scholarly journals Brittle fracture in linearly elastic plates

2021 ◽  
pp. 1-36
Author(s):  
Stefano Almi ◽  
Emanuele Tasso
Keyword(s):  
Thin Film ◽  
Brittle Fracture ◽  
A Priori ◽  
Elastic Plates ◽  
Limit Model

In this work we derive by $\Gamma$ -convergence techniques a model for brittle fracture linearly elastic plates. Precisely, we start from a brittle linearly elastic thin film with positive thickness $\rho$ and study the limit as $\rho$ tends to $0$ . The analysis is performed with no a priori restrictions on the admissible displacements and on the geometry of the fracture set. The limit model is characterized by a Kirchhoff-Love type of structure.

scholarly journals Nonlocal elastodynamics and fracture

2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Robert P. Lipton ◽  
Prashant K. Jha
Keyword(s):  
Field Theory ◽  
Brittle Fracture ◽  
A Priori ◽  
Nonlocal Model ◽  
Elastic Fields ◽  
Nonlocal Field Theory ◽  
Nonlocal Field ◽  
Increasing Function ◽  
Non Locality

AbstractA nonlocal field theory of peridynamic type is applied to model the brittle fracture problem. The elastic fields obtained from the nonlocal model are shown to converge in the limit of vanishing non-locality to solutions of classic plane elastodynamics associated with a running crack. We carry out our analysis for a plate subject to mode one loading. The length of the crack is prescribed a priori and is an increasing function of time.

Effect of Crystallographic Quality of Grain Boundaries on Both Mechanical and Electrical Properties of Electroplated Copper Thin Film Interconnections

2011 ◽  
Author(s):  
Naokazu Murata ◽  
Naoki Saito ◽  
Kinji Tamakawa ◽  
Ken Suzuki ◽  
Hideo Miura
Keyword(s):  
Thin Film ◽  
Brittle Fracture ◽  
Grain Boundaries ◽  
Fracture Mode ◽  
Electron Back Scatter Diffraction ◽  
Fracture Rate ◽  
Copper Films ◽  
Average Grain Size ◽  
Electroplated Copper

Both mechanical and electronic properties of electroplated copper films used for interconnections were investigated experimentally considering the change of their micro texture caused by heat treatment. The fracture strain of the film annealed at 400°C increased from about 3% to 15% and their yield stress decreased from about 270 MPa to 90 MPa. In addition, it was found that two different fatigue fracture modes appeared in the film. One was a typical ductile fracture mode and the other was brittle one. When the brittle fracture occurred, a crack propagated along weak or porous grain boundaries which were formed during electroplating. The brittle fracture mode disappeared after the annealing at 300°C. These results clearly indicated that the mechanical properties of electroplated copper thin films vary drastically depending on their micro texture. The electrical reliability of the electroplated copper yjin film interconnections was also investigated. The interconnections used for electromigration tests were made using by a damascene process. An abrupt fracture mode due to local fusion appeared in the as-electroplated interconnections. Since the fracture rate increased almost linearly with the square of the applied current density, this fracture mode was dominated by local Joule heating. It seemed that the local current concentration occurred around the porous grain boundaries. The life of the interconnections was improved drastically after the annealing at 400°C. This was because of the increase of the average grain size and the improvement of the quality of grain boundaries in the annealed interconnections. However, the stress-induced migration occurred in the interconnections annealed at 400°C. This was because of the high tensile residual stress caused by the constraint of the densification of the films during annealing by the surrounding oxide film. Therefore, it is very important to control the crystallographic quality of electroplated copper films for improving the reliability of thin film interconnections. The quality of the grain boundaries can be evaluated by applying an EBSD (Electron Back Scatter Diffraction) analysis. New two experimentally determined parameters are proposed for evaluating the quality of grain boundaries quantitatively. It was confirmed that the crystallographic quality of grain boundaries can be evaluated quantitatively by using the two parameters, and it is possible to estimate both the strength and reliability of the interconnections.

scholarly journals Asymptotic theory for thin two-ply shells

2020 ◽  
Vol 42 (3) ◽  
pp. 269-282
Author(s):  
David J. Steigmann
Keyword(s):  
Shell Theory ◽  
Asymptotic Theory ◽  
A Priori ◽  
Small Strain ◽  
Asymptotic Model ◽  
Strain Response ◽  
Pure Bending ◽  
Laminated Shells ◽  
Limit Model ◽  
Interfacial Surface

We develop an asymptotic model for the finite-deformation, small-strain response of thin laminated shells composed of two perfectly bonded laminae that exhibit reflection symmetry of the material properties with respect to an interfacial surface. No a priori hypotheses are made concerning the kinematics of deformation. The asymptotic procedure culminates in a generalization of Koiter's well-known shell theory to accommodate the laminated structure, and incorporates a rigorous limit model for pure bending.

