Strain shear

It will be recalled that a stress state at a point is completely specified by the three magnitudes of the principal normal stresses shear stresses certainly are present but their magnitudes are not additional independent quantities. Similarly with strains, shear strains exist, but they are details inside a state that is already fully specified by the three principal linear strains e, 62, and 83. 

To define a shear strain, we define two small, straight material lines inside a sample that meet at a point and at some moment form a right angle. At a later moment, let the angle they form differ from a right angle by 9 then the shear strain suffered by the pair of lines is tan 6. If the directions of the two lines are named i and j the shear strain can be written eij. If a small strain Ssij occurs in a small period of time St, the shear strain rate at some moment, Cy, can be taken as the limit of Setj/dt as f 0, and 

Here the angle 9 necessarily lies in the ij plane so that a change in d is a rotation about the k direction, and this fact is conveniently recorded by writing 0 is the linear strain rate in the i direction, and we do not use the summation convention. 

A point to be noted about eqn. (7.10) is its resemblance to other phenomenological laws where a flux is driven by a gradient, for example, Fick s first law, 

Diffusive flow rate = (material property) x (concentration gradient) 

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By analogy with Eq. (3.1), we seek a description for the relationship between stress and strain. The former is the shearing force per unit area, which we symbolize as as in Chap. 2. For shear strain we use the symbol y it is the rate of change of 7 that is involved in the definition of viscosity in Eq. (2.2). As in the analysis of tensile deformation, we write the strain AL/L, but this time AL is in the direction of the force, while L is at right angles to it. These quantities are shown in Fig. 3.6. It is convenient to describe the sample deformation in terms of the angle 6, also shown in Fig. 3.6. For distortion which is independent of time we continue to consider only the equilibrium behavior-stress and strain are proportional with proportionality constant G ... 

Criteria of Elastic Failure. Of the criteria of elastic failure which have been formulated, the two most important for ductile materials are the maximum shear stress criterion and the shear strain energy criterion. According to the former criterion, from equation 7... 

If it is assumed that yield and subsequent plastic flow of the material occurs in accordance with the maximum shear stress criterion, then /2 may be substituted for in the above and subsequent equations. For the shear strain energy criterion it may be assumed, as a first approximation, that the corresponding value is G j fz. Errors in this assumption have been discussed (11). 

Machining of metals involves extensive plastic deformation (shear strain of ca 2—8) of the work material in a narrow region ahead of the tool. High tool temperatures (ca 1000°C) and freshly generated, chemically active surfaces (underside of the chip and the machined surface) that interact extensively with the tool material, result in tool wear. There are also high mechanical and thermal stresses (often cycHc) on the tool (3). 

In this case, the shear stress is linear in the shear strain. While more physically reasonable, this is not likely to provide a satisfactory representation for the large deformation shear response of many materials either, since most materials may be expected to stiffen with deformation. Note that the hypoelastic equation of grade zero (5.117) is not invariant to the choice of indifferent stress rate, the predicted response for simple shear depending on the choice which is made. 

A number of other indifferent stress rates have been used to obtain solutions to the simple shear problem, each of which provides a different shear stress-shear strain response which has no latitude, apart from the constant Lame coefficient /r, for representing nonlinearities in the response of various materials. These different solutions have prompted a discussion in the literature regarding which indifferent stress rate is the correct one to use for large deformations. 

Orowan [7] has shown that when disloeations move with average veloeity V under the influenee of a shear stress r, the rate at whieh plastie shear strain is aeeumulated is given by... 

If the maximum resolved shear stress r and the plastic shear strain rate y are defined according to (it is assumed that the Xj and Xj directions are equivalent)... 

Steady-propagating plastic waves [20]-[22] also give some useful information on the micromechanics of high-rate plastic deformation. Of particular interest is the universality of the dependence of total strain rate on peak longitudinal stress [21]. This can also be expressed in terms of a relationship between maximum shear stress and average plastic shear strain rate in the plastic wave... 

We first consider strain localization as discussed in Section 6.1. The material deformation action is assumed to be confined to planes that are thin in comparison to their spacing d. Let the thickness of the deformation region be given by h then the amount of local plastic shear strain in the deformation is approximately Ji djh)y, where y is the macroscale plastic shear strain in the shock process. In a planar shock wave in materials of low strength y e, where e = 1 — Po/P is the volumetric strain. On the micromechanical scale y, is accommodated by the motion of dislocations, or y, bN v(z). The average separation of mobile dislocations is simply L = Every time a disloca-... 

A shear stress induces a shear strain. If a cube shears sideways by an amount w then the shear strain is defined by... 

In the same way, the shear strain is proportional to the shear stress, with... 

F(FG = normal (shear) component of force A = area u(w) = normal (shear) component of displacement o-(e ) = true tensile stress (nominal tensile strain) t(7) = true shear stress (true engineering shear strain) p(A) = external pressure (dilatation) v = Poisson s ratio = Young s modulus G = shear modulus K = bulk modulus. 

Glasses, like metals, are formed by deformation. Liquid metals have a low viscosity (about the same as that of water), and transform discontinuously to a solid when they are cast and cooled. The viscosity of glasses falls slowly and continuously as they are heated. Viscosity is defined in the way shown in Fig. 19.7. If a shear stress is applied to the hot glass, it shears at a shear strain rate 7. Then the viscosity, ij, is defined by... 

Fig. 19.7. A rotation viscometer. Rotating the inner cylinder shears the viscous glass. The torque (and thus the shear stress aj is measured for a given rotation rate (and thus shear strain rate y).

For small shear strains we can define a time-dependent compliance (reciprocal modulus) by the equation... 

Fig. 3.13 Variation in direct and shear strains for unidirectional composite loaded axially...

It is interesting to observe that as well as the expected axial and transverse strains arising from the applied axial stress, we have also a shear strain. This is because in composites we can often get coupling between the different modes of deformation. This will also be seen later where coupling between axial and flexural deformations can occur in unsymmetric laminates. Fig. 3.17 illustrates why the shear strains arise in uniaxially stressed single ply in this Example. 

Assuming that the shear strain, yr, varies linearly with radius, r, then... 

Various types of fluids, known as plastic fluids, may be encountered, which do not start to flow until a certain minimum shear stress is reached. The relationship between shear stress and the rate of shear strain may or may not take a linear form. 

With this type of fluid the viscosity decreases as the shear strain increases, typical cases being mud and liquid cement. 

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