# HOOKE’S LAW IS NOT OHM’S LAW

The title of this note may appear tautologous to the casual reader, but the reference here is to more than the obvious difference between mechanical and electrical phenomena. (Attention: Civil, Mechanical and Electrical engineers)

During their educations, engineers and stress analysts learn a variety of three-component “laws” and relationships among physical variables.

These include **E=I·R**, **F=m·a**, **D=V·t**, **C=f·λ**, etc. They also learn an additional one, which is much less general in its applicability, and therefore more apt to be misused. Ask almost any engineer or stress analyst to give you the formula for **Hooke’s law**, and the immediate reply is apt to be “**σ = E·ϵ**” (or ”**S=E·e**” ).

How did we learn this one-dimensional version (1-D) of the law (analogous to Ohm’s law)? Not from Robert Hooke (1635-1703), certainly. Hooke simply reported his observation that the extension of a helical spring was proportional to the applied axial load on the spring. The concepts of stress, strain, and elastic modulus as we now know them were not even defined at the time. Nor, obviously, was it learned from the 19^{th} century elasticians who formalized and generalized the three - dimensional relationships between stress and strain into what should properly be known as **Hooke’s law**.

The learning had to have occurred during the engineer’s schooling (In High School, College or University) – by either explicit or implicit instruction, or some combination of both routes. Mechanics teachers who loosely refer to **Hooke’s law** as σ = **E**·ϵ are undoubtedly major contributors to the misunderstanding. And the practice of restricting problem assignments to primarily uniaxial stress states serves to confirm, in the student’s mind (set), the 1D, one-dimensional nature of Hooke’s law. The textbook authors, **online videos** and lectures also do their share in teaching the same concept – both explicitly and implicitly.

A number of mechanics of materials textbooks formally define Hooke’s law to be “σ = E·ϵ”. If the generalized Hooke’s law is given at all, it may be relegated to a footnote or a paragraph marked for optional study. In either case, the student is given the subliminal but clear message that the topic is of only peripheral significance. The biaxial stress state, a very common occurrence in practical stress analysis, is often presented with the implication of being a special case. The topic is apt to be treated disjunctively in one or more chapters set aside for this purpose, and then largely ignored in subsequent chapters. In these and similar ways, the textbook author can readily condition the student to a one-dimensional view of Hooke’s law.

Why make an issue of what some might dismiss as an academic trifle? Very simple. We currently have a “small army” of practicing engineers (including some engaged in experimental stress analysis) with the thoroughly mistaken notion that stress can be obtained by multiplying a measured strain by the modulus of elasticity.

As noted above, the source of misunderstanding is dual-faceted. The engineering graduate frequently does not appreciate the real-world ramifications of the **biaxial stress** state, nor does he or she understand the practical application of the generalized Hooke’s law to this stress state. As a result, such an engineer may be unaware, for instance, that a single strain gage sensor mounted in the fillet of a crankshaft does not produce enough information to determine the maximum stress there. Even when the complete biaxial state of strain has been established (with, for example, a C5K Series, **Advanced Sensors Technology** three-element strain gage planar rosette), the engineer may try to calculate the principal stresses by multiplying the principal strains by the modulus of elasticity.

The foregoing and similar errors in experimental stress analysis are not rare as one might wish. Since the tendency for such errors seems to arise during the educational process, this may be the best place to apply remedial measures. Accurate experimental determination of stresses requires not only competence with the measuring technology, but also a working familiarity with the concepts of stress and strain at a point, and with the generalized Hooke’s law, which integrates these two concepts. For an engineer thus equipped, it might be unnecessary to point out that Hooke’s law is not Ohm’s law.

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