Normal stresses deviatoric

This result may be contrasted with potential flow past a sphere, where the streamlines again have fore-and-aft symmetry but p is an even function of 9 so that there is no net pressure force (see Chapter 1). Additional drag components arise from the deviatoric normal stress ... 

Two thirds of this drag arises from skin friction, one third from form drag, and the component due to deviatoric normal stress is zero. The corresponding terminal velocity follows from Eq. (3-15) as ... 

Cd2 Cd3 coefficients of pressure drag, drag due to deviatoric normal stress, and drag due to shear stress... 

It is convenient for subsequent derivations to introduce the notion of deviatoric normal stresses, r/ = m + p, meaning the fluid mechanical normal stress plus the thermodynamic pressure ... 

This drag force consists of contributions from pressure (form drag), deviatoric normal stress, and shear stress (friction drag). 

For a Newtonian fluid in simple shearing motion, the deviatoric normal stress components are identically zero, i.e. 

For equilibrium viscometric flows of incompressible fluids (the latter is a reasonable assumption for polymer melts and solutions in most engineering circumstances), the situation is brighter. With the coordinate directions assigned as described for viscometric flows in the last chapter, the shear stresses T12 and T21 are equal (or there would be a rotational flow component), and ti3 = t3i — T32 = T23 = 0, Because ofthe arbitrary nature of the static pressure definition, two independent diflerences of the deviatoric normal stresses are commonly defined ... 

The presence of deviatoric normal stress components in steady viscometric flow implies fluid elasticity, and they are physically real quantities. The well-known fVeissenberg effect is a good example. When a vertical rotating rod is immersed... 

Normal Stresses in Shear Flow. The tendency of polymer molecules to curl-up while they are being stretched in shear flow results in normal stresses in the fluid. For example, shear flows exhibit a deviatoric stress defined by... 

This generalized yield condition, which is usually referred to as the von Mises yield condition, has a simple geometrical visualization in principal stress space of o i,o 2,o 3 shown in Fig. 3.2. There the line making equal angles with the three principal stress axes represents the locus of pure mean normal stress a, along which all deviatoric stresses vanish and no plastic flow can occur. Thus, plastic flow requires a critical deviation from this line in the radial direction away from it... 

While mechanistic models of plastic flow consider simple shear or pure shear (extensional flow where mean normal stress is absent), in experiments fundamental shear information may need to be extracted from more complex 3D flow fields. This is done through the use of multi-axial flow formalisms that are based on the von Mises approach of relating the 3D response to an equivalent ID response described in Chapter 3. In this formalism the deviatoric shear response of the multi-axial field of stress and plastic strain is taken to represent shear flow in the mechanistic context where the effect of the accompanying mean normal stress is considered through its effect on the plastic resistance. There exists an experimental procedure for extracting deviatoric plastic-response information from a tensile-flow field that accomplishes this through the use of specially contoured bars with pre-machined neck regions where the concentrated extensional flow is monitored under conditions of imposed constant deviatoric strain rates (G Sell et al. 1992). 

Of the tensorial representations of system-stress and system-strain increment, only the two invariants of deviatoric equivalent stress and strain increments and the mean normal stress were retained by this well-established procedure (Mott et al. 1993). 

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. 

Taking a cue from the experiments of Sternstein and Ongchin (1969), which demonstrated that both the deviatoric shear-stress component and the mean normal-stress component of a stress tensor affect craze initiation, in different but complementary ways, craze-initiation experiments on thin-walled tubular specimens of PS were carried out by recording the appearance of crazes as a function of time at different levels of and <7. Figure 11.2 shows the results of one such experiment for increasing surface craze density as a function of time over several thousand seconds, at 293 K under several different pairs of applied stresses consisting, e.g., of = 6.3 MPa and i at levels of 11.7, 13.6,16.2, and 18.3 MPA, with the two different forms of stress having been applied simultaneously at the start. Similar experiments were also carried out under of 4.14 and 2.1 MPa at... 

Fig. 11.2 The increase of surface craze density as a function of time in specimens under different combinations of deviatoric shear stresses and a constant mean normal stress at room temperature, with T — 293 K, <7 = 6.3 MPa, Y = 90.85 MPa, and (/) = 3.45 (from Argon (1975) courtesy of the lUPAC).

K, as well as for mean normal stresses of 4.14, 6.3, and 12.4 MPa at 253 K at different levels of simultaneously applied deviatoric stress s. In all cases craze initiation was considered complete at 10 s, when around 10 crazes per cm had formed. Table 11.1 gives the specific three pairs of a and s for 293 K and 253 K for craze initiation, together with dimensionless representations of stresses by parameters and tj, for and s normalized with appropriate uniaxial compressive yield stresses Tq = 103 MPa at 293 K and 144 MPa at 253 K, respectively. The three combinations of stress parameters and tj for crazing at 293 K and 253 K are given in Fig. 11.3, together with the information on intrinsic crazing in flawless specimens in uniaxial tension. 

Since deviatoric shear stresses and mean normal stresses are expected to play different roles in craze initiation, some conditioning experiments were also performed at 293 K as shown in Fig. 11.4(a) under a pure deviatoric shear stress So = 15.86 MPa for periods of 120, 10, 10", and 8 x 10" s. Then, a standard mean normal stress of = 4.14 MPa was applied, which initiated crazing. The response on craze initiation after the aging under so alone and upon addition of the [Pg.350]

In the experiments two nominal global stresses, namely 022, the tensile axial stress, and an, the torsional shear stress, are applied. They produce a global nominal mean normal stress Oa = t22/3 and a deviatoric shear stress = (012 + 022/3) in the surfaces of the tubular specimens. These define the two dimensionless stress parameters for the mean normal stress and the deviatoric shear stress of i = 3aa/ 2 ) and = s/ , respectively. 

Fig 11.3 Measurements of craze initiation under combinations of normalized mean normal stress and normalized deviatoric shear stress rj aXT — 293 K (O) and T — 253 ( ) with lines giving model predictions from eqs. (11.19) and (11.22) H gives the intrinsic crazing response at 293 K (from Argon (2011) courtesy of Elsevier). 

Fig. 11.5 Loci of cavitation in a porous plastic solid under combined deviatoric shear stress i and mean normal stress a for four different initial levels of porosity P (after Gurson (1977)).

In eq. (11.15) the concentrated local mean normal stress ai and deviatoric shear stress Si are in the plastic enclaves that result in craze initiation (the time taken for the formation of 10 crazes per cm in 10 s in the tension torsion experiments is taken as the mean period r of craze initiation), which needs to be stated in terms of global stresses a and s through the use of the average stress eoneentrations and of surface grooves. How this is done is discussed below in Section 11.6 comparing the model predictions with the results of the tension torsion experiments. 

However, in the mode I field, around an atomically sharp crack the concentrations of the deviatoric shear stress and the mean normal stress a are given as special cases of eqs. (12.4) as... 

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