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diflerence and N2 the second normul stress diflerence. Some authors use the difference TII - T33. However, there are only two independent quantities because TI - T33 is just the sum of the other two. The reader should also be aware that other notations for stress are common: P or II for T and u or T’ for T. Also, several authors use the opposite sign for T and T (See Bird et al., 1987a, p. 7, who consider compression, eq. 1.2.40, to be positive). It is perhaps consoling to the student struggling with the stress tensor to learn that although Hooke wrote his force extension law before 1700, it took many small and painful steps until Cauchy in the 1820s was able to write the full three-dimensional state of stress at a point in a material. 1.3 Principal Stresses and Invariants * Later in this and subsequent chapters we will want to make consti- tutive equations independent of the coordinate system. In particular we will need to make scalar rheological parameters like the modu- lus or viscosity a function of a tensor. How can a scalar depend on a tensor? Let us start by con- sidering a simpler but similar problem: How does a scalar depend on a vector? In particular, consider how scalar kinetic energy de- pends on the vector velocity. Recall the equation for kinetic energy E K = 1/2mu2, where u2 = v v. Kinetic energy is a function of the dot or scalar product of the velocity vector, the magnitude of the vector squared. Thus, v v is independent of the coordinate system; it is the invariant of the vector v. There is only one commonly used invariant of a vector: its magnitude. However there are three possible invariant scalar func- tions of a tensor. For the stress tensor we can give these three invariants physical meaning through the principal stresses. It is always possible to take a special cut through a body such that only a normal stress acts on the plane through the point P. This is called a principal plane, and the stress acting on it is a principal stress u. As demonstrated below, there are three of these planes through any point and three principal stresses. We can visualize the principal stresses in terms of a stress ellipsoid. The surface of th is ellipsoid is found by the locus of the end of the traction vector t, from P when fi takes all possible directions. The three axes of the ellipsoid are the three principal *The reader may skip to Section 1.4 on afirst reading. The concept of invariants is used in Section 1.6. 20 / RHEOLOGY Figure 1.3.1. (a) A section of the stress ellipsoid at P through two principal axes a,& and a&2. (b) The stress ellipsoid for a hydrostatic state of stress. stresses and their directions the principal directions. A section of such an ellipsoid through two of the axes is shown in Figure 1.3.1. Note that in the simplest case all the principal stresses are equal: u1 = u2 = 0 3 = u. This equivalence represents the hydro- static pressure p = -0. As we saw at the end of the Section 1.2.3, a hydrostatic state is the only kind of stress that can exist in a fluid at rest. If we line up our coordinate system with the three principal stresses, all the shear components in the stress tensor will vanish. This is nice because it reduces the stress tensor to just three diagonal components: components of the principal stress tensor = T; 0 0 0 0 u3 (1.3.1) However, in practice it is often difficult to figure out the rota- tions of the coordinate system at every point in the material, so as to line it up with the principal directions. Furthermore, it is usually more convenient to leave the coordinates in the laboratory frame. Thus, we normally do not measure the principal stresses (except for purely extensional deformations) but rather calculate them from the measured stress tensor.* We show this next. Because a principal plane is defined as one on which there is only a normal stress, the traction vector and the unit normal to that plane must be in the same direction: t, =ah (1.3.2) *An exception ispow birefringence where differences in the principal stresses and their angle of rotation are measured directly; see Section 9.4. ELASTICSOLID / 21 Thus, a is the magnitude of the principal stress and h its direction. As we saw earlier (eq. 1.2.10), the stress tensor is the machine that gives us the traction vector on any plane through the dot operation. Thus, t ,= f i .T=af i (1.3.3) This equation can be rearranged to give fi . (T - 01) = 0 or ni(Tij - aZij) = 0 (1.3.4) Since fi is not zero, to solve this equation we need to find values of a such that the determinant of T - a1 vanishes. This is usually called an eigenvalue problem. TI1 - 0 TI2 TI 3 [ T31 T32 T33 - det(T-aI) = det T2l T22 -a T23 ] = 0 Expanding t h i s determinant yields the characteristic equation of the matrix a3 - I T a 2 + IITCJ - IIIT = 0 (1.3.5) where the coefficients are IT is called the first invariant of the tensor T, IIT the second invari- ant, and IIIT the third invariant. They are called invariants because no matter what coordinate systems we choose to express T, they will retain the same value. We will see that this property is par- ticularly helpful in writing constitutive equations. Note that other combinations of I;:j can be used to define invariants (cf. Bird et al., 1987a, p. 568). Equation 1.3.5 is a cubic and will have three roots, the eigen- values 01, a2, and 0 3 . If the tensor is symmetric all these roots will be real. The roots are then the principal values of Tij and ni, the principal directions. With them T can be transformed to a new tensor such that it will have only three diagonal components, the principal stress tensor, eq. 1.3.1. To help illustrate the use of eq. 1.3.5 to determine the principal stresses, consider Example 1.3.1. 22 I RHEOLOGY Example 1.3.1 Principal Stresses and Invariants Determine the invariants and the magnitudes and directions of the principal stresses for the stress tensor given in Example 1.2.2. Check the values for the invariants using the principal stress mag- nitudes. Solution For eq. 1.2.32 we obtain Using eqs. 1.3.6-1.3.8 we can calculate the invariants IT = trT = 7 I ZZT = -(Z; - trT2) = 14 2 111~ = detT = 8 ( 1.3.10) From eq. 1.3.5 we can find the principal stress magnitudes: a3 - 7a2 + 14a - 8 = 0, which factors into (a - l ) (a - 2)(a - 4) = 0 Thus a1=1 a 2 = 2 a 3 = 4 (1.3.11) Clearly most cases will not factor so easily, but the cubic can be solved by simple numerical methods. We can check the values for the invariants using these ai : IT = + f a3 = 7 IIT = a102 + a1a3 -b 0 2 0 3 = 14 (1.3.12) IIIT = a l ~ 2 ~ 3 = 8 To obtain the principal directions, we seek r,(i), which are solutions to For each principal magnitude eq. 1.3.13 results in three equations for the three components of each principal direction. These three sets of equations are solved for directions of unit length as follows: Thus the principal directions are where n(’) is rotated + 45” from the i 2 axis.* 1.4 Finite Deformation Tensors Now that we have a way to determine the state of stress at any point in a material by using the stress tensor, we need a similar meas- ure of deformation to complete our three-dimensional constitutive equation for elastic solids. Consider the small lump of material shown in Figure 1.4.1. We have drawn a cube, but any lump will do. P is a point embed- ded in the body and Q is a neighboring point separated by a small distance dx’. Note that dx’ is a vector. The area vector da‘ repre- sents a small patch of area around Q. We use the ’ to denote the rest or reference state of the material; or, if the material is continually deforming, the ’ denotes the state of the material at some past time, t’. From here on we concentrate on deformations from a rest state. In the following chapters we treat continual deformation with time. Now let the body