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PDF vs. KindleWhile the book itself is designed for Amazon’s Kindle, this PDF “preview” is intended for readers who haven’t used Kindles. If you already have a Kindle, try using the “mobi” preview file (here), or “look inside” the book at Amazon.com (here).

The reading “experience” can be markedly different. In the first place, what you see on the PDF is not quite what you will see when you read the book using either a Kindle or a Kindle-app for the device of your choice.

Will this matter? If you haven’t used a Kindle yet, and are used to working with PDFs on a laptop or desktop computer, it might. Consider these comments from two readers of one of the books in this series:

From a reader who prefers PDFs on a PC:

“[It] was not as convenient as a continuous PDF. Normally (as far as possible) it is best to have the picture and explanation in the same page. So I had to do back and forward many times to read about the image I see in a page. I saw a lot of white space which I think is better to use in some way or the other.”

From a reader who used the Kindle app on an Android device and on a PC:

“I think the Kindle for PC is a little inconvenient since the pics/text are reformatted depending on the screen size. [F]or instance, if I expand the Highlight/Notes Column at the left side of the screen, the images would get a little scattered. Looks far better on smaller screens (read the doc on a 4.3” screen mobile phone, looks great)”

This PDF is laid out to resemble what you can expect to see on a Kindle, but does nothing to reproduce the effect of changing the font size, zooming, etc. that you can do with Kindle or Kindle-apps on different devices. If you want to experiment with the Kindle-experience, you can download the relevant software from Amazon.com – you do not need to purchase or own a Kindle.

Prefer a PDF?We currently sell this book only on the Kindle. If you’re interested in another format (PDF, other eBook, etc.) drop us a line and we’ll see what we can do.

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TABLE OF CONTENTS

Preface

1: Introduction - and Prerequisites

2: The Theory (or How To State The Problem)

3: The Procedures

4: Advanced Analyses: Transient, Non-Linear

5: Assignments

Summing Up

References

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Getting Started With Abaqus

Workbook 2

Thermal and Thermo-mechanical Analyses

by

C.Venkatesh

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Preface These Workbooks consist of a series of examples presented as a mixture of text and images. They're intended to be used along with the "theory" book in this series - Getting Started with Abaqus - Essential Theory - and are therefore deliberately low on text and rich on images. The workbooks are self-contained, but you may want to read the "theory" if you're new to the finite element method itself. The other Workbooks are listed in the references section at the end of this book.

The general presentation approach I have used is to write a few lines of text then follow these up with one or more images. This approach works on all Kindles, even the ones with the smallest black-and-white screens. It certainly looks nicer on color devices (such as Kindle for PC).

Abaqus' interface, of course, looks best on a large screen - and Kindles, of course, have small screens. The convenience of buying and reading a book on the Kindle is of no use if the images are scaled down to the point of illegibility. To get around this, each scaled-down "large image" is immediately followed by a "clipped" image of the highlighted area. This seems like the best solution to the problem. It does take you an extra click to move to the next page, but is worth the added legibility.

Why not drop the larger images and retain only the "zoomed in" ones? Remember that if you're reading this book, you're new to Abaqus/CAE. And one of the problems with using a new software interface is that you can't always locate that pesky menu. The larger image, then, serves to show you where the command / menu is located on the overall screen. The zoomed-in image makes it a little more legible.

In some cases, though, you will find (at least on the 6" Kindles) that you will need to "zoom-in" even on the zoomed-in image. To do this move the Kindle's cursor to the image and click on it for the Kindle to rotate the image by 90-degrees. Then use the Kindle's Back button to restore the earlier view and orientation. I've tried to preempt this by flipping some images in the book itself, which means you have to rotate either the Kindle or your neck, but I figured this is easier than the clicks-to-zoom-in-and-back. It's harder to rotate a desktop or laptop computer, of course, so this does make things a little worse on those devices.

Finally, the Kindle lets you select your font size. At larger font sizes, the captions for some images spill over to the next page - and in some cases, are entirely on the next page. Please keep this in mind when reading the book.

