F Errico

S De Rosa

M Ichchou

F Franco

O Bareille

Dispersion curves of infinite laminate panels through a modal analysis of finite cylinders

Keywords: Wave propagation, Dispersion Curves, Modal Analysis, Composite Structures

This work presents an approach for using a modal analysis on an equivalent finite cylindrical model, to predict the elastic waves in infinite, isotropic or composite, panels. In the description of the infinite paths, an analogy, between the classical topologies of a straight line and a circumference, is exploited and tested. Different aspects, concerning the wave-mode duality and the discretization and the needed radii of curvature, are investigated to frame the problem and test the robustness of the methodology. The analysis presents a well conditioned problem and solution for any propagation wave angle by transforming the original problem into a simple modal analysis.

Introduction

The correct knowledge of how elastic waves freely behave inside a specific structure is mandatory for a correct modelling and analysis of the structure itself. Moreover, the generally frequencydependent wavelengths, to be described in the modelling phase, must be known in advance for a correct discretization of the model. In FEA (Finite Element Analysis) [START_REF] Cook | Concepts and applications of finite element analysis[END_REF], the knowledge of the wavelength is absolutely mandatory for the mesh sizing and for selecting the proper elements in the frequency range of analysis; in SEA, [START_REF] Cremer | Structure-borne sound -structural vibrations and sound radiation at audio frequencies[END_REF], detailed information about the group velocity and the modal density are necessary for the characterization of the specific subsystem and to analyse the energy exchange among them.

For an isotropic and homogeneous panel, three wave types are, present: bending, shear and longitudinal waves. An analytic procedure is available in literature for the corresponding wavenumbers, assuming the Kirchhoff-Love plate theory [START_REF] Love | The small free vibrations and deformation of a thin elastic shell[END_REF][START_REF] Leissa | Vibration of plates[END_REF]. In order to deal with composite laminates, the CLPT (Classical Laminate Plate Theory) [START_REF] Jones | Mechanics of composite materials[END_REF] can be used once the characteristic matrices of the laminate are calculated and is the simplest available theory.

More recently, some authors have proposed alternative methodologies for the calculation of the dispersion curves of more complex composite structures [START_REF] Mace | Finite element prediction of wave motion in structural waveguides[END_REF][START_REF] Shorter | Wave propagation and damping in linear viscoelastic laminates[END_REF][START_REF] Chitnis | Wave propagation in laminated composite plates using higher order theory[END_REF][START_REF] Baz | Spectral nite-element modelling of the longitudinal wave propagation in rods treated with active constrained layer damping[END_REF][START_REF] Chakraborty | A spectral finite element model for wave propagation analysis in laminated composite plate[END_REF]. Among the finite-element based methods, the SFEM (Spectral Finite Element Method) is a wavenumber-based procedure which reformulates the wave propagation problem through a linear algebraic eigenproblem in the wavenumber space, assuming a three-dimensional displacement field within the plate [START_REF] Chitnis | Wave propagation in laminated composite plates using higher order theory[END_REF][START_REF] Baz | Spectral nite-element modelling of the longitudinal wave propagation in rods treated with active constrained layer damping[END_REF][START_REF] Chakraborty | A spectral finite element model for wave propagation analysis in laminated composite plate[END_REF][START_REF] Barbieri | Waveguides of a composite plate by using the spectral finite element approach[END_REF].

The WFEM (Wave Finite Element Method) is also used similarly to the SFEM for obtaining the dispersion curves of homogeneous and periodic structures. The method makes use of the Bloch-Floquet theory, [START_REF] Brillouin | Wave propagation in periodic structures: Electric filters and crystal lattices[END_REF], and analyses the wave propagation in the media imposing the periodicity conditions to a single repetitive cell [START_REF] Brillouin | Wave propagation in periodic structures: Electric filters and crystal lattices[END_REF][START_REF] Waki | Numerical issues concerning the wave and finite element method for free and forced vibrations of waveguides[END_REF][START_REF] Droz | A hybrid wave-mode formulation for the vibroacoustic analysis of 2d periodic structures[END_REF][START_REF] Droz | A reduced formulation for the free-wave propagation analysis in composite structures[END_REF][START_REF] Manconi | Modelling wave propagation in two dimensional structures using finite element analysis[END_REF][START_REF] Mead | Wave propagation in continuous periodic structures: research contributions from southampton[END_REF][START_REF] Morsbol | Elastic wave propagation in curved flexible pipes[END_REF].

