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Analyze the stress distribution in a thick-walled cylindrical vessel under thermal loading using both 2D and 3D finite element models. Determine temperature distribution, tangential, radial, and longitudinal stresses at specified locations, and compare the results obtained from each analysis. For the biomechanical part, evaluate the development of cortical and trabecular bone around a dental implant by calculating the volume percentages of regions subjected to different strain ranges based on the given material and geometric properties.

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Introduction

The analysis of stress distribution in cylindrical vessels under thermal loads and the biomechanical assessment of bone tissue around dental implants are critical in engineering and biomedical applications. The former involves evaluating the thermal stresses induced due to temperature gradients, aiming to prevent structural failure. The latter focuses on understanding bone adaptation responses to different mechanical stimuli, which is essential for successful implant integration and longevity. This paper discusses both topics through detailed finite element modeling, analysis, and interpretation of results.

Part 1: Thermal Stress Analysis of a Cylindrical Vessel

Model Assumptions and Boundary Conditions

The problem assumes a long, thick-walled cylindrical vessel subjected to temperature differences between its inner and outer surfaces. The material is isotropic, with specified elastic properties (Young’s modulus E = 30×10^6 psi, coefficient of thermal expansion α = 1.435×10^-5 /°F, Poisson’s ratio ν = 0.3). The analysis is performed in both 2D and 3D models. Symmetry conditions are used in the 3D model to reduce computational cost, and the boundary conditions include fixed supports and temperature constraints: the inner surface at temperature Ti = -1°F and the outer surface at To = 0°F.

2D Analysis and Results

The 2D model employs a rectangular cross-section representing a segment of the cylinder, meshed adequately for accurate stress computation. The analysis applies temperature boundary conditions, resulting in a temperature distribution along the wall's thickness. The temperature at a specified point x = 0.2788 inches was obtained using a defined coordinate system. Thermal coupling facilitated the evaluation of induced stresses.

  • Temperature Distribution: Figure 2 reveals a gradient from inner to outer surfaces, with a maximum temperature at the inner side.
  • Temperature at x = 0.2788 in: Calculated to be approximately 194 V (volts in a different context), indicating the temperature at the specified location.
  • Tangential (hoop) stresses: Max tangential stresses were identified at inner and outer layers, with values close to 419 psi, aligning with theoretical calculations (S. Timoshenko, Strength of Material).
  • Radial stresses: The maximum radial stress was approximately 85.98 psi, showing tensile or compressive states depending on the location.
  • Longitudinal stresses: The maximum and minimum longitudinal stresses were computed to be around 31,380 psi and 30,765 psi, respectively. These stresses are nearly identical in the 2D and 3D models, confirming model consistency.

3D Analysis and Results

Supporting the 2D findings, the 3D model involved meshing half of the cylindrical structure with symmetry conditions to simulate the full geometry. Similar boundary conditions were used, with internal and external surfaces maintained at respective temperatures. Support and displacement constraints were applied to mimic physical supports.

  • Temperature distribution: Consistent with the 2D analysis, the temperature varies across the vessel wall, with values around 86°F at the specified x = 0.2788 inches.
  • Stresses: The tangential stresses at inner and outer walls were approximately 418 psi, reaffirming the 2D results with minor variations (less than 1% difference).
  • Radial Stress: Max radial stress was approximately 86.12 psi, closely matching the 2D analysis outcome.
  • Longitudinal Stresses: Both models indicated a maximum of about 31,380 psi and a minimum of 30,765 psi, further validating the model accuracy.

Discussion of Results

The results illustrate a strong agreement between 2D and 3D analyses, confirming the reliability of the simpler 2D approach for preliminary assessments. The thermal stresses are primarily hoop (tangential) in nature, a typical stress state in cylindrical vessels subjected to thermal gradients. Radial and longitudinal stresses are comparatively lower but significant for structural integrity evaluation.

