PSI - Issue 6

V.D. Kharlab / Procedia Structural Integrity 6 (2017) 286–291

289

Kharlab V.D./ StructuralIntegrity Procedia00 (2017) 000 – 000

4

Fig. 1

Formula for rate of energy input of Q . Now the energy of rod unit length related to the action of the vertical tangent stresses only (the formula for factor  is obtained with this limit) can be determined using the formula

2 GA GA   2 2

(14)

(1) Q U Q Q  

2 0 (1 0,8 ), k 

2

more suitable than the general one

2 ( , ) y z dydz G     . / 2 xz z y

(1)

U

(15)

Q

If the horizontal stress energy is to be taken into account, this can be done using the formula (15) with substitution of stress xz  for stress xy  Involvement of minimum energy theorem. In the linear elasticity there is the Catigliano theorem "Leibenzon L.S. (1948)": among all statically probable stress states the real is the one to which the minimum potential deformation energy corresponds. Let ’ s try to use this theorem for obtaining additional information on parameter k . Potential deformation energy of our object as a function of parameter k is determined by formula (14) . The studied stress state is statically probable as the integral equilibrium condition is fulfilled at any value of k

xz z y dydz Q     .

(16)

Additionally, differential equilibrium equations and boundary conditions are fulfilled. The minimum energy condition (14) provides:

(1) 2 Q dU Q dk 

2 GA 

0 1.6 0, k

k  

0.

(17)

This result meaning that the Zhuravsky ’ s theory corresponds to the minimum energy is unexpected. It should seem, (17) says that true stress state of our object is always one corresponding to the Zhuravsky ’ s theory but this conclusion cannot obviously be true. The violation is cancelled if it is taken into account that parameter k is not a free variable, but depends on the Poisson's ratio (see the example above (5)). It turns out that as applied to the rectangular cross-section, the equality 0 k  means 0   . In other words, the true stress state corresponding to the minimum energy (14) for the rectangular cross section is the Zhuravsky state at zero Poisson ’ s ratio. The same can be said about the round and triangular cross-sections, but with the correction: 0.5   . It may seem that the obtained information does not contain anything new, since it was previously known that for certain values of the Poisson's ratio the theory in question degenerates into the Zhuravsky's theory. However, this is not the case: it was not known previously that the obtained degenerated solutions are accurate (true). The result (17) has identified a certain class of solutions of our theory that are accurate. However, it does not