I recommend that you page through the sets of images more than once, then return to the preceding text and re-read it. This is not a very easy way to read on the Kindle, which is designed more for sequential paging back and forth. I've used a number of hyperlinks to

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make it easier to jump around between pages. You will probably use the Kindle's "Previous / Next" section controls a lot more than with a "serial" book.

Your goal, of course, is not just to reproduce the illustrated procedures on Abaqus. You should be able to extend the documented procedures to your own scenarios. Note that all the examples presented in this series can be worked out using even the student version of Abaqus.

The models used in this book can be downloaded from www.kfourmetrics.com – email the author if you have any trouble finding the files or using them.

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1: Introduction - and PrerequisitesTo make the most of Abaqus, you're likely to be better served by not concentrating on any single task - such as thermal analysis - but by broadening your perspective as you proceed. The approach adopted here is an alternative to reading the entire User's Manuals, since the group-all-related-topics-together-in-one-location approach adopted by the User's Manuals is ideal as a reference, but it's hard to read them from cover to cover.

Each Workbook in this series uses a different theme, but is it essential that they be read strictly in serial order?

In Workbook 0 - User Interface and Modeling Overview we took a look at the logical structure of the interface of Abaqus/CAE, constructed a simple model and traced the route from initial-geometry to final-results. Workbook 1: Linear Static Analyses and Basic Mesh Generation looked at mesh generation and solution-verification.

Can you skip them and go straight to this Workbook?

To use this book, you need to be familiar enough with Abaqus/CAE's interface to create-and-edit sketches, parts, instances, meshes, materials, sections, loads, restraints and jobs. All these were covered in the first Workbook and are not repeated here. In other words, Workbook 0 - User Interface and Modeling Overview is a prerequisite.

It's a little different for Workbook 1: Linear Static Analyses and Basic Mesh Generation. You will probably be able to complete and comprehend all the topics covered here without having worked your way through it. That said, you may find Workbook 1 useful, for four reasons. First, the quality-of-solution section is omitted completely here though the methods discussed are as relevant to thermal analyses as to stress-analysis. Second, some of the user-interface aspects (such as selection of element types) will be covered rapidly here, omitting the discussions contained in that Workbook. Third, we will also discuss thermo-mechanical stresses here, so comfort with linear-static-stress analyses is useful. Finally, you will sooner or later find it useful to work directly with INP files - which were covered in that Workbook, and are hence omitted here. Workbook 1, therefore, is not a prerequisite, but is recommended.

None of the other Workbooks need to be read before this one. Some of the techniques introduced there (such as explicit time integration, the choice of time step-size for transient analyses, etc.) are relevant, but not essential, to the theme of this Workbook.

One final note about this Workbook: in addition to showing how to use Abaqus to model and solve problems involving the transfer of heat energy, we also introduce methods to check your modeling process itself - which is different from, and complements, the quality-of-solution methods discussed in Workbook 1.

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2: The Theory (or How To State The Problem)

Heat, as a form of energy, has a remarkable capability to confound the engineer. The maths is no worse than that of forces and accelerations. It's easier, in fact, given that temperature is a scalar quantity while deformation and displacement are vectors. The physics should be no harder either, since the effects of heat are as familiar as those of forces. But the fact is that stress is more familiar to many than heat - perhaps because heat energy is closely related to concepts in thermodynamics, fluid flow, etc. We'll start, therefore, by reviewing a few terms and comparing them with their equivalents in stress analysis.

Note that the narrative here is intentionally brief. For a longer discussion on the items covered here, see Getting Started With Abaqus: Essential Theory.

Solids vs. Fluids

A Quick Look at Heat Energy

Thermal Analysis vs. Stress Analysis

Steady-state vs. Transient Equilibrium

Linear vs. Non-linear

Coupled vs. Decoupled Fields

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Solids vs. Fluids

The critical idea here is convection: what it is, how it affects the behavior of heat energy, what data you need to model it - and, most important - when it can be neglected.

Conservation of energy in a solid does not involve "flow". In fluids, it usually does. The effects of flow show up as convection - the combined effect of advection and diffusion. The terms - advection and diffusion - are potentially confusing. One easy way to remember the difference between the two is to note that advection is "one way" (i.e. moves energy downstream) while diffusion is "two way" (independent of the direction of flow of the fluid). In fluid mechanics, the Peclet number is used to compare the relative importance of the two: if the Peclet number is much less than 1, diffusion is much more significant than advection.

