Postbuckling behaviour of a growing elastic rod
Abstract
We consider mechanicallyinduced pattern formation within the framework of a growing, planar, elastic rod attached to an elastic foundation. Through a combination of weakly nonlinear analysis and numerical methods, we identify how the shape and type of buckling (super or subcritical) depend on material parameters, and a complex phasespace of transition from super to subcritical is uncovered. We then examine the effect of heterogeneity on buckling and postbuckling behaviour, in the context of a heterogeneous substrate adhesion, elastic stiffness, or growth. We show how the same functional form of heterogeneity in different properties is manifest in a vastly differing postbuckled shape. Finally, a fourth form of heterogeneity, an imperfect foundation, is incorporated and shown to have a more dramatic impact on the buckling instability, a difference that can be qualitatively understood via the weakly nonlinear analysis.
Keywords
Pattern formation Weakly nonlinear analysis Morphoelasticity Bifurcation and buckling Elastic rodsMathematics Subject Classification
35B36 74G60 74K101 Introduction
Mechanicallyinduced pattern formation is a phenomenon prevalent in the morphogenesis of many biological structures, from airway wall remodelling (Moulton and Goriely 2011b), to wrinkling of skin (Efimenko et al. 2005), to blades of grass (Dervaux et al. 2009). The prevailing feature in such systems is the deformation from a ‘trivial’ base state to a more complex geometry, with buckling induced by mechanical stress. This feature is not unique to biological systems; indeed the same basic wrinkling pattern found in an elephant’s skin can be induced by compressing a sheet of rubber. What is unique to the biological world is that such patterns tend to form without any external influence, rather the stress needed for mechanical instability is produced internally. Stress can be introduced internally via different mechanisms, including apical purse string contraction (Sawyer et al. 2010), muscle contraction (Dick and Wakeling 2018), and uniform growth in a confined geometry. A primary origin for stress, and the focus of this paper, is differential growth, i.e. different parts of a tissue growing at different rates. Perhaps the simplest example is a tissue layer that grows relative to an underlying substrate to which it is adhered. The growth induces a compressive stress in the growing layer, and at some critical threshold the tissue buckles, exchanging compression energy for bending energy. This situation underlies the formation of numerous biological patterns, including brain development—induced by the differential growth between the cortex and subcortex (Budday et al. 2015); intestinal crypt fission, in which epithelial tissue grows but is tethered to underlying tissue stroma (Wong et al. 2002); even seashell ornamentation, characterised by the adhesion of the growing mantle tissue to the rigid shell that it secretes (Chirat et al. 2013). Aside from differential growth, inherent geometrical constraints can also induce buckling, for instance a row of cells growing uniformly but within a closed space will similarly develop compressive stress and ultimately buckle.
A mathematical description of growthinduced mechanical buckling can take a variety of forms. A number of discrete cellbased models have been devised, in which mechanical interactions between individual cells, coupled with cell growth or proliferation, can lead to deformation in the form of folds (Drasdo and Loeffler 2001), invaginations (Odell et al. 1981), or protrusions (Buske et al. 2012; Langlands et al. 2016). This approach is more amenable to the inclusion of celllevel biological detail. Continuum modelling, while less amenable to this level of detail, allows one to utilise analytical tools for differential equations, which may improve insight and reveal parametric relationships that are more difficult to attain from discrete models. Within 3D elasticity, growth is naturally incorporated via decomposition of the deformation gradient tensor into a growth tensor describing the local change of mass and an elastic tensor accounting for the elastic response (Rodriguez et al. 1994). However, beyond simple geometries such as the buckling of a sphere (Ben Amar and Goriely 2005), a 3D description of buckling typically requires fully computational techniques such as finite element methods (Ambrosi et al. 2011). In many cases, the geometry under consideration is wellsuited for a reduced dimensional analysis. This is clearly true in the case of filaments. Filaments by definition have one length scale much longer than the other two and hence are well suited to a 1D description. Kirchhoff theory for elastic rods has been applied to a diverse range of filamentary systems, such as DNA coiling (Thompson et al. 2002), neurite motility (Recho et al. 2016), plant tendril twisting (Goriely and Tabor 1998), and many more. A planar rod description may also be relevant even in an inherently 2D system: for instance when a sheet of tissue deforms approximately uniformly in a transverse direction, or when a crosssection of tissue deforms such as in the circumferential wrinkling of a tube (Moulton and Goriely 2011a; Ben Amar and Jia 2013; Balbi and Ciarletta 2013).
