PSI- Issue 9

I. Shardakov et al. / Procedia Structural Integrity 9 (2018) 199–206 Author name / Structural Integrity Procedia 00 (2018) 000–000

201 3

U x

2



;

(1)

div

,

V

  



2

t



– physical relations:





  

  

  

  

  E 2     

  

2

E   

G I

G I

 ;

 

(2)

1 2

1 2

 

 

– geometrical relations:

  1 2      U U 

 T

.

(3)

Here , ,     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. Relations (1) - (3) are supplemented with the boundary and initial conditions. The boundary conditions are formulated in accordance with the experimental conditions: force   F t is applied to the contact surface S  in the direction of the normal vector. 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 ; (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 striker acceleration was registered by the acceleration sensor fixed on the striker. The value of the impact force was calculated as     * n F t M w t  , where * M is the mass of the striker. Fig. 1b shows the characteristic force-time curve describing the variation of the force F with time t, where  denotes the time of striker-specimen contact. The initial conditions for velocities and displacements are assumed to be zero. The numerical solution of the initial boundary value problem was developed in the three dimensional formulation by the finite element method using the ANSYS software. 3. The scheme of experiment and algorithm for data processing The test specimen was a concrete prism with the following dimensions: 400mm L  and 100mm A  (Fig. 1). The impulse excitation of natural vibrations was provided by a striker equipped with an accelerometer. This accelerometer made it possible to measure the time dependence of acceleration of the striker at the time of its contact with the specimen. The impulse excitation of vibrations was realized by two schemes, represented in Fig. 2. The use of the two experimental schemes is explained by the need to determine two elastic characteristics – Young's modulus E and elastic shear modulus G. The impulse generated by the striker along the longitudinal axis of the specimen (Fig. 2a) inevitably leads to the excitation of natural vibrations, with the dominant modes corresponding to the longitudinal eigenmodes. The eigenfrequencies of these particular modes will be determined to a large extent by Young's modulus E. The impulse action of the striker on the edge of the lateral surface (Fig. 2b) leads to the excitation of vibrations with a dominance of flexural-torsional eigenmodes. Registration of vibrograms of displacement along 0, / S S      n x ,