chapter one stress, strain, energy and failuremeqpsun/notes/chapter1.pdf · 2011. 9. 15. ·...

CHAPTER ONE STRESS, STRAIN, ENERGY and FAILURE* Review important concepts and equations in MECH 101** Introduce useful extensions of MECH 101

• 1.1 The Mechanics of Materials (MECH 101) Method• 1.2 Elementary Formulas for Stress and Deflection• 1.3 Stress-Strain-Temperature Relations, Plasticity• 1.4 Maximum Normal and Shear Stresses, Mohr's Circle• 1.5 Energy of Strain and Distortion• 1.6 Failure and Theories of Failure• 1.7 Stress Concentration• 1.8 Members with Cracks; Fracture Mechanics

• Structure

Review and Summary1.1 THE MECHANICS OF MATERIAL(MECH 101) METHOD

Review and Summary1.2 ELEMENTARY FORMULAS FOR STRESS AND DEFLECTION IN MECH 101

The method of derivation of formula in MECH 101

(a) (b) (c) FIGURE 1.1.2. Brief summary of how the flexure formula σ = My/I is derived. (a) Plane sections remain plane after loading. (b) Linear variation of axial normal strain e and a linear stress-strain relation. (c) Axial normal stress distribution and the pertinent equilibrium equations.

(1) Establish the geometry of deformation by experiments(2) Determine the strain distribution by the analysis of the geometry of deformation(3) Determine the stress distribution from the strain distribution by using the Hook's law(4) Relates stress to load by equilibrium (free body diagram)

• Principal of superposition• Concept of safety factor

Review and Summary1.3 STRESS-STRAIN-TEMPERATURE RELATIONS, PLASTICITY

• Plain stress state

Review and Summary1.4 MAXIMUM NORMAL AND SHEAR STRESSES, MOHR'S CIRCLE

(a) (b)

(c) (d)

• Strain energy density – the work done per unit volume during stressing an elastic body

Review and Summary1.5 ENERGY OF STRAIN AND DISTORTION

(a) (b)

FIGURE 1.6.1. Unit volumes of linearly elastic material, with distorted shapes shown bydashed lines. (a) Uniaxial stress σ. (b) Shear stress τ.

(2) straight circular shaft of constant cross section under torque T applied at each end

• The total strain energy in a body of volume V is

(1) straight bar under axial load P at each end

(3) straight beam of constant cross section under moment M applied at each end

Review and Summary1.6 FAILURE AND THEORIES OF FAILURE

• Two major forms of failure:

(1) failure due to brittle fracture, brittle materials

(2) failure due to yielding, ductile materials

Review and Summary1.7 STRESS CONCENTRATION

Figure 1.8.2. Selected stress concentration factors Kt for plane and cylindrical geometries, from[1.8]. Maximum stresses computed from these Kt factors are at the edges of holes or notches (see sketches on page 28).

Review and Summary1.8 MEMBERS WITH CRACKS; FRACTURE MECHANICS

• Stress distribution when there is a crack

FIGURE 1.9.1. (a) Flat sheet of thickness t in uniaxial plane stress, with a sharp internalcrack of length 2a. (b) Qualitative depiction of how normal stresses are distributed near acrack tip. (c) Influence of specimen thickness on fracture toughness.

(a) (b) (c)

• Stress intensity factor

• Critical stress intensity factor KIC fracture toughness (material constant like strength)

critical condition for fracture: KI = KIC

If stress σ and KIC are given, the maximum crack length (ac) can be determined by KI = KIC