# Fall 2006

## Final Exam: ES 240 Solid Mechanics

Notes for students who are preparing for the final.

1. Time: 9:15 am, Thursday, 18 January 2006. Place: Sever Hall 206. No notes or books. Calculators are allowed.
2. There will be 3 hours and 5 problems.
3. Exam problems will mostly draw upon homework and parts of the lecture notes covered in class. The exam intends to test your understanding of the material covered in the course, not your creativity.
4. For the last two topics covered in class, finite deformation and strings and elastica, there was no homework, but some exercises are scattered in the notes. They may appear in the final.
5. For equations, you will need to memorize the most basic ones, such as equilibrium equations, Hooke's law, and strain-displacement relations. But for anything that you cannot remember, you should be able to derive.

## Powerpoint + Report

Find attached my powerpoint presentation and my report.

## EM 397 Term Paper: Stress-Induced Voiding in Dual-Damascene Cu Interconnects

Stress-induced voiding (SIV) is investigated in Cu-based, deep-submicron, dual damascene technology. Two failure modes are revealed by TEM failure analysis. For one mode, voids are formed under the via when the via connects a wide metal lead below it. For the via which is instead under a wide metal line, voids are formed right above the via bottom. The void source results from the supersaturated vacancies which develop when Cu is not properly annealed after electroplating and before being constrained by dielectrics. The driving force comes from the stress built up due to grain growth and the thermal expansion mismatch (CTE) between Cu interconnect and dielectrics. A diffusion model is introduced to investigate the voiding mechanism primarily for the vias connected to wide metal leads.

## ES 246 Project: Saint-Venant Torsion Problem

I plan to explore the Saint-Venant torsion problem applied to prismatic bars with elastic-plastic behavior. Wagner and Gruttmann have developed a finite element method to obtain the elastic/plastic stresses of a bar using a single load step. In particular, I will present the constitutive model that they have developed, and then use ABAQUS to apply Wagner and Gruttmann’s model to various cross-sections.

## EM 397 Term Paper: Dislocations in Epitaxial Thin Films

Dislocations are common in epitaxial systems. For a thin film epitaxially grown on a substrate with coherent interface, it may have spontaneously-formed dislocations when its thickness is larger than certain value, i.e. critical thickness. The presence of dislocations can have an adverse effect on electrical performance of semiconductor materials, providing easy diffusion paths for dopants to lead to short circuits, or recombination centers to reduce carrier density. And, formation of dislocations is one of the most observed mechanisms of relaxation of mismatch strain. However, in optoelectric applications, strain alters the electronic bandgap and band edge alignment, and should be maintained. So, controlling formation of dislocations is very important in the manufacture of microelectronic and optoelectronic devices.

This term paper will review some basic concepts and try to produce some understanding about the control dislocation formation.

## EM 397 Term Paper: Channeling crack of low-k dielectric films

Today low-k dielectric materials are integrated into computer chips to improve the operation speed and reduce the cross-talk noise. Due to weak mechanical properties of low-k dielectric materials, cohesive failure is subjected to occur. Channel cracking is one common mode of cohesive failure. In this term paper, several potential issues relevant to channel cracking of low-k dielectric thin films are reviewed.

## ES 246 project: Planar Composite under Plastic Deformation

The mechanical performance of a homogeneous material can be varied by the addition of second-phase particles. In this project, we will model a planar composite under plastic deformation. As shown on the following figure, the composite consists of matrix material and randomly-distributed inclusion particles. The matrix is assumed to be an elastic-plastic material with isotropic or kinematic hardenings, and the inclusion particle pure elastic with a higher Young’s modulus. The stress/strain field throughout the composite will be calculated numerically with finite element method.

## EM 397 Term Paper: Effects of Substrate Compliance on Buckling Delamination of Thin Films under Compression

For films or coatings deposited on substrate at high temperature, residual compressive stresses are often induced in the surface layers because of the mismatch in the thermal expansion coefficients. Under such compressive residual stresses, the surface thin film is susceptible to buckling-driven delamination. Various shapes of buckled region are observed, including long straight-sided blisters, circular and the ‘telephone cord’ blister.

## ES 246 project: Plane Strain Extrusion - Slip-line Field Solution vs. FEM Solution

Due to maturity of FEM package, slip-line field theory is not widely used these days. However, we shall keep in mind that slip-line field analysis can provide analytical solutions to a number of very difficult problem which may involve huge deformations or velocity discontinuities, e.g. many metal forming processes. To evaluate these two analytical and numerical methods for plasticity I will try a simple example, compare these two solutions and finally get into a conclusion of my own.

