Distinction between uniaxial stress and uniaxial strain

Uniaxial stress is a form of loading in which the 11 (axial) component of stress is nonzero, while all other components of stress are zero.

Uniaxial strain is a form of loading in which the 11 (axial) component of strain is nonzero, while all other components of strain are zero.

For an isotropic material, axial loading (whether uniaxial stress or uniaxial strain) gives stress and strain matrices of the form
\left[\sigma\right]=\begin{bmatrix}  \sigma_A & 0 & 0 \\  0 & \sigma_L & 0 \\  0 & 0 & \sigma_L \\  \end{bmatrix}   and   \left[\varepsilon\right]=\begin{bmatrix}    \varepsilon_A & 0 & 0 \\  0 & \varepsilon_L & 0 \\  0 & 0 & \varepsilon_L \\  \end{bmatrix} ,

Here, the subscript “A” stands for Axial and “L” stands for Lateral. Thus, \sigma_L=0 for uniaxial stress, whereas \varepsilon_L for uniaxial strain. In either case, you can generate a plot of \sigma_A vs. \varepsilon_A=0 , but the slopes of these plots are going to be different. As a rule, since uniaxial strain provides greater confinement, the slope of the axial stress-strain plot is typically steeper in uniaxial strain than it is in uniaxial stress.  For instance, the slope of the Hooke’s law stress-strain line is typically LARGER than Young’s modulus if the loading is uniaxial strain.

Consider an isotropic linear-elastic (Hooke’s law) material, which is described in general by the following set of equations:
\begin{aligned}  \varepsilon_{xx}&=\frac{1}{E}\left[\sigma_{xx}-\nu(\sigma_{yy}+\sigma_{zz})\right]\\  \varepsilon_{yy}&=\frac{1}{E}\left[\sigma_{yy}-\nu(\sigma_{zz}+\sigma_{xx})\right]\\  \varepsilon_{zz}&=\frac{1}{E}\left[\sigma_{zz}-\nu(\sigma_{xx}+\sigma_{yy})\right]\\  \varepsilon_{xy}&=\frac{\sigma_{xy}}{2G}\\  \varepsilon_{yz}&=\frac{\sigma_{yz}}{2G}\\  \varepsilon_{zx}&=\frac{\sigma_{zx}}{2G}\\  \end{aligned}
Here, E is Young’s modulus, \nu is Poisson’s ratio, and G=\frac{E}{2(1+\nu)} is the shear modulus.  Be advised: you might need to make the following notation substitutions if you want these equations to look the same as they do in undergraduate textbooks:

\begin{aligned}  \sigma_{xx}&=\sigma_{x}&,\qquad \sigma_{yy}&=\sigma_{y}&,\qquad \sigma_{zz}&=\sigma_{z}\\  \sigma_{xy}&=\tau_{xy}&,\qquad \sigma_{yz}&=\tau_{yz}&,\qquad \sigma_{zx}&=\tau_{zx}\\  \varepsilon_{xx}&=\varepsilon_{x}&,\qquad \varepsilon_{yy}&=\varepsilon_{y}&,\qquad \varepsilon_{zz}&=\varepsilon_{z}\\  \varepsilon_{xy}&=\gamma_{xy}/2&,\qquad \varepsilon_{yz}&=\gamma_{yz}/2&,\qquad \varepsilon_{zx}&=\gamma_{zx}/2\\  \end{aligned}

For axial loading, we know that \sigma_{xx}=\sigma_A, ~\sigma_{yy}=\sigma_{zz}=\sigma_L, ~\sigma_{xy}=0 , etc., with similar substitutions for the strains. Accordingly, under conditions of axial loading, the general Hooke’s law equations specialize to
\begin{aligned}  \varepsilon_A&=\frac{1}{E}\left[\sigma_A-2 \nu \sigma_L\right]\\  \varepsilon_L&=\frac{1}{E}\left[(1-\nu)\sigma_L-\nu \sigma_A\right]\\  \end{aligned}

For uniaxial stress, where \sigma_L=0 , these give the familiar formulas
\begin{aligned}  \sigma_A&=E \varepsilon_A\\  \varepsilon_L&=-\nu \varepsilon_A\\  \end{aligned}

For uniaxial strain, where \varepsilon_L=0 , Hooke’s law reduces to a very different result
\begin{aligned}  \sigma_A&=C \varepsilon_A ~~~~\text{where}~~~~C=\frac{E(1-\nu)}{(1+\nu)(1-2\nu)}\\  \sigma_L&=\beta \sigma_A ~~~~\text{where}~~~~\beta=\frac{\nu}{1-\nu}\\  \end{aligned}
For ordinary values of Poisson’s ratio (i.e., values falling betwee 0 and 1/2), the uniaxial strain modulus C (also called the “constrained modulus”) is larger than Young’s modulus (i.e., C>E ). This means that it is more difficult to achieve a given increment in strain under conditions of uniaxial strain, which is unsurprising in light of the fact that the the lateral confinement of uniaxial strain causes extra resistance to deformation.

So-called p-waves in geomechanics are uniaxial strain waves, which means they are wide planar waves in which confinement of surrounding material disallows lateral material motion. Acoustic waves (like the sound of a bat hitting a ball which arrives at your ear after a delay) are also uniaxial strain waves, not uniaxial stress waves. In an isotropic linear-elastic idealization, uniaxial strain waves travel with speed given by \sqrt{C/\rho} , where \rho is the density and C is the isentropic constrained modulus defined above. Wave speeds are often miscalculated for the following reasons: (1) people incorrectly use isothermal moduli instead of isentropic (which works only if the thermal expansion coefficient is negligble), (2) people incorrectly use E when they should use C (which works only for uniaxial stress waves, such as on a thin bar, not uniaxial strain), (3) people use the bulk modulus K when they should use C (which works only for inviscid materials like gasses, but not solids), (4) they apply a wave speed formula derived for isotropic materials inappropriately to layered material like geological deposits.