The theory of heat transfer is strongly connected to "numbers" such as the Peclet Number, and the profusion of such numbers is perhaps one reason the science can be hard to get a grasp of. The actual formula for each number is certainly important, but the physical significance of each number is more so. If you're well versed with fluid flow, you're probably quite comfortable with these. If you've been inclined more towards solid mechanics, then a review of the theory of fluid flow is essential.

The equations of fluid flow can get quite complicated. What's more, the numerical solution of the governing equations, usually labeled Computational Fluid Dynamics (CFD), often uses methods other than the finite element method.

Under some conditions solids too can exhibit fluid-like behavior. This behavior is significant if the load-application is very rapid (see, for instance, Blast & Explosive Modeling or Introduction to Hydrocodes, an excellent book for the advanced reader) or if temperatures are high enough to cause creep. In the latter case some materials are better modeled as visco-elastic or visco-plastic, meaning that they exhibit viscous behavior that is characteristic of fluids.

For the most part, though, models that assume "pure" solid behavior are acceptable for stress analysis. For such models, the absence of convection makes solids easier to work with.

But there's a wrinkle to this: heat energy, which we'll take a look at next, can involve convection in more ways than one.

A Quick Look at Heat Energy

A complete study of the transfer of heat starts with thermodynamics and involves both solid and fluid mechanics. As we noted above, restricting our attention to solid domains can make things easier, but such restrictions involve some caveats. Let's walk through the physics of the flow of heat in different media to see what these are.

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The transfer of heat within a solid occurs through conduction. Governed by Fourier's Law of Heat Conduction, the flow of heat energy through the domain depends on the thermal conductivity of the material. The equations of heat conduction are relatively easy to solve using the finite element method.

In a fluid, convection is the means for the transfer of heat within the domain. Even if the fluid is initially at rest, a difference in temperature gives rise to convective currents. And as pointed out above, modeling convection is not only computationally demanding, it is also not well addressed by finite element methods.

Heat energy can also be transmitted in the absence of a solid or fluid medium - in or through a "vacuum". Such radiative heat transfer can be significant at high temperatures or for structures like spacecraft.

Heat transfer can also cause (or be caused by) phase-change in the material, involving aspects such as the latent heats of fusion, solidification, vaporisation, etc. Phase change involves a change in the energy (or enthalpy) of the material without a change in the temperature.

Remember that temperature is different from heat. Temperature is a measure of heat, which is a form of energy. Being a form of energy, heat is conserved - i.e., governed by the law of conservation of energy.

The "problem" in thermal analysis involves calculating the temperature over space and/or time given the sources and sinks of heat energy. This is analogous to stress-strain problems, which involve calculating the deformation (or displacement vector) over space and time given the sources and sinks of "mechanical" energy (i.e. applied forces and supports / restraints).

Temperature is a measure of the kinetic energy of the molecules, while the total heat energy also includes potential energy. Combustion, for example, involves the conversion of such "potential" energy to kinetic energy, leading to a change in the temperature.

Now note that restricting our attention to solids means radiation and convection can be - and, usually, are - omitted. This is because much more heat is transferred within the solid by conduction than by either convection or radiation.

But this is only true within the solid.

How about at the boundaries of the solid? Beyond the boundaries of the solid region that you are modeling must lie either another solid, another fluid, or a vacuum. This means the physics at the boundary of the solid may involve either conduction, convection or radiation - or a combination of these.

In other words, even if we restrict our attention to the conduction of heat in solids, chances are we'll have to deal with convection at the boundaries. To include the effect of convection without directly having to solve the flow problem, we resort to an artifice: the somewhat awkward approach of a film coefficient or convective coefficient. We'll come back to this below.

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Thermal Analysis vs. Stress Analysis

There are several differences and similarities between stress-strain problems and heat-transfer problems. We'll review these, since our study in this book involves both temperature and deformation. Unless we're clear on the statement of each problem, combining them can lead to errors.

