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Overview of Mechanical Behavior of Materials
Published in Heather N. Hayenga, Helim Aranda-Espinoza, Biomaterial Mechanics, 2017
Radu Reit, Matthew Di Prima, Walter E. Voit
Before moving past the concept of viscoelasticity, a last measure has to be introduced: the idea of a complex modulus. Until now, the calculation of a modulus of elasticity presupposed that all energy used to stress the sample elastically deformed the specimen. However, some energy is dissipated as heat or other forms of inelastic loss within the specimen. For example, while measuring the Young’s modulus, E, the measurement is in fact the combination of real (the storage modulus E′) and complex (the inelastic loss modulus E″) components of a complex modulus, E*, such that E∗=E′+iE″ This analogy holds true for different moduli as well, including bulk modulus (K*), axial modulus, Lamé’s first parameter (λ), Lamé’s second parameter (μ)—more commonly referred to as shear modulus (G*)—and the P-wave modulus (M).
A design procedure for evaluation and prediction of in-situ cemented backfill performance
Published in Ferri Hassani, Jan Palarski, Violetta Sokoła-Szewioła, Grzegorz Strozik, Minefill 2020-2021, 2021
In order to obtain the P-wave modulus of the cemented backfill, the density and the P-wave velocity were known. Aiming at the dimension consistency of P-wave modulus and UCS, the UCS prediction formula was established: σc=aP+b
Using P-wave propagation velocity to characterize damage and estimate deformation modulus of in-situ rock mass
Published in European Journal of Environmental and Civil Engineering, 2022
The P-wave modulus is defined as the ratio of axial stress to axial strain in a uniaxial strain state (Mavko et al., 1998). It can be deduced from the theory of elastic wave propagation, which is based on elastic mechanics. The dynamic effects of an elastic wave on the uniform, isotropic, and ideal elastic medium can be described by the wave equation, which takes the form of either (i) a wave equation with potential displacement as the disturbance or (ii) a wave equation with a volume strain and rotation component as the disturbance. The wave equation describes the response of the medium to various physical quantities, not just displacement perturbations, and the three-dimensional wave equations with respect to the former description method are expressed as follows: where u, v, and w are the displacements in the x-, y-, and z-directions, respectively; ρ is the medium density; λ and ν are the Lamé constants; θ is the bulk strain; and ▽2 is the Laplace operator.