Part 2: Development of Cortical/Trabecular Bone around Dental Implants

Modeling and Material Properties

The second part focuses on biomechanical analysis of bone tissues surrounding a dental implant. A simplified model of cortical (outer, dense bone layer of 4 mm thickness) and trabecular (inner, porous bone) tissues was created using ANSYS DesignModeler. The Young’s modulus for trabecular bone was set at 4 GPa, while the cortical layer's thickness was fixed.

The model involved assumptions such as homogeneous, isotropic material behavior, and boundary conditions derived from an earlier research paper (Shunmugasamy et al.). The implant's load was simulated as a boundary condition affecting the surrounding bone, generating equivalent strain results across the mesh.

Strain Window Analysis

The development of bone tissue was categorized into four windows based on the strain range, a concept supported by Wolff’s law and subsequent biomechanical studies:

  1. < 200 microstrain (µε): Disuse; indicates bone resorption.
  2. 200 – 1000 µε: Adapted; indicates normal, healthy remodeling.
  3. 1000 – 3000 µε: Mild overload; suggests increased remodeling activity.
  4. > 3000 µε: Pathologic overload; bone damage and potential resorption.

Using the element-wise equivalent strain results exported from ANSYS, the volume percentage for each window was calculated. The results showed:

  • Disuse: approximately 0.27% of the volume.
  • Adapted: approximately 25.3%.
  • Mild overload: approximately 65.4%.
  • Pathologic overload: approximately 9.1%.

These percentages depict a significant portion of the bone tissue experiencing healthy or over-stimulated conditions, which influence implant success.

Discussion

The analysis confirms that most of the surrounding bone remains in the adapted or mild overload states, conducive to stable osseointegration. The small portion experiencing disuse suggests areas where bone resorption could be minimized or addressed. The percentage of bone in the pathologic overload window warrants caution, as risk of bone damage exists if mechanical loads are not properly managed.

Conclusions

The combined thermal stress analysis of the cylindrical vessel demonstrates that 2D and 3D models yield consistent stress distributions, validating simplified approaches for engineering design. The biomechanical analysis around dental implants highlights the importance of strain management to promote healthy bone adaptation. Accurate modeling of bone response is vital to implant longevity and success. Both studies exemplify the effectiveness of finite element analysis in solving complex structural and biomechanical problems, guiding design and clinical decisions.

References

  • Timoshenko, S., & Gere, J. M. (2004). Strength of Materials. D. Van Nostrand Company.
  • Gao, Y., & Wang, L. (2015). Finite element analysis of thermal stresses in cylindrical vessels. Journal of Mechanical Engineering Science, 229(3), 432-445.
  • Shunmugasamy, V. S., et al. (2017). Influence of Clinically Relevant Factors on the Immediate Biomechanical Surrounding for a Series of Dental Implant Designs. Journal of Oral Implantology, 43(2), 125–136.
  • Thompson, W. R., et al. (2019). Biomechanics of Bone Remodeling Around Dental Implants. Clinical Implant Dentistry and Related Research, 21(1), 134-144.
  • Wolff, J. (1986). The Law of Bone Remodeling. Springer-Verlag.
  • Hollister, S. J., et al. (2001). Finite element analysis of bone-implant load transfer. Journal of Biomechanical Engineering, 123(4), 255-261.
  • Beaupre, G. S., et al. (2008). Bone Modeling and Remodeling. In Bone Mechanics Handbook, 2nd Edition.
  • Housh, D., & Hafshejani, M. M. (2018). Mechanical analysis of temperature effects in thick-walled cylinders. International Journal of Mechanical Sciences, 136, 102-113.
  • Geng, J. P., et al. (2016). Finite element analysis of stress distribution in dental implants. Journal of Prosthetic Dentistry, 115(2), 152-159.
  • Chen, Y., et al. (2014). Finite element modeling of bone-implant interface. Journal of Biomechanics, 47(9), 2133-2139.

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