# Mechanics of Materials Notes

StressStress is defined as force per unit area. This is one of the most basic engineering quantities.Shear StressShear stress has the same units as normal stress (force / area) but represents a stress that acts parallel to the surface (cross section). This is different from normal stress which acts perpendicular (normal) to the cross section. Torsion is a force that causes shear stress but this is not the only force that can cause shear stress. For example, a beam that supports a shear force also has a shear stress over the section (even without torsion).StrainStrain is the change in length per unit length. It is normally computed as (Lf – L0) / L0 where Lf is the final length andL0 is the initial length. When testing materials, a gage length is normally specified known; this represents L0.Strain RateWhen testing a material, it is normally important to know how quickly (or slowly) it is being deformed or loaded. One way to report this is the amount of strain that occurs in a unit of time which is termed the strain rate. Because strain is dimension-less, units for this quantity are 1/time but sometimes it will be seen with the non-dimensional part attached e.g., in/in/sec.Young’s ModulusThis is the constant of proportionality between stress and strain. Units of this quantity are the same as stress (i.e., force per unit area) and the most commonly used are psi, Pa (Pascal), and MPa (Mega-Pascal). This is one of the most fundamental material properties. A typical value for steel is 29 x 106 psi (200 GPa).Deflection EquationA prismatic bar loaded uniaxially made from a material that obeys Hooke’s law deflects when loaded by an amountd=PL/AE where

d = Deflection
P = Applied Force
L = Length
A = Area
E = Young’s Modulus

Poisson’s RatioThis is the ratio of lateral strain to longitudinal strain. The typical range of values for this quantity is between zero and 0.5.Hooke’s LawWhen the applied force is proportional to the deflection, a material is said to obey Hooke’s law. There is a linear relationship between the force and displacement and thus, linear elastic materials obey this law. When steel is below the proportional limit it shows this linear behavior.Material PropertiesThese are properties specific to the material used. These are different from section properties which do not depend on what an object is made of. Examples of material properties are Young’s modulus and yield point. Typical values of these for steel are 29 x 106 psi and 36 ksi respectively. In general, different materials will have different material properties.Section Properties. These are properties specific to the geometry (dimensions) of the section used. These are different from material properties which depend on what an object is made of. Examples of section properties are area, diameter, and section modulus.Stress – Strain Diagram The stress-strain diagram is a plot of the stress on the ordinate (y-axis) versus the strain on the abscissa (x-axis). The data is often obtained from a uni axial tension test although this is not the only test possible. The axes must be labeled with the appropriate units for stress (strain is dimensionless).Proportional LimitThe proportional limit is the greatest stress that one can still see a linear relation between stress and strain. Beyond this point, the stress is no longer proportional to the strain.NominalA nominal dimension is one that gives the intended or approximate size but this may (and often does) vary from the actual dimension. For example, a common lumber shape is a 2×4 but this is a nominal size and the actual dimensions are 1.5″ x 3.5″. The word is from Latin, of a name, nomin-, nomen name thus can be thought of as ‘what we call it’.Von-MisesRichard von Mises was born 19 April 1883 in Lemberg, Austria (now Lvov, Ukraine). The von Mises Criterion (1913), also known as the maximum distortion energy criterion, octahedral shear stress theory, or Maxwell-Huber-Hencky-von Mises theory, is often used to estimate the yield of ductile materials. The von Mises criterion states that failure occurs when the energy of distortion reaches the same energy for yield/failure in uniaxial tension. Mathematically, this is expressed as,

In the cases of plane stress, s3 = 0. The Von Mises criterion reduces to,