Processing-Induced Stresses and Curvature in Patterned Lines on Silicon Wafers

1996 ◽  
Vol 436 ◽  
Author(s):  
Y. -L. Shen ◽  
S. Suresh ◽  
I. A. Blech
Keyword(s):  
Finite Element ◽  
Stress Field ◽  
Plane Strain ◽  
Thermal Loading ◽  
Model Systems ◽  
Experimental Measurements ◽  
Uniform Stress ◽  
Uniform Stress Field ◽  
Induced Stresses ◽  
Generalized Plane Strain

AbstractThe evolution of stresses due to the patterning and thermal loading of thin lines on Si wafers, and the consequent changes in the overall curvature of the wafer are studied theoretically and experimentally. The analysis involves finite element simulations within the context of generalized plane strain models. The analysis is capable of predicting the wafer curvature in directions parallel and perpendicular to the lines. These predictions compare reasonably well with experimental measurements of curvature made on model systems. The thickness, width and spacing of the patterned lines have been varied systematically, and the associated changes in the evolution of stresses and curvature have been determined. The non-uniform stress field within the fine lines is also analyzed.

Processing-Induced Stresses And Curvature In Patterned Lines On Silicon Wafers

1996 ◽  
Vol 428 ◽  
Author(s):  
Y-L. Shen ◽  
S. Suresh ◽  
I. A. Blech
Keyword(s):  
Finite Element ◽  
Stress Field ◽  
Plane Strain ◽  
Thermal Loading ◽  
Model Systems ◽  
Experimental Measurements ◽  
Uniform Stress ◽  
Uniform Stress Field ◽  
Induced Stresses ◽  
Generalized Plane Strain

AbstractThe evolution of stresses due to the patterning and thermal loading of thin lines on Si wafers, and the consequent changes in the overall curvature of the wafer are studied theoretically and experimentally. The analysis involves finite element simulations within the context of generalized plane strain models. The analysis is capable of predicting the wafer curvature in directions parallel and perpendicular to the lines. These predictions compare reasonably well with experimental measurements of curvature made on model systems. The thickness, width and spacing of the patterned lines have been varied systematically, and the associated changes in the evolution of stresses and curvature have been determined. The non-uniform stress field within the fine lines is also analyzed.

Thermo-Mechanical Response of Multilayered Cylinders Under Pressure and Thermal Loading With Generalized Plane Strain Condition

2019 ◽  
Vol 141 (2) ◽  
Author(s):  
Sang-Guk Kang ◽  
Kuao-John Young
Keyword(s):  
Finite Element ◽  
Plane Strain ◽  
Thermal Loading ◽  
Mechanical Response ◽  
Axial Direction ◽  
Plane Strain Condition ◽  
Strain Condition ◽  
Efficient Design ◽  
Element Analysis ◽  
Generalized Plane Strain

Multilayered cylindrical structures subject to pressure and thermal loading are commonly seen in many industries. In this study, the formulas for multilayered cylinders under pressure and thermal loading are derived with an assumption that the cylinders meet generalized plane strain condition, i.e., there is no external constraint in the axial direction and the axial growths of the cylindrical layers are the same. A numerical solution procedure for double-layered cylinders subject to both pressure and thermal load is developed and implemented in a mathcad program. To validate the solution, a finite element model for a double-layered cylinder is prepared with abaqus, and its responses under pressure and thermal loading are compared to those from the mathcad program. The algorithm of the method can be extended to three or more layered cylinders. The method developed in this study allows quick optimization and efficient design refinement for multilayered cylinders without running finite element analysis (FEA).

Elastic Design Methods for Perforated Plates

1978 ◽  
Vol 100 (2) ◽  
pp. 356-362 ◽  
Author(s):  
J. S. Porowski ◽  
W. J. O’Donnell
Keyword(s):  
Finite Element ◽  
Thermal Loading ◽  
Solid Material ◽  
Perforated Plates ◽  
Finite Element Stress Analysis ◽  
Effective Elastic Constants ◽  
New Methods ◽  
Circular Holes ◽  
Elastic Design ◽  
Generalized Plane Strain

Methods for performing finite element stress analysis of perforated plates under pressure and complex thermal loading conditions are described. The concept of the equivalent solid material of anisotropic properties is employed to define the elasticity matrices to be used for axisymmetric analysis of plates containing triangular and square patterns of circular holes. Generalized plane strain effective elastic constants are used for better approximation of the overall plate behavior. New methods and curves for obtaining local ligament stresses from the nominal stresses in the equivalent solid material are given.

