Overview of Solid Mechanics
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 on 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 constitutivelaw 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.
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1.1.1 Deciding What to Calculate
Th is question seems rather silly, but at some point in their careers, most engineers have been asked by their manager, “Why don’t you just set up a fi nite element model of our (crank-case, airframe, material, etc.) so we can stop it from (corroding, fatiguing, fractur-ing, etc.)?” If you fi nd yourself in this situation, you are doomed. Models can certainly be helpful in preventing failure, but, unless you have a very clear idea of why the failure is occurring, you won’t know what to model.
Here is a list of things that can be calculated accurately using solid mechanics:
1. Th e deformed shape of a structure or component subjected to mechanical, thermal, or electrical loading
2. Th e forces required to cause a particular shape change 3. Th e stiff ness of a structure or component
4. Th e internal forces (stresses) in a structure or component
5. Th e 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 diffi cult, however, because the physics of failure can only be modeled using approximate constitutive equations. Th ese mathematical relationships must be calibrated experimentally and do not always perfectly characterize the failure mecha-nism. Nevertheless, there are well-established procedures for each of the following:
1. Predicting the critical loads to cause fracture in a brittle or ductile solid containing a crack
2. Predicting the fatigue life of a component under cyclic loading
3. Predicting the rate of growth of a stress-corrosion crack in a component 4. Predicting the creep life of a component
5. Finding the length of a crack that a component can contain and still withstand fatigue or fracture
6. Predicting the wear rate of a surface under contact loading 7. Predicting the fretting or contact fatigue life of a surface
Solid mechanics is increasingly being used for applications other than structural and mechanical engineering design. Th ese are active research areas, and some are better devel-oped than others. Applications include the following:
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
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2. Predicting the infl uence 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, including casting and solidifi cation, alloy heat treat-ments, and thin-fi lm and surface-coating deposition (e.g., by sputtering, vapor depo-sition, or electroplating)
5. Modeling biological phenomena and processes, such as bone growth, cell mobility, cell wall/particle interactions, and bacterial mobility
1.1.2 Defi ning the Geometry of the Solid
Again, this step seems rather obvious: surely the shape of the solid is always known? True, but it is usually not obvious how much of the component to model and at what level of detail. For example, in a crash simulation, must the entire vehicle be modelled, or just the front part? Should the engine block be included? Th e driver? Th e cell phone that distracted the driver into crashing in the fi rst place?
At the other extreme, it is oft en 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.)? Th e following guidelines might be helpful:
1. For modeling brittle fracture, fatigue failure, or for calculating critical loads required to initiate plastic fl ow in a component, it is very important to model the geometry in great detail, because geometrical features can lead to stress concentrations that initi-ate damage.
2. For modeling creep damage, large-scale plastic deformation (e.g., 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 oft en only infl uence local stresses: they do not have much infl uence far away. Saint Venant’s principle, which will be discussed in more detail in Chapter 5, suggests that a geometrical feature with characteristic dimension L (e.g., the dimension of a hole in the solid) will infl uence stresses over a region with dimension around 3L surrounding the feature. In other words, if you are interested in the stress state at a particular point in an elastic solid, you don’t need to worry about geometrical features that are far from the region of interest. Strictly speaking, Saint Venants’ principle only applies to elastic solids, although it can usually also be applied to plastic solids that strain harden.
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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 refi ning the calculation.
1.1.3 Defi ning Loading
Th ere are six ways that mechanical loads can be induced in a solid:
1. Th e boundaries can be subjected to a prescribed displacement or motion.
2. Th e boundaries can be subjected to a distribution of pressure normal to the surface or frictional traction tangent to the surface, as shown in Figure 1.1.
3. A boundary may be subjected to a combination of displacement and traction (“mixed”) boundary conditions; for example, you could prescribe horizontal dis-placements, together with the vertical traction, at some point on the boundary.
4. Th e interior of the solid can be subjected to gravitational or electromagnetic body forces.
5. Th e 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 three-dimensional (3D) analysis, you must specify three components of either displacement(u1,u2,u3) or traction (t1,t2,t3) (but not both) at each point on the bound-ary. You can mix these: so for example, you could prescribe (u1,t2,t3) or (u1,u2,t3), but exactly three components must always be prescribed. Th is rule also applies to free surfaces, where the tractions are prescribed to be zero.
