elastic and, thus, Hooke s Law can be used to describe the stress-displacement behavior in this region. Using sinx x for small angles and the shear strain= / /, one obtains 

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The edge dislocation moves easily on its glide plane perpendicular to s under the influence of a shearing force. This force is well below the theoretical shear strength of a perfect crystal since not all of the atoms of a glide plane perform their slip at... 

Dislocations as introduced in Section 3.2 have been postulated in order to explain the low yield strength of a crystal (if we compare it to the theoretical shear strength). Likewise, cracks are postulated in order to explain fracturing of crystals well below the theoretical tensile strength of the atomic bonds. Pre-existing cracks can easily magnify low applied stresses at their tips to the maximal atomic bond strength. 

If a crystal lattice were perfect (i.e., contained no defects), slip would have to occur by the simultaneous movement of an entire plane of atoms over another plane of atoms. Frenkel (1926) developed a simple model that calculates the yield stress in a perfect crystal as G/2tt, where G is the shear modulus of the material. Actual yield stresses are two to four orders of magnitude lower than this. The concept of a dislocation was later introduced to explain the discrepancy between the measured and theoretical shear strengths of material. The dislocation... 

Later calculations (MacKenzie 1949) show that the theoretical shear strength is probably about t/15. 

Theoretical calculations have shown that the ratio of theoretical shear strength to tensile strength diminishes as one proceeds from covalent to ionic to metallic bonds. For metals, the intrinsic shear strength is so low that flow at ambient temperatures is almost inevitable. Conversely, for covalent materials such as diamond and SiC, the opposite is true the exceptionally rigid tetrahedral bonds would rather extend in a mode 1 type of crack than shear. 

For the low-temperature region, several dislocation-related hypotheses were proposed to explain the mechanism of plastic deformation. Trefilov and Mil-man [76] suggested that during indentation the theoretical shear strength was... 

Figure 6.3 The stress variation for shear in a perfect crystal can be assumed to be approximately sinusoidal. The maximum shear stress is termed the theoretical shear strength.

Theoretical shear strength decreases with increasing unit slip distance and decreasing interplanar spacing. True or False ... 

Further on, the rightfulness of application of the structural defect concept to polymers yielding process description will be considered. As a rule, previously assumed concepts of defects in polymers were primarily used for the description of this process or even exclusively for this purpose [4—11]. Theoretical shear strength of crystals was first calculated by Frenkel, basing on a simple model of two atoms series, displaced in relation to one another by the shear stress (Fig. 4.1a) [3]. According to this model, critical shear stress Tg is expressed as follows [3] ... 

A Starting point for the discussion is the Frenkel argument for predicting the maximum theoretical shear strength of a crystal [73,76], For a simple lattice of identical atoms with a repeat distance b in the direction of shear on planes separated by a distance h it can most simply be considered that the shear stress r follows a sine curve with shear displacement x to give... 

The shear modulus for A1 is 40 GPa which translates into an estimated theoretical shear strength of 10 GPa. The actual yield strength of 7075 T-6 A1 is 500 MPa, about 1/20 of its estimated theoretical value. For pure A1 (1100-0 annealed) the yield strength is only... 

In a face-centred cubic metal h = a/VS and b = /V6 where a is the lattice parameter and so the theoretical shear strength is predicted to be (Tu—G/9. For chain direction slip on the (020) planes in the polyethylene crystal b = 2.54 A and h is equal to the separation of the (020) planes which is 2.47 A. The theoretical shear stress would be expected to be of the order of G/6. However, more sophisticated calculations of Equation (5.31) lead to lower estimates of which come out to be of the order of G/30 for most materials. Even so, this estimate of the stress required to shear the structure is very much higher than the values that are normally measured and the discrepancy is due to the presence of defects such as dislocations within the crystals. The high values are only realized for certain crystal whiskers and other perfect crystals. 

It is well established that slip processes in atomic solids are associated with the motion of dislocations. The theoretical shear strength of a metal lattice has been shown (Section 5.1.4) to be of the order of G/10 to G/30 where G... 

Frenkel S computed the force required to shear two planes of atoms past each other in a perfect crystal and showed that the critical yield stress (or elastic limit) is of the order G/ln, where G is the shear modulus. Experimental values of the elastic limit are 1(X)-1(K)0 times smaller than the above estimate. By considering the form of the interatomic forces and other configurations of mechanical stability, the theoretical shear strength could be reduced to G/30, still well above the observed values in ordinary materials. It is now firmly established that crystalline imperfections, such as dislocations, microscopic cracks, and surface irregularities, are primarily the reasons for the observed mechanical weakness of crystalline solids. This aspect of the mechanical behavior of solids, including a discussion of strengthening mechanisms, is discussed in Volume 2, Chapter 7. In this section the chemical and structural aspects of mechanical behavior, i.e., bonding and crystal structure, are emphasized. 

The theoretical shear strength of crystals is G/3 to G/30, where G is the shear modulus the actual shear strength of pure metal... 

The theoretical shear strength of ice is about 14 x lO psi, while glaciers deform at stresses as low as 2 psi. However, a number of non-metallic crystals with high or rather high melting points are ductile at room temperature. Examples are NaCl, AgCl, and MgO. The critical resolved shear stress of NaCl is about only 100 psi, while the theoretical strength is about 10 x lO" psi. 

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