Does this stress tensor satisfy the equations of equilibrium? Assume there is no body force. 0
2. In the absence of body forces, do the following stress components satisfy the equations of equilibrium? 011 = a [až +v (zł – za)], 022 = a [xí +v (zá – z)] , 033 1,033 = av (2ỉ + x3) -2av2122,013 = 031 = 0,023 = 032 = 0 012 = 021 =
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...
4 Stress Tensor 1. Given: T=14 1 01 MPa. 0 0 1 (a) find the stress vector on the plane whose normal is in the direction of +ý (b) Find the normal stress on the same plane. (c) Find the magnitude of the tangential stress on the same plane. (d) Find the three principal directions of stress. Hint: the determinant of a 3 x3 11 a12 13 det a a2 a23a11 (a22033-a3223) a12 a1a33-a31a23)+a13(a2132-a31022) 031 a32 33 Given the following...
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), )...
Question 1: Stress tensor for a parallel plate capacitor Consider an infinite parallel-plate capacitor, with the lower plate (at = = -d/2) carrying the charge density - and the upper plate (at z =d/2) carrying the charge density o. (a) Determine all nine elements of the Maxwell stress tensor (i) for -1 <d/2 and (ii) for : > d/2. (b) Use the stress tensor to find the force per unit area acting on the top plate. (C) Find the electromagnetic...
Derive the equations (velocity and shear stress equations) for flow through a cylindrical tube, but only use the cylindrical equation of motion equation (B.5-6). List all assumptions you make. Equation B.5-6: find both equations and list all the assumptions you make in order to derive both equations dz (B.5-5) (B.5-6) These equations have been written without making the assumption that r assumption is made that the stress tensor is symmetric, Tre Tor 0. is symmetric. This means, for example, that...
2. Using Maxwell's stress tensor, calculate the force per unit length of electric attraction between the two parallel wires that each carry a linear charge density separated by a distance 2a. (Hint. Integrate along a plane equidistant between the wires.]
Question 3 If the state of stress (in unit of MPa) at a point in a body is given as follows Figure 1 95x + 2y, , 10x + 10y2: o, y2 + 2 + 50; y= x+yIya = 0, T determine the body forces to satisfy the equilibrium equations ºyº *) Determine the direction cosines at a point P (2,121 h) Determine the normal stress resultant stress and shear stress on the plane whose normal is 21.2) (6 Marks)...
6. Let the components of the Cauchy stress tensor for an object B at time t be 0 CX3 T = [Tj] = | bx3 ax2 cx1 MPa, where b, c, and d are constants . - 0 -bx10 (a) Determine the body forces required if the object B is at rest for all times t. (b) At the location x - (4,7,-4), calculate the traction vector acting on the surface defined by -X1 + 2x2 + 2x3-2-0 and the...
2.8 For the stress matrix I [o] = ray (a? – y). 0 (a? - y). (73 – 3a²y/ 3 0 0 0 2az2 L where a is a constant, obtain the components of the body force vector necessary for equilibrium.