The 〈111〉 Elliptic Inclusion in an Anisotropic Solid of Cubic Symmetry

1982 ◽  
Vol 49 (2) ◽  
pp. 353-360 ◽  
Author(s):  
H. C. Yang ◽  
Y. T. Chou
Keyword(s):  
Stress Field ◽  
Uniform Stress ◽  
Elliptic Inclusion ◽  
Cubic Symmetry ◽  
Elastic Fields ◽  
Closed Form Solutions ◽  
Strain Transformation ◽  
Anisotropic Solid ◽  
Uniform Stress Field ◽  
Free Strain

This paper deals with a generalized plane problem in which a uniform stress-free strain transformation takes place in the region of an elliptic cyclinder (the inclusion) oriented in the 〈111〉 direction in an anisotropic solid of cubic symmetry. Closed-form solutions for the elastic fields and the strain energies are presented. The perturbation of an otherwise uniform stress field due to a 〈111〉 elliptic inhomogeneity is also treated including two extreme cases, elliptic cavities and rigid inhomogeneities.

Research on Nonlinear Damping Characteristics of Damping Alloy With Uniform Stress Field

Volume 1 ◽  
2004 ◽  
Author(s):  
Hongzhao Liu ◽  
Ziying Wu ◽  
Lilan Liu ◽  
Daning Yuan ◽  
Zhongming Zhang
Keyword(s):  
Stress Field ◽  
Nonlinear Function ◽  
Nonlinear Damping ◽  
Damping Capacity ◽  
Internal Damping ◽  
Uniform Stress ◽  
Nonlinear Relation ◽  
Uniform Stress Field ◽  
Half Power ◽  
Versus Strain

For the high damping metal material like damping alloy, the damping capacity usually changes with the strain amplitude and frequency nonlinearly. First, to extract the pattern of the internal damping versus strain, two time-domain calculation methods are presented in this paper. One is the moving exponent method (MEM for short) based on FFT (MEM+FFT) and the other is the moving autoregressive model method (MARM). The computing accuracy of the two methods has been compared through numerical simulations. The nonlinear relation curve of loss factor versus strain is achieved by the impulse excitation experiment employing uniform stress field. Then, to extract the pattern of the internal damping versus vibrating frequency, the sine sweep-frequency excitation experiment based on the half-power bandwidth method is carried out. The resulting curve indicates that the internal damping is also a nonlinear function of frequency.

Atomic transport via point defects in crystals III. Migration in a stress field

1985 ◽  
Vol 398 (1814) ◽  
pp. 203-208 ◽  
Keyword(s):  
Stress Field ◽  
Point Defects ◽  
Strain Tensor ◽  
Three Dimensional ◽  
Molecular Volume ◽  
Uniform Stress ◽  
Atomic Transport ◽  
Elastic Dipole ◽  
Uniform Stress Field ◽  
Defect Species

The kinetic theory of isothermal atomic transport via point defects that was presented in two previous papers (Franklin, A. D. & Lidiard, A. B. Proc. R. Soc. Lond . A 389, 405–431 (1983) and Franklin, A. D. & Lidiard, A. B. Proc. R. Soc. Lond . A 392, 457–473 (1984)) has been expanded into a three-dimensional formulation to analyse transport in an applied non-uniform stress field. The fluxes of the various defect species take the general form familiar from non-equilibrium thermodynamics, while the contribution to the force on defect species Y arising from the stress σ αβ is confirmed to be v ∇(λ (Y) αβ σ αβ ), where v is the molecular volume of the solid and λ (Y) αβ is the elastic-dipole strain tensor of the defect species Y (summation over repeated Cartesian indices α, β is here assumed). Full details of these calculations are presented in Lidiard, A. B. A. E. R. E. Rep . no R. 11367 (1984).

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