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Chapter 1
Introduction
Solid Mechanics is a collection of physical laws, mathematical techniques, and computer algorithms that can be used to predict the behavior of a solid material that is subjected to mechanical or thermal loading. The field has a wide range of applications, including
1. Geomechanics 2. Civil engineering 3. Mechanical engineering 4. Manufacturing engineering 5. Biomechanics 6. Materials Science 7. Microelectronics 8. Nanotechnology
This chapter describes how solid mechanics can be used to solve practical problems. The remainder of the book contains a more detailed description of the physical laws that govern deformation and failure in solids, as well as the mathematical and computational methods that are used to solve problems involving deformable solids. Specifically,
Chapter 2 covers the mathematical description of shape changes and internal forces in solids; Chapter 3 discusses constitutive laws that are used to relate shape changes to internal forces; Chapter 4 contains analytical solutions to a series of simple problems involving deformable solids; Chapter 5 provides a short summary of analytical techniques and solutions for linear elastic solids; Chapter 6 describes analytical techniques and solutions for plastically deforming solids; Chapter 7 gives an introduction to finite element analysis, focusing on using commercial software; Chapter 8 has a more complete discussion of the implementation of the finite element method; Chapter 9 describes how to use solid mechanics to model material failure; Chapter 10 discusses solids with special geometries (rods, beams, membranes, plates and shells).
Solid mechanics is incomprehensible without some background in vectors, tensors and index notation. These topics are reviewed briefly in the appendices.
1.1 Defining a Problem in Solid Mechanics
Regardless of the application, the general steps in setting up a problem in solid mechanics are always the same: 1. Decide upon the goal of the problem and desired information; 2. Identify the geometry of the solid to be modeled; 3. Determine the loading applied to the solid; 4. Decide what physics must be included in the model; 5. Choose (and calibrate) a constitutive law that describes the behavior of the material; 6. Choose a method of analysis; 7. Solve the problem. Each step in the process is discussed in more detail below.
1.1.1 Deciding what to calculate
This seems a rather silly question
Here is a list of of some of the things that can typically be calculated very accurately using solid mechanics: 1. The deformed shape of a structure or component subjected to mechanical, thermal or electrical loading; 2. The forces required to cause a particular shape change; 3. The stiffness of a structure or component; 4. The internal forces (stresses) in a structure or component; 5. The critical forces that lead to failure by structural instability (buckling);
6. Natural frequencies of vibration for a structure or component.
In addition, solid mechanics can be used to model a variety of failure mechanisms. Failure predictions are more difficult, however, because the physics of failure can only be modeled using approximate constitutive equations. These must be calibrated experimentally, and do not always perfectly characterize the failure mechanism. Nevertheless, there are well established procedures for each of the following:
1. Predict the critical loads to cause fracture in a brittle or ductile solid containing a crack; 2. Predict the fatigue life of a component under cyclic loading; 3. Predict the rate of growth of a stress-corrosion crack in a component; 4. Predict the creep life of a component; 5. Find the length of a crack that a component can contain and still withstand fatigue or fracture; 6. Predict the wear rate of a surface under contact loading; 7. Predict the fretting or contact fatigue life of a surface.
Solid mechanics is increasingly being used for applications other than structural and mechanical engineering design. These are active research areas, and some are better developed than others. Applications include
1. Calculating the properties (e.g. elastic modulus, yield stress, stress-strain curve; fracture toughness, etc) of a composite material in terms of those of its constituents. 2. Predicting the influence of the microstructure (e.g. texture; grain structure; dispersoids; etc) on the mechanical properties of metals such as modulus, yield stress, strain hardening, etc. 3. Modeling the physics of failure in materials, including fracture, fatigue, plasticity, and wear, and using the models to design failure resistant materials 4. Modeling materials processing 5. Modeling biological phenomena and processes, such as bone growth; cell mobility; cell wall/particle interactions; and bacterial mobility.
1.1.2 Defining the geometry of the solid
Again, this seems rather obvious
At the other extreme, it is often not obvious how much geometrical detail needs to be included in a computation. If you model a component, do you need to include every geometrical feature (such as bolt holes, cutouts, chamfers, etc)? The following guidelines might be helpful
1. For modeling brittle fracture, fatigue failure, or for calculating critical loads required to initiate plastic flow in a component, it is very important to model the geometry in great detail, because geometrical features can lead to stress concentrations that initiate damage. 2. For modeling creep damage, large scale plastic deformation (eg metal forming), or vibration analysis, geometrical details are less important. Geometrical features with dimensions under 10% of the macroscopic cross section can generally be neglected. 3. Geometrical features often only influence local stresses
As a general rule, it is best to start with the simplest possible model, and see what it predicts. If the simplest model answers your question, you’re done. If not, the results can serve as a guide in refining the calculation
There are five ways that mechanical loads can be induced in a solid: 1. The boundaries can be subjected to a prescribed displacement or motion; 2. The boundaries can be subjected to a distribution of pressure normal to the surface, or frictional traction tangent to the surface; 3. A boundary may be subjected to a combination of displacement
and traction (“mixed”) boundary conditions 4. The interior of the solid can be subjected to gravitational or electromagnetic body forces; 5. The solid can contact another solid, or in some cases can contact itself. 6. Stresses can be induced by nonuniform thermal expansion in the solid, or some other materials process such as phase transformation that causes the solid to change its shape.
