publications

2024

1. 

   Elastocapillarity-driven surface growth in tumour spheroids

   Davide Riccobelli

   arXiv preprint arXiv:2410.03344, 2024

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   Growing experimental evidence highlights the relevant role of mechanics in the physiology of solid tumours, even in their early stages. While most of the mathematical models describe tumour growth as a volumetric increase of mass in the bulk, in vitro experiments on tumour spheroids have demonstrated that cell proliferation occurs in a thin layer at the boundary of the cellular aggregate. In this work, we investigate how elasticity and surface tension interact during the development of tumour spheroids. We model the tumour as a hyperelastic material undergoing boundary accretion, where the newly created cells are deformed by the action of surface tension. This growth leads to a frustrated reference configuration, resulting in the appearance of residual stress. Our theoretical framework is validated through experimental results of tumour spheroid cutting. Similar to fully developed tumours, spheroids tend to open when subject to radial cuts. Remarkably, even newly formed spheroids, which lack residual stress, exhibit this behaviour. Through both analytical solutions and numerical simulations, we show that this phenomenon is driven by elastocapillary interactions, where the residual stress developed in grown spheroids amplifies the tumour opening. Our model’s outcomes align with experimental observations and allow us to estimate the surface tension acting on tumour spheroids.

   @article{riccobelli2024elastocapillarity,
     author = {Riccobelli, Davide},
     journal = {arXiv preprint arXiv:2410.03344},
     title = {Elastocapillarity-driven surface growth in tumour spheroids},
     year = {2024}
   }

2. 

   Modelling of initially stressed solids: structure of the energy density in the incompressible limit

   Marco Magri, and Davide Riccobelli

   SIAM Journal on Applied Mathematics, 2024

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   This study addresses the modelling of elastic bodies, particularly when the relaxed configuration is unknown or non-existent. We adopt the theory of initially stressed materials, incorporating the deformation gradient and stress state of the reference configuration (initial stress tensor) into the response function. We show that for the theory to be applicable, the response function of the relaxed material is invertible up to an element of the material symmetry group. Additionally, we establish that commonly imposed constitutive restrictions, namely the initial stress compatibility condition and initial stress reference independence, naturally arise when assuming an initial stress generated solely from elastic distortion. The paper delves into modelling aspects concerning incompressible materials, showcasing the expressibility of strain energy density as a function of the deviatoric part of the initial stress tensor and the isochoric part of the deformation gradient. This not only reduces the number of independent invariants in the energy functional, but also enhances numerical robustness in finite element simulations. The findings of this research hold significant implications for modelling materials with initial stress, extending potential applications to areas such as mechanobiology, soft robotics, and 4D printing.

   @article{magri2024modelling,
     author = {Magri, Marco and Riccobelli, Davide},
     journal = {SIAM Journal on Applied Mathematics},
     title = {Modelling of initially stressed solids: structure of the energy density in the incompressible limit},
     volume = {accepted},
     year = {2024}
   }

3. 

   Elastic Instability behind Brittle Fracture

   Davide Riccobelli, Pasquale Ciarletta, Guido Vitale, Corrado Maurini, and Lev Truskinovsky

   Physical Review Letters, 2024

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   We argue that nucleation of brittle cracks in initially flawless soft elastic solids is preceded by a continuum instability which cannot be captured without accounting for geometrically and physical nonlinearities of the constitutive response. To corroborate this somewhat counterintuitive claim, we present a theoretical and numerical study of the simplest model where a homogeneous elastic body subjected to tension is weakened by a free surface which then serves as a site of crack nucleation. We show that in this prototypical setting, brittle fracture starts as a symmetry breaking elastic instability activated by softening and involving large elastic rotations. The implied bifurcation of the homogeneous elastic equilibrium is highly unconventional due to its extraordinary sensitivity to geometry, reminiscent of the transition to turbulence. We trace the development of the instability beyond the limits of continuum elasticity by using quasi-continuum theory allowing one to capture the ultimate strain localization indicative of the formation of actual cracks.

   @article{riccobelli2024fracture,
     author = {Riccobelli, Davide and Ciarletta, Pasquale and Vitale, Guido and Maurini, Corrado and Truskinovsky, Lev},
     doi = {10.1103/PhysRevLett.132.248202},
     issue = {24},
     journal = {Physical Review Letters},
     pages = {248202},
     publisher = {American Physical Society},
     title = {Elastic Instability behind Brittle Fracture},
     volume = {132},
     year = {2024}
   }

4. 

