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Nonlinear Spherically-Symmetric Wave Propagation in a Ball with Inclusion Region Which is Composed of a Blatz-Ko Material

Sarsangi Aliabad, Hamed | 2025

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 57834 (09)
  4. University: Sharif University of Technology
  5. Department: Civil Engineering
  6. Advisor(s): Mofid, Massood; Toufigh, Vahab; Emad Motaghian
  7. Abstract:
  8. The primary objective of this fundamental research is to examine the nonlinear spherically symmetric wave propagation and the dynamic response of a sphere with an inclusion region having symmetrically distributed eigenstrain. The sphere is assumed to be composed of a specific compressible nonlinear elastic material whose behavior is described by the practical Blatz-Ko model. To achieve this, the study is divided into two main parts. In the first part, the governing dynamic equilibrium equations are derived using established methods in nonlinear elasticity theory. This approach begins by transforming the initial configuration into a manifold with Riemannian geometry, referred to as the material manifold, which is identified by eigenstrain values. This transformation is defined so that the structure remains strain-free after enforcing eigenstrains. Subsequently, a second deformation is needed to place the material manifold into Euclidean space. Unlike the static solution, the second deformation is time-dependent. After this procedure the governing dynamic equilibrium equations are developed. The second part is devoted to solving these nonlinear partial differential equations. For this reason, Finite Difference Method with definite increment and time step is employed. the two required initial conditions include , representing the initial position of the geometry, and , indicating zero initial velocity. The Leapfrog method is utilized to determined the values of for subsequent time steps, while the Newton's method is used for solving the dynamic equilibrium equations. Obviously, the ratio significantly influences the convergence of the solutions. In other words, due to the complexity of the dynamic equilibrium equations, achieving convergent results is not feasible for all values of and . Finally, the responses of three cases including (1) sudden loading, (2) sudden unloading, and (3) sinusoidal periodic loading are fully investigated and compared with similar cases in an inclusion-free sphere. The results demonstrate that the inclusion region and eigenstrain values impact the convergence rate and wave amplitude. Specifically, increasing eigenstrain values leads to a decrease in the convergence rate. Moreover, the effect of the radius of inclusion region shows different behavior in loading and unloading cases. To be more detailed, increasing the inclusion radius raises the convergence rate during loading, while the smaller radius results in decresing it during unloading. Another significant finding reveals a direct relationship between eigenstrain values and oscillation amplitude under sinusoidal periodic loading, where higher eigenstrain values cause increased oscillation amplitude
  9. Keywords:
  10. Eigenstrain ; Finite Difference Method ; Spherical Ball ; Nonlinear Wave Propagation ; Blatz-Ko Material ; Dynamic Equivalent Inclusion Method (DEIM)

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