Both these procedures are affected by numerical conditioning when an heading angle, different from a few specific values, is imposed. For example, in the case of the WFEM, the polynomial eigenproblem, to be solved, when an general angle is imposed, might turn to a transcendental eigenvalue problem which is characterized by numerical instabilities and eigenvalue tracking issues, [START_REF] Manconi | Modelling wave propagation in two dimensional structures using finite element analysis[END_REF]. Moreover, it is fundamental determining which solutions of the eigenvalue problem are artefacts of the spatial discretisation and which are valid estimates of wavenumbers in the continuous structure, [START_REF] Droz | A hybrid wave-mode formulation for the vibroacoustic analysis of 2d periodic structures[END_REF][START_REF] Chronopoulos | Modeling the response of composite panels by a dynamic stiffness approach[END_REF][START_REF] Chronopoulos | Computing the broadband vibroacoustic response of arbitrarily thick layered panels by a wave finite element approach[END_REF].

The tracking of eigenwaves, performed through a Wave Assurance Criterion, [START_REF] Droz | A hybrid wave-mode formulation for the vibroacoustic analysis of 2d periodic structures[END_REF][START_REF] Magliacano | Validation of the shift cell approach for the modelling of acoustic properties of porous materials embedding periodic inclusions[END_REF], is a timeconsuming task and it does not assure correct and robust results for two dimensional waveguides.

Other issues of the method are associated with the periodicity effects: the solution scheme gives the same result for the wave modes and the frequencies of the propagation constants with period 2π, because of the spatial periodicity, giving rise to aliasing effects, [START_REF] Manconi | Modelling wave propagation in two dimensional structures using finite element analysis[END_REF].

On the other hand, the wave propagation in curved waveguides, i.e. cylinders and cones, is deeply analysed in the literature [START_REF] Mead | Free vibration of a thin cylindrical shell with periodic circumferential stiffeners[END_REF][START_REF] Langley | Wave motion and energy flow in cylindrical shells[END_REF][START_REF] Renno | Calculating the forced response of cylinders and cylindrical shells using the wave and finite element method[END_REF][START_REF] Markus | Axisymmetric and asymmetric wave motion in orthotropic cylinders[END_REF][START_REF] Honarvar | Wave propagation in transversely isotropic cylinders[END_REF]. The propagation of longitudinal and flexural waves in axial-symmetric circular cylindrical shells with periodic circular axial curvature is studied using a finite element method in [START_REF] Ghoshal | Wave propagation in circular cylindrical shells with periodic axial curvature[END_REF]. The waves in thin uniform cylindrical shells, periodically stiffened by uniform circular frames of general cross-section, is analysed in [START_REF] Mead | Free vibration of a thin cylindrical shell with periodic circumferential stiffeners[END_REF]. Modern methods appeared continuously in the literature, for curved structures, both for the free wave propagation and the forced response [START_REF] Liu | Exact characteristic equations for free vibrations of thin orthotropic circular cylindrical shells[END_REF][START_REF] Errico | The modelling of the flow-induced vibrations of periodic at and axial-symmetric structures with a wave-based method[END_REF].

A wave-based methodology for the free and forced analysis of the circumferential wave propagation of axial-symmetric structures, whatever they complexity and tapering, is proposed in [START_REF] Errico | The modelling of the flow-induced vibrations of periodic at and axial-symmetric structures with a wave-based method[END_REF].

Differently, some attempts to use a finite model to gather informations of an infinite waveguide are present in literature, with success in the identification of the periodic structural band-gaps [START_REF] Brun | Asymptotics of eigenfrequencies in the dynamic response of elongated multi-structures[END_REF][START_REF] Brun | Phononic band gap systems in structural mechanics: Finite slender elastic structures and infinite periodic waveguide[END_REF].

The issues of having a correct mathematical formulation for the out-of-resonance wave-mode duality is analysed and discussed by Langley in [START_REF] Langley | Some perspectives on wave-mode duality in sea[END_REF]. It is stated and proved that, at least in the case of a 1D wave propagation, in a resonance condition a mode can be represented using a wave description [START_REF] Langley | Some perspectives on wave-mode duality in sea[END_REF].

To authors knowledge, very few works, dealing with a full representation of the dispersion curves of the periodic (or homogeneous) waveguide, are present in literature and, in this work, an alternative approach is proposed which allows to overcome all the numerical instabilities. The approach produces results for every heading angle, through a simple real modal analysis of a cylindrical equivalent finite element model, performable with any available in-house or commercial code.

