On the evolution of intrinsic curvature in rod-based models of growth in long slender plant stems

Abstract

a b s t r a c tIn many applications of rod theories as models for plant stem growth and development, it is necessary to allow the intrinsic curvature and flexural stiffness of the rod to evolve. In the present paper, the application of evolution equations for these quantities is examined and a new evolution equation for the intrinsic curvature is proposed. To illustrate the new evolution equation, several examples of the evolution of rods in the presence of external forces and tip growth are presented. Growth in plants frequently results in residual stresses and the compatibility of rod models with these fields is also discussed.

On the growth and form of shoots

“…, DB ¼ B max À B 0 and B max is the maximum value of the bending rigidity of fully lignified wood, dt is the age of a material point outside the growth zone (dt ¼ [21][22][23][24][25] that assume that the actual curvature tries to relax to the natural curvature. We differ by accounting for the experimentally demonstrated facts that a shoot grows to change its curvature in response to its local orientation with respect to the direction of gravity while also trying to maintain straightness [11] noting that the actual (sensed) curvature is different from the (actuated) growth curvature.…”

“…By treating the growing rod as a one-dimensional continuum that can change its reference length, diameter and stiffness, various authors have formulated a theory for how its shape can vary with time [10,[21][22][23][24]. A general discussion of possible evolution equations for the natural curvature has also been considered [22,25], with autotropism being the primary driver. These theoretical studies do not consider the role of multiple stimuli that include, for example, autotropic, gravitropic or phototropic responses, nor do they explore the large parameter space of variables to understand the qualitative nature of the solutions or their implications for the observed morphological diversity of plant shoots.…”

Tree growth stress and related problems

“…A cantilever or curved beam model was used to describe the negative (plagio-) gravitropic behavior of woody plant stems [17,[19][20][21][22][23]. Through simulations of observable shapes of various tree stems, they concluded that, in reaction wood, the surface growth stress is sufficiently large to bend the inclined stem upward against the weeping caused by the increasing weight.…”

On morphoelastic rods

“…The literature on three-dimensional continuous bodies which can change their relaxed shape by growth and remodeling is extremely developed, as is evident from recent surveys on the subject [11][12][13]. This is not the case for morphoelastic structures that is to say, thin bodies, such as rod and shells, which can change their relaxed shape [14][15][16][17]. This, despite the fact that models of thin structures, being more accessible to analytical investigation because of their simplicity, seem to be better suited to explaining certain qualitative features of several biological systems [18][19][20].…”