Conservation of energy is universal, but can be stated differently for different applications. For stress analysis, it involves the conservation of linear and angular momentum. The momentum of a body depends on its inertial properties - mass and the moment of inertia. The density of the body is its "specific mass" - the mass per unit volume. Given the density of a material, we can calculate the mass if we know the volume.

In the context of heat transfer, the heat capacity of the body is important. This is usually provided as the specific heat capacity - heat capacity per unit mass per unit change in temperature. Just as the mass relates the energy supplied to the change in momentum (or the deformation), the heat capacity relates the energy supplied to the change in temperature.

In SI units, specific heat capacity is measured in Joules/kilogram/degree-Kelvin.

The Thermal Conductivity of the material is analogous to the Elastic Modulus of a solid. It relates the heat energy to the temperature gradient, just as the Elastic Modulus relates the "mechanical" energy to the deformation gradient (i.e. the strain).

In SI units, thermal conductivity is specified in Watts/meter-Kelvin.

In stress analysis, the material's Elasticity Modulus is the basis for the Stiffness Matrix (in finite element terms). Similarly, in thermal analysis the material's coefficient of thermal conduction is the basis for the Conduction Matrix - though the latter term is rarely used. Conventionally, the matrix is referred to as the "stiffness" matrix even though it represents the "conductance" of the domain.

From a finite element modeling perspective, recall that elements serve two purposes: they define both the domain (or region being modeled) and the variation of the items of interest within each element. For stress analysis, the basic "item of interest" is the deformation (or displacement) vector. From this, we can calculate the strain and stress. For thermal analysis, the item-of-interest is the temperature, from which we can calculate the heat flux.

This means that element shapes are similar for both stress and thermal analysis (since the domain is similar in both), but element formulations differ (since formulations represent the differential equations involved and the variation of the items-of-interest). For the same reason, the degrees-of-freedom per node differ. Thermal problems have only 1 dof per node: the temperature, which is a scalar value.

Unlike the deformation vector, the temperature can be discontinuous in space (and, therefore, across element boundaries) as shown below.

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Discontinuity in temperatureJust as the gradient of the deformation field is strain (which is related to stress by the elasticity constants), the gradient of the deformation field is the heat flux, often referred to simply as the flux. Heat flux is the heat-rate-per-unit-area, so the units are power-per-unit-area (watts/m2 in SI units).

While stress and strain are tensors (i.e. have components that are vectors), the heat flux is a vector.

To test your understanding of the physics, try to figure out on your own whether the heat flux can be discontinuous in space.

In stress analysis, triangular and tetrahedral elements are notorious for being less "efficient" than quadrilateral and hexahedral elements. This notoriety, though, is restricted to stress analysis. Triangles and tetrahedra work perfectly fine for thermal analyses.

Just as in stress analysis, materials can be isotropic, orthotropic or anisotropic. The data required, of course, is different. For example, orthotropic materials are common in stress-analysis: many fibre reinforced composites can be modeled adequately with orthotropic data. This involves providing material constants along "principal" directions (which are usually normal to each other, hence the term "ortho-tropic"). Similar behaviors can apply to heat transfer too, but can be more complicated, particularly if the model is to avoid excessive detail. For instance a printed-circuit board consists of an insulator with strips of conductors embedded in it. Capturing the thermal behavior without representing every strip of the conductor can be a challenge.

Next, we'll review how boundary conditions are related to the "type" of problem - static or transient.

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Steady State and Transient Equilibrium

In stress-analysis, a model for static analysis has to have some displacement constraints (which must be enough to remove any "rigid" body motions). Unless there's a bug in the software, whatever forces you specify, the applied forces will match the computed reactions at the constraints. If you specify the "wrong" forces, the computed reactions will be "wrong" - but the body will still be in equilibrium, so you will get a solution even if it's the wrong one.

It's important to remember that the way you "pose" the problem affects the "output" of your model. If you pose the above problem as a static problem, you're telling the software that the reactions have to match applied forces. If you pose the problem as a dynamic one, you're telling the software to include inertial forces (mass-times-acceleration and damping-forces) when balancing the applied forces.