The paper is structured as follows: Section 2 gives an overview of all the reference solutions adopted; Section 3 describes the analogy proposed and Section 4 contains the analytic and numerical validations for all the analysed test-cases.

Reference Solutions

In the following section an overview of the adopted reference solutions is given for isotropic beams and plates and for composite laminates.

Isotropic beam and plate

Considering a beam with E as the Young modulus, I the area moment of inertia, ρ the mass density and A the cross-sectional area; The phase wave speed in a flexural beam can be expressed a follows, [START_REF] Leissa | Vibration of plates[END_REF][START_REF] Mead | Wave propagation in continuous periodic structures: research contributions from southampton[END_REF][START_REF]Investigation and assessment of the wave and finite element method for structural waveguides[END_REF]:

c b = EI ρA 1/4 2πf (1)
Similarly a uniform thin and flat plate is here considered, made of an homogeneous material.

From classical thin plate theory, [START_REF] Leissa | Vibration of plates[END_REF], three wave types propagate in the material of thickness h: longitudinal, shear and bending waves. Each of these is associated with the respective wavenumber:

k l , k s , k b . k l = 2πf ρ(1 -ν 2 ) E k s = 2πf 2ρ(1 + ν) E k b = 2πf ρh D 1/4 ( 2
)
where D is the well known flexural stiffness of the plate and ν the Poisson ratio. By using these relationships, any information for a predictive methodology can be gathered. For example, the discretization of the predictive finite element model could be designed to work up to a given excitation frequency, once the wavelengths are known.

The Classical Lamination Theory (CLPT)

The basic analytic equations for a thin composite plate are here summarized. For a laminate, the relations between forces/moments and strain/curvatures can be written as, [START_REF] Jones | Mechanics of composite materials[END_REF][START_REF] Barbieri | Waveguides of a composite plate by using the spectral finite element approach[END_REF]:

N = A 0 + Bγ 0 M I = B 0 + Dγ 0 (3)
where 0 are the strains of the middle plane and γ 0 the curvatures of it.

A, B and D are the matrices which compose the well known matrices which depends on stressstrain relation of each lamina and the chosen layup sequence [START_REF] Jones | Mechanics of composite materials[END_REF][START_REF] Barbieri | Waveguides of a composite plate by using the spectral finite element approach[END_REF].

0 =        ∂u ∂x ∂v ∂y ∂u ∂y + ∂v ∂x        (4)
γ 0 =        - ∂ 2 w ∂x 2 - ∂ 2 w ∂y 2 -2 ∂ 2 w ∂y∂x        (5)
where u and v are the in-plane displacements, while w is the out-of-plane displacement. Following the procedure explained in [START_REF] Jones | Mechanics of composite materials[END_REF], a 3D displacements wave is assumed to propagate along the plate:

     u v w      =      U V W      e j[k(cos(θ)x+sin(θ)y)-ωt] ( 6
)
where θ is the heading angle of the wave and U, V, W the displacement field variables. It is possible to converge to a polynomial problem in k (the wavenumber).

The following expressions can be used to compute the wavenumbers of the laminated waveguide, once the material is chosen, [START_REF] Jones | Mechanics of composite materials[END_REF][START_REF] Barbieri | Waveguides of a composite plate by using the spectral finite element approach[END_REF].

k b (θ, f ) = 2πf ρ s P T DP 1/4 (7) k s (θ, f ) = 2πf ρ s η s (θ) (8)
k l (θ, f ) = 2πf ρ s η l (θ) (9)
where:

P T = cos(θ) 2 sin(θ) 2 2sin(θ) 2 cos(θ) 2 (10)
L =      cos(θ) 0 0 sin(θ) sin(θ) cos(θ)      (11)
η l and η s eigenvalues of the matrix L T AL.