   Reconstructing relaxed configurations in elastic bodies: Mathematical formulation and numerical methods for cardiac modeling

   Nicolas A. Barnafi, Francesco Regazzoni, and Davide Riccobelli

   Computer Methods in Applied Mechanics and Engineering, 2024

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   Modeling the behavior of biological tissues and organs often necessitates the knowledge of their shape in the absence of external loads. However, when their geometry is acquired in-vivo through imaging techniques, bodies are typically subject to mechanical deformation due to the presence of external forces, and the load-free configuration needs to be reconstructed. This paper addresses this crucial and frequently overlooked topic, known as the inverse elasticity problem (IEP), by delving into both theoretical and numerical aspects, with a particular focus on cardiac mechanics. In this work, we extend Shield’s seminal work to determine the structure of the IEP with arbitrary material inhomogeneities and in the presence of both body and active forces. These aspects are fundamental in computational cardiology, and we show that they may break the variational structure of the inverse problem. In addition, we show that the inverse problem might have no solution even in the presence of constant Neumann boundary conditions and a polyconvex strain energy functional. We then present the results of extensive numerical tests to validate our theoretical framework, and to characterize the computational challenges associated with a direct numerical approximation of the IEP. Specifically, we show that this framework outperforms existing approaches both in terms of robustness and optimality, such as Sellier’s iterative procedure, even when the latter is improved with acceleration techniques. A notable discovery is that multigrid preconditioners are, in contrast to standard elasticity, not efficient, where a one-level additive Schwarz and generalized Dryja–Smith–Widlund provide a much more reliable alternative. Finally, we successfully address the IEP for a full-heart geometry, demonstrating that the IEP formulation can compute the stress-free configuration in real-life scenarios where Sellier’s algorithm proves inadequate.

   @article{barnafi2024reconstructing,
     author = {Barnafi, Nicolas A. and Regazzoni, Francesco and Riccobelli, Davide},
     doi = {10.1016/j.cma.2024.116845},
     journal = {Computer Methods in Applied Mechanics and Engineering},
     pages = {116845},
     title = {Reconstructing relaxed configurations in elastic bodies: Mathematical formulation and numerical methods for cardiac modeling},
     volume = {423},
     year = {2024}
   }

5. Oberwolfach Report

   Buckling behind brittle fracture in soft solids

   Davide Riccobelli

   In Fracture as an Emergent Phenomenon , 2024

2023

1. 

   Flattened and wrinkled encapsulated droplets: Shape-morphing induced by gravity and evaporation

   Davide Riccobelli, Hedar H. Al-Terke, Päivi Laaksonen, Pierangelo Metrangolo, Arja Paananen, Robin H. A. Ras, Pasquale Ciarletta, and Dominic Vella

   Physical Review Letters, 2023

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   We report surprising morphological changes of suspension droplets (containing class II hydrophobin protein HFBI from Trichoderma reesei in water) as they evaporate with a contact line pinned on a rigid solid substrate. Both pendant and sessile droplets display the formation of an encapsulating elastic film as the bulk concentration of solute reaches a critical value during evaporation, but the morphology of the droplet varies significantly: for sessile droplets, the elastic film ultimately crumples in a nearly flattened area close to the apex while in pendant droplets, circumferential wrinkling occurs close to the contact line. These different morphologies are understood through a gravito-elastocapillary model that predicts the droplet morphology and the onset of shape changes, as well as showing that the influence of the direction of gravity remains crucial even for very small droplets (where the effect of gravity can normally be neglected). The results pave the way to control droplet shape in several engineering and biomedical applications.

   @article{riccobelli2023flattened,
     author = {Riccobelli, Davide and Al-Terke, Hedar H. and Laaksonen, P{\"a}ivi and Metrangolo, Pierangelo and Paananen, Arja and Ras, Robin H. A. and Ciarletta, Pasquale and Vella, Dominic},
     doi = {10.1103/PhysRevLett.130.218202},
     journal = {Physical Review Letters},
     number = {21},
     pages = {218202},
     title = {Flattened and wrinkled encapsulated droplets: Shape-morphing induced by gravity and evaporation},
     volume = {130},
     year = {2023}
   }

2. 

