Mathematical modelling and experiments on the shape morphing of active bodies

by Dario Andrini & Valentina Damioli

Shape transformations are ubiquitous in nature as they are exploited by living organisms to sustain fundamental aspects of life. These shape changes are typically driven by the energy that the very special architecture of biological tissues can extract from the surroundings. This feature is commonly referred to as internal activity, a concept that has recently been extended to designate those materials that are capable of drawing non-mechanical energy from the environment and converting it into mechanical energy. These special properties have been recently exploited to design a new generation of materials whose activity pattern can be programmed in space and time, so that shape changes can be attained in a controllable manner. These properties have implications for the emerging field of soft robotics, where soft active materials are employed in the design of novel actuation and sensing devices, often drawing inspiration from nature. Besides its scientific relevance per se, it is then clear that a deeper understanding of how nature exploits activity has a strong impact on technology through a process of reverse engineering. Moreover, understanding the role of activity in the definition of the mechanical properties and morphology of biological tissues has important implications for the study of pathological conditions.

Planar morphing vs three-dimensional morphing

Many studies focused on the morphing of thin active plates due to their capability to attain large shape transitions. Indeed, the thinness of such structures allows planar geometrical incompatibilities to produce out-of-plane deformations thus resulting in non-flat complex shape transformations. By planar geometrical incompatibility, we mean a condition in which the rest distances between material points, namely the distances ensuring the absence of mechanical stress, are not compatible with a flat configuration. According to the theory developed by Sharon and coworkers in [1], such rest distances can be encoded in a metric tensor often referred to as target metric. Then, the authors postulate an elastic energy of the body which depends on the difference between the target metric and the actual metric, the latter being the one related to the visible geometric configuration of the body. Specifically, the more similar the target and actual metric are, the lower the accumulated elastic energy, hence the lower the residual stresses. More in detail, such elastic energy is given by the sum of two contributions, one related to in-plane stretches of the plate (depending on the difference between the actual and target metrics) and the other related to the bending deformations.

In the case of very thin plates, bending deformations only marginally contribute to the total elastic energy accumulation. Hence, bending turns out to be energetically convenient with respect to stretching. This fact explains why in thin structures planar incompatibilities are resolved through bending deformations, giving rise to non-flat configurations. The opposite happens for thick (stocky) bodies for which planar stretching is energetically preferable to bending. Hence planar geometrical incompatibilities generally result in a planar morphing in the case of thick structures, while giving rise to non-flat configurations in the case of thin structures. In the following, we will present two instances of morphing in thick and thin structures.

Fig 1. Planar geometric incompatibility. On the left (a) is illustrated a flat circle onto which a set of concentric ones are drawn. According to classic Euclidean geometry, the circumference 𝒞 of such circles is related to their radius r through the formula 𝒞=2𝜋r. Specifically, such a condition is necessary to guarantee the flatness of the disk. Instead, on the right (b) the circumference of the red circles satisfies the condition 𝒞<2𝜋r, that is incompatible with a flat configuration. Mathematically, the notion of distances between points is encoded into the so-called metric tensor: the disk on the left (a) features a flat metric tensor while the one on the right (b) features a non-flat one. Adapted from [2]).

Planar morphing of thick (stocky) bodies

As already mentioned, programming a shape transformation of an active structure requires either the control of the spatial distribution of the material architecture or the external stimuli triggering the active response [3,4]. For instance, in hydrogels, the amount of swelling can be controlled by locally prescribing the degree of polymerisation of the matrix. Regarding liquid crystal elastomers, temperature-driven molecular reorientations cause active strains along mutually orthogonal directions that can be encoded in the material during fabrication. In practice, each active material features a specific active mechanism whose convenience in applications strongly depends on the desired shape change. Then, a relevant and timely question is to devise new strategies and tools for identifying the active mechanism that is most efficient for the attainment of a specific shape change. In [5] we address this question in the case of a tall (i.e., thick) cylindrical body of which we consider only shape changes of its cross section (see Fig. 2). Hence, we restrict to planar morphing since, as already mentioned in the previous section, thick structures prefer to resolve geometrical incompatibilities through planar deformations, thus remaining flat.

In order to select the most effective active mechanism, in [5] we set up an optimal control problem that selects, among the activation patterns producing a prescribed shape change, the one minimising an objective functional.

Fig.2. Kinematics of the shape optimisation problem. Here, 𝜔 is the cross section of the cylindrical body which, thanks to internal activity undergoes a deformation denoted by f. The activity pattern is engineered so that the deformed cross section closely matches a “target” shape that we highlight in orange. Adapted from [5]).

In the following, we will refer to it as the activation complexity functional, since it is designed so that the lower its value is the “simpler” the activation pattern. As regards the description of activity, differently from previous works on the topic ([6,7,8]), we do not restrict the study of a specific material architecture and adopt the target metric as its unifying descriptor. The intent is to develop a design tool that applies to a broad set of materials and may inspire new efficient morphing strategies.

We first focus on the case of shape changes attainable through an affine map (i.e., affine shape changes), for which we characterise analytically some of the optimal solutions, thus providing insight into the problem. Indeed, affine shape changes always guarantee the existence of a spatially homogeneous activation pattern (i.e., a uniform target metric) as an admissible competitor of the optimal control problem.