When specifying boundary conditions, you must follow these rules: 1. In a 3D analysis, you must specify three components of either displacement 2. Similarly, in a 2D analysis you must prescribe two components of displacement or traction at each point on the boundary. 3. If you are solving a static problem with only tractions prescribed on the boundary, you must ensure that the total external force acting on the solid sums to zero (otherwise a static equilibrium solution cannot exist).
In practice, it can be surprisingly difficult to
find out exactly what the loading on your system looks like. For example, earthquake loading on a
building can be modeled as a prescribed acceleration of the building’s base
This is where standards are helpful. For example, building codes regulate civil engineernig structures; NHTSA specify design requirements for vehicles, and so on.
You can also avoid the need to find exactly what loading a structure will experience in service by simply calculating the critical loads that will lead to failure, or the fatigue life as a function of loading. In this case some other unfortunate engineer will have to decide whether or not the failure loads are acceptable.
1.1.4 Deciding what physics to include in the model
There are three decisions to be made here:
1. Do you need to calculate additional field quantities, such as temperature, electric or magnetic fields, or mass/fluid diffusion through the solid? Temperature is the most common additional field quantity. Here are some rough guidelines that will help you to decide whether to account for heating effects.
As a rough guide, the stress induced by temperature variation in a component is
To decide whether you need to do a transient heat conduction analysis,
note that the temperature rise at a distance r from a point source of heat of intensity
Finally to decide whether you need to account for heat generated by plastic flow, note
that the rate of heat generation per unit volume is of order
2. Do you need to do a dynamic analysis, or a static analysis? Here are some rough guidelines that will help you to decide: The speed of a shear wave propagating through an
elastic solid is The stress induced by acceleration (e.g. in a
rotating component) is of order
3. Are you solving a coupled fluid/solid interaction problem? These arise in aeroelasticity (design of flexible aircraft wings or helicopter rotor blades; or very long bridges); offshore structures; pipelines; or fluid containers. In these applications the fluid flow has a high Reynold’s number (so fluid forces are dominated by inertial effects). Coupled problems are also very common in biomedical applications such as blood flow or cellular mechanics. In these applications the Reynolds number for the fluid flow is much lower, and fluid forces are dominated by viscous effects. Different analysis techniques are available for these two applications. Such problems are beyond the scope of this book.
1.1.5 Defining material behavior
Choosing the right equations to describe material behavior is the most critical part of setting up a solid mechanics calculation. Using the wrong model, or inaccurate material properties, will always completely invalidate your predictions. Here are a few of your choices, with suggested applications:
1. Isotropic
linear elasticity (familiar in one
dimension as
2. Anisotropic linear elasticity (similar to isotropic linear elasticity, but models materials which are stiffer in some directions than others). Good for reinforced composites; wood; single crystals of metals and ceramics. Between 3 and 21 material properties must be determined. Material data is highly accurate and readily available.Isotropic or anisotropic linear elasticity is good for the vast majority of engineering design calculations, where components cannot safely exceed yield. It can be used for deflection calculations, fatigue analysis, and vibration analysis.
3. `Hyperelasticity’
4. Viscoelasticity. Used to model materials which exhibit a gradual
increase in strain when loaded at constant stress (with stress rate-v-strain
rate
5. Rate
independent metal plasticity. Used to model permanent deformation in
metals loaded above their yield point.
A wide range of models are available.
The simplest is a rigid perfectly plastic solid, which
changes its shape only if loaded above its yield stress
6. Viscoplasticity: similar in structure to metal plasticity, but
accounts for the tendency of the flow stress of a metal to increase when deformed
at high strain rates. These would be
used in modeling high-speed machining, for example, or in applications
involving explosive shock loading.
Viscoplastic constitutive equations are also used to model creep
7.
8. Strain
Gradient Plasticity: A formulation
developed in the last 5-10 years to model the behavior very small volumes of
a metal (i.e. less than 100
9. Discrete Dislocation Plasticity: A technique to model plastic flow in very small volumes of material by tracking the nucleation, motion and annihiliation of individual dislocations in the solid. DDP models contain a large number of material parameters that are very hard to calibrate. Currently a research tool.