Within a continuum formalism, a commonlyconsidered problem is a planar rod on an elastic foundation or substrate, typically with the ends fixed. The basic premise is that growth (axial elongation) of the rod generates compressive stress, creating buckling from a flat to a curved state, but this deformation is resisted by elastic tethering to a fixed substrate. This basic setup, or very similar, has been studied for many years in an engineering context (where ‘thermal expansion’ typically plays the role of ‘growth’), dating back to classic works of Timoshenko (1925) and Biot (1937), and continues to find new interest and applications.
It is only more recently that the relation to biological pattern formation has become clear and similar systems have been specialised for biological problems. An important aspect that distinguishes biological systems from the above examples is the various ways in which growth can occur; a key challenge here is connecting a continuum level description of growth to underlying celllevel processes. This has stimulated extensive mathematical modelling development and has created the need for a systematic framework. There are three such works of particular relevance for the present paper. In Moulton et al. (2013), the theory of Kirchhoff rods was extended by incorporating growth in a manner inspired by the morphoelastic decomposition. This is the framework upon which our analysis is built. Also of note are recent descriptions of buckling in the context of intestinal crypt formation, invaginations that are present throughout the intestines. Edwards and Chapman (2007) applied a continuum mechanics approach to the formation of a single crypt. They modelled the crypt epithelium and its underlying tissue stroma as a beam upon a viscoelastic foundation. By performing a linear stability and eigenvalue analysis of buckling, they examined the effect of changes to proliferation, cell death, adhesion, or motility. Nelson et al. (2011) complemented this analysis with a ‘bilayer’ model representing an epithelial layer connected to a flexible substrate. Nelson et al. also conducted an eigenvalue analysis similar to Edwards and Chapman and combined this with a numerical analysis of the full, nonlinear model, demonstrating the influence of heterogeneity in both growth and bending stiffness on the resulting buckled crypt shape. Similar systems, but in 2D using plate theory, have also explored pattern formation due to growth instabilities. Hannezo et al. (2011) characterised transitions from cryptlike to herringbone and labryinth patterns; Nelson et al. extended their 1D models in Nelson et al. (2013) and found that crypt patterning could be most strongly controlled through heterogeneity in growth.
Our objective in this paper is to use the morphoelastic rod framework of Moulton et al. (2013) to extend the results of Edwards and Chapman (2007) and Nelson et al. (2011) and analyse unexplored features that are of general relevance. A focal point for our analysis is the behaviour of the system beyond the initial buckling. In an engineering context, buckling will typically signify failure, and so the threshold value to induce buckling may be the most relevant quantity. For pattern formation in biology, on the other hand, the shape evolution well beyond the initial instability is often critical to the final pattern (and its biological functionality), and hence analysis only of the onset of instability is insufficient.
Also of relevance in many biological systems is understanding the role of heterogeneity in mechanical pattern formation. Heterogeneity can arise in three main forms: growth,^{1} the mechanical properties of the rod, or substrate adhesion. Here it is important to distinguish between growth and remodelling. Growth refers to an increase (or decrease) in mass, i.e. a change in size of a tissue layer without any change in its material properties. Remodelling, on the other hand, refers to a change in material properties without any change in mass, e.g. due to fibre reorientation or cell differentiation. In a growing tissue, both of these processes occur and will commonly occur nonuniformly. The crypt, for instance, is not a layer of uniform cells, but rather consists of a clear proliferative hierarchy of cells with varying rates of division as one moves up the crypt axis (Wright and Alison 1984). Heterogeneity in adhesion may occur due to nonuniform changes in the substrate layer, or in a biological context due to changes in the cells, or may occur due to buckling itself, for instance due to viscoelastic effects.
