COMSOL 6.4 - Elasto-Acoustic Effect in Rail Steel

The elasto-acoustic effect is the change in the speed of elastic waves that propagate in a structure undergoing static elastic deformations. The effect is used in many ultrasonic techniques for nondestructive testing of prestressed states within structures.

This example model studies the elastoacoustic effect in steels typically used in railroad rails. The analysis is based on the Murnaghan material model, which represents a hyperelastic isotropic material, and is based on an expansion of the elastic potential in terms of displacement gradients keeping the terms up to the third order. This material model can be used to study various nonlinear effects in materials and structures, of which the elasto-acoustic effect is an example.

Model Definition

The geometry represents a head of a railroad rail. It is a beam with a length of L0 = 0.607 m and a cross section of 0.0624 m by 0.0262 m. The rail is made of steel with the following properties (taken from Ref. 1):

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Density: ρ = 7800 kg/m3

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Lamé elastic moduli: λ = 11.58·1010 Pa and μ = 7.99·1010 Pa.

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Murnaghan third-order elastic constants: l = −24.8·1010 Pa, m = −62.3·1010 Pa, and n = −71.4·1010 Pa.

To create a prestressed state, the beam is stretched to a length L = (1 + ε)L0, where ε = 5·10−4. The model computes the eigenfrequencies of the beam for the free and prestressed states and the relative change in the speed of propagating elastic waves. Since only the axial waves are of interest here, symmetry conditions are used along two planes (xy and xz), so that bending is suppressed. The model is shown in Figure 1.

Figure 1: Geometry and mesh.

Results and Discussion

Instead of directly computing the wave speed, which can be a difficult task, the eigenfrequencies of the structure are used in order to implicitly obtain the wave speed. The relative change in the axial wave speed per unit strain can be estimated by the following formula:

(1)

Here the letter c denotes axial wave speed, and f the natural frequency for an axial mode. The subscript 0 refers to the unstrained state. This equation takes the elongation of the rod into account through the last two terms.

For a stress free sample, the computed eigenfrequency is f0 = 4242.34 Hz, which is shown in Figure 2. For prestressed sample, the computed eigenfrequency is f = 4234.66 Hz shown in Figure 3.

Figure 2: Eigenfrequency and normalized eigenfunction for a stress-free rail.

Figure 3: Eigenfrequency and normalized eigenfunction for a prestressed rail.

The deformation in the prestressed state is shown in Figure 4 below.

Figure 4: The static displacement field in the prestressed beam.

When the piece of rail is stretched as shown in Figure 4, the length of the rail changes, but most importantly, we get a prestressed state, which changes the wave speed in the material. Therefore, the eigenfrequency changes to f = 4234.66 Hz. The resulting value of the relative change in wave speed using Equation 1 is −2.62, which is in a good agreement with the experimental value of −2.52 reported in Ref. 1 (Table III, specimen 1). It is noted that if a linearly elastic material model is used, the predicted change in wave speed instead shows an increase.

Notes About the COMSOL Implementation

The eigenfrequency computation is performed using boundary conditions in the axial direction in both ends. One of them is implemented using a symmetry condition, which is just a simple way of prescribing the displacement in the normal direction to zero. In the other end, the displacement is prescribed to the value that gives the intended axial strain. In the first analysis, where the unstrained eigenfrequency is studied, this boundary condition acts as if the prescribed value is zero, since no static analysis precedes the eigenfrequency calculation.

Reference

1. D.M. Egle and D.E. Bray, “Measurement of Acoustoelastic and Third-order Elastic Constants for Rail Steel,” J. Acoust. Soc. Am., vol. 60, no. 3, p. 741, 1976.

Nonlinear_Structural_Materials_Module/Hyperelasticity/rail_steel

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

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In the tree, select > .

Click .

Click

In the tree, select > .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
eps0	5e-4	5E-4	Prescribed strain
L0	0.607[m]	0.607 m	Length of the beam

Geometry 1

Block 1 (blk1)

In the toolbar, click

In the window for , locate the section.

In the text field, type L0.

In the text field, type 0.0624*0.5.

In the text field, type 0.0262*0.5.

Click

Click the

button in the toolbar.

Form Union (fin)

In the window, click .

In the window for , click

Solid Mechanics (solid)

Hyperelastic Material 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Materials

Material 1 (mat1)

In the window, under right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Lamé parameter λ	lambLame	11.58e10	N/m²	Lamé parameters
Lamé parameter μ	muLame	7.99e10	N/m²	Lamé parameters
Murnaghan third-order elastic moduli	l	-24.8e10	N/m²	Murnaghan
Murnaghan third-order elastic moduli	m	-62.3e10	N/m²	Murnaghan
Murnaghan third-order elastic moduli	n	-71.4e10	N/m²	Murnaghan
Density	rho	7800	kg/m³	Basic