The Proposed Analogy

The method, presented here, is based on the assumption that the free wave propagation in an infinite flat media, i.e. a panel, can be also described considering a free wave propagation along a circumferential path. In fact, in a local reference system which follows the geometrical (circumferential; Fig. 1) path, the waves are free to propagate in analogy to what they do in the case of the equivalent infinite flat structure. The finiteness of the circular structure does not affect the wave propagation since no impedance variations are encountered running in loop along the curved path. Equivalently, this is a way to describe an infinite periodic condition, in a local coordinate system, as resulting from the Bloch-Floquet theorem [START_REF] Brillouin | Wave propagation in periodic structures: Electric filters and crystal lattices[END_REF][START_REF] Mead | Wave propagation in continuous periodic structures: research contributions from southampton[END_REF]. The difference stands in its implicit geometrical definition through a finite circular structure, instead of explicitly defining a propagation relation on a reference line. In a periodic waveguide, the link between a wavefield Γ (displacement, velocity, force, etc.), at two points x and x + ∆ x , is given by complex propagating constants, [START_REF] Brillouin | Wave propagation in periodic structures: Electric filters and crystal lattices[END_REF][START_REF] Mead | Wave propagation in continuous periodic structures: research contributions from southampton[END_REF]:

Γ(x + ∆ x , f ) = Γ(x, f )e j∆xkn(f ) (12)
where k n is the wavenumber associated with the wavetype n and f the frequency. On the other hand, for a circumferential mode of order i, the field in two points is related, similarly, as follows:

Γ(θ + ∆ θ , f ) = Γ(x, f )e j∆ θ i/2πR (13)
where i/2πR is the circumferential wavenumber associated with the mode of order i.

In a set of discrete (natural) frequencies, each specific circumferential mode is representative of the corresponding wavemode and Eq. ( 12) and ( 13) are equivalent. Figure 1 and 2 illustrate the previous concepts.

It is worth to emphasize that the aim of the approach is not related to the analysis of the waves in a cylindrical waveguide. The equivalent circular geometry is here used only to simulate a wave propagation in an infinite domain: in this way, the wave properties in an infinite panel (or beam) can be calculated working on the circumferential wave propagation in a finite circle.

A modal analysis (Eq. 14) of cylinders or rings, built from the reference structure, gives a discrete set of frequencies (natural modal frequencies) where the modal circumferential wavelength represents the wave propagation in the infinite media. The dispersion curves of the laminate can be thus calculated in discrete points.

By using the equation of motion in discrete coordinates and assuming no external forces are applied to the system, the eigenvalue problem can be written as:

[K -ω 2 M]q = 0 ( 14
)
where q is the nodal vector of degrees of freedom (DoFs); K and M are the stiffness and mass matrices. Damping can be modelled by including, in Eq. 14, appropriate complex matrices and/or coefficients. The operation is easily performable using any commercial FE software.

It is useful to remind the role of the modal wavelengths in rings; a purely circumferential mode for a circular structural model is in Fig. 2. Given a certain radii (R), in general, the wavelength associated to a given mode is given by:

λ i = 2πR i ( 15
)
where i is the order of the circumferential mode and can be rawly identified in the number of lobes present in the modal deformed shape.

A key parameter is, here, the modal angular sampling factor, given by the ratio of the modal wavelength and the radii of the circumference: 2π/i. Its importance, within the present method, is discussed in next section.

The modes of a finite structure describe a steady wave condition while the dispersion curves, for any structure, describe a wave propagation [START_REF] Langley | Some perspectives on wave-mode duality in sea[END_REF]. Any wave type propagating in an infinite media, can be described using this analogy. However, to correctly and easily identify the bending, shear and longitudinal wavemodes singularly, the circular waveguide can be constrained, in a Finite Element Solution, in order to show only the desired circumferential wave types. For example, in the case of bending waves, the constraints of the cylindrical domain must allow ony out-of-plane displacements and rotations. Similar considerations are applied for shear and longitudinal waves. In summary, the proposed method is composed by the following sequential passages:

1. Generate a finite element model of the unit cell of the structure to be analysed, as usually done in a SFEM or WFEM framework (Fig. 1(a))

2. Generation of a circular topology using the elementary cell as a base. The radius and the number of elements has to be chosen in accordance to the target frequency band (Fig. 1(b))

3. Perform a modal analysis of the circular structure (Eq. 14)

4. Identify the wave branches and evaluate the circumferential wavenumbers (Eq. 15) of the structural eigenmodes calculated at the previous step 5. Plotting the λ i for each natural frequency and for each branch.

Validations

In the following section the validation of the proposed approach is performed for different test-cases.