The distinction between uniaxial stress and uniaxial strain is even more dramatic when you compare the stress-strain predictions for a simple plasticity model, like von Mises theory shown below:

Here, the blue dashed line is probably what you recall from undergraduate strength of materials. The solid red line is the response in uniaxial strain. Note an important distinction: an implicit relationship does NOT exist between stress and strain increments in uniaxial stress (i.e., the blue dashed line has a zero slope, implying a zero stress increment can be paired with an arbitrary the strain increment), but an implicit relationship DOES exist between stress and strain increments in uniaxial strain (i.e., a given axial stress increment corresponds to a unique axial strain increment). The fact that there are two distinct slopes in uniaxial strain is the reason why elastic-plastic waves always split into a fast-moving elastic precursor followed later by a slower-moving plastic wave. In stress space, uniaxial stress pushes the material up to the von Mises cylinder, where it gets “stuck” at that initial contact point (causing stress to stay constant even though strain is ever increasing). The stress space path for uniaxial strain loading allows the stress state to slide along the von Mises cylinder parallel to the hydrostat, which is why the post-yield response in uniaxial strain has a slope given by the bulk modulus.

See below to understand that changing the plasticity model also produces significant changes in how the stress state moves through stress space. Below, instead of the non-moving von Mises cylinder, the yield surface is the classical “evolving teardrop” shape used in geomechanics. In that case, uniaxial stress states no longer get “stuck” but instead evolve from one part of the yield surface to another, though still in a radically different way than the motion of the stress state in uniaxial strain.

Uniaxial stress loading also goes by the name unconfined compression (assuming that the axial stress is compressive). Unconfined compression is a special case of so-called triaxial compression in which the stress is still of axial form (i.e., one axial component different from the two equal lateral components), but with the lateral stress is held at a fixed nonzero value, often called the bath pressure since this value is typically imposed by first subjecting the material to a hydrostatic loading leg, thereafter changing only the axial component while the lateral stress is held fixed.

As a rule, material behavior in uniaxial stress is very different from that in uniaxial strain. Shown below, for example, are stress paths predicted by a geomechanics plasticity model showing very different trends in uniaxial stress in comparison to uniaxial strain. These plots depict a so-called meridional profile of stress space in which the ordinate is a measure of shear stress and the abscissa is a measure of pressure. The teardrop shape is the evolving yield surface according to this model, where the curved part of the surface at the right-hand-side represents the effect of pores. The rightward motion of this surface represents inelastic pore collapse. The other line in these plots is the evolving stress state.

LOADING WITH LATERAL STRAIN HELD CONSTANT (AT ZERO)


In this plot and the one below, the abcissa is the first invariant of stress I_1 , and the ordinate is the equivalent shear given in terms of the second stress invariant as \sqrt{J_2} . Above, the black line represents an evolving stress state induced during uniaxial strain loading. The green (teardrop shaped) line is the evolving yield surface of a geomechanics model that is run with pressure-dependent strength (which is why the initial slope of the green line is positive) as well as kinematic hardening (which is why the green line moves upward) and porosity (which is why the green line has a cap at the right-hand-side allowing inelastic pore collapse). Rightward motion of the green yield represents the fact that increasingly large pressures are required to continue the process of pore collapse. Of particular note in this plot is that uniaxial strain loading provides lateral confinement that produces increasing pressures that bend the stress path to the right and furthermore keep the stress state always on the “cap” part of the green yield surface. Contrast these features with triaxial stress loading shown below, where the possibility of unlimited (outward moving) lateral strains can produce material dilatation even when all stresses are compressive.

LOADING WITH LATERAL STRESS HELD CONSTANT

Above, the evolving stress state (red) shows a hydrostatic leg where the path is horizontal followed by a triaxial leg in which lateral stress is held constant while the axial stress is increased. The teardrop-shaped yield surface (blue) evolves such that the cap moves outward during the initial hydrostatic leg to account for pore collapse. During the initial part of the triaxial compression phase, the stress state is initially still on the cap (where the outward normal to the yield surface points toward the right in the the compression direction). This particular model evolves the yield surface in such a way that the stress state eventually resides on a part of the yield surface where the outward normal points in towards the left, allowing this model to give very accurate predictions of dilatation in this mode of loading. It is unlikely, however, that this model would be predictive under a change in the loading trajectory. In particular, note that this model moves the cap continuously outward, which means that it predicts the unlikely result that dilatation would make the material stronger in hydrostatic compression. The CSM group is exploring alternative plasticity theories that will allow prediction of dilatation in triaxial compression without anomalous hydrostatic strengthening.

Summary: This post has examined Hooke’s law to demonstrate that uniaxial strain provides confinement that makes the slope of the axial stress-strain line higher than it is for uniaxial stress. This also implies that acoustic waves travel faster than the often misquoted thin-rod speed given by \sqrt{E/\rho} . Images were also shown for a geomechanics model subjected to conditions of constant lateral strain (producing a bend in the stress path towards increasing pressures) and conditions of constant lateral stress (producing a straight path through meridional stress space that eventually allows the stress to reach a dilatational portion of the yield surface).

BOTTOM LINE: Uniaxial stress is very different from Uniaxial strain!

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