Maloney Home Page  |  Let’s not overrate Young’s modulus: How materials resist deformation in 1D, 2D, and 3D

Motivation

Don’t get me wrong—I’m a big fan of Young’s modulus. This material property is almost universally used to characterize the stiffness of solids, or the amount of resistance when one compresses or elongates something. Check out the variety of contexts in which Young’s modulus has arisen in the recent research literature:

  • Measurements and modeling of the stiffness of a microscale tensioned membrane over a large temperature range (Storch et al., “Young’s modulus and thermal expansion of tensioned graphene membranes” Physical Review B (2018)).

  • Resting steel spheres on thin compliant layers and laterally observing the deformation to characterize the film stiffness (Gross et al., “Simultaneous measurement of the Young’s modulus and the Poisson ratio of thin elastic layers” Soft Matter (2017)).

  • Using speckle patterns and machine vision to characterize ceramic stiffness from deflection under a three-point bending test (Farsi et al., “Full deflection profile calculation and Young’s modulus optimisation for engineered high performance materials” Scientific Reports (2017)).

  • Determining the bulk stiffness of a composite material containing unusually shaped nanoparticles embedded within a block copolymer matrix (Raja et al., “Influence of three-dimensional nanoparticle branching on the Young’s modulus of nanocomposites” PNAS (2015)).

Young’s modulus is so widely used that I’ll sometimes see an article drop “the elastic modulus” (meaning Young’s elastic modulus) as if there’s only one:

  • “Fiber diameter and orientation along with polymeric composition are the major factors that determine the elastic modulus of electrospun nano- and microfibers” (from the abstract) Vatankhah et al. Acta Biomaterialia 10(2) (2014).

  • The elastic modulus of multilayer graphene is found to be more robust to damage created by high-energy α-particle irradiation as compared to monolayer graphene” (from the abstract) Liu et al. Advanced Materials 27(43) (2015).

  • “Our previous study showed that embryonic tendon elastic modulus increases as a function of developmental stage” (from the abstract) Schiele et al. Journal of Orthopaedic Research 33(6) (2015).

and so on. Sometimes the meaning of the specific elastic modulus is clear from the context (or is later clarified), sometimes not.

But anyone interested in the mechanics of materials or material physics needs to know about other elastic moduli (at a minimum, the shear modulus and the bulk modulus, as addressed in this note) to understand the origin, nature, and implications of elasticity in materials.

So let’s not overrate Young’s modulus; instead, let’s appreciate its advantages while understanding its limitations and what insights come with a familiarity with other types of stiffness.

From this discussion, and as the culmination of this note, derived below, we’ll obtain the equation $$\epsilon_{ij}=\frac{1}{2G}\sigma_{ij}-\left(\frac{1}{6G}-\frac{1}{9K}\right)\delta_{ij}\sigma_{kk},$$ which powerfully describes, using contracted notation, the strain \(\epsilon\) resulting from any stress \(\sigma\) in any real (isotropic, uniform) material. The essential material properties here are the shear modulus \(G\) and the bulk modulus \(K\), to be discussed in detail below. Notably, Young’s modulus doesn’t appear at all in this expression! The reason, as we’ll discover, is that Young’s modulus can be decomposed into other moduli that are arguably more fundamental in the context of general 3D deformation of materials.

The 1D framework

To get started, recall that an elastic modulus, in general, describes the relationship between a stress (i.e., a force applied per unit area) and a strain (i.e., a deformation normalized to an original dimension). An elastic modulus, as a material property, is useful because to a certain degree it’s situation and geometry independent: an idealized spring can be cut in half, for example, to obtain two springs with double the spring constant (that is, the halves are twice as stiff because there’s less spring to extend) but the exact same elastic moduli, as the material is unchanged. In the McMaster-Carr catalog, for example, we find that the quarter-inch-long springs have approximately double the specified spring constant of the half-inch-long springs:

Elastic moduli are also generally useful because we wish to understand how a solid pulls or pushes back in response to (suitably small) forces. (At large forces, the predominant mechanism or the geometry may change; this the realm of nonlinear elasticity. Alternatively, the sample may not recover; this is the realm of plasticity.) And the elastic moduli are correlated in interesting ways to other material properties such as the melting temperature, density, heat capacity, and thermal expansion, providing insight into material physics.

