AbstractThe response of an inhomogeneous soil layer with exponentially varying stiffness with depth is explored using one-dimensional viscoelastic wave propagation theory. The governing equation is treated analytically, leading to an exact harmonic solution of the Bessel type. Both positive and negative velocity gradients are examined using a pertinent dimensionless parameter. It is shown that (1) for positive stiffness gradients, strains attenuate with depth faster than displacements that, in turn, attenuate faster than stresses; and (2) close to the soil surface, curvatures are controlled by acceleration, whereas they are controlled by strain at depth. The fundamental natural frequency of the layer compares well against approximations on the basis of the Rayleigh quotient. Novel asymptotic and ad hoc approximate solutions for the base-to-surface transfer function are proposed, providing good alternatives to the complex exact solution at both high and low frequencies. New expressions are derived relating (1) shear strain and peak particle velocity; and (2) curvature and peak ground acceleration close to the soil surface. A full-domain approximation is provided, allowing the practical implementation of the specific velocity model. Numerical examples are presented.IntroductionAvailable analytical solutions for the seismic response of continuously inhomogeneous media reveal the possibility of higher amounts of seismic energy reaching the ground surface over media with discontinuous variations in elastic properties with depth (Towhata 1996). Problems involving continuous inhomogeneity have been treated for different velocity models (Ewing et al. 1957; Ambraseys 1959; Toki and Cherry 1972; Schreyer 1977; Gazetas 1982; Dakoulas and Gazetas 1985; Aki 1993; Towhata 1996; Semblat and Pecker 2009; Paraschakis et al. 2010; Rovithis et al. 2011; Mylonakis et al. 2013; Vrettos 2013; Garcia-Suarez 2020) without necessarily acknowledging the importance of continuous versus discontinuous variations in soil material properties with depth. For the common case of a parabolic (power law) velocity model, the special condition of vanishing stiffness at the soil surface has been examined by Dobry et al. (1971), Towhata (1996), Travasarou and Gazetas (2004), Rovithis et al. (2011), and Kausel (2013). More recently, Mylonakis et al. (2013) demonstrated analytically that under viscoelastic conditions and regardless of excitation frequency and material damping, the near-surface shear strain may be either zero or infinite (but never finite) depending solely on rate of inhomogeneity. This result is in contrast to surface displacements that, naturally, are always finite.The effect of soil inhomogeneity has been studied in multiple dimensions for different types of stress waves. Vrettos (1990) tackled the problem of dispersive horizontally polarized shear surface waves propagating in an inhomogeneous half-space for both bounded and unbounded variations in the shear modulus with depth. A collection of available solutions (not restricted to elastic waves) for wave propagation in inhomogeneous media is available in Manolis and Shaw (1997, 1999, 2000) and Brekhovskikh and Bayer (1976).The seismic response of inhomogeneous media modeled using “equivalent” homogeneous counterparts, that is, defined by means of an average wave propagation velocity, has been explored against exact solutions (Idriss and Seed 1967; Dobry et al. 1971; Zhao 1997; Rovithis et al. 2011; Garcia-Suarez 2020). Additionally, simple analytical formulae have been proposed for the fundamental dynamic characteristics of inhomogeneous soils at resonance conditions using Rayleigh approximations (Mylonakis et al. 2013) and asymptotic analysis (Garcia-Suarez 2020; Garcia-Suarez et al. 2020).Despite research on the subject for more than half a century, nearly all analytical solutions for inhomogeneous media are limited to cases in which stiffness and shear wave propagation velocity vary as a power of depth. Although some dependable solutions exist for other variations in stiffness (e.g., the constant minus the exponential function considered in the seminal study by Vrettos 2013), these solutions are semianalytical in nature because they are expressed in the form of an infinite power series and not special functions (e.g., of the Bessel type) associated with a wealth of mathematical properties that facilitate their analytical treatment. Closed-form analytical solutions, including relevant asymptotic expressions and approximations pertaining to low and high frequencies, can shed light on the physics of the problem, which is often not possible with other methods. The gap in knowledge regarding the dynamic behavior of more general velocity profiles provided the motivation for the herein reported work.In this paper, the seismic response of an inhomogeneous soil layer resting on a rigid base is investigated in the realm of one-dimensional viscoelastic wave propagation theory. An exponential function is adopted, which can describe both positive and negative velocity gradients. The latter may be relevant for cases of large confining stresses under building foundations, stiff overconsolidated surface crusts, or ground improvement near the soil surface, which may result in soil stiffness decreasing with depth (Dobry et al. 1976; Vrettos 2013). For simplicity and as a first approximation, soil mass density and material damping are assumed to be constant with depth. The problem is treated analytically, leading to an exact solution expressed in terms of Bessel functions for the vibrational characteristics of the system, the base-to-surface transfer function, and the variation in ground displacements, shear strains, and curvatures with depth. The solution is compared against Rayleigh approximations for the fundamental frequency of the layer by implementing different shape functions for the corresponding mode shapes. Asymptotic and approximate solutions are proposed for the response in low and high frequencies, which are combined through a smooth transition function to provide a full-domain approximation. The model is validated against a numerical layer transfer-matrix (Haskell-Thomson) formulation for two actual soil profiles. Apart from its intrinsic theoretical interest, the proposed exact solution can be used to assess other relevant solutions for wave propagation and site response analysis.Problem StatementThe problem considered in this work refers to a continuously inhomogeneous viscoelastic soil layer of thickness H, excited by vertically propagating seismic shear (SH) waves applied at its base (Fig. 1).Soil mass density, ρ, and hysteretic damping ratio, ξ, are assumed constant with depth, whereas the shear wave propagation velocity follows the exponential variation (1) where V0 = shear wave propagation velocity at the ground surface; zr = reference depth; and α = dimensionless inhomogeneity parameter.Rearranging Eq. (1), parameter α can be defined as (2) where VH(=V0eα) = shear wave propagation velocity at the base (zr=H). Evidently, positive and negative values of α correspond to stiffness increasing (V0

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