T. B. Krit

Shear standing waves in a resonator with a rubberlike inhomogeneous medium

A modification of the finite-element method is proposed for calculating shear standing waves in a resonator filled with an incompressible elastic medium with allowance for the finite dimensions of the resonator and inhomogeneities of the shear modulus. Resonance curves are calculated for resonators with inhomogeneities in the form of cavities and elastic inclusions. Numerical calculations are compared with experimental data.

Nonlinear moduli estimation for rubber-like media with local inhomogeneities elastography

Static shear deformations of a plane-parallel layer of rubber-like material created simultaneously with the uniaxial compression are considered. The layer is fixed between the rigid plates. Displacement of one plate relative to the other resulted in shear strain of the layer. This strain could reach 0.6 of the layer thickness. At such strain effects due to the cubic nonlinearity arise. It is shown that measuring the dependence of the shear stress on the shear strain along one axis at different compression along the perpendicular axis one could determine nonlinear Landau parameters. The measurements were performed in two layers of polymeric material plastisol of 7 mm thickness with a rectangular base 8.9x8.9 cm, mounted between three aluminum plates. The upper plate was loaded with masses ranging from 0 to 25 kg and was fixed in each series of the stress-strain measurements. The values of the Landau coefficient A were measured in layers with different value of linear shear modulus.

Shear waves in a resonator with cubic nonlinearity

Shear waves with finite amplitude in a one-dimensional resonator in the form of a layer of a rubber-like medium with a rigid plate of finite mass at the upper surface of the layer are investigated. The lower boundary of the layer oscillates according to a harmonic law with a preset acceleration. The equation of motion for particles in a resonator is determined using a model of a medium with a single relaxation time and cubical dependence of the shear modulus on deformation. The amplitude and form of shear waves in a resonator are calculated numerically by the finite difference method at shifted grids. Resonance curves are obtained at different acceleration amplitudes at the lower boundary of a layer. It is demonstrated that, as the oscillation amplitude in the resonator grows, the value of the resonance frequency increases and the shape of the resonance curve becomes asymmetrical. At sufficiently large amplitudes, a bistability region is observed. Measurements were conducted with a resonator, where a layer with the thickness of 15 mm was manufactured of a rubber-like polymer called plastisol. The shear modulus of the polymer at small deformations and the nonlinearity coefficient were determined according to the experimental dependence of mechanical stress on shear deformation. Oscillation amplitudes in the resonator attained values when the maximum shear deformations in the layer were 0.4–0.6, which provided an opportunity to observe nonlinear effects. Measured dependences of the resonance frequency on the oscillation amplitude corresponded to the calculated ones that were obtained at a smaller value of the nonlinear coefficient.

Standing waves in an elastic layer loaded with a finite mass

Standing shear waves in a plane-parallel rubberlike layer fixed without slippage between two rigid plates with finite masses are investigated. The lower plate, which underlies the layer, oscillates in the direction parallel to its surface under an external harmonic force, whereas the upper plate freely overlies the layer. It is shown both theoretically and experimentally that such a system exhibits resonances at frequencies the values of which depend on the mass of the free plate and the shear modulus of the layer. The shapes of the resonance curves are calculated and measured for different values of parameters of the layer and different masses of the upper plate. From the measured resonance curves, it is possible to determine the dynamic shear modulus and the shear viscosity of the rubberlike material.

Standing shear waves in rubberlike layered media

Standing shear waves arising in layered media the shear modulus of which varies in a stepwise manner at the plain boundaries between the layers are considered. A general solution is obtained for the shear wave amplitudes in a resonator with an N-layer structure the lower boundary of which performs harmonic vibrations while a finite-mass plate is attached to the upper boundary. Results of calculations and measurements are presented for a resonator with a structure in which nondeformable metal layers alternate with elastic rubberlike polymer layers. It is shown that the resonance frequencies of such a resonator can be controlled by changing the number of layers and their thicknesses. It is demonstrated, both experimentally and theoretically, that, from the resonance curve of a resonator with a two-layer structure, it is possible to determine the shear modulus of one of the layers under the condition that the elasticity of the other layer is known. The method of separation into a finite number of layers is used to analyze the resonance characteristics of a one-dimensional resonator filled with a rubberlike medium the properties of which continuously vary in the direction perpendicular to the shear displacements. The choice of the number of layers depending on the type of inhomogeneity is analyzed.