2. Similarly, in a two-dimensional (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 and moment acting on the solid sum to zero (otherwise, a static equilibrium solution cannot exist).
e1 t1 t2 e2
FIGURE 1.1 Tractions acting on the boundary of a solid.
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In practice, it can be surprisingly diffi cult to fi nd out exactly what the loading on your system looks like. For example, earthquake loading on a building can be modeled as a pre-scribed acceleration of the building’s base, but what acceleration should you apply? Pressure loading usually arises from wind or fl uid forces, but you might need to do some sophisticated calculations just to identify these forces. In the case of contact loading, you’ll need to be able to estimate friction coeffi cients. For nanoscale or biological applications, you may also need to model attractive forces between the two contacting surfaces. Here, standards are help-ful. For example, building codes regulate civil engineernig structures, the National Highway Traffi c Safety Administration specifi es design requirements for vehicles, and so on.
You can also avoid the need to fi nd 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 Th ere are four decisions to make here:
1. Do you need to calculate additional fi eld quantities, such as temperature, electric or magnetic fi elds, or mass/fl uid diff usion through the solid? Temperature is the most common additional fi eld quantity. Here are some rough guidelines that will help you to decide whether to account for heating eff ects.
Th e stress induced by temperature variation in a component can be estimated from the formula σ = E(αT)max − (αT)min), where E is the Young’s modulus of the material, α is its thermal expansion coeffi cient, and T is temperature. Th e symbols (αT)max,(αT)min denote the maximum and minimum values of the product αT in the component. You need to account for temperature variations if σ is a signifi cant fraction of the stress induced by mechanical loading.
2. 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 Q! in an infi -nite solid is ∆T Q= ! erfc ( /r 2 αt)/(4πκr), where erfc( ) denotes the complementary error function, κ is the material’s thermal conductivity, and χ = κ/(ρcp) is its thermal diff usivity, with ρ the mass density and cp the specifi c heat capacity. Th is equation suggests that a solid with dimension L will reach its steady-state temperature in time t ≈ 25L2/χ. If the timescale of interest in your problem is signifi cantly larger than this and heat fl ux is constant, you can use the steady-state temperature distribution. If not, you must account for transients.
Finally, to decide whether you need to account for heat generated by plastic fl ow, note that the rate of heat generation per unit volume is of order q!=σ εY!p, where σY is the material yield stress, and ε!p is the plastic strain rate. Th e temperature rise attrib-utable to rapid (adiabatic) plastic heating is thus of order ΔT = σYΔεp/(ρcp), where Δεp is the strain increment applied to the material.
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3. Do you need to do a dynamic analysis or a static analysis? Here are some rough guide-lines that will help you to decide. Th e speed of a shear wave propagating through an elastic solid is c = µ ρ/ , where ρ is the mass density of the solid, and μ is its shear modulus. Th e time taken for a wave to propagate across a component with size L is of order t = L/c. In many cases, stresses decay to their static values aft er about 10L/c. If the loading applied to the component does not change signifi cantly during this time period, a quasi-static computation (possibly including accelerations as body forces) should suffi ce.
Th e stress induced by acceleration (e.g., in a rotating component) is of order Lρa, where L is the approximate size of the component, ρ is its mass density, and a is the magnitude of the acceleration. If this stress is negligible compared with other forces applied to the solid, it can be neglected. If not, it should be included (as a body force if wave propagation can be neglected).
4. Are you solving a coupled fl uid/solid interaction problem? Th ese situations arise in aeroelasticity (design of fl exible aircraft wings or helicopter rotor blades, or very long bridges), off shore structures, pipelines, or fl uid containers. In these applications, the fl uid fl ow has a high Reynold’s number (so fl uid forces are dominated by inertial eff ects). Coupled problems are also very common in biomedical applications, such as blood fl ow or cellular mechanics. In these applications, the Reynold’s number for the fl uid fl ow is much lower, and fl uid forces are dominated by viscous eff ects. Several analysis techniques are available for solving such coupled fl uid/structure interaction problems but are beyond the scope of this book.
1.1.5 Defi ning 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 invalidate your predictions. Here are a few of your choices, with suggested applications:
1. Isotropic linear elasticity (familiar in one dimension as σ = Eε): Th is constitutive