Young’s modulus is arguably the preeminent elastic modulus (and thus a candidate for being overrated). It couples, very specifically, the axial stress and axial strain of a rod or bar subject to uniaxial loads only. (There’s no distinction between a rod and a bar in this context; the criterion is only that the geometry is long and narrow. This condition ensures that every point in the sample is close to the stress-free surface; thus, no lateral stresses exist.) Those uniaxial qualifiers tell us that this stress-strain relationship is a very 1D type of relationship (albeit including beam bending, which still involves axial elongation and contraction). As a result, naive applications of Young’s modulus to other configurations—such as shorter bars, plates, thin films, or constrained objects—are likely to be incorrect to a certain degree.

Under the 1D framework, we might apply a tensile or a compressive force \(F\) to the ends of a long, narrow object (let’s call it a rod) with cross-sectional area \(A\). The dogbone sample, for example, allows a firm grip while clearly defining the region to be stretched, and Young’s modulus is ideal for expressing this relationship:


Aluminum dogbone specimens intended for tensile testing (Pasco).

The resulting axial stress \(\sigma\) in the rod is \(\sigma=F/A\). The constitutive equation that quantifies the stress–strain coupling is simple Hooke’s Law: $$\sigma=E\epsilon,$$ where \(E\) is Young’s modulus and \(\epsilon\) is the engineering strain, or normalized deformation, or finite elongation per unit length: \(\epsilon=\Delta L/L\). (I discuss another useful definition of strain here.) So this method offers a straightforward way to measure material stiffness in a 1D context. Fair enough. To understand why an elastic modulus other than Young’s modulus would ever be needed, we need to incorporate additional dimensions.

The 2D framework

Real objects exist in 3D, of course, but it’s instructive to consider a 2D case first to define a new set of terms and to appreciate the differences that arise when we extend our thinking to an arbitrary 3D chunk of material.

Let’s first draw a square in a fixed location and orientation in 2D and consider how it might be deformed. Specifically, let’s distinguish loads that tend to distort the square into a smaller or larger square or a rectangle (normal loads, those that come in from the sides and top and bottom and point directly at the faces) from loads that tend to distort the square into a rhombus or diamond (shear loads, which act tangentially to the faces):


One can load the side of a fixed 2D square in two different ways: perpendicular or parallel to the side. The former requires a partner load on the opposite side to keep the square from accelerating away. The latter requires four (!) equal-strength components to keep the square from translating or rotating.

These two types of loads represent a dichotomy; they’re fundamentally different and produce different outcomes. When applied to a 2D square, we obtain the following distinct results: normal loads cause changes in lengths, whereas shear loads cause changes in angles:


Deformation resulting from normal and shear loading.

(Case I is drawn to incorporate the Poisson effect, or the lateral strain that occurs when one stretches or compresses a material. This relationship is mediated by Poisson’s ratio \(\nu\), which normalizes the contraction relative to the axial strain: \(\nu=-\frac{\epsilon_y}{\epsilon_x}\). Poisson’s ratio is typically positive; that is, we generally observe lateral contraction when we stretch a material.)

Right away we encounter a new type of deformation (namely, shear) that requires us to now distinguish our elastic moduli:

  • Young’s modulus mediates how much the side length of the square changes in the direction we pull (as an extension of the relationship of uniaxial stress vs. uniaxial strain described earlier). If the normal loading is along more than one axis, then we need to also consider Poisson effects.

  • The shear modulus mediates how much the corner angle changes as a result of a shear load.