The inclusion or exclusion of inertial forces in the equilibrium equation can seem obvious, but it's a frequently-made mistake in modeling. Stress analysis has so many different types of analyses - static, quasi-static, harmonic, random, shock spectrum ... - that even experienced analysts can go wrong. Things are a little easier with heat transfer, but the importance of "posing" the problem correctly is an aspect you should not neglect.

For instance, a model for dynamic analysis need not have any displacement constraints - consider "free-free" structures, such as an aircraft in flight.

Let's look at the corresponding scenario in thermal problems.

From a mathematical point of view, temperature is the "equivalent" of displacement (or deformation). Next, flux is the "equivalent" of stress. Finally, convective boundaries are the equivalent of "spring-supports". (These are well-known in civil engineering models as "elastic foundations", but are less widely used in mechanical engineering models.)

So just as you must specify some displacement constraints for a static-model in stress analysis, you must specify some temperature constraints in thermal analysis for a static solution to proceed.

If the previous sentence seemed ambiguous to you, you're on the right track: it's not just enough to say "some constraints". We need to know precisely how many constraints we must specify so that the problem has a unique solution.

Most engineers who work with problems involving forces and deformation are familiar with computing the "rigid-body-motions" of the structure: for a static analysis, enough constraints should be specified to eliminate (i.e. prevent) any such rigid body motions, but specifying unnecessary constraints can alter the solution. (If you're new to the field, you should review the differences between structures and mechanisms, over-constrained and under-constrained structures, etc. - topics covered in textbooks such as Engineering Mechanics of Solids).

When translating the "rigid body motions" idea to thermal problems, keep in mind the fact that temperature is a scalar quantity.

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In stress-analysis, a stress-free boundary is one that has no applied forces (no applied traction, strictly speaking) and therefore has no energy "flowing" across it. In heat transfer, a flux-free boundary is an insulated boundary - no heat flow across it. Stress boundary conditions are rarely used in stress analysis - but flux boundary conditions may be the only way to model a thermal problem. The reverse is true of concentrated forces. They are commonly used in stress analysis, but concentrated sources of heat are less widely used in thermal models. If you think about it, you'll realize that this is a result of modeling decisions, not one imposed by the fundamental equations.

In a thermal problem, if the heat fluxes at the boundary are not balanced, the body cannot be in thermal equilibrium. Until it reaches a steady-state, the temperature has to change with time - that is, the problem must involve transient heat transfer.

We noted above that the heat capacity of the material is the thermal "equivalent" of the inertial properties. The higher the inertial properties of a body (mass or moments of inertia) the more energy it takes to change its position in space. Similarly, the higher the heat capacity of a material, the more heat energy it takes to change its temperature. For dynamic stress analysis, you must specify the density of a body so that its mass can be calculated using the element's volume. Similarly, for transient thermal analyses, you must specify the heat capacity. Since this is commonly tabulated as the "specific" heat capacity, you usually provide both the density and the specific heat capacity.

The conditions at the boundary can, to summarize, be any of the following:

1. specified temperature - referred to as an "Essential" boundary condition, or a "Dirichlet" boundary condition

2. specified heat flux - referred to as a "Natural" boundary condition, or a "Neumann" boundary condition

3. convection, which consists of a film coefficient and an ambient temperature - referred to as a "mixed" boundary condition

4. radiation, the specification of which can be fairly complex as we'll see below

Linear vs. Non-linear

A large majority of stress-analysis problems can be modeled reasonably well using linear models. Heat transfer is not as forgiving.

First, let's take radiation. As we mentioned above, a hot body radiates heat to its surroundings. And, conversely, hot surroundings radiate heat to a colder body. If the body's in a vacuum, such radiation may be the only mode of heat transfer. This is the case for spacecraft, for example.

Even if the body's in a fluid, radiation may well be the dominant mode of heat transfer between the body and its surroundings (or "ambience"). This is usually the case at high temperatures - strictly speaking, at large differences in temperature, which means radiation can be important in applications like cryogenics. Why? Because the heat energy radiated is proportional to the 4th power of the temperature (Stefan's Law of radiation),

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which means that it varies exponentially with temperature (strictly speaking, the 4th

power of the temperature difference). Convection, in contrast, depends only on the 1st power of the temperature (i.e. the temperature itself). As the temperature difference at the boundary rises, the energy transferred through radiation increases faster than that transferred by convection.

