Abstract
The objective of this study is to obtain the propagation velocity of an elastic wave in a loaded isotropic solid and to show the usefulness of the third-order elastic constant in determining properties of practical materials. As is well known, the infinitesimal elastic theory is unable to express the influence of stress on elastic wave propagating in loaded materials. To solve this problem, the authors derive an equation of motion for elastic wave in a finitely deformed state and use the Lagrangian description where the state before deformation is used as a reference, and Murnaghans finite deformation theory for the unidirectional deformed isotropic solid. Ordinary derivatives were used for the mathematical treatment and although the formulas are long the content is simple. The theory is applied to the measurement of the third-order elastic constants of common steels containing carbon of 0.22 and 0.32 wt%. Care is taken in preparing specimens to precise dimensions, in properly adhering of transducer to the surface of the specimen, and in having good temperature control during the measurements to obtain precise data. As a result, the stress at various sites in the structural materials could be estimated by measuring the elastic wave propagation times. The results obtained are graphed for illustration.
Highlights
There is little discussion of the practical application of the third-order elastic constants from the viewpoint of engineering
In this paper we introduced the formulas to show the relationship between the velocity of the elastic wave propagation and the stresses under the assumption that the elastic waves propagate in the unidirectional loaded isotropic materials
Three coordinate systems were used to treat the elastic waves in the finitely deformed solid; the first coordinate system corresponds to the non-deformed state, the second to Takahashi and Motegi. SpringerPlus (2015)4:325 the statically finitely deformed state, and the third to the state where an infinitesimal dynamical deformation is superposed on the finite deformation of the second state
Summary
There is little discussion of the practical application of the third-order elastic constants from the viewpoint of engineering. The third-order elastic constants and its mathematical procedure of practical materials were first reported by Hughes and Kelly (1953), their mathematical treatments were difficult to understand. We use Murnaghans finite elastic theory (Murnaghan 1951) combined with the Lagrangian description for a simpler description. Three coordinate systems were used to treat the elastic waves in the finitely deformed solid; the first coordinate system corresponds to the non-deformed state, the second to Takahashi and Motegi. SpringerPlus (2015)4:325 the statically finitely deformed state, and the third to the state where an infinitesimal dynamical deformation is superposed on the finite deformation of the second state.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