Both of these features—large deformation beyond buckling and heterogeneity—pose significant mathematical challenges. To capture postbuckling behaviour requires analysis of a nonlinear system of equations, as opposed to the linear stability analysis that can be used to detect buckling. Furthermore, including heterogeneity complicates the use of many analytical tools, either rendering the system analytically intractable or complicating attempts to unfold bifurcations. Here, rather than rely fully on computational techniques, our approach is to analyse postbuckling behaviour and the effect of heterogeneity through a combination of a weakly nonlinear analysis and numerical solution. This approach yields a broad understanding of the role of heterogeneities in growth, material properties, and adhesion, and reveals features of postbuckling pattern formation not described in previous analyses.
We consider a 1D model system of a growing planar rod on an elastic foundation, serving both as an extension of the classical setup and as an abstracted framework for several of the aforementioned biological systems. The rod is subject to growth in the axial direction and clamped boundary conditions, which drive buckling at a critical growth. The goal of this paper is to understand the factors driving the onset of buckling and the postbuckling behaviour. In particular, we investigate how the buckling and postbuckling behaviour changes in the presence of spatial heterogeneity in material properties, obtaining explicit relations for how the pitchfork bifurcation that arises is impacted by heterogeneity, and exploring the shape evolution in the nonlinear postbuckled regime.
The remainder of this paper is structured as follows. In Sect. 2, we outline the general theory for Kirchoff rods and incorporation of growth, as developed in Moulton et al. (2013). Then, in Sect. 3 we summarise the linear stability analysis before extending to a weakly nonlinear analysis. The results of the weakly nonlinear analysis and numerical analysis of the full nonlinear model are presented in Sect. 4, first in a homogeneous setting, then with the addition of different material heterogeneities. Finally, we close by discussing the implications of our results and directions for future model extensions.
2 Model setup
2.1 Nondimensionalisation
3 Stability analysis
In this section, we present the analytical tools that we will use to investigate the buckling and postbuckling behaviour of the morphoelastic rod. We first adapt and summarise the linear stability analysis from Moulton et al. (2013), used to calculate the growth bifurcation value, \(\gamma ^*\), before unfolding the bifurcation with a weakly nonlinear analysis.
3.1 Linear stability analysis
3.2 Weakly nonlinear analysis
4 Buckling and postbuckling behaviour
Having established a relationship for the postbuckling amplitude, we now explore the effect of material parameters and heterogeneity on the buckling and postbuckling behaviour. First we examine the form of bifurcation in a homogeneous setting, effectively by analysing how the critical buckling growth \(\gamma ^*\), the buckling mode, and the pitchfork constants \(K_1\) and \(K_2\) vary with the two free parameters in the nondimensional system, \(\widehat{L}_0\) and \(\hat{k}\). We then adapt the weakly nonlinear analysis to incorporate heterogeneity and investigate the impact of nonuniformity in foundation stiffness, rod stiffness, and growth. In each case, we complement the analytical work by solving the full nonlinear system (10)–(14), using the numerical package AUTO07p (Doedel et al. 2007). AUTO07p uses pseudoarclength continuation to trace solution families and solves the system with an adaptive polynomial collocation method.
4.1 Effect of length and foundation stiffness
In Figs. 2 and 3, we plot bifurcation diagrams for varying values of the dimensionless foundation stiffness \(\widehat{k}\) and the dimensionless rod length \(\widehat{L}_0\), respectively, for increasing growth. The horizontal axis in each case is the growth parameter \(\gamma \), and the vertical axis plots the nontrivial branches, \(\pm \Vert y\Vert :=\pm \text {max}_{S_0}\vert y(S_0)\vert \), which is closely related to the value of the constant \(C_1\) but more representative of the postbuckling amplitude. The solid lines are determined from the weakly nonlinear analysis, while the dashed lines are numerical results. We also plot the buckled shape \((x(S_0), y(S_0))\) at the specified points for each branch.
