Scope:

ASTM E1876 test method is used for material development, characterization, design data generation, and quality control purposes. This test is used for elastic, isotropic, and homogeneous materials.Specimens of these materials possess specific mechanical resonant frequencies. Resonance frequency depends upon the mass, elastic moduli, and geometry of a suitable test specimen (rectangular or cylindrical geometry), and knowledge about these parameters helps to calculate the resonance frequency of a material. Resonance frequency values can be used to calculate dynamic elastic properties. 

Dynamic Young’s modulus is determined using the resonance frequency in the flexural or longitudinal mode of vibration. The dynamic shear modulus, or modulus of rigidity, is determined using torsional resonance vibrations. Young’s modulus and shear modulus are used to compute Poisson’s ratio.

The ASTM E1876 test method is also performed at extremely low and high temperatures with suitable equipment modifications and appropriate modifications to the calculations to compensate for thermal expansion.

Test Procedure:

In the ASTM E1876 test method, elastic properties are determined by calculating resonance frequencies. Resonance frequencies of test specimens are determined by exciting the specimens at different frequencies to find a particular frequency that matches the specimen’s natural resonance frequency. It is a hit-and-trial method. 

This test method measures the fundamental resonant frequency of test specimens of suitable geometry by exciting them mechanically by a singular elastic strike with an impulse tool. The signals are analyzed, and the fundamental resonant frequency is isolated and measured by the signal analyzer, which provides a numerical reading that is (or is proportional to) either the frequency or the period of the specimen vibration. The dimensions of the specimen, its resonance frequency, and its mass are used to calculate Young’s modulus and shear’s modulus. 

Video 01: What is resonance?

Specimen size:

For ASTM E1876, the specimens should be either rectangular or circular. Both can be used to calculate elastic properties. 

Data:

The following data is calculated in ASTM E1876:

1. Young modulus: 

Y = (38.3 × f² × ρ × ℓ⁴)/d²

where:

Y = Young’s modulus

f = resonance frequency of oscillations

ρ = density

ℓ = length

d = width

2. Shear modulus:

E = 2G(1 + ν)

Where:

E =  Young’s modulus 

G =  shear modulus 

ν =  Poisson’s ratio

3. Poisson’s ratio:

µ = Poisson’s ratio

E = Young’s modulus

G = shear modulus

4. Moduli at elevated and cryogenic temperatures:

M_T = modulus at temperature T 

M₀ = modulus at room temperature 

f_T = resonant frequency in a furnace or cryogenic chamber at temperature T

f_O = resonant frequency at room temperature in furnace or cryogenic chamber, 

α = average linear thermal expansion from room temperature to test temperature

∆T = temperature differential in °C between the test temperature T and room temperature

Conclusion:

ASTM E1876 test method is used for material development, characterization, design data generation, and quality control purposes.