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Full Title

Material and Structural Characterization of Shock Impingement on Elastic Surfaces

Names

Wang, Jingfan (author)

Kumar, Rajan (professor co-directing dissertation)

Oates, William (professor co-directing dissertation)

Shanbhag, Sachin (university representative)

Krick, Brandon (committee member)

Shoele, Kourosh (committee member)

Florida State University (degree granting institution)

FAMU-FSU College of Engineering (degree granting college)

Department of Mechanical Engineering (degree granting department)

Date Issued

2022

Format

text

doctoral thesis

Abstract

Fluid dynamic characterization of shock impingement on a flat surface has been studied extensively, whereas the material behavior induced by the shock has largely been ignored. The focus of this study is mainly on the material and structural characteristics of the deformable flat surfaces in the presence of shock impingement. Two structures, including a semi-infinite half-space and a flat plate, have been investigated. This study first investigates the elastic surface of a semi-infinite half-space subjected to surface tractions across an ideal inviscid shock. Anisotropic elasticity of material under the shock-induced discontinuous surface loading is evaluated using Stroh formalism. We find that a shear stress jump across an idealized shock impingement causes a stress singularity, while a pure pressure jump without shear does not produce a singularity. Theoretical solutions of the idealized discontinuous surface loads are compared to finite element modeling. Besides the stress components induced by the surface tractions, these analyses also include the investigation of the contributions of an arbitrary vector h, which is constrained by the prescribed far-field conditions, to the solutions of Stroh formalism. In the cases of uniform loads and a pure pressure jump acting on the surface, the stress components using Stroh formalism have good agreements with those by the finite element method. However, in the case of a shear stress jump acting on the surface, there are some stress mismatches between the results of the two techniques. Additional constraints are needed to solve the vector h in the Stroh method. Up to this point, Stroh formalism is further employed to study the case of an ideal Mach 5 shock by neglecting the stress mismatches caused by the jump in the shear components across the shock since those values are much smaller compared to the associated pressure components. The results illustrate that the highly localized stress singularity in the resultant von Mises stress \sigma_H can be treated as negligible, and the aluminum structure remains well within the elastic regime given inviscid Mach 5 flow conditions. Although the aluminum surface remains elastic under the selected Mach 5 conditions, potential stress concentration or large deformation may occur in the structure involving more complex geometries, i.e., elastic plates. Keep this in mind, further investigations of the characterization of a flat plate surface are carried out when the plate is subjected to a typical wall pressure with no intermediate plateau using the techniques of two-dimensional isotropic elastic theory and Fourier sine series expansion. Various shock impingement points are studied. The results indicate that the normal stress in the span direction is the dominant stress component contributing to the resultant von Mises stress \sigma_H. As the shock impingement point is near the two ends, the wall pressure acts more like a uniform load, which causes a more symmetric distribution of the \sigma_H. In these cases, the maximum of \sigma_H is also near the edge midpoint. More asymmetric distributions of \sigma_H are obtained as the shock impinges at the positions away from the ends. The \sigma_{Hmax} moves towards the right end of the plate until its location reaches an upper bound. Meanwhile, the \sigma_{Hmax} keeps decreasing due to the fact that the average pressure loading goes down as the shock impingement point moves from the left end of the plate to the right. However, the isotropic elastic theory is limited to the isotropic material properties and lacks the convenience to calculate the deflection of the plate. An extension of Stroh formalism is further introduced to investigate the deflections of the plates with relatively general anisotropic elastic properties. This work also provides the derivation process of the extended Stroh formalism by taking advantage of the material eigenrelation and other elegant mathematical treatments involved in the Stroh formalism, as well as the previous isotropic model. Notably, choosing appropriate functions for the stress potentials is critical when applying the Stroh formalism to the elastic problems. By taking the geometric boundary conditions of the plate problems into consideration, an exponential function of the thickness coordinate has been chosen. Furthermore, by employing the stress-strain constitutive law and the equations of equilibrium, a modified eigenrelation can be obtained with a relatively general anisotropic elastic tensor. Finally, the material and structural characterization of an aluminum plate subjected to the shock impinging at various locations on its upper edge are studied using the Stroh-type formalism. Three back pressures are also involved. Here the von Mises stress \sigma_H and the transverse displacement {\widetilde{u}}_3 of the flat plate are employed to evaluate these effects of the different shock impingement points. In general, the shock-induced pressure loading on the plate acts more like a uniform load when the shock impingement point is near the two ends of the plate. As the shock impinges away from the plate ends, the distributions of \sigma_H and {\widetilde{u}}_3 become more asymmetric. Particularly, there is a mismatch between the locations of the maximum of \sigma_H and {\widetilde{u}}_3. When p_{back} is no greater than the mean pressure value after the shock, the associated \sigma_H and {\widetilde{u}}_3 of the plate gradually decrease as the shock impingement point shifts from the left end to the right. In contrast, in the case of p_{back}=1 atm, the associated \sigma_H and {\widetilde{u}}_3 of the plate gradually increase. Variations of the safety factors of the plate are further investigated with various span-to-thickness ratios P under the combined conditions with different back pressures p_{back} and the shock impingement positions {\widetilde{x}}_0. The cases with p_{back}=1 atm have the lowest safety factors compared to other cases with the same P. The associated safety factors are below 1 when the ratios P increase to 100 or greater. When p_{back}=0 and p_1, a smaller {\widetilde{x}}_0 causes a lower safety factor. Notably, the combination of p_{back}=p_1 and {\widetilde{x}}_0=0.75 leads to the highest safety factors compared to other cases with the same P. Even when P = 300, the safety factor is still greater than 1. However, a high span-to-thickness ratio may lead to a violation of the small strain assumption of the elastic model. Therefore, further investigations are suggested on the limitations of the elastic model in this Stroh-type formalism.

Topics

Keywords

Anisotropic Elasticity, Shock Impingement, Stroh Formalism, Stroh-type Formalism, Structural and Material Characterization, Von Mises Stress

Date of Defense

March 29, 2022.

Submitted Note

A Dissertation submitted to the Department of Mechanical Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy.

Bibliography Note

Includes bibliographical references.

Advisory Committee

Rajan Kumar, Professor Co-Directing Dissertation; William S. Oates, Professor Co-Directing Dissertation; Sachin Shanbhag, University Representative; Brandon Krick, Committee Member; Kourosh Shoele, Committee Member.

Publisher

Florida State University

Identifier

2022_Wang_fsu_0071E_17127

Wang, J. (2022). Material and Structural Characterization of Shock Impingement on Elastic Surfaces. Retrieved from https://purl.lib.fsu.edu/diginole/2022_Wang_fsu_0071E_17127