Analytic Validation: Infinite Beam and Ring

A first test-case is the one of a beam. This is particularly convenient since both the bending wave speed of an infinite beam and the natural frequencies of a beam-ring can be calculated using analytic formulas. For every ring wavelength λ i , where i is the circumferential order, the natural frequencies for bending, shear and longitudinal modes, respectively f b,i , f s,i and f n,i , are given by Blevins in [START_REF] Blevins | Formulas for natural frequency and mode shape[END_REF]:

f b,i = i(i 2 -1) 2πR 2 i EI ρA 1/4 ; f s,i = √ i 2 + 1 2πR G/ρ; f l,i = √ i 2 + 1 2πR E/ρ; (16)
with E as the Young modulus, G the shear modulus, I the moment of inertia, ρ the mass density and A the cross-sectional area.

In Fig. 3, a beam with rectangular section is used as a test-case and the results obtained with Eq. ( 1) and ( 16) are compared. The results show a very good agreement which validates the analogy proposed. The results introduce a range of validity of the approach starting from a specific value of the modal sampling factor (modal order). The accuracy of the proposed method is high enough starting from the sixth modal order and it is independent on the radii and length of the ring considered, as expected (Fig. 4).

Analytic Validation: Infinite Cylinder Ovaling Modes

Another analytic validation is here proposed, using the natural ovaling modes of an infinite cylinder. The natural frequencies, for this specific modes of the infinite cylinder, can be calculated using the formulas proposed by Blevins in [START_REF] Blevins | Formulas for natural frequency and mode shape[END_REF], here reported for completeness in Eq. ( 17). For every ring wavelength λ i , where i is the circumferential order, the natural frequency f n,i is:

f n,i = h 2πR 2 √ 12 E ρ(1 -ν 2 ) i(i 2 -1) √ i 2 + 1 (17)
In Fig. 5, an infinite plate of thickness h is used as a test-case and the results obtained with Eq.

(1) and ( 17) are compared for bending waves. The results show a very good agreement, validating the analogy proposed also for a higher order model. Moreover, again, the range of validity depends on the modal sampling factor. In Fig. 4, the relative error for this specific case is shown for different radii. The error, in percentage, is independent on the radii of the cylinder and is inferior to 2% starting from the sixth modal order, as found in the previous section.

Numerical Validation: Isotropic Panel

A first validation with an isotropic aluminium material is performed: E = 7.0x10 10 Pa, ρ = 2750 kg/m 3 , ν = 0.33. The analytic solution is used as a reference. In Fig. 6 a comparison of the infinite plate bending wavelength (versus frequency) with the results of a modal analysis on cylinders of different radii is shown. As in the case of the beam, the accuracy of the method is only dependent on the modal angular sampling factor; as soon as it reaches the unitary value, thus the sixth modal order is reached, the relative error of the present method is inferior to 2%. This is illustrated more specifically in the case of laminates, comparing the error for every heading angle.

Depending on the value of R, the region of accuracy moves with the frequency, approaching higher bands when R lowers, and vice-versa, since any modal order shifts with the frequency.

Depending on the discretization adopted, the accuracy method can strongly vary. Two cases are here analysed: fixed radii R (Fig. 7) and fixed element size ∆ X (Fig. 8).

In the first case the question moves to a classic sampling problem. Given a number of elements (N el ), through which the circumference is discretized, assuming, as a rule of thumb, at least four elements per wavelength as an acceptable approximation (six or ten are, in general, preferred), the maximum modal order predicted is given by the integer of the ratio N el /4. The error starts increasing up to modal order N el /2 before diverging (Fig. 9).

On the other hand, keeping the element size fixed and increasing the number of elements means increasing the circumference radii R (Fig. 10). The expected effect is a frequency shift of the validity region, in the wavenumber domain, when moving from one case to the other, as in Fig. 10.

, 0 • , 0 • , 0 • , 90 • ]

Laminated Panel

An heading angle can be specified in order to analyse the wave propagation in a specific direction (Fig. 11). In this solutions scheme, the problem is never ill-conditioned and can be normally solved with a simple real modal analysis, independently on the value of the angle itself. The graphiteepoxy lamina elastic properties, used in the present work, are in Table 1. The thickness of each lamina is 1mm.

In the framework of the present approach, a cut-on wavenumber can be defined; below this value, a cylinder with radii R, has no predictive capability. In general, given a certain radii R the cut-on wavenumber can be defined from the smallest modal order (i = 2): 2/R. This aspect has an impact only for bending waves, as evident in Fig. 12, 13 and 14, where, in addition, the accuracy of the method is shown also for shear and longitudinal wave types. Increasing the size of the radii, this limit moves to lower wavenumbers and frequencies.

