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Solid Mechanics 3

Assessment Task 1a - 2013

Milestone 1 (Due by week 3)

Question 1.

For each of the plane-stress conditions given below, construct a Mohrs circle of stress, find the principalstresses and the orientation of the principal axes relative to the x,y axes and determine the stresses on anelement, rotated in the x-y plane 30

oanticlockwise from its original position:

(a) x = 500 MPa

y = 300 MPa

xy = - 60 MPa

(b) x = 200 MPa

y = -50 MPa

xy = 300 MPa

Question 2.

For each of the plane-stress conditions given in Question 1, using the matrix transformation law, determine thestate of stress at the same point for an element rotated in the x-y plane 30

oanticlockwise from its original

position.

Question 3

The state of stress at a point of an elastic solid is given in the x-y-z coordinates by:

[ ] MPa

=

902300

2301600

00200

a. Using the matrix transformation law, determine the state of stress at the same point for an element

rotated about the x-axis (in the y-z plane) 60oclockwise from its original position.b. Calculate the stress invariants and write the characteristic equation for the original state of stress,c. Calculate the deviatoric invariants for the original state of stress,d. Calculate the principal stresses and the absolute maximum shear stress at the point.e What are the stress invariants and the characteristic equation for the transformed state of stress,

Question 4

The 3-D state of stress is given by

[ ] MPa

=

603040

304020

402080

Determine

The total stress (magnitude and direction with x, y, z axes) on a plane described by direction cosines:l = 0.6, m = negative, n = 0.4.

Magnitude of normal and shear stresses on this plane.

Principal stresses and direction cosines of the principal planes.

Maximum shear stress.

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Milestone 2 (Due by week 4)

Question 1Write a matrix giving the components of the hydrostatic (mean) strain tensor. Evaluate the first, second and thirdinvariants of the hydrostatic strain tensor.

Question 2

Determine the engineering strain e and the true strain for each of the following situations of extending a bar oflength L or compressing a billet of height h:Extension from L to 1.001 LCompression from h to 0.999 hExtension from L to 1.6LCompression from h to 0.4hCompression from h to zero height

Question 3A 50 mm-diameter-forging billet is decreased in height from 100 to 60. Assuming constant volume for plastic

deformations,

a. Determine the average axial strain and the true strain in the direction of compression.b. What is the final diameter of the forging?c. What are the transverse strains?

Question 4A 50 mm-thick plate is decreased in thickness according to the following schedule: 30, 15, 6 mm. Calculate thetotal strain once on the basis of initial and final dimensions and once as the summation of the incrementalstrains, using

a.Conventional strain andb.True strain.

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Milestone 3 (Due by week 5)

Question 1

Show that the constancy of volume results in e1+e2+e3=0 and 1+2+3=0. Why the relationship for conventionalstrain is valid only for small strains but the relationship for true strain is valid for all strains.

Question 2The state of stress at a point of an elastic solid is given in the x-y-z coordinates by:

[ ] MPa

zzzyzx

yzyyyx

xzxyxx

=

=

120500

50800

00220

If E=200 GPa, = 0.3, calculate the strain tensor.

Question 3It was found experimentally that a certain material does not change in volume when subjected to an elastic stateof stress. Calculate Poissons ratio for this material?

Question 4Determine the volume of a solid Aluminum sphere which is subject to a fluid pressure of 120 MPa. E=65 Gpa,

= 0.26. The original volume of the sphere is 1000 mm3.