   Tunable morphing of electroactive dielectric-elastomer balloons

   Yipin Su, Davide Riccobelli , Yingjie Chen, Weiqiu Chen, and Pasquale Ciarletta

   Proceedings of the Royal Society A, 2023

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   Designing smart devices with tunable shapes has important applications in industrial manufacture. In this paper, we investigate the nonlinear deformation and the morphological transitions between buckling, necking and snap-through instabilities of layered dielectric elastomer (DE) balloons in response to an applied radial voltage and an inner pressure. We propose a general mathematical theory of nonlinear electro-elasticity able to account for finite inhomogeneous strains provoked by the electro-mechanical coupling. We investigate the onsets of morphological transitions of the spherically symmetric balloons using the surface impedance matrix method. Moreover, we study the nonlinear evolution of the bifurcated branches through finite-element numerical simulations. Our analysis demonstrates the possibility to design tunable DE spheres, where the onset of buckling and necking can be controlled by geometrical and mechanical properties of the passive elastic layers. Relevant applications include soft robotics and mechanical actuators.

   @article{su2023tunable,
     author = {Su, Yipin and Riccobelli, Davide and Chen, Yingjie and Chen, Weiqiu and Ciarletta, Pasquale},
     doi = {10.1098/rspa.2023.0358},
     journal = {Proceedings of the Royal Society A},
     number = {2276},
     pages = {20230358},
     publisher = {The Royal Society},
     title = {Tunable morphing of electroactive dielectric-elastomer balloons},
     volume = {479},
     year = {2023}
   }

2022

1. 

   The Föppl–von Kármán equations of elastic plates with initial stress

   Pasquale Ciarletta, Giulia Pozzi, and Davide Riccobelli

   Royal Society Open Science, 2022

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   Initially stressed plates are widely used in modern fabrication techniques, such as additive manufacturing and UV lithography, for their tunable morphology by application of external stimuli. In this work, we propose a formal asymptotic derivation of the Föppl-von Kármán equations for an elastic plate with initial stresses, using the constitutive theory of nonlinear elastic solids with initial stresses under the assumptions of incompressibility and material isotropy. Compared to existing works, our approach allows to determine the morphological transitions of the elastic plate without prescribing the underlying target metric of the unstressed state of the elastic body. We explicitly solve the derived FvK equations in some physical problems of engineering interest, discussing how the initial stress distribution drives the emergence of spontaneous curvatures within the deformed plate. The proposed mathematical framework can be used to tailor shape on demand, with applications in several engineering fields ranging from soft robotics to 4D printing.

   @article{ciarletta2022foppl,
     author = {Ciarletta, Pasquale and Pozzi, Giulia and Riccobelli, Davide},
     doi = {10.1098/rsos.220421},
     journal = {Royal Society Open Science},
     number = {5},
     pages = {220421},
     publisher = {The Royal Society},
     title = {The {F}{\"o}ppl--von {K}{\'a}rm{\'a}n equations of elastic plates with initial stress},
     volume = {9},
     year = {2022}
   }

2. 

   Mathematical modelling of axonal cortex contractility

   Dario Andrini, Valentina Balbi, Giulia Bevilacqua, Giulio Lucci, Giulia Pozzi, and Davide Riccobelli

   Brain Multiphysics, 2022

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   The axonal cortex is composed of a regular structure of F-actin and spectrin able to contract thanks to myosin II motors. Such an active tension is of fundamental importance in controlling the physiological shape of axons. Recent experiments show that axons modulate the contraction of the cortex when subject to mechanical deformations, exhibiting a non-trivial coupling between the hoop and the axial active tension. However, the underlying mechanisms are still poorly understood. In this paper, we propose a continuum model of the axon based on the active strain theory. By using the Coleman–Noll procedure, we shed light on the coupling between the hoop and the axial active strain through the Mandel stress tensor. We propose a qualitative analysis of the system under the simplifying assumption of incompressibility, showing the existence of a stable equilibrium solution. In particular, our results show that the axon regulates the active contraction to maintain a homeostatic stress state. Finally, we propose numerical simulations of the model, using a more suitable compressible constitutive law. The results are compared with experimental data, showing an excellent quantitative agreement.