As previously stated, the accuracy of the method is only dependent on the modal angular sampling factor, and, when it reaches the unitary value, the accuracy is acceptable as in Fig. 15, where the relative errors are compared. For a modal order superior to the fifth, the relative error is inferior to 2% and keeps lowering for increasing the frequency, as found in all previous test-cases.

The great advantage is in the ease of use and the possibility to arbitrarily choose an heading angle, without taking care about the eventual ill-conditioning of the mathematical problem, [START_REF] Barbieri | Waveguides of a composite plate by using the spectral finite element approach[END_REF][START_REF] Manconi | Modelling wave propagation in two dimensional structures using finite element analysis[END_REF][START_REF] Droz | A hybrid wave-mode formulation for the vibroacoustic analysis of 2d periodic structures[END_REF]. Moreover, the eigenvalue tracking is no more needed since any wave type evolves singularly if specific boundary conditions are imposed to the the finite cylinder before the modal analyses, as previously discussed. In Fig. 16 a pattern of bending wavenumbers is calculated, for a specific Once a modal analysis is performed, the modal shapes can be extracted straightforwardly. A better way to visualize the mode shapes, will be the stretching of the circumferential modes along a rectilinear path. In Fig. 17 a fourth order mode has been chosen and the mode shapes are visualized for bending, shear and longitudinal wave types, along the circular model. On the other hand, in Fig. 18, the same modes are visualised in a rectified model.

Complex Structural Shapes

In order to prove the accuracy of the analogy, even for non-homogeneous structures and large heterogeneity scales, an aluminium-made double-wall panel with mechanical connections is used as a test-case (Fig. 19). A wave finite element method (WFEM) is used for obtaining the reference results [START_REF] Manconi | Modelling wave propagation in two dimensional structures using finite element analysis[END_REF][START_REF] Morsbol | Elastic wave propagation in curved flexible pipes[END_REF][START_REF] Errico | The modelling of the flow-induced vibrations of periodic at and axial-symmetric structures with a wave-based method[END_REF]. Figure 20 shows the comparison among the modes generated by the cylindrical model (R = 0.4 m), built from the elementary cell representing the reference waveguide, and the flat infinite reference waveguide. In this case only bending waves have been analysed. Even in this complex case the agreements is excellent above the sixth modal order.

Conclusions

The proposed analogy proved to be accurate and robust in predicting the dispersion curves of beams and plates through a simple modal analysis of a cylindrical finite element model. The calculations are always well-conditioned for every heading angle chosen, differently from most of the methods present in literature. The results are accurate and the relative error is independent on the cylinder axial length and radii (R). A test on a double-wall panels with mechanical connections shows how the features of the method are applicable also to complex structural shapes.

The predictive region can be easily trimmed using different values of the curvature or, instead, using a high value of R and a fine mesh in order to be able to move from low (low cut-on wavenumber) to high frequencies. The computational cost associated with such meshes, especially for a simple real modal analysis, is not a relevant parameter and , moreover, the numerical conditioning is null.

A generic FE-based code can be used for the purpose and both plate and solid elements have been tested and validated for the accuracy of the results. Again, it must be highlighted that for bending motion, the results start to have an acceptable accuracy (lower than 2%) starting from the fifth modal order. A correct mesh size is required to avoid aliasing problems at higher frequencies.

Further developments might be related to the possibility of predicting also coupled wave-types which arise to higher frequencies and the possibility to include global and through-thickness damping.

A

  Matrix of the extensional stiffnesses of the plate B Matrix of the in-plane/flexural coupling laminate moduli D Matrix of the flexural stiffenesse of the plate K Stiffness matrix M Mass matrix M I Matrix of the moments per unit length N Matrix of the shearing forces per unit length q Nodal vector of degrees of freedom Γ Wavefield linked to the target field variable ρ s Density per unit area 0 Strains of the middle plane of the plate γ 0 Curvatures of the plate λ i Wavelength of the circumferential mode of order i ν

Figure 1 :

 1 Figure 1: The analogy proposed: (a) Free wave propagation along an infinite flat waveguide; (b) equivalence in a circular topology generated from the original model.

Figure 2 :

 2 Figure 2: Purely circumferential bending modes in a circular structure: (a) Circumferential Order = 2; (b) Circumferential Order = 3; (c) Circumferential Order = 4; (d) Circumferential Order = 5.