The dependence on the 4th power means radiation is inherently non-linear: the flux (i.e. heat lost or gained) is not a linear function of the variable of state (i.e. temperature). Since this is only on the boundary, it's often called a "boundary non-linearity", similar to contact boundaries in stress analysis.

In some materials, the coefficient of heat conduction varies significantly with temperature. This too makes the problem non-linear. Since the coefficient is a material property, the term "material non-linearity" is widely used - similar to stress-analysis problems involving plasticity (or other forms of inelastic behavior of the material itself).

Our study will include this, but will exclude radiative boundaries. Modeling radiation usefully requires discussing things like view factors and surface emissivity - which will take us well beyond "beginners" realms. If you're interested in this area, the Abaqus Theory Manual is not a bad place to start from.

The Theory Manual on cavity-radiation and view-factors.

How about convective heat transfer at the boundary?

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Flow in a fluid can be remarkably complicated: boundary layers, turbulence, boiling, etc. are essential for even reasonably accurate models in many situations. Fortunately for us, the use of "film coefficients" means we can often neglect these effects. Strictly speaking, we're not neglecting them - all we're doing is lumping all these effects into the coefficient, on the grounds that we're not interested in the fluid itself. There is a price to pay for this approach, though, as we'll see when we review convective boundaries. A convective boundary, then, will be non-linear only if the film-coefficient itself has to be modeled as dependent on the temperature.

Coupled vs. Decoupled Fields

What is the dependence of the temperature field on the deformation field, and vice versa? If they're completely independent, they are "decoupled" - else they're "coupled".

For a solid, the coefficient of thermal expansion tells us the dependence of deformation on the temperature. If the coefficient is significant (for the expected range of variation in temperature), then we cannot assume they're decoupled. If such a body is restricted from expanding or contracting while the temperature changes, the result will be a stress field in the body.

The restrictions on expansion or contraction may be for a variety of causes - thermal gradients, restraints, differential expansion, etc. A welding jig, for example, holds the structure in place while heat flows in from the welding electrode or flame. The designer may need to consider how the structure behaves both while it's being welded, and after it's cooled and the clamps have been removed. Another scenario is a bimetallic strip, where the differences in coefficients of thermal expansion cause the strip to change shape. Yet another example is a solidifying ingot or casting. The outer layers cool faster than the inner ones leading to various effects, most of which are considered defects in the resulting solid.

Tensile and compressive stresses across the cross-section of a cooled-ingot.

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The most common scenario is "one way coupling": the temperature field is independent of the stress-field, but the stress-field depends on the temperature field.

There are some scenarios where this is not realistic. One example is friction-stir welding. Here, the source of heat is the friction caused by the movement of the components. That is, the source of heat itself depends on the deformation field. Such coupled problems are hard to model both because the physics is complicated and because reliable data is hard to obtain. Simulation of such problems often relies on empirical data rather than "fundamental" mechanics.

Another example of two-way coupling is materials that have significant "internal" friction. A rubber tyre, for example, heats up both because of external friction (contact with the road) and internal friction (within the tyre's material itself, since the tyre is "cyclically" deforming as the vehicle rolls forward). The viscous behavior of the tyre's material, which affects how it deforms, depends on the temperature of the tyre. So the heat generated within the tyre depends on the flexing of the plies - and the flexing of the plies depends on the temperature. That's why it's a "coupled" problem: the stress field and the temperature field are interdependent. The mechanics of such problems is reasonably well understood. As a result, finite element models based on fundamental mechanics are feasible, but require fairly cutting-edge material models. Such models - involving things like visco-elasticity and incompressibility - are one of Abaqus' strengths, though these are well beyond our scope in this series.

In this Workbook we'll consider one-way coupling only: thermo-mechanical analyses, where the deformation and stress field depend on the temperature field.

If you know the temperature field itself, of course, you need not solve the thermal problem at all: you can specify the temperature field in the statement of the stress-analysis problem itself. But if the geometry is complicated and the temperature is non-uniform, it can be faster, easier, and more convenient to solve the temperature problem and apply these values to the stress-strain model. As we'll see later, Abaqus/CAE makes this approach particularly easy.