   @article{andrini2022mathematical,
     author = {Andrini, Dario and Balbi, Valentina and Bevilacqua, Giulia and Lucci, Giulio and Pozzi, Giulia and Riccobelli, Davide},
     doi = {10.1016/j.brain.2022.100060},
     journal = {Brain Multiphysics},
     pages = {100060},
     publisher = {Elsevier},
     title = {Mathematical modelling of axonal cortex contractility},
     volume = {3},
     year = {2022}
   }

2021

1. 

   Active elasticity drives the formation of periodic beading in damaged axons

   Davide Riccobelli

   Physical Review E, 2021

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   In several pathological conditions, such as coronavirus infections, multiple sclerosis, Alzheimer’s and Parkinson’s diseases, the physiological shape of axons is altered and a periodic sequence of bulges appears. Experimental evidences suggest that such morphological changes are caused by the disruption of the microtubules composing the cytoskeleton of the axon. In this paper, we develop a mathematical model of damaged axons based on the theory of continuum mechanics and nonlinear elasticity. The axon is described as a cylinder composed of an inner passive part, called axoplasm, and an outer active cortex, composed mainly of F-actin and able to contract thanks to myosin-II motors. Through a linear stability analysis we show that, as the shear modulus of the axoplasm diminishes due to the disruption of the cytoskeleton, the active contraction of the cortex makes the cylindrical configuration unstable to axisymmetric perturbations, leading to a beading pattern. Finally, the nonlinear evolution of the bifurcated branches is investigated through finite element simulations.

   @article{PRE_2021,
     author = {Riccobelli, Davide},
     doi = {10.1103/PhysRevE.104.024417},
     journal = {Physical Review E},
     number = {2},
     pages = {024417},
     publisher = {APS},
     title = {Active elasticity drives the formation of periodic beading in damaged axons},
     volume = {104},
     year = {2021}
   }

2. 

   Rods coiling about a rigid constraint: Helices and perversions

   Davide Riccobelli, Giovanni Noselli, and Antonio DeSimone

   Proceedings of the Royal Society A, 2021

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   Mechanical instabilities can be exploited to design innovative structures, able to change their shape in the presence of external stimuli. In this work, we derive a mathematical model of an elastic beam subjected to an axial force and constrained to smoothly slide along a rigid support, where the distance between the rod midline and the constraint is fixed and finite. Using both theoretical and computational techniques, we characterize the bifurcations of such a mechanical system, in which the axial force and the natural curvature of the beam are used as control parameters. We show that, in the presence of a straight support, the rod can deform into shapes exhibiting helices and perversions, namely transition zones connecting together two helices with opposite chirality. The mathematical predictions of the proposed model are also compared with some experiments, showing a good quantitative agreement. In particular, we find that the buckled configurations may exhibit multiple perversions and we propose a possible explanation for this phenomenon based on the energy landscape of the mechanical system.

   @article{Riccobelli_2021,
     author = {Riccobelli, Davide and Noselli, Giovanni and DeSimone, Antonio},
     doi = {10.1098/rspa.2020.0817},
     journal = {Proceedings of the Royal Society A},
     number = {2246},
     pages = {20200817},
     publisher = {The Royal Society},
     title = {Rods coiling about a rigid constraint: Helices and perversions},
     url = {https://doi.org/10.1098%2Frspa.2020.0817},
     volume = {477},
     year = {2021}
   }

2020

1. 

   Surface tension controls the onset of gyrification in brain organoids

   Davide Riccobelli, and Giulia Bevilacqua

   Journal of the Mechanics Physics of Solids, 2020

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   Understanding the mechanics of brain embryogenesis can provide insights on pathologies related to brain development, such as lissencephaly, a genetic disease which cause a reduction of the number of cerebral sulci. Recent experiments on brain organoids have confirmed that gyrification, i.e. the formation of the folded structures of the brain, is triggered by the inhomo-geneous growth of the peripheral region. However, the rheology of these cellular aggregates and the mechanics of lissencephaly are still matter of debate. In this work, we develop a mathematical model of brain organoids based on the theory of morpho-elasticity. We describe them as non-linear elastic bodies, composed of a disk surrounded by a growing layer called cortex. The external boundary is subjected to a tissue surface tension due the intercellular adhesion forces. We show that the resulting surface energy is relevant at the small length scales of brain organoids and significantly affects the mechanics of cellular aggregates. We perform a linear stability analysis of the radially symmetric configuration and we study the post-buckling behaviour through finite element simulations. We find that the process of gyrification is triggered by the cortex growth and modulated by the competition between two length scales: the radius of the organoid and the capillary length due to surface tension. We show that a solid model can reproduce the results of the in-vitro experiments. Furthermore, we prove that the lack of brain sulci in lissencephaly is caused by a reduction of the cell stiffness: the softening of the organoid strengthens the role of surface tension, delaying or even inhibiting the onset of a mechanical instability at the free boundary.