10. Critical state plasticity (cam-clay). Used to model soils, whose behavior depends on moisture content. Somewhat similar in structure to metal plasticity, except that the yield strength of a soil is highly pressure dependent (it increases with compressive pressure). Simple models contain only 3 or 4 material parameters that can be calibrated quite accurately.
11. Pressure-dependent viscoplasticity. Similar to critical state plasticity, in that these models account for changes in flow stress of a material with confining pressure. Used to model granular materials, and some polymers and composite materials (typically in modeling processes such as extrusion or drawing).
12. Concrete models. Intended to model the crushing (in compression) or fracture (in tension) of concrete (obviously!). The mathematical structure resembles that of pressure dependent plasticity.
13. Atomistic
models. Replace traditional
stress-strain laws with a direct calculation of stress-strain behavior using
embedded atomic scale simulations. The
atomic scale computations use empirical potentials to model atom
interactions, or may approximate the Schrodinger wave equation directly.
Techiques include the `Quasi-Continuum’ method, and the
Coupled-Atomistic-Discrete Dislocation Method. Their advantage is that they
capture the physics of material behavior extremely accurately; their
disadvantage is that they currently can only model extremely small material
volumes (20-100nm or so). Atomistic models based on empirical potentials
contain a large number of adjustable parameters
This is by no means an exhaustive list
These material models are intended primarily to
approximate stress-strain behavior.
Special constitutive equations have also been developed to model the
behavior of contacting surfaces or interfaces between two solids (Coulomb
friction is a simple example). In addition, if you need to model damage
(fracture or fatigue), you may need to select and calibrate additional
material models. For example, to model
brittle fracture, you would need to know the fracture toughness of the
material. To model the growth of a
fatigue crack, you would probably use
1.1.6 A representative initial value problem in solid mechanics
The result of the decisions made in Sections 1.1.1-1.1.5 is a Boundary value problem (for static problems) or initial value problem (for dynamic problems). This consists of a set of partial differential equations, together with initial and boundary conditions, that must be solved for the displacement and stress fields, as well as any auxiliary fields (such as temperature) in the solid. To illustrate the structure of these equations, in this section we list the governing equations for a representative initial value problem.
As a representative example, we state the initial value problem that governs elastic wave propagation in a linear elastic solid.
Given: 1.
The shape of
the solid in its unloaded condition 2.
The Young’s
modulus E and Poisson’s ratio 3.
The thermal
expansion coefficient 4.
The initial
displacement field in the solid 5.
A body force
distribution 6.
Boundary
conditions, specifying displacements
Calculate displacements 1. The strain-displacement (compatibility) equation 2. The linear elastic stress-strain law
4.
The fields must
satisfy initial conditions 5.
As well as
boundary conditions
1.1.7 Choosing a method of analysis
Once you have set up the problem, you will need to solve the equations of motion (or equilibrium) for a continuum, together with the equations governing material behavior, to determine the stress and strain distributions in the solid. Several methods are available for this purpose.
Exact solutions: There is a good chance that you can find an exact solution for:
1. 2D (plane stress or plane strain) linear elastic solids, particularly under static loading. Solution techniques include transforms, stress function methods, and complex variable methods. Dynamic solutions are also possible, but somewhat more difficult. 2. 2D viscoelastic solids. 3. 3D linear elasticitity problems can be solved, usually using integral transforms, if they are simple enough. 4. 2D (plane strain) deformation of rigid plastic solids (using slip line fields)
Naturally, analytical solutions are most easily found for solids with a simple geometry (e.g. an infinite solid containing a crack; loading applied to a flat surface, etc). In addition, special analytical techniques can be used for problems where the solid’s geometry can be approximated in some way. Examples include membrane theory; shell and plate theory; beam theory; and truss analysis.
Even when you can’t find an exact solution to the stress and strain fields in your solid, you can sometimes get the information you need using powerful mathematical theorems. For example, bounding theorems allow you to estimate the plastic collapse loads for a structure quickly and easily.
Numerical Solutions: These are used for most engineering design calculations in practice. These include 1. The finite element method 2. Finite difference methods 3. The boundary integral equation method (or boundary element method) is a more efficient computer technique for linear elastic problems, but is less well suited to nonlinear materials or geometry. 4. Free volume methods: Used more in computational fluid dynamics than in solids, but good for problems involving very large deformations, where the solid flows much like a fluid. 5. Atomistic methods: used in nanotechnology
applications to model material behavior at the atomic scale. Molecular Dynamic techniques integrate the
equations of motion (
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(c) A.F. Bower, 2008 |