I. (50%) For an isotropic material, use E, v, and G as the starting point, where...
1. For an isotropic material, E and v are often chosen as the two independent engineering constants. There are other elastic constants that we have introduced in Lecture 5: the shear modulus G, the bulk modulus K, and the Lames, constants, μ and λ·Suppose that Prove that μ and λ, and K, satisfy the following relations: 2vG 2 3(1- 2v) To prove it, apply pure shear stress t, and connect it to pure shear strain γ by G, ie. τ_Gy....
For an isotropic material, (a) Calculate the components of the strain tensor and the stress tensor for the following set of given displacements for an isotropic material: Uj = - X1 , U2 = -V – X2, U3 = -V – X3 , E E E where o is a constant. (b) Check the equilibrium equations to see if they are satisfied for zero body forces. (c) Show the edge tractions on a diagram of the body0 S XL SL,0...
graph G, let Bi(G) max{IS|: SC V(G) and Vu, v E S, d(u, v) 2 i}, 10. (7 points) Given a where d(u, v) is the length of a shortest path between u and v. (a) (0.5 point) What is B1(G)? (b) (1.5 points) Let Pn be the path with n vertices. What is B;(Pn)? (c) (2 points) Show that if G is an n-vertex 3-regular graph, then B2(G) < . Further- more, find a 3-regular graph H such that...
Let G = (V, E) be a finite graph. We will use a few definitions for the statement of this problem. The Tutte polynomial is defined as the polynomial in 2 variables, 2 and y, given by: Definition 1 Tg(x,y) = (x - 1)*(A)-k(E)(y - 1)*(A)+|A1-1V1 ACE where for A CE, k(A) is the number of connected components of the graph (V, A). For this problem we will need the following definition: Definition 2 (Acyclic Graph) A graph is called...
Problem 1: (a)A thick-walled cylinder, made from a homogeneous and isotropic elastic material, has an inner radius a and outer radius b. The cylinder is subjected to an internal pressure pi, and is under plane stress conditions. In this case the displacement field is of interest is given by and the stress field of interest is given by C2 C2 where, with (E and v) denoting the Young's modulus and the Poisson's Ratio. EA EB Show that b2 b2 (b/a)2-1)...
1. Consider the following displacement field in an isotropic linearly elastic solid descri terms of the Young's modulus E and Poisson's ratio v: (a) Determine the stress tensor (matrix), ) Is the state of stress a possible equilibrium stress field? Neglect body forces. 10 point 2. The assemblv shown in 1. Consider the following displacement field in an isotropic linearly elastic solid descri terms of the Young's modulus E and Poisson's ratio v: (a) Determine the stress tensor (matrix), )...
4. R(A, B, C, D, E, F, G, H, I, J) where A → B, C, D BE F→ G, H, I (A, F) → B, C, D, E, G, H, I, J For each of the following relations, normalize it into a set of BCNF relations.
Automata: solve a - e 2. (10+10+10+10+10-50 points) Agrammar is a 4-tuple G, G-ON,E,11,L$) where N is a finite set of nonterminal symbols Σ is a finite set of terminal symbols is a finite set of rules S is the starting symbol Let N- (S, T s-{a, b, c} s-> ab aT >aaTb aT-ac S is the starting symbol. (a 10 points) Prove that the given grammar G is a context sensitive grammar. (b-10 points) What is the language L-...
Use source transformation to find I, in the given circuit where V = 20 V. 12 192 w h 422 4.2 + V- { 32 V The current 1, in the given circuit is
#4 (ii) Use (i) (otherwise no credit will be given!) to show that cos(20) = 2 cos - 1. 3. Prove that if z + is real, then z is real or |z| = 1. 4. Identify all the point in the complex plane which satisfy the following relations: (i) z2 = 2(2-1), 12-11 212 +11, (iii) z +11 <4 - 12 - 11.