9.1. Use Equation (9.2.) for oS in terms of oxx, oyy, and 9 to do the following.

(a) Determine the angle 9 of the planes on which oS is a maximum.

(b) Determine the orientation of these planes relative to those on which on is a maximum.

(c) Determine the normal stress, on, on the plane on which oS is a maximum.

(d) Determine the magnitude of oSmax. Express all answers in terms of oxx, oyy, and oxy.

9.2. Show that J1 and J2 in three dimensions (Equations 9.8a) reduce to Equations (9.5) in two dimensions.

9.3. We have shown (Chapter 6) that 2Sxzoxz + |Szxozx is the total work done per unit time in a unit volume of ice subjected to simple shear. It is also true that 2exxo'xx is the work done by a normal stress. Thus the total work done is

9.4. A laboratory ice deformation experiment is run using biaxial compression with applied stresses o 1 and o2 on the faces of a cube. Stresses in the third direction are atmospheric. Strain rates are S1 and s2 in the o 1 and o2 directions. Determine the effective stress and the effective strain rate.

Figure P3.

9.5. An experimental system is designed to run tests in combined uniaxial compression and simple shear (Figure P3). Determine the effective stress and effective strain rate for this stress configuration.

9.6. The third invariant of the stress tensor, J3, can be interpreted in terms of the stress configuration. To do this, we define a stress configuration parameter, Ç, by Ç = J3*. Here, J3* is a normalized value of J3 obtained by dividing all of the stresses by a constant factor, c, which you will derive below. The motivation for doing this is that it simplifies the expression for Ç. We proceed as follows.

The octahedral shear stress is defined by: a^ = 3aja^. Let us normalize ao by dividing all stresses by a constant factor, c, thus: a2 = (1 /3c2)(a/7a/7). (Here, the * is used to indicate the normalized value.) Let us further select c such that a0* = 21/6. Obtain an expression for c in terms of the second invariant of the stress tensor, J2. Because the normalized stresses must retain the sign of the original stresses, use |c| where necessary. Show the following.

(a) The deviatoric stresses in uniaxial compression under a compressive stress, a3, are: (—a3/3, —a3/3, 2a3/3), and (remembering that a3 is negative) that Ç = — 1 for this case.

(b) The deviatoric stresses in pure shear under stresses —a1, a3, are: (—a1, 0, a3), and that Ç = 0 for this case.

(c) For the stress configuration in Problem 9.5 above, obtain an expression for Ç in terms of axx and axz, and evaluate this for:

axx = -0.1 MPa, oxz = 0.1 MPa, and axx = -0.1 MPa, axz = 0MPa.

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