Resolved shear stress defined

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)... 

The critical resolved shear stress, Terss, is the minimum shear stress required to initiate slip for a particular slip system defined when a = 0/. 

Because of chain inextensibility, the shear rate of any slip system is not dependent on the normal-stress component in the chain direction (Parks and Ahzi 1990). This renders the crystalline lamellae rigid in the chain direction. To cope with this problem operationally, and to prevent global locking-up of deformation, a special modification is introduced to truncate the stress tensor in the chain direction c. Thus, we denote by S° this modification of the deviatoric Cauchy stress tensor S in the crystalline lamella to have a zero normal component in the chain direction, i.e., by requiring that 5 c,c = 0, where c,- and c,- are components of the c vector (Lee et al. 1993a). The resolved shear stress in the slip system a can then be expressed as r = where R is the symmetrical traceless Schmid tensor of stress resolution associated with the slip system a. The components of the symmetrical part of the Schmid tensor / , can be defined as = Ksfw" + fs ), where if and nj are the unit-vector components of the slip direction and the slip-plane normal of the given slip system a, respectively. 

The authors [6] suggest the term shearability , Sm, for the maximum shear strain that a homogeneous crystal can withstand. It is defined by Sm = argmax o-(s), where a(s), is the resolved shear stress and s is the engineering shear strain in a specified slip system. The relaxed shear stress, (7 in Table 4.2 is normalized by Gr. In this table, experimental and calculated values of the relaxed shear vales of Gr are given. For details on these calculations, refer to the work of Ogata et al. [6]. 

Plastic deformation is produced by the shear stresses set up in the material and another basic law of plastic deformation, applicable most simply to single crystals, is that the plastic strain depends only on the shear stress in the slip plane, resolved parallel to the slip direction. Further, appreciable plastic deformation by slip starts when this resolved shear stress reaches a fairly well-defined value called the critical resolved shear stress (c.r.s.s.). This is Schmidt s law and it embodies the result that the yield stress is a characteristic of a given material, other conditions being the same. 

Normal stresses For the exact definition of shear stresses and normal stresses, we use the illustration of the stress components given in Fig. 15.3. The stress vector t on a body in a Cartesian coordinate system can be resolved into three stress vectors h perpendicular to the three coordinate planes In this figure t2 the stress vector on the plane perpendicular to the x2-direction. It has components 21/ 22 and T23 in the X, x2 and x3-direction, respectively. In general, the stress component Tjj is defined as the component of the stress vector h (i.e. the stress vector on a plane perpendicular to the Xj-direction) in the Xj-direction. Hence, the first index points to the normal of the plane the stress vector acts on and the second index to the direction of the stress component. For i = j the stress... 

A close look at the stress state indueed within this joint indicates clearly a complex interaction of strain, and there are many useful commentaries on the limitations of this test(4, 5. 25, 31). If the adherend and adhesive moduli are very different (e.g. the ratio of epoxy to steel moduli, E E = 40), the axial strains in the adhesive will be about 40 times greater than those in the adherend with a similar ratio for the lateral (Poisson s) strains. Where the two materials join, this conflict is resolved by generating large interfacial radial shear stresses (Fig. 4.11(c)). Joint strength inereases with a decrease in adhesive thickness, and in a thin bondline affected completely by adherend restraint a eomplex stress arises. The ratio of the applied stress to the strain across the adhesive is then defined as the apparent or constrained Young s modulus, a(5)-... 

The state of stress or strain within a test piece may be defined by considering an infinitesimally small cubic volume element with edges parallel to orthogonal axes x, y and z. The components of stress (defined as force per unit area) acting normally on planes perpendicular to the directions x, y and z are customarily represented by yy and respectively. The stresses acting on planes perpendicular to x and in the direction of z are the shear stress components = a y and The stress-induced displacement of a point (x,y,z) within a material can be resolved into components M, V and w parallel to the axes x, y and z. The normal strains are defined by... 

The force acting on any differential segment of a surface can be represented as a vector. The orientation of the surface itself can be defined by an outward-normal unit vector, called n. This force vector, indeed any vector, has direction and magnitude, which can be resolved into components in various ways. Normally the components are taken to align with coordinate directions. The force vector itself, of course, is independent of the particular representation. In fluid flow the force on a surface is caused by the compressive (or expansive) and shearing actions of the fluid as it flows. Thermodynamic pressure also acts to exert force on a surface. By definition, stress is a force per unit area. On any surface where a force acts, a stress vector can also be defined. Like the force the stress vector can be represented by components in various ways. 

The forces and stresses applied to a body may be resolved in three vectors, one normal to an arbitrarily selected element of area and two tangential. For the yz plane, the stress vectors are a, and on, a, respectively. Six analogous stresses exist for tile other orthogonal orientations, giving a total of nine quantities, of which three exist as commutative pairs (arl = crSr). The state of stress, therefore, is defined by three tensiic or normal components (pxx,oyy. o--f) and three shear or tangential components (crIy,cr.I ,CTy,), The shear components are most readily applicable to the determination of jj and G,... 

The mechanism(s) of a particulate fluid electroviscous effect is still not fully resolved and quantified. It is not strictly relevant to this work and is therefore not dealt with in detail. At this stage it can only be said that it is a very multi-parameter and multidisciplinary event and, secondly, it should be understood that there is little change in the viscosity p of the fluid as it is normally defined in its continuum context save for a derived effective or non Newtonian viscosity sense. The term electroviscous, which has often been used to describe the present class of fluids, is misleading in this case. Rather, the held imposes a yield stress type of property on the fluid which is similar to, but not the same as, that which is a feature of the ideal Bingham plastic. This can readily be seen by referring to Figs. 6.63 to 6.66 inclusive. It is alternatively possible to claim that either the plastic viscosity changes with shear rate or the electrode surface yield stress does. 

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