Considering length \(\widehat{L}_0\) leads to similar changes in both critical buckling growth and mode number. However, while the buckling mode increases for increasing length, the critical buckling growth decreases. Recalling the scaling \(\widehat{L}_0 = L_0(A/I)^{1/2}\), this reflects the notion that a short or thick rod can endure more growth before buckling, and will buckle at lower mode. We note also that as \(\widehat{L}_0\rightarrow \infty \), \(\gamma ^*\rightarrow \gamma ^*_{\mathrm {inf}}\), the critical growth value for buckling on an infinite domain (29).
Perhaps most notable is the transition from supercritical to subcritical bifurcation evident in both diagrams. We find that subcritical bifurcations occur for large enough \(\widehat{k}\) or small enough \(\widehat{L}_0\). We find through numerical continuation that the subcritical branches then fold back, a feature not captured by the weakly nonlinear analysis at order \(O(\delta ^3)\). A linear stability analysis (see “Appendix B”) confirms that the portion of the subcritical branch before folding back is unstable, while the portion after the foldback is stable. (As would be expected, the curved branches are stable in the supercritical case.) This implies that a subcritical bifurcation signifies a discontinuous jump from the trivial flat state to the finite amplitude stable branch, as well as the presence of a hysteresis loop if \(\gamma \) is subsequently decreased.
4.2 Locating the pitchfork transition
For given parameters \(\widehat{k}\) and \(\widehat{L}_0\), the weakly nonlinear analysis enables us to determine the type of pitchfork bifurcation simply by computing the sign of \(K_1\). Figure 4 shows the regions in \(\widehat{k}\)–\(\widehat{L}_0\) space where supercritical and subcritical pitchforks occur. Note that despite having an explicit expression for \(K_1\), the actual computation of its value was done numerically as it requires root finding for the eigenvalue \(\gamma ^*\). Hence, to produce Fig. 4 we computed \(K_1\) over a discrete grid in the \(\widehat{k}\)–\(\widehat{L}_0\) plane. The transition boundary was then verified at several points through numerical path following in AUTO07p.
Unexpectedly, we do not find a simple monotonic transition boundary, as seen in similar studies (Hutchinson 2013), but rather an intricate pattern with an oscillatory structure. This structure implies that multiple transitions between super and subcritical buckling can occur for a fixed \(\widehat{k}\) and varying \(\widehat{L}_0\) (or vice versa); that is, simply increasing the length of the rod monotonically can create repeated transitions between super and subcritical bifurcation. For an infinite rod, the transition can be computed as \(\widehat{k} \approx 0.38196\) (see “Appendix B”); this point is included as a dashed, horizontal line in Fig. 4, and it appears that as \(\widehat{L}_0\rightarrow \infty \), the oscillations dampen and the transition boundary approaches this constant value. By contrast, as \(\widehat{L}_0\) decreases, the oscillation amplitude increases, although the slenderness assumption of the rod breaks down as \(\widehat{L}_0\sim O(1)\).
4.3 Parameter heterogeneity
Thus far, we have assumed spatial homogeneity in model parameters. However, in many biological systems, heterogeneities are inherent in the system. This raises the question of how a given heterogeneity is manifest in the buckled pattern. To explore this, we initially consider three distinct forms of heterogeneity in: the foundation stiffness, the rod stiffness, and the growth. Each heterogeneity has a clear biological interpretation. For example, in the intestinal crypts, these heterogeneities would correspond to: the different types of extracellular matrix secreted by the cells comprising the underlying tissue stroma (foundation heterogeneity), the mechanical properties of epithelial cells in the crypt (stiffness heterogeneity), and the variations in the proliferative capacity of these cells (growth heterogeneity). In order to understand the effect of each type of spatial heterogeneity, we examine heterogeneity for each parameter in isolation.
With parameter heterogeneity, it becomes increasingly difficult to obtain analytically tractable results with a weakly nonlinear analysis, especially if the amplitude of the heterogeneity is pronounced. Nevertheless, when the parameters are close to homogeneous, we can extend the weakly nonlinear analysis and, in particular, ask how the heterogeneity impacts the pitchfork bifurcations observed in the homogeneous case. We complement this analysis with numerical solutions of the full system defined by Eqs. (10)–(14). For computational convenience, heterogeneity is incorporated via a sequence of numerical continuations in the growth and heterogeneity parameters.
