MEEN 5330 Continuum Mechanics , all references from Continuum Mechanics by Frederick & Chang unless noted

MEEN 5330 Continuum Mechanics,all references from Continuum Mechanics by Frederick & Chang unless noted Chapter 2 – Stress Prof. Larry Peel MEEN 5330

Chapter 2 - Stress Continuum = ? (Homogenous, continuous, uniform, constant density?) Our Definition covers most solids, liquids, fluids at speeds less than the speed of sound What are some materials that meet the definition? That do not meet the definition? MEEN 5330

Chapter 2 – Definition of Density • r = density • = dM/dV (Instantaneous density) • r = DM/DV • (average density) MEEN 5330

Chapter 2 – Types of Continuums Fluid - Solid - MEEN 5330

Chapter 2 – Forces Body Forces - Surface Forces Point - Distributed - MEEN 5330

Stress Tensor - symmetric MEEN 5330

Definitions of Stress and StrainSymmetric MEEN 5330

Converting a stress matrix to a vector MEEN 5330

Converting a stress matrix to a vector cont’d MEEN 5330

Stress and Strain MEEN 5330

Equilibrium of stress and internal forces MEEN 5330

Equilibrium equations MEEN 5330

Example of stress on a plane MEEN 5330

Example of stress on a plane cont’d MEEN 5330

Transformation Law for stress tensor MEEN 5330

Transformation Law for stress tensor cont’d MEEN 5330

All references from Continuum Mechanics by Frederick unless notedPrinciple Stresses and Principle Axes (1st 4 pages from Higdon) MEEN 5330

Mohr’s Circle and Principle Axes / Stresses MEEN 5330

Mohr’s Circle and Principle Axes / Stresses cont’d MEEN 5330

Principle Axes / Stresses - Frederick The stress acting on a surface is give by: If the stress is normal to the area then: Or s is the principle stress and ni is the principle direction: MEEN 5330

Principle Axes / Stresses - Frederick, cont’d Also realizing that: If we take the following determinant, we get three related stress invariants: MEEN 5330

Stress Invariants - Frederick, cont’d Where: or: Does this look familiar to a problem in chapter 1? MEEN 5330

Principle Stresses - Frederick, cont’d • From Equation 2.49 we get 3 principle stresses. • If s(1), s(2), s(3) are different, then the principle directions are orthogonal to each other. • The principle stresses are always real. MEEN 5330

Principle Stresses - Example (Schaum’s outline): MEEN 5330

Principle Stresses - Example, cont’d MEEN 5330

The Stress vector and normal Stresses, related to the principle Axes The max / min normal stress = the max/min principle stress MEEN 5330

Other Stress Invariants The stress invariants can be used to obtain principal stresses, and are important for such things as yield conditions, ie Octahedral Shear. MEEN 5330

Stress Invariants Example (Schaum’s Outline) MEEN 5330

Stress Deviator Tensor Many times it is necessary to decompose the stress tensor into a uniform normal stress and a state of pure shear. Used in Plasticity, thermo-elasticity, fluid dynamics, and other areas. The stress deviator tensor is defined as: We can verify that it is a state of pure shear by looking at normal stresses: MEEN 5330

Stress Deviator Tensor Example (Schaum’s Outline) MEEN 5330

Maximum Shearing Stresses Desired: The maximum value of the shearing component S of the stress vector si. Looking at the principal stresses and axes: Where N is the max normal stress. and.. then MEEN 5330

Maximum Shearing Stresses The only non-zero solutions are, relative to the principal axes: MEEN 5330

Maximum Shearing Stresses Example MEEN 5330

Maximum Shearing Stresses Example cont’d MEEN 5330

Questions and Conclusions • Questions? • The summary on pages 61 and 62 is very useful. • Most important items: definitions, components of a stress tensor, stress on a plane, equations of equilibrium, transformation of stress tensor, principal stress and axes, stress invariants, stress deviator, and max shearing stress. • Review and sample problems on Tuesday. MEEN 5330

MEEN 5330 Continuum Mechanics , all references from Continuum Mechanics by Frederick & Chang unless noted