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reflection and transmission coefficients in long elastic bars

Frank Richter's picture

 

Dear iMechanica Community,

I'd like to ask for advice on modelling reflection and transmission coefficients when two bars with different cross-sectional areas contact each other and are exposed to pulse loading.

I have modelled this process in Abaqus/Explicit as a two-dimensional simulation. The investigation was conveniently done as a parametric study, defining the lateral dimension as parameter. Both bars are slim (length 2 m, diameter 2 cm) and elastic.

Two material combinations are defined:
identical materials: 2 steel bars
and different materials: first bar is steel, second bar is aluminum

Theory gives a negative reflection coefficient R and a positive transmission coefficient T.

Elements:
CAX4R is the only axisymmetric solid element available in Abaqus/Explicit with four nodes
CPE4R is the only plane strain solid element available in Abaqus/Explicit with four nodes
CPS4R is the only plane stress solid element available in Abaqus/Explicit with four nodes

Contact is defined as CONTACT PAIR with the penalty constraint. Surfaces are element-based. Abaqus was run in double precision.

R and T from FEM data are computed as the maximum of the signal with respect to the maximum of the incident wave.

Observation:
the FEM data for identical materials (large dots on the red and blue curves) and the ones for the reflection coefficient in case of different materials (small dots on the green curve) match the theoretical result.

A discrepancy arises for the transmission coefficient in case of different materials (small dots with positive T, which should lie on the orange curve).

The result is alike for axisymmetric, plain strain and plane stress elements.

The different formulas for computing the area ratio for axisymmetric and rectangular elements is properly accounted for.

The graphs show:
R and T for the axisymmetric case, theory and FEM data
the superposition of all signals from one of the parametric studies

Could someone please advise me ? Is someone aware of
1) an effect not taken into account in the theoretical formulas ?
2) a compilation of measured data ?
3) FEM data published elsewhere ?

Thank you for generously sharing your expertise,

Frank Richter

 

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