With heterogeneity introduced, the weakly nonlinear analysis can be viewed as a twoparameter unfolding, both in the distance from the critical buckling growth, via \(\gamma = \gamma ^* + \varepsilon \gamma ^{(1)}\), and in the distance from homogeneity, characterised by \(\widehat{\epsilon }\). We are thus faced with balancing three small parameters: \(\varepsilon \), \(\widehat{\epsilon }\), and the order of the expanded variables, which we denoted by \(\delta \), e.g. as in \(y = \delta y^{(1)} + O(\delta ^2)\). In the homogeneous case the correct balance is given by \(\epsilon =\delta ^2\). With \(\widehat{\epsilon }> 0\), numerous balances could be sought, and a full analysis of the twoparameter unfolding is beyond the scope of this paper. Our approach involves starting from homogeneity, and increasing the order of \(\widehat{\epsilon }\) to see when and how it first impacts on buckling. Hence, we again take \(\epsilon =\delta ^2\), and consider \(\widehat{\epsilon }=\delta ^\beta \eta \), with \(\beta >1\) and \(\eta \) an O(1) control parameter.

Foundation heterogeneity, for which \(\mu =\widehat{k}\);

Rod stiffness heterogeneity, for which \(\mu =E\), the Young’s modulus^{3};

Growth heterogeneity, for which \(\mu =\gamma \).
In order to investigate greater amplitudes of heterogeneity and the postbuckling shape evolution, we perform numerical continuation on the full model. As an illustrative example, we apply the same form of heterogeneity for each of the three parameters: \(\xi (S_0) = \cos (2\pi S_0/\widehat{L}_0)\) and \(\widehat{\epsilon } = 0.9\), characterising a significant decrease in the middle region and increase in the outer regions. Figure 5 depicts the resultant rod shapes. As evident in Fig. 5, the heterogeneity has a markedly different effect for each material parameter.
The heterogeneity thus acts as an amplifying force where \(\xi (S_0) < 0\), and a resistive force where \(\xi (S_0) >0\). This is apparent in Fig. 5b: the magnitude of \(y^{(1)}\) is largest in the middle, with \(\xi (S_0) < 0\), reducing the effects of \(H^\mathrm {old}_{y^{(3)}}\) and leading to an increase in amplitude.
Rod stiffness heterogeneity In the case of rod stiffness, the dominant trend is compression in the middle region, leading to a significant decrease in amplitude and arclength, and formation of a near cusplike point, reflecting the reduced energy cost of both bending and stretching in the middle region. We have also examined the competing energies within the system: bending versus stretching versus foundation (explicitly defined in “Appendix E”). We find that both the bending and foundation energy are reduced as \(\widehat{\epsilon }\) increases, despite the cusplike formation, while the stretching energy increases (see Fig. 10). The total energy remains roughly constant through most of this tradeoff, but eventually, at large values of \(\widehat{\epsilon }\), the stretching penalty outweighs the benefit to the bending and foundation energies and a sharp rise in the total energy occurs for \(\widehat{\epsilon }\gtrsim 0.7\).
It is worth comparing these results to similar studies. In Nelson et al. (2011), bending stiffness heterogeneity and growth heterogeneity were considered. In the case of growth heterogeneity, Nelson et al. concluded that net growth affects the postbuckling behaviour more than heterogeneity, whereas we have found a significant change in morphology due to growth heterogeneity, even with no change in net growth. This discrepancy may be partially due again to the inextensibility assumption present in Nelson et al. (2011). More likely though, the behaviour may be attributable to viscous relaxation. Nelson et al. have modelled the foundation as viscoelastic springs, thus incorporating a stress relaxation not present in our model. Indeed, they presented an example (see Fig. 11 of Nelson et al. 2011) in which a change in morphology does initially occur due to growth heterogeneities, but the difference is then lost once stresses are allowed to relax. Some form of viscous relaxation is almost certainly present in development of the colorectal crypt, and incorporating such effects in our framework is the subject of ongoing work.
