49 
absorption in the elastic range being small compared with the energy 
absorbed plastically. Ths full curve relates to a normal static tension 
test at a low rate of strain, For higher rates of strain due to impulsive 
loading both yield stress and ultimate stress tend to increase, the former 
to the greater relative extent, and under high rates of strain the limited 
available evidence*suggests that the ourve tends to be of the form AKF'. 
The elastic absorption of energy is still small compared with ths plastic 
absorption.® Since it is not practicable to construct vessels with hulls 
which oan absorb sufficient energy elastically to withstand any likely 
scale of attack, the mich greater energy absorption in plastic yielding 
is of paramount importance. 
430. Summing up, therefore, whereas the traditional design of engineering 
structures under static loads is based essentially on the theory of 
elasticity, the present problem of structures submitted to underwater 
explosions is primarily a question of the theory of plasticity. The 
basic laws and simplifying assumptions involved in the theory of the plastic 
yielding of steel will, therefore, now be discussed, 
Pundamental assumptions for plastic yielding of steel 
431. As a first simplification, yielding in a simple tension test will be 
assumed to take place at a constant stress. For static loads this 
involves neglecting the difference between the upper and lower yield points 
and limiting consideration to strains up to the point D in fig.22, For 
high rates of strain where the 
true curve becomes more of the 
k £ shape AKF', the assumption of 
a flat topped stress-strain 
curve tends to be relatively 
accurate for strains right up 
to failure. Ths second 
Simplification is to negleot 
the initial elastic portion so 
> that finally the assumed curve 
is that shown in fig, 23. 
This refers to the condition 
of simple tension and it is 
now necessary to consider the 
general case of any system of 
stresses, 
A STRAIN 1326 let 6, , Ss Pe 
denote the three principal 
stresses and 6,, ©, @2 the 
Fig. 23 - Simplified stress-strain curve corresponding principal strains, 
Tensile stresses will be taken 
as positive and ¢, will refer to the greatest algebraic stress and 64 
to the least algebraic stress, that is, 6, >62%6,- For the simple 
tension test the assumed relation of fig. 23 then corresponds to 
STRESS 
6, = constant = B» 
6,=0 eal? MSO ae ED) 
63= 0 
For more general conditions of stress, several different theories have been 
advanced for the condition of plastic yielding, but only two of these simple 
theories have survived the test of time and experiment. 
x Provided brittle fracture does not take place. 
