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In general, residual stress refers to the internal stress distribution present in a material system when all external boundaries of the system are free of applied traction. Virtually any thin ¯lm bonded to a substrate or any

individual lamina within a multilayer material supports some state of residual stress over a size scale on the order of its thickness. The presence of

residual stress implies that, if the ¯lm would be relieved of the constraint of the substrate or an individual lamina would be relieved of the constraint of its neighboring layers, it would change its in-plane dimensions and/or would become curved. If the internal distribution of mismatch strain is incompatible with a stress-free state, then some residual stress distribution will remain even under these conditions.

1.7.1 Classification of film stress

Film stresses are usually divided into two broad categories. One category

is growth stresses, which are those stress distributions present in ¯lms following growth on substrates or on adjacent layers. Growth stresses are

strongly dependent on the materials involved, as well as on the substrate temperature during deposition, the growth °ux and growth chamber conditions. Advances in nonintrusive observational methods for in-situ stress

measurement and growth surface monitoring have made it possible to follow the evolution of growth surface features and the corresponding evolution of average stress levels in the course of ¯lm formation. These capabilities have provided new insights into the origins of ¯lm stress and have led to a subdivision of the category of growth stresses into those stresses which arise during various phases of the growth process and those which are present at the end of the growth process. Usually, growth stresses are reproducible for a given process and the values at the end of growth persist at room temperature for a long time following growth. Growth stresses are also commonly called intrinsic stresses, a term of limited use as a descriptor of what is represented.

A second category of ¯lm stress represents those stress conditions arising from changes in the physical environment of the ¯lm material following its growth. Such externally induced stresses are commonly called extrinsic stresses. In many cases, these stresses arise only when the ¯lm is bonded to a substrate, and the distinction between growth stresses and induced

stresses becomes hazy at times. The classi¯cation scheme has no fundamental signi¯cance, and the lack of a clear distinction between categories is largely immaterial. The purpose in this section is to identify examples of both types of stress.

The development of growth or intrinsic stresses for a particular material system, substrate temperature and growth °ux depends on many factors. Perhaps the most important among these are the bonding of the deposit to the substrate (epitaxial or not, for example), the mobility of adatoms on the ¯lm material itself, and the mobility of grain boundaries formed during growth. Except for the case of ideal epitaxy, the ¯nal growth structure

is inevitably metastable. Because of the huge number of degrees of freedom involved in establishing this metastable structure, the degree of departure from a completely stable equilibrium structure can be signi¯cant.

Many mechanisms for stress generation during ¯lm material deposition have been proposed, and the more common among days have been reviewed by

(Doerner and Nix 1988). These mechanisms include:

Modeling & Simulation

An Introduction

After some consideration regarding a meaningful way of putting System, Model, and Simulation in an appropriate perspective I arrived at the following distinction.

System

 A system exists and operates in time and space.

Model

 A model is a simplified representation of a system at some particular point in time or space intended to promote understanding of the real system.

Simulation

 A simulation is the manipulation of a model in such a way that it operates on time or space to compress it, thus enabling one to perceive the interactions that would not otherwise be apparent because of their separation in time or space.

Modeling and Simulation is a discipline for developing a level of understanding of the

interaction of the parts of a system, and of the system as a whole. The level of understanding which may be developed via this discipline is seldom achievable via any other discipline.

A system is understood to be an entity which maintains its existence through the interaction of its parts. A model is a simplified representation of the actual system intended to promote understanding. Whether a model is a good model or not depends on the extent to which it

promotes understanding. Since all models are simplifications of reality there is always a trade-off as to what level of detail is included in the model. If too little detail is included in the model one runs the risk of missing relevant interactions and the resultant model does not promote understanding. If too much detail is included in the model the model may become overly complicated and actually preclude the development of understanding. One simply cannot

develop all models in the context of the entire universe, of course unless you name is Carl Sagan.

A simulation generally refers to a computerized version of the model which is run over time to study the implications of the defined interactions. Simulations are generally iterative in there development. One develops a model, simulates it, learns from the simulation, revises the model, and continues the iterations until an adequate level of understanding is developed.

Modeling and Simulation is a discipline, it is also very much an art form. One can learn about riding a bicycle from reading a book. To really learn to ride a bicycle one must become actively engaged with a bicycle. Modeling and Simulation follows much the same reality. You can learn much about modeling and simulation from reading books and talking with other people. Skill and talent in developing models and performing simulations is only developed through the

building of models and simulating them. It's very much a learn as you go process. From the interaction of the developer and the models emerges an understanding of what makes sense and what doesn't. What is Simulation?

There are many different definitions of simulation. SIMUL8 software specializes in discrete event simulation described below.

Discrete Event Simulation

A simulation is a computer model that mimics the operation of a real or proposed system, such as the day-to-day operation of a bank, the running of an assembly line in a factory, or the staff assignment of a hospital or call center.

The model is time based, and takes into account all the resources and constraints involved, as well as the way these things interact with each other as time passes. Simulation also builds in the randomness you would see in real life. For example, it doesn't always take exactly 5 minutes for a customer to be served and a customer doesn't always arrive every 15 minutes. This means that the model really can match reality, so when you make changes to the model it will demonstrate exactly how the system would behave in real life.

With simulation you can quickly try out your ideas at a fraction of the cost of trying them in the real organization. And, because you can try ideas quickly, you can have many more ideas, and gain many insights, into how to run the organization more effectively.

