PSI - Issue 13

I. Shardakov et al. / Procedia Structural Integrity 13 (2018) 1342–1346 Author name / Structural Integrity Procedia 00 (2018) 000 – 000

1343

2

natural vibrations, ASTM C215-14 (2016), Zheng et al (2008), Ignat'kov (2011) . This paper is devoted to the determination of parameters of viscoelastic model, which is based on the interpretation of the results of recording the natural vibrations of a concrete specimen under impulse loading generated by an impact of the striker on the specimen. Experimental registration of velocity vibrograms on the specimen surface is performed using a laser vibrometer. The interpretation of experimental measurements is realized through the numerical solution of a three-dimensional initial boundary problem of natural vibrations of a concrete specimen, the physical properties of which are described in the framework of a viscoelastic model. The required values of elastic and viscous properties of concrete are determined from the proposed iterative sequence of numerical solutions.

2. A mathematical model of natural vibrations of concrete specimen

The model describes an impact load experiment, which is aimed at registering the natural vibrations of a concrete specimen under impulse load produced by a striker. The concrete specimen is shaped as a parallelepiped of length L, and square cross section with the side A The volume of the specimen is denoted as V, and the surface of the striker- specimen contact is denoted as S  . The stress-strain state of a specimen hit by a striker is described by the equilibrium equations, physical relations and geometrical relations as follows:

2

U x



V

div

,

  



;

(1)

2

t







  

  

  

  

  E 2     

 

G I

G I

2

E   

 

;

(2)

1 2

1 2

 

 

  1 2      U U 

 T

.

(3)

Here   1 2 3 , , x x x  x are the Cartesian coordinates, , ,    are the stress, strain and strain rate tensors, U is the velocity vector;  is the nabla operator,   I  ,   I  are the first invariants of the strain and strain rate tensors, E is the unit tensor, ρ is density, G,  are the shear modulus and Poisson's ratio, β is the parameter that determines the contribution of viscous forces to the stress tensor. The boundary conditions are formulated in accordance with the experimental conditions: force   F t is applied to the end of beam in the direction of the normal vector at the contact surface S  . During the implementation of the experiment, the specimen was hung by thin threads, which did not affect the natural deformation modes. With this in mind, the boundary conditions are set in the following form:   , 0, F t S       n n n n x ; 0, / S S     n x (4) where n is the vector of the normal to the surface. The form of the function   F t was determined based on the results of measuring the acceleration   n w t of the striker during the time of its contact with the specimen surface. The value of the impact force was calculated as     * n F t M w t  , where * M is the mass of the striker. The initial conditions for velocities and displacements are assumed to be zero. A numerical solution of the initial boundary value problem was developed in the three dimensional formulation by the finite element method using the ANSYS software. The scheme of experiment is represented at Fig. 1. The test specimen was a concrete prism 1with the following dimensions: 400mm L  and 100mm A  . The impulse excitation of natural vibrations was provided by a striker equipped with an accelerometer. The impulse excitation of vibrations was realized by two schemes. The impulse gene rated by the striker 2 along the longitudinal axis of the specimen (Fig. 1a) inevitably leads to the excitation of natural vibrations, with the dominant modes corresponding to the longitudinal eigenmodes. The eigenfrequencies of these par ticular modes will be determined to a large extent by Young's modulus E. The impulse action of the striker on the edge 3. The scheme of experiment and algorithm for data processing