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AE 321 Homework 7

This document provides the details for homework assignment #7 which is due on April 5, 2019. It covers chapters 4 and 5 on material behavior and formulation. There are 4 problems included in the homework. Problem 1 asks about the volume change of a cube submerged in water based on water pressure. Problem 2 asks about determining boundary conditions on a hollow circular object under different loading conditions. Problem 3 asks to determine constants to satisfy equilibrium for an elastic object under given body forces. Problem 4 asks to show that for uniform body forces, the Laplacian of volumetric deformation is zero.

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124 views1 page

AE 321 Homework 7

This document provides the details for homework assignment #7 which is due on April 5, 2019. It covers chapters 4 and 5 on material behavior and formulation. There are 4 problems included in the homework. Problem 1 asks about the volume change of a cube submerged in water based on water pressure. Problem 2 asks about determining boundary conditions on a hollow circular object under different loading conditions. Problem 3 asks to determine constants to satisfy equilibrium for an elastic object under given body forces. Problem 4 asks to show that for uniform body forces, the Laplacian of volumetric deformation is zero.

Uploaded by

bob toodle
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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AE321 – Homework #7

Due: Friday April 5, 2019


Chapters 4 & 5: Material Behavior and Formulation

1- A cube made of a linearly elastic isotropic and homogeneous material is submerged to the
bottom of the ocean. If the dimensions of the body are small so that one can neglect the variation
of hydrostatic pressure across it
a) What is the volume change of the body in terms of the water pressure P?
b) Discuss how this result would be affected if the material were NOT isotropic?
c) Discuss how this result would be affected if the material were NOT homogeneous?

2- Find all boundary conditions on all surfaces of the hollow circular (inner radius a, outer
radius b) shown below in both (i) Cartesian coordinates and (ii) cylindrical coordinates, where P,
Po and Pi are normal tractions and q is a shear traction. Note: the loading shown on the cross-
sectional figure acts over the entire length L and over the entire circumference.

Po
q

P
Pi
L

3- Consider an elastic object under body forces given by


−6𝐺𝐺𝐺𝐺𝐺𝐺
𝒇𝒇 = � 2𝐺𝐺𝐺𝐺𝐺𝐺 �
10𝐺𝐺𝐺𝐺𝐺𝐺
𝐸𝐸
where 𝐺𝐺 = and 𝜗𝜗 = 1/4. Assume displacements are given by
2(1+𝜗𝜗)
𝐶𝐶1 𝑥𝑥 2 𝑦𝑦𝑦𝑦
𝒖𝒖 = �𝐶𝐶2 𝑥𝑥𝑦𝑦 2 𝑧𝑧�
𝐶𝐶3 𝑥𝑥𝑦𝑦𝑧𝑧 2
Determine constants 𝐶𝐶1 , 𝐶𝐶2 , and 𝐶𝐶3 such that body remains at equilibrium.

4- Show that when body forces are uniform the Laplacian of the volumetric deformation 𝑒𝑒 = 𝜀𝜀𝑘𝑘𝑘𝑘
vanishes (in other words, it is a harmonic function).

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