How Does Simulation Work?

When you click the run button in a simulation model you see the work you do (products, patients, paper work etc.) move around the organization. The clock in the corner of the screen tells you what the equivalent time would be in the real system.

Simulation is animated. This enables visualization of a new facility and a greater ability to visualize the impact of experiments in an existing facility. You can see key bottlenecks, over-utilized resources and under resourced elements of a system.

The software automatically collects performance measures as the model runs so that you can not only see visually what will happen, you can also get accurate numerical results to prove your case.

Typical outputs include:

   

Inventory Throughput

Bottleneck utilization Productivity

Typical inputs include:

   

Cycle time Staff levels

Arrival/order rates Average order size

What Can Be Simulated?

There are many scenarios that can be simulated. As a general rule systems that involve a process flow with discrete events can be simulated. So any process you can draw a flowchart of you should be able to simulate.

The processes you'll gain most benefit from simulating are those that involve change over time and randomness. For example a gas station. Nobody can guess at exactly which time the next car will arrive at the station, whether they'll decide to purchase gas only etc. Modeling complex dynamic systems like this effectively in any other way isn't possible.

Why Simulate?

There are many process improvements you can make using simulation: higher quality and

efficiency from capital assets, better management of inventory, higher return on assets this list is endless. But some of these improvements could be made without simulation, so the real question is 'Why use simulation instead of another method?'

Simulation vs Real Life Experimentation

Cost:

Experimenting in real life is costly. Its not only the capital expenditure of hiring new staff or purchasing new equipment its the cost of the ramifications of these decisions. What if you fire 3 staff and then find you can't cope with the workload and you loose customers? The only cost

with simulation is the software and the man hours to build the simulation.

Repeatability:

In real life its really difficult to repeat the exact circumstances again so you only get 1 chance to collect the results and you can't test different ideas under the exact same circumstances. So how do you know which idea is really the best. With simulation you can test the same system again and again with different inputs.

Time:

If you want to know whether hiring another 3 doctors will reduce patient waiting lists over the next 2 years you'll actually have to wait 2 years. With simulation you can run 2, 10 or even 100 years into the future in seconds. So you get the answer now instead of when its too late to do anything about it.

Simulation vs Other Mathematical Modeling Techniques Interaction of Random Events:

Some other mathematical tools can manage to effectively model a steady state scenario but only simulation lets you build in random occurrences like a machine breaking down and see the effects of this further down the line. The more complex the scenario is the more these tools fall down and simulation is the only answer.

Non Standard Distributions:

Many mathematical techniques force the model builder to describe a situation as an

approximation, it takes and average of 5 minutes to serve each customer. In real life this isn't the case. It takes 3 minutes to serve the customer if they have 4 items, it takes 7 minutes if they have 20 items. Approximating means results such as resource utilization time, customer waiting time are all inaccurate. Only simulation gives you the flexibility to describe events and timings as they actually are in real life.

Makes you think:

Simulation provides a vehicle for a discussion about all aspects of a process. The rule and data collection forces you to consider why elements work in a certain way, if they could work better. It also brings to the surface inconsistencies and inefficiencies especially between different

sections of a process who work independently. Sometimes the simulation doesn't even have to be finished the framework it has provided to think through the issues reveals the solution. Communication:

Because simulation is visual and animated it lets you clearly describe your proposal to others. Its more convincing that just displaying the end results as people can't see where these came from. Simulation is so effective at communicating ideas that many companies now use it as a sales tool to sell their products.

Finite-element method

W. Robert J. Funnell

Dept. BioMedical Engineering, McGill University

1. Introduction

This is a very brief introduction to some of the concepts involved in the finite-element method. For more details, many text books are available (e.g., Fundamentals of the Finite Element Method, H. Grandin Jr., Macmillan, New York, xvi + 528 pp, 1986).

In the finite-element method, a distributed physical system to be analysed is divided into a number (often large) of discrete elements.

The complete system may be complex and irregularly shaped, but the individual elements are easy to analyse.

The division into elements may partly correspond to natural subdivisions of the structure. For example, the eardrum may be divided into groups of elements corresponding to different material properties.

Most or all of the model parameters have very direct relationships to the structure and material properties of the system.

A finite-element model generally has relatively few free parameters whose values need to be adjusted to fit the data. This assumes, of course, that the parameters are known a priori from other measurements.

The elements

 

may be 1-D, 2-D (triangular or quadrilateral), or 3-D (tetrahedral, hexahedral, etc.); and may be linear or higher-order.

The elements may model mechanics, acoustics, thermal fields, electromagnetic fields, etc., or coupled problems.

In a mechanical problem, the elements may model membranes, beams, thin plates, thick plates, solids, fluids, etc.

The behaviour of a particular type of element is analysed in terms of the loads and responses at discrete nodes. This analysis is often based on the Ritz-Rayleigh procedure, which is discussed in Section 2.

An example of a particularly simple element formulation is presented in Section 3.

The result of the analysis of a typical element type is a small matrix relating a vector of nodal displacements to a vector of applied nodal forces.

The components of the matrix can be expressed as functions of the shape and properties of the element, and the values of the components for a particular element can then be obtained by substituting the appropriate shape and property parameter values into the formulæ.

Once the element matrices have been calculated, they are all combined together into one large matrix representing the whole complex system, as discussed in Section 4