In vivo measurements of muscle elasticity applying shear waves excited with focused ultrasound

The common algorithm of shear waves excitation for diagnostical ultrasonic devices was modified for measurements in muscles. We measured the speed of shear waves, excited by a focused ultrasound at a frequency of 5 MHz in the muscles of the volunteers. Siemens Acuson S2000 was used for in vivo measurements. The suggested algorithm was tested on the muscle mimicking phantoms. The values of shear wave velocities in the same areas of studied phantoms at the same angles measured with Siemens Acuson S2000 system corresponded to the values obtained by Verasonics, where the region of shear wave excitation had a form of “blade” of thickness less than 0.5 mm, length and width of 1.5-2 mm. Due to this form of the region, the excited shear wave has propagated codirectional with the long side of the ultrasonic medical probe. Thus, the direction of propagation of the shear wave with respect to the phantom fibers, became dependent on the position of the probe. [The reported study was funded by RFBR and Moscow city Gove...

Standing shear waves in nonlinear gel-like media under static shear stress

Standing shear waves in a resonator in the form of layer of gel-like medium placed between two rigid plates are studied. The bottom plate is fixed to the vibrator and oscillates in the horizontal direction with a preset amplitude. Two rubber threads attached to the upper plate can displace the plate by a specified value in the horizontal direction. The change in the tension of the threads creates an additional static deformation of the elastic layer resulting in the effective shear elasticity increase. The measured static stress-strain dependence of the elastic layer can be described by the cubic parabola. We measured the dependences of the resonance frequency on the static deformation of the layer. For static deformations of the layer less than 0.3 h (h—is the layer thickness), the resonance frequency increases linearly, that can be explained by a linear growth of the elastic force of rubber threads. In the deformation range of 0.3-1 h, an additional shift of the resonance frequency caused by the nonline...

Standing shear waves in anisotropic viscoelastic media

We studied standing shear waves in anisotropic resonator represented by a rectangular parallelepiped (layer) fixed without slipping between two wooden plates of finite mass. The viscoelastic layer with edges of 70 mm × 40 mm × 15 mm was made of a rubber-like polymer plastisol with rubber bands inside. The bands were placed vertical between the top and the bottom plate. Mechanical properties of the plastisol itself were carefully measured previously. It was found that plastisol shows a cubic nonlinear behavior, i.e. the stress-strain curve could be represented as: σ = μe + βμe3, where e stands for shear strain and σ is an applied shear stress. The value of shear modulus μ depends on frequency and was found to be several kilopascals which is common for such soft solids. Nonlinear parameter β is frequency dependent too and varies in range from tenths to unity at 1-100 Hz frequency range, decreasing with frequency growth. Stretching the rubber bands inside the layer leads to change of elastic properties in re...

Estimation of nonlinear parameters applying uniaxial and shear stress to inhomogeneous viscoelastic media

Static shear deformations of a plane-parallel layers of several viscoelastic media created simultaneously with the uniaxial compression are considered. Each layer is fixed between two rigid plates. Displacement of one plate relative to the other resulted in shear strain of the layer. This strain could reach 0.6 of the layer thickness. At such strain, effects due to the cubic nonlinearity arise. It is shown that measuring the dependence of the shear stress on the shear strain along one axis at different compression along the perpendicular axis one could determine nonlinear Landau parameters. The measurements were performed in layers of plastisol, gelatin, and farina-gelatin of 7 mm thickness with a rectangular base 8.9 × 8.9 cm, mounted between three aluminum plates. The upper plate was loaded with masses ranging from 0 to 25 kg and was fixed in each series of the stress-strain measurements. The values of the Landau coefficient A were measured in layers with different value of linear shear modulus. The dif...

Shear waves in a cubic nonlinear inhomogeneous resonator

We study finite-amplitude shear waves in one-dimensional resonator represented by a layer of rubber-like medium with inhomogeneities in the form of through holes made on the side face. The holes are parallel to the bases and perpendicular to the direction of vibrations. Two different configurations of the resonator: with holes at the bottom and at the top are studied. A rigid plate of finite mass is fixed on the upper surface. The lower boundary of the layer oscillates harmonically with a given acceleration. The equation of motion of particles in the resonator was found using the model of medium with one relaxation time, and a cubic dependence of the shear modulus of deformation. The measurements were performed in a resonator in the form of a rectangular parallelepiped of 15 mm thickness made of a rubber-like polymer plastisol. The linear shear modulus and shear viscosity of the polymer at the first resonant frequency were determined using the finite element method. The amplitudes of the oscillations in t...