However, this rule applies most reliably in our well-defined 2D world. Since there’s no guarantee that an arbitrary load would have entirely normal or entirely shear character, a whole segment of elasticity theory has arisen to interpret loads of intermediate character; the most well-known approach is Mohr’s circle, which provides a geometrical interpretation of how to decompose an arbitrary load into a set of normal loads:


An arbitrary collection of normal and shear loads can always be decomposed into a set of solely normal stresses (the so-called principal stresses) if one allows the 2D element to be rotated.

Importantly, the transformation requires a change in orientation. As we proceed to considering real 3D objects, we immediately see that this ambiguity in orientation presents a problem with the classification of normal vs. shear loads.

The 3D framework

The dichotomy of normal vs. shear stresses for infinitesimal elements can’t be extended in general to real objects; here’s why. Consider a material undergoing uniaxial loading, and let’s position our 2D element in the middle of the material as shown below:


Images based on a photograph by Nielson Fitness.

Following the reasoning above, we’d interpret the axial load as exerting a normal stress on the element, thus elongating its side lengths. No problem, right? Such configurations are well handled using Young’s elastic modulus (with Poisson's ratio describing the lateral contraction):

But consider: If we simply rotated our conceptual element by 45°, we’d see an entirely different stress state. We’d see some degree of normal stress, but we’d also see substantial shear stress that would distort our original square into a diamond:

We encounter a paradox regarding whether either normal or shear deformations are being induced within the material. After all, Nature has no preferred orientation, so the appearance of different results for our different rotations becomes problematic. We’re left to conclude that we can’t generally transfer a certain loading configuration on an object to an arbitrary internal element in the material. This is the reason that the division of normal vs. shear stress and strain runs into problems in real materials (although these notions are totally appropriate for a well-defined surface because a surface has a built-in orientation).

Since we’re now thinking in 3D, let’s consider a stress state that acts on all three directions equally—and of course the most familiar example is a compressive equitriaxial stress, also known as a hydrostatic stress, also known as pressure. It seems natural to have a modulus describing a material’s resistance to a change in volume from a uniform application of pressure. This new modulus, the bulk modulus, truly brings us into the 3D realm. Now we have three types of elastic moduli:

  • Young’s modulus mediates changes in a single side length.

  • The shear modulus mediates changes in a single angle.

  • The bulk modulus mediates changes in volume.

And we have a new dichotomy, as there are two ways to deform any 3D object: we could change its size or we could change its shape. The loading that changes an object’s size is known as a dilatational load. (The compressive version is pressure.) The loading that changes an object’s shape is known as deviatoric stress. (Think of this as a sort of 3D version of shear, although it doesn’t consist solely of shear loads.) These two different stress states are distinct but can be applied in the response of any real material—be it solid, liquid, or gas—to a mechanical load.

Notably, the uniaxial loading state that’s so useful in measuring Young’s modulus produces a mix of internal dilatational and deviatoric stress states, making this loading type and Young’s modulus slightly… unsatisfying for considering the general loading of 3D objects.

It turns out that the fundamentally different responses to deviatoric and dilatational loads provide substantial insight into material differences and practical use, including the potential for material failure under a load. Consider these values and correlations for various types of matter:

Features of the shear modulus


Approximate shear moduli of a variety of material types and the corresponding origins. The bonds in solids generally resist rearrangement; notably, however, the 1D bonding in polymers and most animal tissue allows substantial rearrangement (depending on the degree of crosslinking; non-crosslinked elastomers such as rubber are particularly compliant in shear). Moreover, fluids (liquids and gases) shear effortlessly.

The shear modulus is amazing! It’s got some neat properties:

  • The shear modulus governs resistance to molecular rearrangement (as opposed to the bulk modulus, which governs resistance to changes in molecular spacing). Solids can sustain a deviatoric load, whereas simple fluids such as idealized liquids and gases cannot. As a result, it works as a first pass to assume that fluids have zero shear modulus (and zero Young’s modulus). Fluids simply don’t exhibit the relatively strong intermolecular bonds that resist shear rearrangement in solids.