   @article{riccobelli2019surface,
     author = {Riccobelli, Davide and Bevilacqua, Giulia},
     journal = {Journal of the Mechanics Physics of Solids},
     pages = {103745},
     title = {Surface tension controls the onset of gyrification in brain organoids},
     volume = {134},
     year = {2020}
   }

2. 

   Mechanics of axisymmetric sheets of interlocking and slidable rods

   Davide Riccobelli, Giovanni Noselli, Marino Arroyo, and Antonio DeSimone

   Journal of the Mechanics Physics of Solids, 2020

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   In this work, we study the mechanics of metamaterial sheets inspired by the pellicle of Euglenids. They are composed of interlocking elastic rods which can freely slide along their edges. We characterize the kinematics and the mechanics of these structures using the special Cosserat theory of rods and by assuming axisymmetric deformations of the tubular assembly. Through an asymptotic expansion, we investigate both structures that comprise a discrete number of rods and the limit case of a sheet composed by infinitely many rods. We apply our theoretical framework to investigate the stability of these structures in the presence of an axial load. Through a linear analysis, we compute the critical buckling force for both the discrete and the continuous case. For the latter, we also perform a numerical post-buckling analysis, studying the non-linear evolution of the bifurcation through finite elements simulations.

   @article{Riccobelli_2020,
     author = {Riccobelli, Davide and Noselli, Giovanni and Arroyo, Marino and DeSimone, Antonio},
     doi = {10.1016/j.jmps.2020.103969},
     journal = {Journal of the Mechanics Physics of Solids},
     pages = {103969},
     title = {Mechanics of axisymmetric sheets of interlocking and slidable rods},
     url = {https://doi.org/10.1016%2Fj.jmps.2020.103969},
     volume = {141},
     year = {2020}
   }

2019

1. 

   Activation of a muscle as a mapping of stress–strain curves

   Davide Riccobelli, and Davide Ambrosi

   Extreme Mechanics Letters, 2019

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   The mathematical modeling of the contraction of a muscle is a crucial problem in biomechanics. Several different models of muscle activation exist in literature. A possible approach to contractility is the so-called active strain: it is based on a multiplicative decomposition of the deformation gradient into an active contribution, accounting for the muscle activation, and an elastic one, due to the passive deformation of the body. We show that the active strain approach does not allow to recover the experimental stress–stretch curve corresponding to a uniaxial deformation of a skeletal muscle, whatever the functional form of the strain energy. To overcome such difficulty, we introduce an alternative model, that we call mixture active strain approach, where the muscle is composed of two different solid phases and only one of them actively contributes to the active behavior of the muscle.

   @article{riccobelli2018activation,
     author = {Riccobelli, Davide and Ambrosi, Davide},
     journal = {Extreme Mechanics Letters},
     pages = {37--42},
     publisher = {Elsevier {BV}},
     title = {Activation of a muscle as a mapping of stress{\textendash}strain curves},
     volume = {28},
     year = {2019}
   }

2. 

