Poissons Ratio () = - lateral strain / Axial strain = - x/z = -y/z= - [d / do] / [l/ lo]

x =y = - z The negative sign is added to yield a positive value for

For many metals and alloys the poissonsratio range between 0.25 and 0.35

Shear stress strain diagram

Elastic behavior : = G

G = E / 2(1+ ) for isotropic material, for which the properties are the same in all directions.

True stress and strain

T = F / Ai , load divided by the instantaneous cross section area

T = ln (li / lo), li: instantaneous length ---- lo: original lengthFor plastic deformation (>y) there is conservation of volume: Aolo = Ai li Ao / Ai = li/lo

Relations between true and engineering stress and strain:

T = F / Ai = F / Ao x Ao/ Ai= x Ao/Ai = x li/lo = li-lo / lo = li/lo 1 li/lo = 1+ Thus T = (1+ ) This equation is valid from yielding to the on set of necking y < < u

T = ln (li / lo) = ln (1+), This equation is valid from yielding to the on set of necking y < < u

For some metals and alloys the region of the true stress strain curve from the onset of plastic deformation to the point at which necking begins (y < < u) may be approximated by:

T = C T( )ntrue stress (F/A) true strain: ln(L/Lo)

hardening exponent: n=0.15 (some steels) to n=0.5 (some copper)K

K and n are constants that vary from alloy to alloy.

Taking logarithm of both sides yields a straight line:

Log T = n log T + log K (y = mx +c)n (strain hardening exponent) defines the slope of the straight line

K: strength coefficient

large hardening

small hardening

un

loa

dre

loa

dy 0y

1

An increase in y due to plastic deformation.HARDENING

Since n is the slope of the straight line higher values of n corresponds to higher strain hardening rate.

High values of n

In case where n and K are to be found for a particular alloy, two points on the stress strain curve that defines and in the region where y < < u has to be given.Log T1 = n log T1 + log KLog T2 = n log T2 + log KBy subtracting the two equations:

n = (Log T2 - Log T1 ) / (log T2 - log T1 )Then substitute in any equation to get K.

Construct a table of F , L, and F L(l-lo) = L/lo =F/Ao0 0

7330 50.851-50.8 0.051/50.8 7330/ [(/4) d2]15100 50.902-50.8

0.05 0.165

lastic strainrecovered 0.005

To Generate a True Stress Strain Response