  • Another key aspect: Shear is what causes every ductile material to fail. This aspect is important because ductile materials are essential in engineering practice—they tend to fail slowly and with accumulating evidence, as opposed to brittle materials that fail with little warning.

    Shear is a convenient way of coming apart under excessive loading because it’s so easy! Shear doesn’t require the creation of a new large surface (as cracking does), which costs substantial energy. Shear doesn’t even require an entire plane of atoms to shift at once. Instead, the shear we observe occurs through dislocation glide, or the movement of 1D defects, which involves rows of atoms slipping past each other.


    Process of shear through edge dislocation glide in an idealized crystal. A 1D defect perpendicular to the page carries plasticity through the crystal. Adapted from Shackelford’s
    Introduction to Materials Science for Engineers (Pearson 2015).

    This mechanism has been compared to moving a carpet by kicking a hump down its length or to the way a caterpillar moves by elongating one section at a time.

    So it doesn’t matter that uniaxial stretching with a dogbone specimen is a convenient experimental framework; shear (or the more sophisticated version, deviatoric stress) is the most important factor regarding failure in a ductile engineering material. Even when uniaxial tension is applied to a ductile dogbone specimen, the resulting plastic deformation is driven by shear stress. In fact, for very ductile metals, the failure surfaces prominently display a 45° angle, the orientation of greatest shear stress on a polycrystal when one applies a uniaxial load (as we might have determined from Mohr’s circle).

    I describe a surprising consequence in my note on interpreting the Tresca criterion. In brief, let’s say that one side of a solid cube of ductile material is loaded by a normal tensile stress of 50 MPa. If we add a normal tensile stress of 20 MPa applied to another axis, does that additional load bring the material closer to failure? On the contrary; it does nothing (because it doesn’t change the maximum deviatoric stress). How about if we change that tensile stress of 20 MPa to a compressive stress of -20 MPa; does that somehow offset or lessen the first stress of 50 MPa? On the contrary; it can lead to failure because the combination of 50 MPa tension along one axis and 20 MPa compression along another axis jacks up the deviatoric stress. Deviatoric stress isn’t kidding.

  • Thermoset polymers, rubber, and our biological tissues all consist substantially of carbon. But although the bulk moduli of these materials are largely similar, the shear moduli differ by many orders of magnitude. More generally, the carbon bond offers tremendous 3D stiffness in diamond (with its 3D network covalent structure), 2D stiffness in graphite, and 1D stiffness in rubber and, for example, our tissue (which manifests only after the polymer chains unkink and straighten). Thus, the shear modulus is strongly sensitive to the dimensionality of the deformation mechanism, providing insight into the influence of material structure on stiffness.

Given all this, why not love the shear modulus?

Features of the bulk modulus

Oh wow, that was great. But also:


Approximate bulk moduli of a variety of material types and the corresponding origins. For ceramics and solids, the bulk modulus is generally only slightly higher than the shear modulus and retains essentially the same ordering, as mediated by the relation \(K=\frac{2G(1+\nu)}{3(1-2\nu)}\). A more startling transition comes about with the other material classes: the bulk stiffness of polymers, biological tissue, and liquids largely converges around 1–10 GPa, and gases assume a finite stiffness that’s roughly equal to their pressure. The stiffness of condensed matter is driven primarily by energy minimization, whereas that of gases is driven primarily by entropy maximization.

The bulk modulus is amazing! It’s got its own neat properties:

  • The bulk modulus couples a dilatational load (such as pressure) to a volume change. All stable materials can support hydrostatic pressure, and solids and liquids support such loading easily, with very little deformation, whereas gases dramatically contract or expand. In fact, the bulk modulus across condensed matter doesn’t vary by all that much—a few orders of magnitude, corresponding to the particular strength of the cohesive bonds (covalent, ionic, or metallic). In contrast, the bulk modulus of gases is primarily of entropic origin, the ideal gas featuring no such electrostatic interactions.

    As a result, it works as a first pass to assume that solids and liquids have very high bulk moduli (ten thousand to ten million atmospheres, say), whereas gases have very low bulk moduli that’s similar to their pressure.