## citation

I guess it's time that I cite some papers that are relevant to what I am looking at. A paper byL.Mahadevan et al.: Elements of draping
and another one
Confined elastic developable surfaces: cylinders, cones and the elastica,

## ES 246 Project: Large Deformation Plasticity of Amorphous Solids, with Application and Implementation into Abaqus

node/add/imageI propose to investigate an elastic-viscoplastic constitutive model proposed by Anand and Gurtin [1] for the large deformation of amorphous solids.  Specifically, I will present the constitutive framework proposed for elastic-plastic amorphous materials, I will implement the constitutive equations into Abaqus/Explicit, and I will compare numerical results with experimental results for polycarbonate [2].

## Solid Mechanics Homework 43-46

43. Energy loss
44. Zener model and relaxation test
45. Zener model and cyclic-load test
46. Vibration of a viscoelastic rod

## Viscoelasticity

A word file is attached.

## EM 397 Thin Film Mechanics Term Paper

Each student completes a term paper of selected topics that (a) addresses a phenomenon in thin film materials, and (b) involves analyses using mechanics. The project contributes 25% of the grade, distributed as follows:

• 5%: November 30 (Thursday). Post your title and abstract in iMechanica, formated as below
1. Title (EM 397 Term Paper: e.g., Dislocations in Epitaxial Thin Films).
2. Tags (EM 397, Fall 2006, University of Texas at Austin, thin films, term paper)
3. Body: (i) Describe the phenomenon. (ii) Explain how mechanics is relevant. (iii) Cite at least 1 journal article.
• 10%: December 12 Tuesday (2:00-4:00 pm). 30 minute presentation. Use power point slides.
• 10%: December 18 Monday.

## ES240: Papers for Will Adams' final project

3 Papers that might be useful for Will Adam's final project (attached)

## Homework Q39 ~ Q42 is due Friday, Dec. 1

As promised, no homework for Thanksgiving :)

## Solid Mechanics Homework 39-42

39. A circular transverse wave
40. Creep and recovery
41. Temperature dependence and Mr. Arrhenius
42. A loose nylon bolt

## ES 240 project: Analysis of Resonance in Wine Glasses

We studied in class the phenomenon of resonance in forced, damped oscillators.  The mass and stiffness of a one-dimensional oscillator give rise to a natural frequency of oscillations known as the resonance frequency.  With no damping, energy input at this frequency accumulates and the amplitude of vibrations increases.

The phenomenon of resonance generalizes to linear elastic materials with many more (ie infinite) degrees of freedom: energy input at a natural frequency of vibration will accumulate and result in increasing amplitude of vibration.  The natural frequency in this case is determined by material properties (ie Young's modulus) and the geometry and dimensions of the object (ie a wine glass).  With so many degrees of freedom, the resonance frequency of common objects may be impossible to calculate exactly and it may be necessary to use the finite element method to investigate resonance.

## ES 240 project: Deformation of the Sarcolemma

The cardiac myocyte is the basic contractile unit of the heart. In addition to potentiating contraction through chemical and electrical means, each myocyte is a complex sensor that monitors the mechanics of the heart. Through largely unknown means, mechanical stimuli are transduced into biochemical information and responses. Such mechanotransduction has been implicated in the etiology of many cardiovascular pathologies [1]. One such mechanical parameter that the myocyte most likely monitors is the hydrostatic pressure in the myocardium.

## ES 240 Project: Finite-element modeling of nano-indentation of thin-film materials

Measuring mechanical properties of materials on a very small scale is a difficult, but increasingly important task. There are only a few existing technologies for conducting quantitative measurements of mechanical properties of nanostructures, and nano-indentation is the leading candidate. In this project, we simulate the nano-indentation tests of thin film materials using finite element software ABAQUS. The materials properties and test parameters will be taken from references on nano-indentation experiments [1, 2]. Therefore, the model can be validated by comparing its predictions with experiment results. In addition, we will change 1) the thickness of the thin film and 2) the material of the substrate (for the thin film) in the model, in order to study substrate's effects on nano-indentation tests.

## ES 240 Project: Draping of a thin elastic sheet

Everyone has seen how a table cloth hangs over the edge of the table. The way in which the excess material is accomodated, that is, the nature of the wrinkles, may depend on the material properties of the table cloth, the angle which the edge of the table is making (a right angle in the case of most tables but one can imagine the wrinkles of a table cloth draped over a circular table, or for that matter any shaped table).

If you aren't quite sure what I am talking about then take a scarf or any isotropic homegenous material and just susupend it of the corner of your desk.

I don't have any article to cite. I don't know if any work has been done on this. My aim is to read Landau Lifshitsz and attack this problem from first principals.

I would also like to use Abaqus to see if I can simulate the system. And then vary things likes E and poisson's ratio etc. And also the angle of the corner makes etc.

## ES 240 Project: Numerical calculation of stresses and displacements on buckled square thin membranes with FEM

Please see the attached PDF document for ES240 project proposal.

Please see the attached documents for the presentation and report files for this project (updated on 12/16/2006).