Radial force development during root growth measured by photoelasticity

Abstract

Background and aims

The radial growth of roots largely affects and reorganizes the porous or crack networks of soils and substrates. We studied the consequences of a radial steric constriction on the root growth and the feedback force developed by the root on the solid phase.

Methods

We developed an original method of photoelasticity to measure in situ root forces. By changing the gap width (0.5 to 2.3 mm) between two photoelastic disks we applied variable radial constrictions to root growth and simultaneously measured the corresponding radial forces. Changes in morphology and forces of primary roots of chick pea (Cicer arietinum L.) seedlings were recorded by time-lapse imaging every 24 min up to 5 days.

Results

The probability of root entering the gap depended on the gap size but was also affected by circumnutation. Compared to non-constrained root controls, no significant morphological change (elongation, diameter) was measured outside the gap zone. Inside the gap zone, outer cortex cells were compressed, the central cylinder was unaffected. Radial forces were increasing with time but no force levelling was observed even after 5 days.

Conclusions

Radial constrictions applied to roots did not significantly reduce their growth. The radial force was related to the root strain in the gap.

Appendix

We first recall the main features of photoelasticity for the reader who is not accustomed to this technique. Then in a second part we present the way we analysed the photoelastic pattern of fringes in the particular case of a photoelastic disk compressed diametrically.

Photoelasticity is the property of some materials to become birefringent under mechanical load, i.e. the refraction index is no longer a scalar constant but has a tensorial form (Majmudar and Behringer 2005 and references therein). A plane polarized light of wavelength λ passing through a 2D sheet of photoelastic material of thickness t will be decomposed along the two local principal stress directions 1 and 2 and each of these components will experience different refractive indices, n ₁ and n ₂ according to the principal values σ ₁ and σ ₂ of the local stress tensor at the position (x,y) of the sheet where the light ray arrives. Both quantities are related through a linear relation (the stress optic law of equation (2)) by introducing the photoelastic constant C:

$$ {n_1} - {n_2} = C\left( {{\sigma_1} - {\sigma_2}} \right) $$

(2)

The difference in the refractive indices leads to an optical path difference δ and therefore to a relative phase retardation ϕ (equation (2)) between the two component waves.

$$ \varphi = \frac{{2\pi }}{\lambda }\delta = \frac{{2\pi }}{\lambda }({n_1} - {n_2})t $$

(3)

This phase retardation gives interference patterns for the emerging light according to the values of ϕ. By combining equations (2) and (3), one obtains the value of the light intensity I(x,y) as a function of the local deviatoric stress.

$$ I(x,y) = {I_0}{\sin^2}({{\varphi } \left/ {2} \right.}) = {I_0}{\sin^2}\left( {\frac{{\pi Ct}}{\lambda }({\sigma_1} - {\sigma_2})} \right) $$

(4)

The goal of photoelasticity is to extract data on the stresses (or forces) from the positions of (black) fringes. In particular the black fringe (destructive interference) of order m will be obtained for the positions (x,y) where the principal stress difference (σ ₁ – σ ₂) verifies:

$$ {\sigma_1} - {\sigma_2} = \frac{{m\lambda }}{{Ct}} $$

(5)

In our case, the photoelastic material is a disk of thickness t and radius R. A preliminary experimental calibration was used to determine the photoelastic constant C in a simple loading test. During calibration, a diametrical compression of the photoelastic disk was exerted vertically by compressing the disk diametrically along the y axis with a micrometric displacement. The vertical force was measured directly, the disk being placed on an electronic scale (Fig. 2). We thus relate the observed positions of the black fringes to the measured applied force F. Note that for a point force f = F/t per unit length t, the elastic theory provides a simple analytical solution for the stress tensor for every point (x,y) of the disk. In particular it gives the positions of the black fringes x _m of order m along the x-axis perpendicular to the y-compression axis according to the equation (6):

$$ F = m\frac{\lambda }{C} \cdot \frac{{\pi R}}{4} \cdot \left[ {\frac{{{{\left( {1 + {{\left( { \frac{{{x_m}}}{R} } \right)}^2}} \right)}^2}}}{{1 - {{\left( { \frac{{{x_m}}}{R} } \right)}^2}}}} \right] $$

(6)

This relation directly links the compression force F to the positions of the black fringes x _m . The expression still holds for the case of a compression with an extended contact zone of size a (instead of a point force), when R > > a. For more complicated geometries of loading (normal and tangential forces applied at more than 2 contact points), it is still possible to extract data on forces from optical fringes by using numerical solutions of elastic theory and/or solve the inverse problem as was done by (Majmudar and Behringer 2005). For simplicity in this article, we limit our analysis to the cases in which the root induced a diametrical compression of the photoelastic disk.