CHAPTER ONE STRESS, STRAIN, ENERGY and FAILURE
CHAPTER ONE STRESS, STRAIN, ENERGY and FAILURE
CHAPTER ONE STRESS, STRAIN, ENERGY and FAILURE
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<strong>CHAPTER</strong> <strong>ONE</strong><br />
<strong>STRESS</strong>, <strong>STRAIN</strong>, <strong>ENERGY</strong> <strong>and</strong> <strong>FAILURE</strong><br />
* Review important concepts <strong>and</strong> equations in MECH 101<br />
** Introduce useful extensions of MECH 101<br />
• 1.1 The Mechanics of Materials (MECH 101) Method<br />
• 1.2 Elementary Formulas for Stress <strong>and</strong> Deflection<br />
• 1.3 Stress-Strain-Temperature Relations, Plasticity<br />
• 1.4 Maximum Normal <strong>and</strong> Shear Stresses, Mohr's Circle<br />
• 1.5 Energy of Strain <strong>and</strong> Distortion<br />
• 1.6 Failure <strong>and</strong> Theories of Failure<br />
• 1.7 Stress Concentration<br />
• 1.8 Members with Cracks; Fracture Mechanics
Review <strong>and</strong> Summary<br />
1.1 THE MECHANICS OF MATERIAL(MECH 101) METHOD<br />
• Structure
Review <strong>and</strong> Summary<br />
1.2 ELEMENTARY FORMULAS FOR <strong>STRESS</strong> AND DEFLECTION IN MECH 101
The method of derivation of formula in MECH 101<br />
(a) (b) (c)<br />
FIGURE 1.1.2. Brief summary of how the flexure formula σ = My/I is derived. (a) Plane sections remain plane after loading.<br />
(b) Linear variation of axial normal strain e <strong>and</strong> a linear stress-strain relation. (c) Axial normal stress distribution <strong>and</strong> the<br />
pertinent equilibrium equations.<br />
(1) Establish the geometry of deformation by experiments<br />
(2) Determine the strain distribution by the analysis of the geometry of deformation<br />
(3) Determine the stress distribution from the strain distribution by using the Hook's law<br />
(4) Relates stress to load by equilibrium (free body diagram)<br />
• Principal of superposition<br />
• Concept of safety factor
Review <strong>and</strong> Summary<br />
1.3 <strong>STRESS</strong>-<strong>STRAIN</strong>-TEMPERATURE RELATIONS, PLASTICITY
• Plain stress state
Review <strong>and</strong> Summary<br />
1.4 MAXIMUM NORMAL AND SHEAR <strong>STRESS</strong>ES, MOHR'S CIRCLE<br />
(a)<br />
(b)<br />
(c)<br />
(d)
Review <strong>and</strong> Summary<br />
1.5 <strong>ENERGY</strong> OF <strong>STRAIN</strong> AND DISTORTION<br />
• Strain energy density – the work done per unit volume during<br />
stressing an elastic body<br />
(a)<br />
FIGURE 1.6.1. Unit volumes of linearly elastic material, with distorted shapes shown by<br />
dashed lines. (a) Uniaxial stress σ. (b) Shear stress τ.<br />
(b)
• The total strain energy in a body of volume V is<br />
(1) straight bar under axial load P at each end<br />
(2) straight circular shaft of constant cross section under torque T applied at each end<br />
(3) straight beam of constant cross section under moment M applied at each end
Review <strong>and</strong> Summary<br />
1.6 <strong>FAILURE</strong> AND THEORIES OF <strong>FAILURE</strong><br />
• Two major forms of failure:<br />
(1) failure due to brittle fracture, brittle materials<br />
(2) failure due to yielding, ductile materials
Review <strong>and</strong> Summary<br />
1.7 <strong>STRESS</strong> CONCENTRATION
Figure 1.8.2. Selected stress concentration factors K t for plane <strong>and</strong> cylindrical geometries, from<br />
[1.8]. Maximum stresses computed from these K t factors are at the edges of holes or notches<br />
(see sketches on page 28).
Review <strong>and</strong> Summary<br />
1.8 MEMBERS WITH CRACKS; FRACTURE MECHANICS<br />
• Stress distribution when there is a crack<br />
(a) (b) (c)<br />
FIGURE 1.9.1. (a) Flat sheet of thickness t in uniaxial plane stress, with a sharp internal<br />
crack of length 2a. (b) Qualitative depiction of how normal stresses are distributed near a<br />
crack tip. (c) Influence of specimen thickness on fracture toughness.
• Stress intensity factor<br />
• Critical stress intensity factor K IC fracture toughness (material<br />
constant like strength)<br />
critical condition for fracture: KI = K IC<br />
If stress σ <strong>and</strong> K IC are given, the maximum crack length (a c ) can<br />
be determined by KI = K IC
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