The image below shows what you will be able to calculate by the time you're done with this Workbook - the temperature distribution and the corresponding stress-state.

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The upper image shows the temperature distribution, the lower shows the corresponding stress-distribution.

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3: The ProceduresWhen going over this section, make sure you focus on the logical structure of the user-interface. If you're already familiar with static stress analysis, you will find that the setup is similar enough that you'll be tempted to speed through the steps. This is OK for the areas related to creation of the parts, assemblies, etc., but I recommend you make sure you correlate the material data, steps and boundary conditions with the physics of heat transfer.

Also pay attention to the verification methods presented here, since they're intended to prevent (as opposed to catch and correct) mistakes. An ounce of prevention is worth a pound of cure - but only if you're preventing the right things.

It's important to note that we're now introducing verification before we even start modeling. This is in contrast to the after-the-solution approach we presented in the Workbook 1. Remember that these two approaches are complementary, not mutually exclusive: you should make it a practice to prepare both pre- and post-verification plans for your models.

Verification

The "Exact" Solution

Geometry and Mesh

Temperature BCs

Flux BCs

Convective BCs

Thermo-Mechanical Stresses

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ReferencesHeat TransferJ.P.Holman, Mcgraw-Hill

This is widely used in many undergraduate courses, where the worth of some of its contents are often missed - data that you will find useful to estimate quantities such as convection coefficients. The table-of-contents could (should?) have been more streamlined, so if you haven't used this book earlier you may need to be persistent in your search. But do persevere.

A First Course in Finite ElementsJ.Fish, T.Belytschko, John Wiley and Sons

This is a comprehensive text-book and is a good addition to your reference-shelf. Chances are you won't read it completely unless it's a text-book for a course you're doing. But if you do choose to pursue a career in FEA, chances are very good that you will end up reading every bit of it sooner, rather than later.

An Analysis of the Finite Element MethodW.G.Strang, G.J.Fix, Wellesley-Cambridge Press

While not directly related to Abaqus, this is an outstanding book by an outstanding teacher (you can, and should, view Dr.Strang's video lectures off the MIT website). Immensely readable, invaluable, well worth the price of purchase and the time to read it.

Python Scripts for Abaqus. Learn by ExampleG.Puri, Wellesley-Cambridge Press

Considering I'm writing for Abaqus-beginners, why am I recommending a book on scripting (i.e. customising or programming) Abaqus? The book does require that you know a fair deal about Abaqus. However, the author has created some nice videos on some common applications. You could do worse than check these out.

The 3DS "Community" - http://iam.3ds.com

Licensed users can access paid support, but how about academic users or others? This is a recent (June 2011 onwards) attempt by Dassault and / or Simulia to address such needs. You can use any email address to create an account, then (in theory) tap into the wisdom of the crowds. Once you login, your browser is redirected (at the time this book was published) to https://swym.3ds.com.

There are some tutorials, etc. so perhaps you'll find what you want. But I suspect the password-protected approach is too heavy-handed for light-fingered internet users.

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The Yahoo-group for Abaqus

This is a relatively lively (by CAE standards) group, averaging a few posts a day. It doesn't seem to have any affiliations with Dassault, but does have some pretty erudite members. However, on the downside, many of the questions tend to be advanced: well beyond the profile of the intended reader of this book. Also, perhaps because of their erudition, some of the gurus can be quite acid at times. "Stupid" questions are sometimes snapped at, sometimes ignored. But if you do your homework and establish this fact when you ask your question, chances are you'll get an answer.

Finally, there are the other books in this Getting Started With Abaqus series (clicking on a link takes you to the corresponding Amazon page):

• Essential Theory • Workbook 0: User Interface and Modeling Overview • Workbook 1: Linear Static Analyses and Basic Mesh Generation • Workbook 3: Mode Shapes • Workbook 4: Non-linear Analyses • Workbook 5: Explicit Analysis

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About the AuthorC.Venkatesh lives in Hyderabad, India. His email address is venkatesh@kfourmetrics.com