PSI - Issue 59

Olena Romashko-Maistruk et al. / Procedia Structural Integrity 59 (2024) 352–359 O. Romashko-Maistruk and V. Romashko/ Structural Integrity Procedia 00 (2023) 000 – 000

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part of the potential energy is nevertheless lost or "dissipated" in the form of heat during the course of various processes associated with the strain of the material of the sample itself. Usually these losses are insignificant and therefore they are completely neglected. Consider the general deformation case of some reference sample in the form of an elementary parallelepiped with dimensions dx dy dz   (Fig. 2), made of a completely elastic material. The full potential energy of its deformation, which accumulates at the same time, can be calculated using a well-known expression:

2 1

2 1

2 1

dU

dzdy

dx

dzdx

dy

dxdy

dz

























,

(1)

x

x

y

y

z

z

where x  , y  , z  , x  , y  , z  are the longitudinal stresses and the longitudinal strains of materials caused by them in the direction of the axes x , y , z respectively.

 y

dx dx(1-  ) x

 x

 z

 z

dy(1-  ) y

dy

 x

dz(1-   ) z

 y

dz

Fig. 2. General case of the reference sample elastic deformation.

It is important to remember that the potential energy of elastic deformation is always accumulated in the reverse form: when the sample is unloaded, it again turns into the energy of external forces or is transformed into kinetic energy. At the same time, the value of the specific potential energy deformation u accumulated in a unit volume of the sample material dV , can be calculated by the formula:

2 1

dV dU

(

)

u

 









      

.

(2)

x x

y y

z z

In the case of a uniaxial stress-strain state, the specific potential energy of concrete elastic deformation (Fig. 3) can be determined by the expression:

2 ,

f

2 1

2 1

2 1

dV dU

co c du

2

u

E

 



  



 



,

(3)

2

co c du ,

z z

E

where co E is the initial modulus of concrete elasticity; c du f , and c du ,  - the strength of compressed concrete and the corresponding concrete critical strains under the dynamic loads action. In general, such concrete deformation is possible only with its instantaneous loading    , that is, when plastic strain do not have time to manifest themselves. In all other cases, the concrete deformation potential energy must be calculated according to the actual concrete stress-strain diagrams (Romashko and Romashko (2017), Romashko and Romashko (2019), Romashko (2021)).