Assignment 1 (elasticity)

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McMaster University *

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3M03

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Mechanical Engineering

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Feb 20, 2024

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MATLS 3M03: Mechanical Behaviour of Materials Assignment 1, 2024 WS Student full name: Student ID number: MATLS 3M03: MECHANICAL BEHAVIOUR OF MATERIALS Assignment 1: Elasticity Instructor: Oleg Rubel Instructions: • It is sufficient to mark your choice (or choices) for multiple choice questions (justifica- tion is not required). • Provide sufficient details to justify the solution for long answer questions (not just a final answer). • Provide a complete reference to external resources when used (including pages for a book or URL for web sites). Question 1: [1 point] Which of the following are material properties? ◦ Young modulus ◦ Electrical resistance ◦ Melting temperature ◦ Density ◦ Weight Question 2: [1 point] How many independent components are there in the stress tensor generally? ◦ 3 ◦ 9 ◦ 6 ◦ 1 Page 1 of 4

MATLS 3M03: Mechanical Behaviour of Materials Assignment 1, 2024 WS Question 3: [1 point] All diagonal components vanish when the strain tensor is transformed into its principal axes. ◦ True ◦ False Question 4: [1 point] What is the main factor that determines the strength of metallic bonds? ◦ number of electrons in the atom ◦ attractive Coulomb interaction between adjacent ions ◦ fluctuating dipole moments on adjacent atoms ◦ number of valence electrons contributed to the “sea” of free electrons Question 5: [1 point] A bulk member experiences hydrostatic pressure of 2 GPa. List all components of the stress tensor σ ij ( i , j = 1 , 2 , 3)? Question 6: [2 points] Determine the pure shear strain ϵ 12 for the shape sketched below. 1 2 before deformation after deformation Page 2 of 4

MATLS 3M03: Mechanical Behaviour of Materials Assignment 1, 2024 WS Question 7: [5 points] A composite material is to be made of alternating sheets of polyethylene (core matrix) and aluminum (reinforcement). The individual materials (polyethylene and aluminum) have elastic moduli of 1 and 70 GPa, respectively. We aim to achieve an elastic modulus E x of 30 GPa for the composite material along the sheets. (a) What should be the volume fraction of aluminum reinforcement needed to achieve the desired E x ? (b) How is the volume fraction related to the thickness of the layers (specifically, the L m /L r ratio)? (c) What would be the elastic modulus E y of the composite material in the transverse direction (perpendicular to the sheets)? Page 3 of 4

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MATLS 3M03: Mechanical Behaviour of Materials Assignment 1, 2024 WS Question 8: [5 points] A rod with a cross-section of a × a is compressed along the vertical axis 3, resulting in a strain of ϵ 3 = − 0 . 001. The specimen is constrained from expanding or contracting along the other two axes. Determine the resultant stress tensor σ i ( i = 1 , . . . , 6) assuming that the material is isotropic with a Young’s modulus of 200 GPa and a Poisson’s ratio of 0.3. Page 4 of 4