   On the existence of elastic minimizers for initially stressed materials

   Davide Riccobelli, Abramo Agosti, and Pasquale Ciarletta

   Philosophical Transactions of the Royal Society A, 2019

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   A soft solid is said to be initially stressed if it is subjected to a state of internal stress in its unloaded reference configuration. Developing a sound mathematical framework to model initially stressed solids in nonlinear elasticity is key for many applications in engineering and biology. This work investigates the links between the existence of elastic minimizers and the constitutive restrictions for initially stressed materials subjected to finite deformations. In particular, we consider a subclass of constitutive responses in which the strain energy density is taken as a scalar valued function of both the deformation gradient and the initial stress tensor. The main advantage of this approach is that the initial stress tensor belongs to the group of the divergence-free symmetric tensors satisfying the boundary condition in any given reference configuration. However, it is still unclear which physical restrictions must be imposed for the well-posedness of this elastic problem. Assuming that the constitutive response depends on the choice of the reference configuration only through the initial stress tensor, under given conditions we prove the local existence of a relaxed state given by an implicit tensor function of the initial stress distribution. This tensor function is generally not unique, and can be transformed accordingly to the symmetry group of the material at fixed initial stresses. These results allow to extend Ball’s existence theorem of elastic minimizers for the proposed constitutive choice of initially stressed materials.

   @article{riccobelli2018constitutive,
     author = {Riccobelli, Davide and Agosti, Abramo and Ciarletta, Pasquale},
     journal = {Philosophical Transactions of the Royal Society A},
     number = {2144},
     pages = {20180074},
     publisher = {The Royal Society},
     title = {On the existence of elastic minimizers for initially stressed materials},
     volume = {377},
     year = {2019}
   }

3. Ph.D. Thesis

   Mathematical modelling of soft and active matter

   Davide Riccobelli

   Politecnico di Milano , 2019

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   This thesis focuses on the mathematical modelling of soft and active solid matter using continuum mechanics. An elastic body is said to be soft if it can undergo large deformations; it is said to posses an active behaviour when it can rearrange its micro-structure in presence of external stimuli, not necessarily of mechanical nature. Examples of active processes are biological growth or the contraction of dielectric elastomers provoked by an electromagnetic field. The research activities undertaken concerned both analytical and numerical tasks to solve some physical problems in this field. In particular, we focused on: - the constitutive theory of soft materials with initial stresses, - the mathematical modelling of active phenomena in biological matter, - the formation of patterns in soft solids due to a mechanical instability. The thesis is organized as follows. In Chapter 1, we briefly expose some basic notions of non-linear elasticity. We review the fundamental literature on the mathematical modelling of biological growth and muscle contraction, and on an emerging field in mechanics, called morpho-elasticity. In Chapter 2, we investigate the mathematical description of elastic bodies possessing a non-vanishing distribution of initial stress, i.e. the Cauchy stress in the undeformed reference configuration. We provide new mathematical and physical interpretations of the required constitutive restrictions, proving the existence of energy minimizers in the framework of the theory of initially stressed materials. In Chapter 3, we propose new mathematical models of active processes in soft biological matter, particularly focusing on tumour growth and muscular contraction. We show that it is not possible to recover the experimental stress-stretch curve corresponding to a uniaxial deformation of a skeletal muscle using the active strain method, based on a multiplicative decomposition of the deformation gradient. Instead, we propose an alternative model based on a mixture approach, called “mixture active strain”. Moreover, we show that solid tumours behave as growing poroelastic materials, where the growth is modulated by a chemo-mechanical feedback. The results of our model are in very good agreement with both “in-vitro” and “ex-vivo” experimental data. In Chapter 4, we model morpho-elastic phenomena in both living and inert soft matter. First, we investigate the mechanics of tumour capillaries, showing that the incompatible axial growth of the straight vessel can trigger an elastic instability, generating a tortuous shape. Second, we study how residual stresses can induce mechanical instabilities in soft spheres, e.g. in growing tumour masses. Considering several spatial distributions of the residual stress field, we prove that different topological transitions occur in the sphere where the hoop residual stress reaches its maximum compressive value. Third, we show that gravity bulk force can cause an elastic instability in soft elastic bilayers. We show that the non-linear elastic effects saturate the dynamic instability of the bifurcated solutions that characterize fluid-like matter, displaying a rich morphological diagram where both digitations and stable wrinkling can emerge. Finally, the results of this thesis prove how the combination of nonlinearities and nonconvexity in elastic mixed boundary value problems may emerge as complex physical phenomena, whose understanding requires the development of novel mathematical tools (Chapter 5).

   @phdthesis{riccobellithesis,
     author = {Riccobelli, Davide},
     school = {Politecnico di Milano},
     title = {Mathematical modelling of soft and active matter},
     year = {2019}
   }

4. 