4.3.1 The role of extensibility
It is important to note that many of the above trends are reliant on the assumption of rod extensibility. In an inextensible model, axial compression is not permitted, as the arclength is fixed, which is equivalent to the geometric constraint \(\alpha \equiv 1\). For explicit comparison, we consider the same stiffness heterogeneity in an inextensible rod. Consequently, only bending is affected by the heterogeneity (40). In Fig. 6, we compare the shape evolution with increasing \(\widehat{\epsilon }\) in both inextensible and extensible models, for \(\xi (S_0) = \cos (\pi S_0/\widehat{L}_0)\) and \(\xi (S_0) = \cos (2\pi S_0/\widehat{L}_0)\). For the inextensible models, we take an equivalent foundation stiffness, \(\widehat{k} = 0.04\), but set \(\gamma = 1.1\) to obtain a similar initial amplitude. In an inextensible rod, the arclength is fixed and thus the response to heterogeneity is to alter the shape towards aligning points of minimal and maximal curvature with material points of maximal and minimal stiffness, respectively. Hence in Fig. 6a the inextensible rod shifts to have maximal amplitude on the soft region on the right side, whereas the extensible rod (Fig. 6b) compresses on the right side, thus producing a completely different morphology with minimal amplitude. In Fig. 6c, the inextensibility leads to a localisation of curvature in the soft middle region, as opposed to the strong compression in the extensible case, shown in Fig. 6d. These simulations illustrate the dramatic effect that extensibility can have on shape morphology and the response to material heterogeneity.
4.3.2 Foundation imperfection
In Fig. 8, we compare the net axial stress with increasing growth for each of the four heterogeneities considered. In each case we have imposed \(\xi =\hat{y}\), and due to the different nature of the perturbation schemes, we have chosen the scale factors such that the total perturbation from the uniform state is equivalent across the four cases. The perfect buckling case appears as the solid blue line, with the sharp cusp appearing at \(\gamma ^*\) and signifying that buckling occurs at a critical compressive stress, which is relieved partially through the buckling. Each of the heterogeneities produces a similar curve, though we see that the additive heterogeneity in foundation shape just considered has the most significant effect, followed by growth heterogeneity. Both foundation and stiffness heterogeneity follow the perfect case very closely; zooming in on the cusp region (see inset) shows that these forms lead to a delayed bifurcation, and, in the case of foundation stiffness, the bifurcation occurs at slightly larger stress before subsequently compensating and dipping below the perfect case.
5 Discussion
We have investigated the buckling and postbuckling behaviour of a planar morphoelastic rod attached to an elastic foundation. We extended the original linear stability analysis by Moulton et al. (2013) by conducting a weakly nonlinear analysis, complemented with numerical solutions of the full, nonlinear model. We first considered a homogeneous setting, and then explored the effect of heterogeneity in material parameters.
In the homogeneous case, we obtained a classic pitchfork bifurcation, with buckling occurring at a critical growth. The nature of the bifurcation (its location and type—supercritical or subcritical) could be characterised via two dimensionless parameters, one (\(\widehat{L}_0\)) relating to length of the finite rod, and another (\(\widehat{k}\)) comparing the relative stiffness of foundation and rod. Increasing length was found to destabilise the rod, causing bifurcation at a smaller value of growth and with increased mode number. Increasing the foundation stiffness, on the other hand, stabilises the rod, increasing the critical growth and the mode number. The influence of foundation stiffness on the buckling mode and the onset of instability shows how variations in system parameters, even in a heterogeneous setting, can have a dramatic impact on the resulting morphology. Such results may have strong relevance in biological systems, where the precise form of the structure, e.g. number of folds, is crucial to functionality. A telling example is in the gyrification of the brain, where deviations in the developmental timing or degree of cortical folding have a severe neurological impact (Goriely et al. 2015).
The general trends we have found in the homogeneous case are consistent with previous analyses of a similar nature, e.g. O’Keeffe et al. (2013). The type of bifurcation, however, was nonstandard: the boundary between supercritical and subcritical bifurcations exhibited an unexpected complexity. In a biological context, where monotonically increasing growth is a natural driver of the formation and subsequent evolution of spatial patterns, this transition has critical importance, signifying where a smooth shape evolution (supercritical) can be expected as opposed to a discontinuous jump from a flat state (subcritical). While further work is needed to establish how prevalent the latter may be, the appearance of structural patterns that arise rapidly, such as the sharp spines in certain mollusc seashells (Chirat et al. 2013) that appear directly adjacent to a flat portion of shell, may point to subcritical regimes. Here the effect of a finite domain is also apparent, as the complexity of the transition becomes less pronounced as \(\widehat{L}_0\) increases.