Finally, we discuss non-affine shape changes by investigating the bending of a rectangular block and analyzing the impact on the optimal solutions of different complexity functionals. Some of them are chosen to demonstrate the compatibility of our approach with some classes of existing materials, such as hydrogels and nematic elastomers. Interestingly, the optimal solutions often feature a non-trivial stress pattern, which turns out to be functional to the reduction of the complexity of the activation (see Fig. 3–4). In our opinion, the relevance of the presented case study is twofold. On the one hand, it shows that the appropriate choice of complexity functional leads to the optimal design of target metrics compatible with a specific material class. On the other hand, it highlights that our computational tool may be employed to devise novel morphing strategies or material architectures, where stresses promote a reduction in the complexity of the controls.

Fig 3. Optimal active bending of a block for an activation complexity functional Inspired by hydrogels. The latter are active materials that typically swell or shrink isotropically in response to an external stimulus. In subfigure (a), T denotes a measure of the residual stress emerging from the incompatibility of the target metric. In subfigures (b)-(c), we report the components of the target metric, namely, the activation pattern. Note that a hydrogel-like behaviour is captured by the optimisation procedure since the target metric closely matches a conformal one. Adapted from [5]).

Fig. 4 . Optimal active bending of a block for an activation complexity functional inspired by nematic elastomers. They are a class of active materials in which macroscopic distortions are caused by temperature-driven molecular re-orientation. In subfigure (a), T denotes a measure of the residual stress emerging from the incompatibility of the target metric. In subfigures (b)-(c), we report the components of the target metric, namely the activation pattern. Note that a bi-layer structure naturally emerges from the optimization procedure as characterised by a region of sharp transition of the target metric components that divides the reference domain into two halves. Adapted from [5]).

To emphasise the applicability of the proposed approach in a more general context, we conclude this section by presenting the numerical results relevant to the shape change depicted in Fig. 5 and concerning the morphing of an ellipse into a complex shape resembling a Batman-like logo.

Fig. 5. Optimal solution for the transformation of an ellipse into a shape resembling a Batman-like logo. Prescribed shape change (a) and norm of the Cauchy stress T. In (b)–(d), the contour plots of the optimal target metric components. Despite the presence of sharp corners, the numerical procedure can achieve the target shape with remarkable accuracy, thus confirming the possibility to deal with extreme shape changes. Adapted from [5]).

Transient shape morphing of thin active gel plates

Inspired by the shape morphing observed in Nature, researchers have developed synthetic replicas of soft and smart materials that mimic some biological features. The arrangements of thin structures, such as leaves, petals, or seed pods [9,10,11], are often based on anisotropic swelling and shrinkage, driven by the variation of water content within the soft, porous tissue. Hydrogels are active crosslinked polymers that can change shape because of water absorption, which can be triggered by non-mechanical stimuli, without dissolving. The fabrication of shape-changing materials needs 4D printing techniques, where the form and function of the material evolve after it is 3D-printed in a programmed manner. In our BioMat laboratory at SISSA, we developed a photolithography-based printing procedure and a mixture protocol that allows repeated programming for reversible shape transformation of polyacrylamide-co-sodium acrylate (PAAm-co-SA) hydrogels by prescription of a target metric [12]. Material properties, such as the hydrogel stiffness, are locally prescribed leading to in-plane incompatibilities and out-of-plane deformations. Researchers have already investigated how to obtain equilibrium configurations [3,13] but understanding transient phases is still uncovered. The intent of our study is to comprehend the complexity of the transient configurations. Thus, free-swelling and shrinking experiments have been carried out for the mechanical characterisation of homogeneous specimens and to explore the time evolution of the configuration of non-homogeneous samples during the transient and at equilibrium, as shown in Fig. 6. The target metric is given by well-designed digital patterns (Fig. 6a), projected on the pre-gel solutions by a UV video-projector designed to reach cone-like shape configurations at equilibrium (Fig. 6b). In our case, the shape transformation is triggered by the variation of the salt concentration in the bathing solution. The study of the transient morphing of heterogeneous gel samples involved three types (I, II, III) of annulus-shaped PAAm-co-SA hydrogels, with their radial patterns set to achieve conical surfaces, with different base angles of 45, 60 and 90 degrees. Experimental results on hydrogel samples are compared with those obtained from numerical analysis based on suitable mathematical models that involve aspects of mechanics and differential geometry [1,14]. It has been observed that, in thin structures, the control of the form is complex, especially at the initial transient phase. Water diffusion at the boundary of the sample cannot be neglected, as it is accountable for the generation of non-axisymmetric shapes determined by mechanical instabilities. These aspects are in good agreement with our mathematical model and computational results (Fig. 7). In the simulations, under the constraint of axisymmetric deformation, the realized metric differs from the target one, a fact that implies the occurrence of stresses at early times. In particular, compressive stresses at the inner and outer edges determine the bifurcation of the hydrogel samples towards the non-axisymmetric shapes observed experimentally. Numerical simulations at the cross-sections unveiled relevant stresses at the edges, shown in Fig. 8 as adapted from [12]).

Fig. 6. (a) Non-homogeneous optical patterns. b) Relevant frames of transient shape evolution during a free swelling experiment of types I, II and III up to 120 min. The scale bar is 10 mm. Adapted from [12]

Fig. 7. Results from the finite element numerical simulations of the transient morphing of samples of type I, II, and III. Adapted from [12]

Fig. 8. Results from the numerical simulations revealing the complex state of stress at the cross-section of the samples (type III) due to the constraint of axisymmetric deformation. Adapted from [12]

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