Figure 3 :

 3 Figure 3: Bending, shear and longitudinal wavemodes wavelength in an infinite beam and the corresponding beamring circumferential modes

Figure 4 :Figure 5 :

 45 Figure 4: Relative error (log-scale) between the bending wavelength in an infinite panel and the modal wavelengths of an infinite cylinder

Figure 6 :

 6 Figure 6: Isotropic uniform plate: infinite panel solution and the circumferential modes of a cylindrical finite structure made of plate elements

Figure 7 :Figure 8 : 32 Figure 9 :

 78329 Figure 7: Comparison for increasing number of elements for a given element size . (a) 32, (b) 80, (c) 300 elements

Figure 10 :

 10 Figure 10: Effect of the mesh: comparison for increasing number of elements for a given element size (isotropic panel). Circumference radii R variable: Fig. 7(a) 0.05 m, Fig. 7(b) 0.13 m, Fig. 7(c) 0.48 m

Figure 11 :Figure 12 :

 1112 Figure 11: Scheme of the laminate heading angle with respect to the wave propagation direction.

Figure 13 :

 13 Figure 13: Dispersion curves of the laminate panel; heading angle 90 • . CLPT results used as a reference.

Figure 14 :

 14 Figure 14: Dispersion curves of the laminate panel; heading angle 45 • . CLPT results used as a reference.

igure 15 :Figure 16 :Figure 17 :

 151617 Figure 16: Pattern of the bending wavenumbers for the laminate plate analysed; 3000 Hz

Figure 18 :

 18 Figure 18: Rectified circumferential modes in a laminate infinite panel; X-Y as cross sectional plane. Fourth circumferential mode -(a) Bending: 16.6 Hz; (b) Shear: 1364 Hz; (c) Longitudinal: 4674 Hz

Figure 20 :

 20 Figure 20: Bending Waves for the double-wall panel with mechanical connection. Numerical comparison between the WFEM and the present analogy.

Table 1 :

 1 Lamina elastic properties and stacking sequence

	E 1 1	E 2 2	G 12	ν 12	Layup
	125 GPa 12.5 GPa 6.89 GPa 0.38 [90 •

Acknowledgements

This project has received funding from the European Unions Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No. 675441. The author would like to gratefully acknowledge everyone involved in the VIPER project. Discussions with Dr. Petrone are gratefully acknowledged.


Martine Ben Amar

Pierre Nassoy

Loïc Le Goff

Loic Legoff

Physics of growing biological tissues: the complex cross-talk between cell activity, growth and resistance

Keywords: biomechanics, biophysics, complexity, mechanics, cellular biophysics morphogenesis, tissue mechanics, tissue growth, active and multi-scale rheology, finite elasticity

Physics of growing biological tissues: the complex cross-talk between cell activity, growth and resistance

Introduction

Shapes in nature, from cauliflower to brain, emerge from the growth of tissues. Growth intrinsically generates strains and stresses. Conversely, through mechanisms of mechano-transduction, stresses regulate growth [START_REF] Mirabet | The role of mechanical forces in plant morphogenesis[END_REF][START_REF] Legoff | Mechanical forces and growth in animal tissues[END_REF]. Understanding ultimately the emergence of shapes, 2019 The Author(s) Published by the Royal Society. All rights reserved. in particular in biological tissues, requires capturing the mechanical properties of tissues that are inherent to living matter, and related to growth. As detailed in this review, there is an important body of work that addresses tissue mechanics from the macroscopic view point, trying to analyse whole organ biomechanics. At the other end of the spectrum, cell-based computer modelling approaches the problem from the microscopic point of view. The questions addressed here are: what is an appropriate mechanical description of growing tissues at an intermediate scale between an individual cell and an organ? How can continuum mechanical theories model heterogeneous structures or the impact of isolated active cells on the mechanics of tissues?