   A comparison between active strain and active stress in transversely isotropic hyperelastic materials

   Giulia Giantesio, Alessandro Musesti, and Davide Riccobelli

   Journal of Elasticity, 2019

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   Active materials are media for which deformations can occur in absence of loads, given an external stimulus. Two approaches to the modeling of such materials are mainly used in literature, both based on the introduction of a new tensor: an additive stress \mathsfP_\textact in the active stress case and a multiplicative strain \mathsfF_a in the active strain one. Aim of this paper is the comparison between the two approaches on simple shears. Considering an incompressible and transversely isotropic material, we design constitutive relations for \mathsfP_\textact and \mathsfF_a so that they produce the same results for a uniaxial deformation along the symmetry axis. We then study the two approaches in the case of a simple shear deformation. In a hyperelastic setting, we show that the two approaches produce different stress components along a simple shear, unless some necessary conditions on the strain energy density are fulfilled. However, such conditions are very restrictive and rule out the usual elastic strain energy functionals. Active stress and active strain therefore produce different results in shear, even if they both fit uniaxial data. Our results show that experimental data on the stress-stretch response on uniaxial deformations are not enough to establish which activation approach can capture better the mechanics of active materials. We conclude that other types of deformations, beyond the uniaxial one, should be taken into consideration in the modeling of such materials.

   @article{giantesio2017comparison,
     author = {Giantesio, Giulia and Musesti, Alessandro and Riccobelli, Davide},
     journal = {Journal of Elasticity},
     number = {1},
     pages = {63--82},
     publisher = {Springer Nature},
     title = {A comparison between active strain and active stress in transversely isotropic hyperelastic materials},
     volume = {137},
     year = {2019}
   }

2018

1. 

   Shape transitions in a soft incompressible sphere with residual stresses

   Davide Riccobelli, and Pasquale Ciarletta

   Mathematics and Mechanics of Solids, 2018

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   Residual stresses may appear in elastic bodies, owing to the formation of misfits in the microstructure, driven by plastic deformations or thermal or growth processes. They are especially widespread in living matter, resulting from dynamic remodelling processes aimed at optimizing the overall structural response to environmental physical forces. From a mechanical viewpoint, residual stresses are classically modelled through the introduction of a virtual incompatible state that collects the local relaxed states around each material point. In this work, we employ an alternative approach based on a strain energy function that constitutively depends only on the deformation gradient and the residual stress tensor. In particular, our objective is to study the morphological stability of an incompressible sphere, made of a neo-Hookean material, and subjected to given distributions of residual stresses. The boundary value elastic problem is studied with analytic and numerical tools. Firstly, we perform a linear stability analysis on the prestressed solid sphere using the method of incremental deformations. The marginal stability conditions are given as a function of a control parameter, which is the dimensionless variable that represents the characteristic intensity of the residual stresses. Secondly, we perform finite-element simulations using a mixed formulation in order to investigate the postbuckling morphology in the fully nonlinear regime. Considering different initial distributions of the residual stresses, we find that different morphological transitions occur around the material domain, where the hoop residual stress reaches its maximum compressive value. The loss of spherical symmetry is found to be controlled by the mechanical and geometrical properties of the sphere, as well as the spatial distribution of the residual stress. The results provide useful guidelines for designing morphable soft spheres, for example by controlling residual stresses through active deformations. They finally suggest a viable solution for the nondestructive characterization of residual stresses in soft tissues, such as solid tumours.

   @article{riccobelli2017shape,
     author = {Riccobelli, Davide and Ciarletta, Pasquale},
     journal = {Mathematics and Mechanics of Solids},
     number = {12},
     pages = {1507--1524},
     publisher = {SAGE Publications Sage UK: London, England},
     title = {Shape transitions in a soft incompressible sphere with residual stresses},
     volume = {23},
     year = {2018}
   }

2. 

