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# UG Course on Solid Mechanics

Given below is a sequence that might properly address the question of what to teach in the first (and the only) UG couse on strength of materials or solid mechanics.

**0. Note: **It's a mistake to believe that the contents for such a course can be covered in a linear fashion. Apply the spiral theory of knowledge and revisit certain concepts again and again: e.g., the concepts of stress, strain, fields, BV problems, theoretical structure, etc.

**1. Introduction:**

The range of (stress/strain/displacement analysis) problems to address.

Qualitative and empirical characterization of materials response under tension, compression, shear, fatigue, creep, impact, etc.

Engineering choices and the need to make precise the notions of strength, deflection, deformation etc. under different loadings.

What quantities must be formulated to meet the above objective? The static case as the simplest.

Revision of the simplest 1D case (covered earlier in high-school physics): strain, stress, Hooke's law and Young's modulus.

The notion of stress as internal resistive state. The notion of strain as something to do with (relative) displacement. Highlight that stress-strain can be induced out of temperature and EM fields imposed on a constrained body. Highlight Strains --> Stress.

Arouse curiosity: What would stress and strain look like in 2D? In 3D?

**2. Strain: **

The displacement undergone by a rigid body (particle). The vector *fields* of relative displacement and deformation.

Identifying the components of the deformation field.

Isolation of the strain and rotation tensor fields starting from the displacement vector field. Simple (non-rigorous) demonstration that both are tensor fields.

Introduction of the notion of compatibility. Making it mathematically precise using the differential strain-displacement relations. (May be, a demonstration using a finite differences based model that the compatibility relations indeed ensure compatibility. Touch upon sufficiency.)

Strain as an essentially geometric concept.

Simple examples (e.g. Shames)

**3. Stress:**

Internal resistance to external loading--why the notion matters. Historical evolution. The idea of the mathematical cut and the resistive traction vector.

Introduction of the idea of a stress field. Taylor's series expansion and equilibrium equations (i.e. divergences of the three traction vectors).

Simple (non-rigorous) demonstration that stress is a tensor field.

Point out the similarity with the kinematical (geometric) characterization of strain.

Point out how the complementary property of shear corresponds with dropping the rotational part from the definition of strain.

Introduce the linear stress-strain constitutive relations--first for components, and then, the interrelations.

Simple examples (e.g. Shames)

**4. 2D and 3D Fields: General considerations**

The geometrical definition of the principal quantities. Why have them: simplified presentation or "visualization" of tensor fields. The mathematical jugglary and Mohr's circle.

The plane stress and the plane strain conditions: Why have the condition. Where it applies. The pitfalls.

Analysis of some typical examples.

The tri-axial state of stress. Some typical examples. How and why the complexity increases. Why such a loading is adverse. How and why 2D analysis is not enough or can be misleading.

**5. The Structure of the Theory:**

Bring out the structure of the subject matter:

Relative Displacement <--> Deformation <-> Strain <-> Stress <-> Traction <-> Loads.

Apply the structure to the static case: three laws (in Shames): compatibility, constitutive law, equilibrium relations.

Demonstration (physical arguments) that a vector field cannot take the place of the tensor stress field. Ditto, for stress field.

Stress analysis as a BV problem. Introduce the effect of size (even if typical analytical models are always for infinite domains).

**6. The Application Specifics:**

Point out the typical combinations of member geometry+loading (to be studied next).

Spend some time to establish the relations of 3D stress/strain tensor concepts and the analysis in question for each of these combinations.

Spend less time than is usual on discussing the usual strength of materials kind of analyses (or their proofs). Spend up to half or even less time if they are not civil engineering majors. (Comments: Popov's book, in particular, spends inordinately long time on beams alone. Not necessary for non-civil engineers. Most teachers spend such a long time on these topics and emphasize them on exams primarily because the teachers themselves have come from the civil engg. depts!)

Always trace principal stresses/strains (or pure shear ones) for each and every case of stress analysis that ever gets discussed in the class. Always show Mohr's circles. (No book does this--and I doubt if any teacher does it either.)

The cases to be covered are the usual ones: beams, columns, torsion

Beams: Forces, stresses and displacements of beams.

Torsion: The usual topics.

Columns: The usual topic.

To reiterate: Cover the topics in a way that it is the understanding of stress/strain *tensor* field concepts that gets reinforced, not a reverence for the particular approximations in analysis.

**7. Failure and Fracture:**

-- Distinguish between failure and fracture. (e.g., stiffness as the design criterion.)

-- Failure criteria. Do point out their relevance to design. Point out the physical meanings of each criterion--don't leave anything (in this topic or otherwise) just mathematically dangling abstractly, connecting nowhere to physical reality.

-- Introduction to elastic stability. Do point out how we *begin* by assuming instability in the analysis. Highlight how the simple analysis of column instability is just a begnning.

-- Stress concentration and fracture toughness. (Introduction). Do mention size effect.

**9. Miscellaneous topics:**

Some of them, optional; others, better handled in an accompanying laboratory course.

-- Study of an array of the typical components of machines and structures. (Charts showing stress distributions in such components.)

-- Energy theorems. This is good inasmuch as physics is being tied to. Unfortunately, this also means calculus of variations (CoV). The new movement is towards towards explaining everything as an application of (CoV)--from optics and QM to mechanics of solids. This is very unfortunate. Actually, CoV is just an optional way of viewing physics and often-times not at all at the core of the physics of the specific situations. So, spend time wisely.

-- Introduction to elasticity. If the audience is talented enough, introduce the use of potentials in some simple 2D problems. (Do not emphasize complex number manipulations by themselves--and always remember, the entire theory is only linear elastic and 2D. Mathematical pleasure apart, it has severe restrictions as a theory of engineering.)

-- Introduction to the kind of analysis that is involved in tackling topics like plasticity, metal forming, rheology, etc.

-- A study of the parallels in the theoretical structure of solid and fluid mechanics

-- Stress waves.

-- Impact loading.

-- NDT

-- Experimental stress analysis: Photo-elasticity, brittle coatings.

**General Comments:**

(i) Introduce the field concept as early as possible.

(ii) Discuss each pedagogical (or illustrative) example from all points of view: analytical solution, principal stress contours, computer simulation, photoelasticity results.

(Note, here, analytical solutions have been distinguished from principal stresses. This is not a redundancy. The point is, often times, analytical solutions are expressed in terms that are convenient to analysis, but which may not bring out the principal (or pure) quantities.)

Always discuss variations in boundary conditions and their impact on solution.

(iii) Do not emphasize and do not reward the facility in sheer mathematical manipulations--such a facility can be rather easily developed via sheer pattern-matching, without having developed any real physical understanding.

(iv) For the same reason, do not assign many small programs in C/C++/Java/VB etc. that take away mental energy in simply another kind of drill tasks. Instead, give away working programs to students and ask them to try out some variations.

(Note: This post will probably undergo several revisions.)

- Ajit R. Jadhav's blog
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