  • Every stable material compresses under pressure, but pressure can’t destroy a uniform material. In other words, we can apply a crazy amount of pressure to a material sample and—barring a temperature increase leading to combustion, for example, or a pressure-induced phase transformation, or transformation into a neutron star—we’ll never see anything other than harmless compression. This is why a fragile glass or metal object can fall to the ocean floor and remain unharmed. Recovery expeditions from the RMS Titanic established that even the most fragile solid detritus from the Titanic was structurally undamaged by a pressure of 6000 psi (400 atm):


    Selection of fragile glass and ductile metal objects recovered with essentially no structural damage from the
    Titanic debris field between 1987 and 1994 (http://www.premierexhibitions.com).

  • Consider another example: Submarine hulls don’t fail because any specific material element is subjected to hydrostatic pressure. After all, a depth of 2000 m (well beyond the working depth of submarines) would cause a cubic centimeter of steel to compress by only 0.5 μm on a side. Ultimately, submarines driven past their critical depth fail because their hull material isn’t subjected to hydrostatic pressure: the pressure on the outside is much larger than the pressure on the inside (and both are far less than the in-hull longitudinal and hoop stresses), and these differences produce a deviatoric stress state that induces failure.

    Consider the reverse example: a pipe that failed from excessive internal pressure:


    Failure surfaces of an overpressurized pipe that failed through excessive hoop stress (photographs by R. A. Simonds, from Courtney’s
    Mechanics of Materials). The failure mode of this ductile material is shear related, of course, as clearly indicated by the 45° failure surface.

    Notably, even though this failure ultimately arose from pressure, the failure mode was that of deviatoric stress, as indicated by the 45° angle in the failure surface—the hallmark of shear.

  • The bulk modulus is a great way to start exploring the origin of the elastic moduli in condensed matter and gases. (This section gets a little mathy.) Very broadly, the (isothermal) bulk modulus \(K\) is defined as the pressure \(P\) needed to achieve a certain decrease in volume per unit volume \(V\) at constant temperature \(T\)—succinctly, \(K\equiv -V(\partial P/\partial V)_{T}\). Let’s also pull in the fundamental equation for a closed system $$dU=T\,dS-P\,dV,$$ which tells us, for example, that we can increase a system’s energy \(U\) either by heating it (i.e., transferring entropy \(S\) to it at a certain temperature \(T\)) or by doing work on it (here, by compressing it, i.e., transferring volume \(V\) away from it at a certain pressure \(P\)). To clarify the influence on \(K\), we isolate \(P\) and perform the differentiation to obtain $$K\equiv-V\left(\frac{\partial P}{\partial V}\right)_T=V\left[\left(\frac{\partial ^2 U}{\partial V^2}\right)_T-T\left(\frac{\partial^2 S}{\partial V^2}\right)_T\right].$$

    In condensed matter, with its strong intermolecular bonds, the energy term containing \(U\) dominates, and we find that the bulk modulus scales with the curvature of the bond energy as a function of molecular spacing. This curvature grows more extreme with a stronger bond, and so we observe a correlation between the bulk modulus and the melting temperature (or density), as we can confirm from a trip to Wolfram Alpha.

    Scientists have spent about a century developing pair potential functions to describe the energy associated with the atomic bond. With a potential function in hand, predicting the bulk modulus using the above equation is straightforward.

    In gases, in contrast, strong intermolecular bonds are absent, and the entropy term containing \(S\) dominates. (Gases have an enormous entropy because of the huge number of positions and velocities that a molecule can explore in the gaseous state.) We can simply insert the equation of state for an ideal gas, \(P=nRT/V\), into the above definition of \(K\) to find that the isothermal bulk modulus for an ideal gas at a certain temperature is exactly \(P\).

Given all this, why not love the bulk modulus?

Mathematical framework

All of the concepts discussed in this note can be summarized in one unified mathematical expression—if you’re into that kind of thing.

