Applied mechanics of solids A.F. Bower

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.

Deciding what to calculate This seems a rather silly question but at some point of their careers, most engineers have been told by their manager `Why don’t you just set up a finite element model of our (crank-case; airframe; material., etc, etc) so we can stop it from (corroding; fatiguing; fracturing, etc).’ If you find 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 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); FEA model of rupture during tube hydroforming. –6. Natural frequencies of vibration for a structure or component.

Failure mechanisms 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.

Defining the geometry of the solid Again, this 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? The passengers? 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 they do not have much influence far away. Saint Venant’s principle, suggests that a geometrical feature with characteristic dimension L (e.g. the diameter of a hole in the solid) will influence stresses over a region with dimension around 3L surrounding the feature. This means that if you are interested in the stress state at a particular point in an elastic solid, you need not worry about geometrical features that are far from the region of interest. Saint-Venants principle strictly only applies to elastic solids, although it can usually also be applied to plastic solids that strain harden. 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

Deciding what physics to include in the model 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. 2. Do you need to do a dynamic analysis, or a static analysis? 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.

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: FEA model of a rubber tire, using a hyperelastic constitutive equation. 1. Isotropic linear elasticity 2. Anisotropic linear elasticity 3. `Hyperelasticity’ 4. Viscoelasticity. 5. Rate independent metal plasticity. 6. Viscoplasticity 7. Crystal plasticity 8. Strain Gradient Plasticity 9. Discrete Dislocation Plasticity 10. Critical state plasticity (cam-clay) 11. Pressure-dependent viscoplasticity. 12. Concrete models. 13. Atomistic models.