Multiplicative heterogeneity with respect to three different material properties was then considered: the foundation stiffness, the rod stiffness, and growth. A modified weakly nonlinear analysis showed that in each case the heterogeneity served to translate the bifurcation point, but did not alter its nature. Explicit relations for the shift in bifurcation allowed us to determine how the form of the heterogeneity influences the direction and degree of the translation. For example, the simplest relation appeared with heterogeneity in the foundation stiffness, in which case the greatest effect occurs when the heterogeneity is aligned with the square of the buckling mode. This reflects the intuitive notion that weakening the foundation attachment in regions where the uniform rod deforms maximally has the strongest impact.
Having examined the effect of heterogeneity on postbuckled shape, two natural and related questions follow from this: (i) can one tailor the heteregeneities to achieve a desired shape? And (ii) given a particular shape, can one infer the form and type of any material heterogeneity present? These questions, with significant relevance both from morphogenetic and tissue engineering perspectives, are related to the mathematical inverse problem. Such a problem is inherently complex, as the shapes considered are (partial) solutions of a highorder nonlinear boundary value problem, only achieved in the forward direction through numerical path continuation. In order to develop some intuition, here we provide a simple but illustrative example: we take a candidate shape with embedded heterogeneity—foundation heterogeneity following Fig. 5b—and we try to match that shape, in a trialanderror manner, by varying the heterogeneity in either the rod stiffness or the growth (as well as the net growth), guided by the results of Sect. 4.3; we then consider characteristics other than the shape itself and seek distinctive differences, i.e. signatures of the heterogeneities (more details provided in “Appendix F”).
The result of this exercise is summarised in Fig. 9. In Fig. 9a we plot the ‘matched’ shapes; clearly the match is imperfect, highlighting already the nontrivial nature of tailoring the heterogeneity to achieve a specific shape. Figure 9b shows the axial stress \(n_3\) in the outer region. We observe that growth heterogeneity produces disparate regions of tension, \(n_3 > 0\) (where growth is reduced), and compression, \(n_3 < 0\) (where growth is increased). In contrast, the rod remains in compression for foundation and rod stiffness. In distinguishing foundation stiffness heterogeneity, the foundation energy density \(\mathrm {U}^\mathrm {F}\), defined explicitly in “Appendix E”, provided the clearest indicator. Figure 9c plots \(\mathrm {U}^\mathrm {F}\) in the middle section of the rod, where the shapes are qualitatively most similar, and we find a significant decrease in the case of foundation heterogeneity.
In a thought experiment where the morphology is given and the task is to determine the heterogeneity, these differences could in principle be detected by cutting experiments that release residual stress, as is done for instance in arteries (Chuong and Fung 1986) and solid tumours (Stylianopoulos et al. 2012). However, while this example suggests the possibility of distinguishing between forms of heterogeneity and using heterogeneity to tailor properties, it is clear that this is not a straightforward problem, and a more rigorous treatment would be needed to reach firm conclusions. Moreover, we observed no features that clearly distinguished the case of rod stiffness heterogeneity from the other two heterogeneities. In a 3D setting, more measurable quantities are available, for example, stress in transverse directions, which could potentially yield measurable differences in behaviour. On the other hand, the general complexity of the inverse problem will increase as the number of variables increases. In any case, modelling studies and computational and/or analytical results such as those provided by a weakly nonlinear analysis can provide important insights in a tissue engineering context, e.g. determining the right ‘ingredients’ to generate desired tissue morphologies; as well as for building intuition for how different regions of a heterogeneous elastic tissue with evolving material properties will behave. For instance, this is of particular relevance in brain injury, where morphological heterogeneities have crucial influence in understanding the deformation response to injury (Goriely et al. 2015), and in intestinal tissue health, where deformation plays a significant role in facilitating wound healing (Seno et al. 2009) and tumour expansion (Preston et al. 2003).