It’s called generalized Hooke’s Law, discussed at a slightly more technical level here. That note derives the relationships connecting Young’s modulus, the shear modulus, the bulk modulus, and Poisson’s ratio. But even in that note, I consistently expressed generalized Hooke’s Law in terms of Young’s elastic modulus! What a hypocrite. Here, let’s re-derive generalized Hooke’s Law (for a uniform isotropic material) using those valuable players, the shear and bulk elastic moduli. As shown by the rotated-element example above, all these moduli and Poisson’s ratio are intrinsically connected.

OK, so we already know that the shear modulus \(G\) couples the angular shear strain (this turns out to be \(2\epsilon_{xy}\), as discussed in the other note) to the shear stress \(\sigma_{xy}\). Replacing the \(x\), \(y\), \(z\) coordinate system with a 1, 2, 3 coordinate system, we obtain the equations

$$\epsilon_{12}=\frac{\sigma_{12}}{2G};$$ $$\epsilon_{13}=\frac{\sigma_{13}}{2G};$$ $$\epsilon_{23}=\frac{\sigma_{23}}{2G}.$$

We also know that the bulk modulus \(K\) couples the volumetric strain (which is \(\epsilon_1+\epsilon_2+\epsilon_3\) for small strains, as discussed in the other note) with the dilatational stress or negative pressure \((\sigma_{11}+\sigma_{22}+\sigma_{33})/3=-P\):

$$\epsilon_{11}+\epsilon_{22}+\epsilon_{33}=\frac{(\sigma_{11}+\sigma_{22}+\sigma_{33})}{3K}.$$

We can combine all six equations (in a hand-wavy way) into an elegant single equation with the aid of some handy notation:

  • An index such as \(i\) or \(j\) can range from 1 to 3 independently.

  • A repeated index indicates summation (e.g., \(\sigma_{kk}\) is shorthand for \(\sigma_{11}+\sigma_{22}+\sigma_{33}\)).

  • The Kronecker delta \(\delta_{ij}\) is 1 when \(i=j\) and 0 otherwise.

Thus, we have

$$\epsilon_{ij}=\frac{\sigma_{ij}}{2G}-\left(\frac{1}{6G}-\frac{1}{9K}\right)\sigma_{kk}\delta_{ij}.$$

This equation expands into six (!) independent equations that can be used to calculate the normal stresses \(\sigma_{11}\), \(\sigma_{22}\), and \(\sigma_{33}\) and the shear stresses \(\sigma_{12}\), \(\sigma_{13}\), and \(\sigma_{23}\). Note how the \(\frac{1}{6G}\) term on the right negates the \(\frac{1}{2G}\) coefficient of \(\sigma_{ij}\) under the hydrostatic case of \(\sigma_{ii}\delta_{ij}=\sigma_{11}+\sigma_{22}+\sigma_{33}=-3P\). So now we have another version of generalized Hooke’s Law, one that displays the usefulness of elastic moduli other than Young’s modulus.

For maximum utility, though, I want to show generalized Hooke’s Law for strains \(\epsilon\) in terms of stresses \(\sigma\)—and stresses in terms of strains—using (1) the shear modulus \(G\) and bulk modulus \(K\) and (2) Young’s modulus \(E\) and Poisson’s ratio \(\nu\):

$$\epsilon_{ij}=\frac{\sigma_{ij}}{2G}-\left(\frac{1}{6G}-\frac{1}{9K}\right)\sigma_{kk}\delta_{ij}.$$ $$\epsilon_{ij}=\frac{1+\nu}{E}\sigma_{ij}-\frac{\nu}{E}\sigma_{kk}\delta_{ij}.$$ $$\sigma_{ij}=2G\epsilon_{ij}+\left(K-\frac{2}{3}G\right)\epsilon_{kk}\delta_{ij}.$$ $$\sigma_{ij}=\frac{E}{1+\nu}\epsilon_{ij}+\frac{\nu E}{(1+\nu)(1-2\nu)}\epsilon_{kk}\delta_{ij}.$$

All of these equations tell us the exact same thing—but remember that they apply only to uniform isotropic* materials; all might be useful in various scenarios.

