Transmission coefficients for P and S waves in HardBego stones

The point where the incident ray reaching the S critique angle hits the interface The point where the incident ray reaching the P critique angle hits the interface

Transmitted P ray

Transmitted S ray

Reflected S ray from P ray

Reflected S ray from S ray

Reflected P ray from P ray

Part 1 transmitted S wave

Part 2 transmitted S wave

Ip: Normal component of the energy flux density vector of the transmitted P waveIs: Normal component of the energy flux density vector of the transmitted S waveIi: Normal component of the energy flux density vector of the incident P wave

Ray tracing plots in HardBego Stone

Transmitted P wave wavefront

Transmitted S wave wavefront

Reflected P wave wavefront

Reflected S wave wavefront

Transmitted P tensile tail wavefront

All ray/wave front plots only include transmitted P/S rays which are reflected once at the boundary before advancing to the axis, reflected P and S rays induced by transmitted P rays hitting the boundary. FEM simulations show these rays primarily contribute to tensile stress built in.

Transmitted P wave

Reflected P-S wave

Transmitted S wave

Reflected P wave

Reflected S wave

Reflected P-S wave

Reflected P wave

Transmitted tensile tail

Tensile Stress

Pressure in fluids

Tensile stress

Reflected P waves focus

Pressure in Fluids

Tensile stress

Reflected P-S waves focus

Reflected S waves focus

One sensor is placed at (0, 3) mm where the reflected P and S waves focus to detect tensile stress

The other sensor is placed at (0, 2.4) mm where the reflected P-S waves focus to detect tensile stress

c

Reflected P waves focus

Reflected S waves focus

Reflected P-S waves focus

Curl displacement

Pressure In Fluids

Curl Displacement

Pressure In Fluids

Curl Displacement

Div displacement

Pressure in Fluids

Div Displacement

Pressure in Fluids

Div Displacement

Max shear stress

Pressure inFluids

Max Shear stress

Pressure inFluids

Max Shear stress

Contribution from various types of waves

P and part 1 S waves Part 2 S waves P and S waves Head and surface waves

Max tensile stress 24.76 MPa

Max tensile stress 33.64 MPa

Max tensile Stress 58.30 MPa

Max tensile stress 32.3142 MPa (different locations)

Maximum tensile stress in different sizes stones

Wavefronts when the maximum tensile stress occurs

2 mm 4 mm 6 mm 8 mm

10 mm 12 mm