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20) they can not all be zero. 23) For a given set of the nine stress components, the preceding equation constitutes a cubic equation for the three unknown magnitudes of λ. 23 Cauchy was first to show that since the matrix is symmetric and has real elements, the roots are all real numbers. 24 The three lambdas correspond to the three principal stresses σ(1) > σ(2) > σ(3) . When any one of them is substituted for λ in the three equations in Eq. 19 those equations reduce to only two independent linear equations, which must be solved together with the quadratic Eq.

1. 1: Elongation of an Axial Rod We seek to quantify the deformation of the rod and even though we only have 2 variables (l0 and l), there are different possibilities to introduce the notion of strain. 1) λ≡ l0 3 This stretch is one in the undeformed case, and greater than one when the rod is elongated. 5) we note the strong analogy between the Lagrangian and the engineering strain on the one hand, and the Eulerian and the natural strain on the other. The choice of which strain definition to use is related to the stress-strain relation (or constitutive law) that we will later adopt.

37) Mechanics of Materials II Draft 10 MATHEMATICAL PRELIMINARIES; Part II VECTOR DIFFERENTIATION that is [∇v]ij gives the rate of change of the ith component of v with respect to the jth coordinate axis. Note the diference between v∇x and ∇x v. In matrix representation, one is the transpose of the other. 22 23 The gradient of a vector is a tensor of order 2. We can interpret the gradient of a vector geometrically, Fig. 10. e ∆s is very small), and let the unit vector m points in the direction from a to b.