*Is isotropy a reasonable assumption? Often, yes; amorphous ceramics such as window glass have no preferred direction and are thus isotropic. Metals are generally polycrystalline, and their grain orientation in bulk form is often sufficiently random that any directional influence is negligible, so they can also be considered isotropic.

What materials aren’t isotropic? The single-crystal silicon used in wafers for integrated circuit manufacturing, for example. Single-crystal turbine blades and metal whiskers. Thin polycrystalline films or strongly deformed bulk metals in which the grains have a preferred direction. Composites, in which certain stress-bearing materials are often oriented in specific directions. You get the idea.

For example, if we’re limited to shear stress or strain, then we can discard the second term on the right-hand side because \(\delta_{ij}=0\). And if we’re in a plane-stress or plane-strain situation in which \(\sigma_{33}=0\) or \(\epsilon_{33}=0\), respectively, then we obtain other useful simplifications. We find, for example, that although the bending stiffness of a beam scales with \(E\), the bending stiffness of a plate scales with \(E/(1-\nu^2)\)—in other words, a plate is effectively stiffer because its width doesn’t allow it to easily contract laterally in the manner of a narrow rod.

A plethora of other elastic moduli

Beyond the shear modulus and bulk modulus (which are certainly appealing for characterizing the steady-state elasticity of uniform matter), revel in the many other elastic moduli that practitioners have found useful:

  • The complex modulus \(G^\prime(\omega)+iG^{\prime\prime}(\omega)\), which quantifies the dynamic stiffness as a function of an oscillation frequency \(\omega\) (and can be expressed in terms of the strain rate). The \(G^\prime\) component corresponds to the shear stiffness \(G\) of an ideal solid, whereas \(G^{\prime\prime}\) incorporates the strain-dependent stiffness of an ideal liquid (whose stiffness generally scales up with the strain rate, as mediated by the viscosity). The imaginary unit \(i\) indicates that the deformation of a liquid lags behind that of a solid. (The lag arises because if you shear and release a solid, it springs back; if you shear and release a liquid, it deforms and sits; you’d have to pull it back to restore the original configuration.)
  • Various poroelastic moduli that apply in the context of a solid elastic material containing pores filled with a liquid that can move in response to deformation—a two-phase system that’s exquisitely sensitive to the nature of the surrounding containment.
  • The secant and tangent moduli: For materials such as elastomers that can elastically deform to several times their length (in contrast to metals and ceramics, which generally can’t deform by more than a fraction of a percent without plastically deforming or fracturing), we encounter a strongly nonlinear elastic stress-strain diagram that presents us with a challenge of whether to look at slopes or values.
  • Anisotropic elastic moduli emerge when we move from randomly polycrystalline and amorphous materials to polycrystalline materials with a texture—meaning a prevailing orientation—and single-crystal materials. In this case, generalized Hooke’s Law no longer holds, as there may be more than one independent shear modulus based on the load orientation.
  • Various field specific moduli such as soil and rock moduli are useful to geologists, for example, who need to incorporate creep, or time-dependent deformation of solids under load. Yes, all the elasticity behavior described in this note is only an idealization over relatively short times. (Short times may be a few minutes for a very hot metal, years for a glacier, or hundreds of years for hot rock; this is the realm of the deformation mechanism map.) Ultimately, as noted by Prof. Chris Schuh at MIT, there’s no such thing as elasticity; there’s only negligible plasticity.

With this information in mind, the next time you hear about an “elastic modulus,” consider: are they talking about Young’s elastic modulus or another elastic modulus? And are they using the appropriate framework to analyze and present their results? Again, let’s not overrate Young’s modulus; instead, let’s appreciate its advantages while understanding its limitations and what insights come with a familiarity with other types of stiffness.

 

© Copyright John M. Maloney