An obstacle problem for Koiter’s shells

2019 ◽  
Vol 24 (10) ◽  
pp. 3061-3079 ◽  
Author(s):  
Philippe G Ciarlet ◽  
Paolo Piersanti
Keyword(s):  
Asymptotic Analysis ◽  
A Priori ◽  
Elastic Shell ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Three Dimensional Model ◽  
Limit Model ◽  
Membrane Shells ◽  
Point Of Departure ◽  
Confinement Condition

In this paper, we define, a priori, a natural two-dimensional Koiter’s model of a ‘general’ linearly elastic shell subject to a confinement condition. As expected, this model takes the form of variational inequalities posed over a non-empty closed convex subset of the function space used for the ‘unconstrained’ Koiter’s model. We then perform a rigorous asymptotic analysis as the thickness of the shell, considered a ‘small’ parameter, approaches zero, when the shell belongs to one of the three main classes of linearly elastic shells, namely elliptic membrane shells, generalized membrane shells and flexural shells. To illustrate the soundness of this model, we consider elliptic membrane shells to fix ideas. We then show that, in this case, the ‘limit’ model obtained in this fashion coincides with the two-dimensional ‘limit’ model obtained by means of another rigorous asymptotic analysis, but this time with the three-dimensional model of a ‘general’ linearly elastic shell subject to a confinement condition as a point of departure. In this fashion, our proposed Koiter’s model of a linearly elastic shell subject to a confinement condition is fully justified in this case, even though it is not itself a ‘limit’ model.

scholarly journals On travelling wave solutions of a model of a liquid film flowing down a fibre

2021 ◽  
pp. 1-30
Author(s):  
HANGJIE JI ◽  
ROMAN TARANETS ◽  
MARINA CHUGUNOVA
Keyword(s):  
Thin Film ◽  
Weak Solutions ◽  
A Priori Estimates ◽  
Travelling Waves ◽  
A Priori ◽  
Travelling Wave ◽  
Model Parameters ◽  
Priori Estimates ◽  
Thin Film Model ◽  
Long Time

Abstract Existence of non-negative weak solutions is shown for a full curvature thin-film model of a liquid thin film flowing down a vertical fibre. The proof is based on the application of a priori estimates derived for energy-entropy functionals. Long-time behaviour of these weak solutions is analysed and, under some additional constraints for the model parameters and initial values, convergence towards a travelling wave solution is obtained. Numerical studies of energy minimisers and travelling waves are presented to illustrate analytical results.

scholarly journals Uniqueness in the determination of loads in multi-span beams and plates

2018 ◽  
Vol 30 (1) ◽  
pp. 176-195 ◽  
Author(s):  
ALEXANDRE KAWANO ◽  
ANTONINO MORASSI
Keyword(s):  
Source Term ◽  
A Priori ◽  
Uniqueness Result ◽  
Transverse Load ◽  
Elastic Plates ◽  
Small Interval ◽  
Elastic Beams ◽  
Position Variable ◽  
Interior Points

Most of the results available on the inverse problem of determining loads acting on elastic beams or plates under transverse vibration refer to single beam or single plate. In this paper, we consider the determination of sources in multi-span systems obtained by connecting either two Euler–Bernoulli elastic beams or two rectangular Kirchhoff–Love elastic plates. The material of the structure is assumed to be homogeneous and isotropic. The transverse load is of the form g(t)f(x), where g(t) is a known function of time and f(x) is the unknown term depending on the position variable x. Under slight a priori assumptions, we prove a uniqueness result for f(x) in terms of observations of the dynamic response taken at interior points of the structure in an arbitrary small interval of time. A numerical implementation of the method is included to show the possible application of the results in the practical identification of the source term.

Local Cross-sectional Profiling of Multilayer Thin Films with an Atomic Force Microscope for Layer Thickness Determination

1997 ◽  
Vol 12 (8) ◽  
pp. 1935-1938 ◽  
Author(s):  
J. R. LaGraff ◽  
J. M. Murduck
Keyword(s):  
Thin Film ◽  
Thin Films ◽  
Sample Preparation ◽  
Atomic Force Microscope ◽  
A Priori ◽  
Ion Milling ◽  
Cross Sectional ◽  
Atomic Force ◽  
The Individual ◽  
Additional Sample

A new essentially nondestructive cross-sectional method is described for measuring the individual thicknesses of multilayer YBa2Cu3O7 (YBCO) and SrTiO3 (STO) thin films using off-axis ion milling and the atomic force microscope (AFM). Since the ion-milling is done during routine patterning of a thin-film device and the AFM requires only a small area for imaging, no additional sample preparation is required. This is a significant improvement over traditional cross-sectional techniques which often require lengthy and destructive sample preparation. Also, there is no a priori reason that this technique would not be amenable to other multilayer thin-film systems.

Some a Priori Estimates for a Singular Evolution Equation Arising in Thin-Film Dynamics

1996 ◽  
Vol 27 (3) ◽  
pp. 638-660 ◽  
Author(s):  
Stephen H. Davis ◽  
Emmanuele DiBenedetto ◽  
David J. Diller
Keyword(s):  
Thin Film ◽  
Evolution Equation ◽  
A Priori Estimates ◽  
A Priori ◽  
Priori Estimates
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