   Morpho-elastic model of the tortuous tumour vessels

   Davide Riccobelli, and Pasquale Ciarletta

   International Journal of Non-Linear Mechanics, 2018

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   Solid tumours have the ability to assemble their own vascular network for optimizing their access to the vital nutrients. These new capillaries are morphologically different from normal physiological vessels. In particular, they have a much higher spatial tortuosity forcing an impaired flow within the peritumoral area. This is a major obstacle for the efficient delivery of antitumoral drugs. This work proposes a morpho-elastic model of the tumour vessels. A tumour capillary is considered as a growing hyperelastic tube that is spatially constrained by a linear elastic environment, representing the interstitial matter. We assume that the capillary is an incompressible neo-Hookean material, whose growth is modeled using a multiplicative decomposition of the deformation gradient. We study the morphological stability of the capillary by means of the method of incremental deformations superposed on finite strains, solving the corresponding incremental problem using the Stroh formulation and the impedance matrix method. The incompatible axial growth of the straight capillary is found to control the onset of a bifurcation towards a tortuous shape. The post-buckling morphology is studied using a mixed finite element formulation in the fully nonlinear regime. The proposed model highlights how the geometrical and the elastic properties of the capillary and the surrounding medium concur to trigger the loss of marginal stability of the straight capillary and the nonlinear development of its spatial tortuosity.

   @article{riccobelli2018morpho,
     author = {Riccobelli, Davide and Ciarletta, Pasquale},
     journal = {International Journal of Non-Linear Mechanics},
     pages = {1--9},
     publisher = {Elsevier {BV}},
     title = {Morpho-elastic model of the tortuous tumour vessels},
     volume = {107},
     year = {2018}
   }

2017

1. 

   Rayleigh–Taylor instability in soft elastic layers

   Davide Riccobelli, and Pasquale Ciarletta

   Philosophical Transactions of the Royal Society A, 2017

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   This work investigates the morphological stability of a soft body composed of two heavy elastic layers, attached to a rigid surface and subjected only to the bulk gravity force. Using theoretical and computational tools, we characterize the selection of different patterns as well as their nonlinear evolution, unveiling the interplay between elastic and geometric effects for their formation. Unlike similar gravity-induced shape transitions in fluids, as the Rayleigh-Taylor instability, we prove that the nonlinear elastic effects saturate the dynamic instability of the bifurcated solutions, displaying a rich morphological diagram where both digitations and stable wrinkling can emerge. The results of this work provide important guidelines for the design of novel soft systems with tunable shapes, with several applications in engineering sciences.

   @article{riccobelli2017rayleigh,
     author = {Riccobelli, Davide and Ciarletta, Pasquale},
     journal = {Philosophical Transactions of the Royal Society A},
     number = {2093},
     pages = {20160421},
     publisher = {The Royal Society},
     title = {Rayleigh{\textendash}{Taylor} instability in soft elastic layers},
     volume = {375},
     year = {2017}
   }

2. 

   Solid tumors are poroelastic solids with a chemo-mechanical feedback on growth

   Davide Ambrosi, Simone Pezzuto, Davide Riccobelli, Triantafyllos Stylianopoulos, and Pasquale Ciarletta

   Journal of Elasticity, 2017

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   The experimental evidence that a feedback exists between growth and stress in tumors poses challenging questions. First, the rheological properties (the “constitutive equations”) of aggregates of malignant cells are still a matter of debate. Secondly, the feedback law (the “growth law”) that relates stress and mitotic–apoptotic rate is far to be identified. We address these questions on the basis of a theoretical analysis of in vitro and in vivo experiments that involve the growth of tumor spheroids. We show that solid tumors exhibit several mechanical features of a poroelastic material, where the cellular component behaves like an elastic solid. When the solid component of the spheroid is loaded at the boundary, the cellular aggregate grows up to an asymptotic volume that depends on the exerted compression. Residual stress shows up when solid tumors are radially cut, highlighting a peculiar tensional pattern. By a novel numerical approach we correlate the measured opening angle and the underlying residual stress in a sphere. The features of the mechanobiological system can be explained in terms of a feedback of mechanics on the cell proliferation rate as modulated by the availability of nutrient, that is radially damped by the balance between diffusion and consumption. The volumetric growth profiles and the pattern of residual stress can be theoretically reproduced assuming a dependence of the target stress on the concentration of nutrient which is specific of the malignant tissue.

   @article{ambrosi2017solid,
     author = {Ambrosi, Davide and Pezzuto, Simone and Riccobelli, Davide and Stylianopoulos, Triantafyllos and Ciarletta, Pasquale},
     journal = {Journal of Elasticity},
     number = {1-2},
     pages = {107--124},
     publisher = {Springer Netherlands},
     title = {Solid tumors are poroelastic solids with a chemo-mechanical feedback on growth},
     volume = {129},
     year = {2017}
   }