In the final section, we have examined a fourth type of heterogeneity, with a view to establishing why the impact of heterogeneity on the bifurcation itself was relatively minor in the previous scenarios. Here we made the key distinction between multiplicative and additive heterogeneity. A multiplicative heterogeneity appears in a term that multiplies dependent variables in the system; due to the nature of the perturbative expansion, such terms only serve to shift the pitchfork bifurcation. An additive heterogeneity, for which a perturbation is applied to a term that does not multiply dependent variables, can have a significant effect, creating an imperfect bifurcation (broken pitchfork) and creating a larger deviation from the perfect, homogeneous, case. Here we considered an imperfectlystraight foundation, and showed that the effect is maximal when the form of the imperfection matches that of the buckling mode.
In this paper, we have assumed each model parameter to be independent from the others and, for the sake of clarity, varied each parameter in isolation. A natural extension would be to consider the combined effect of several simultaneouslyvarying parameters via interparameter coupling. For example, one could consider a rod with nonuniform stiffness and a growth evolution law that depends on axial stress. This would naturally induce heterogeneity in multiple parameters, and the resultant competing effects would likely produce a complex solution space. Another natural extension is to consider nonlinear constitutive effects. There would certainly be benefit to considering a nonlinear constitutive relation between axial stress and the elastic stretch \(\alpha \), in particular because many of our simulations featured significant compression potentially beyond the threshold for quantitative validity of the Hooke’s law considered here. Another useful extension is to incorporate nonlinearity in the response of the foundation, a phenomenon that has been studied in great detail in systems without growth (Hutchinson 2013), but whose role in the context of growth remains unclear.
Finally, we note that many of our modelling choices were motivated by observations on the intestinal crypt (and other, physiologically similar structures). Thus while the model represents an idealised version of a crypt, there are several extensions that would render it biologically realistic. For instance, our growth parameter contains no information about the timescale of growth, which is a fundamental aspect of many biological systems, particularly the crypt. Therefore, one could introduce timedependent growth or timedependent mechanical relaxation (for example in the foundation), allowing remodelling to occur over time. Alternatively, the proliferative structure within a crypt suggests the spatial form of the growth stretch should be bimodal (Alberts et al. 2002; Wright and Alison 1984). In the context of mechanosensitive growth, Miyoshi et al. (2012) showed that a specific subset of stromal cells is activated during wound healing to increase stem cell proliferation in the crypt, as one example. The crypt also provides a natural setting to investigate possible feedback mechanisms between growth and the underlying foundation; this work is currently underway.
Footnotes
 1.
Here we refer to nonuniform growth along the axis of the buckling tissue, not the differential growth that is assumed to occur between layers.
 2.
Here we follow standard terminology (Goriely 2016) in referring to \(\gamma \) as the growth ‘stretch’, the rationale being that growth acts to ‘stretch’ the arclength by adding new material. That is, the word ‘stretch’ does not refer to a ‘stretching’ of old material, but rather an increase in reference arclength by the addition of new material.
 3.
In this case heterogeneity is incorporated prior to nondimensionalization, and scaling proceeds using the baseline value. Note also that we do not vary the Young’s modulus E present in the definition of the foundation force (7), so that we may distinguish the material properties in the foundation from material properties of the rod itself.
 4.
Setting \(\xi (S_0) = \widehat{y}(S_0)\) biases the buckled rod to the lower amplitude branch, as seen by setting \(D_1=1\). (Note that numerical continuation, increasing \(\gamma \) from the flat rod state, cannot be used to detect the split branches.)
Notes
Acknowledgements
This work was supported by Cancer Research UK (CRUK) Grant Number C5255/A23225, through a Cancer Research UK Oxford Centre Prize DPhil Studentship. PKM would like to thank the Mathematical Biosciences Institute (MBI) at Ohio State University, for partially supporting this research. MBI receives its funding through the National Science Foundation Grant DMS1440386. The authors thanks A. Goriely for useful discussions.
Supplementary material
References
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