The viewpoint adopted here is that starting with a theory of finite elasticity and refining it by introducing dissipative corrections and mechanical feedbacks could provide a unifying theoretical framework. Indeed, finite elasticity is the theory of large deformations and so concerns soft solids. Usually, this formalism is restricted to the simplest case where internal forces only depend on the deformation [START_REF] Ogden | Non-linear elastic deformations[END_REF][START_REF] Holzapfel | Nonlinear solid mechanics: a continuum approach for engineering[END_REF] but can include growth [START_REF] Goriely | The mathematics and mechanics of biological growth[END_REF], and thus is able to predict shapes adopted by living systems during morphogenesis and embryogenesis. Finite elasticity requires constitutive laws which adjust to the soft body under study and the best choice has always been at the heart of the work of Rivlin. In the context of this thematic issue, the theoretical tools elaborated by Rivlin in the last half century regarding anisotropy [START_REF] Ericksen | Large elastic deformations of homogeneous anisotropic materials[END_REF][START_REF] Rivlin | Stress-deformation relations for anisotropic solids[END_REF] and fibre networks [START_REF] Rivlin | Networks of inextensible cords[END_REF] of rubber are highly relevant to understanding biological tissues. Indeed, he aimed at establishing the most precise constitutive laws for systems which were not completely solid nor completely fluid which is one of the key points of tissue embryogenesis. Thus, it is not surprising that his work serves as a valuable theoretical basis for living systems in their full complexity [START_REF] Saccomandi | Generalised Mooney-Rivlin models for brain tissue: a theoretical perspective[END_REF]. Our aim, here, is not a full review of all the contributions and formalisms developed ever since to describe morphogenetic events but to show better how finite elasticity with growth can apprehend modern models of biology. We illustrate our purpose with four examples which have received attention recently or will receive in the near future. These four examples are fully driven by cells, being either the elementary units of tissues as in epithelia or embedded into tissues. Our challenging perspective is to show that, even in these cases, finite elasticity can offer a valuable analysis.

The paper is organized as follows: §2 presents the basis of microscopic, cell-based models and contrasts it with the continuum approaches at larger scales, with an emphasis on finite elasticity theory. We then expose the challenges to overcome if we want to make finite elasticity a successful description of tissue mechanics and embryogenesis at large. In § §3-6, four selected systems of study are presented in detail: cells embedded in connective tissues, brain tissue, Caenorhabditis elegans embryo epithelial cortex in action and, finally, thin bilayers made of an epithelium and a soft extracellular matrix (ECM). All these systems have been or will be perfect examples of study in the finite elasticity framework. Finally, numerical models well adapted to treat inhomogeneities are presented in §7.

Modeling tissue mechanics at different scales

In the last two decades, numerous efforts have been devoted to the characterization of the material properties of individual cells [START_REF] Kasza | The cell as a material[END_REF]. However, in a living organism, cells are interacting both with neighbouring cells and with the surrounding ECM. Collections of interacting cells may exhibit emergent properties, which do not necessarily mirror the properties of their constitutive elements. This prompts mechanical modelling endeavours at the tissue level.

(a) From cells to tissue: the cell-based, microscopic view Considerable advances were recently made in live imaging technology. Ever-improving confocal microscopes and genetically encoded fluorescent proteins allow us to image entire embryos at a subcellular resolution [START_REF] Mavrakis | Lighting up developmental mechanisms: how fluorescence imaging heralded a new era[END_REF][START_REF] Pantazis | Advances in whole-embryo imaging: a quantitative transition is underway[END_REF]. Imaging combined with laser ablation to infer tension at a subcellular scale [START_REF] Hutson | Forces for morphogenesis investigated with laser microsurgery and quantitative modeling[END_REF][START_REF] Vuong-Brender | The interplay of stiffness and force anisotropies drives embryo elongation[END_REF][START_REF] Amar | Assessing the contribution of active and passive stresses in C. elegans elongation[END_REF] or opto-genetics to turn on/off cell contractility [START_REF] Izquierdo | Guided morphogenesis through optogenetic activation of Rho signalling during early Drosophila embryogenesis[END_REF][START_REF] Guglielmi | An optogenetic method to modulate cell contractility during tissue morphogenesis[END_REF] to tissue scale mechanics. Investigation of developmental processes from this microscopic point of view has been done in systems such as Drosophila gastrulation and metamorphosis [START_REF] Bertet | Myosin-dependent junction remodelling controls planar cell intercalation and axis elongation[END_REF][START_REF] Zallen | Patterned gene expression directs bipolar planar polarity in Drosophila[END_REF][START_REF] Aigouy | Cell flow reorients the axis of planar polarity in the wing epithelium of Drosophila[END_REF][START_REF] Bosveld | Mechanical control of morphogenesis by Fat/Dachsous/Four-jointed planar cell polarity pathway[END_REF], zebrafish gastrulation [START_REF] Behrndt | Forces driving epithelial spreading in zebrafish gastrulation[END_REF] or chick neural-tube closure [START_REF] Nishimura | Planar cell polarity links axes of spatial dynamics in neural-tube closure[END_REF]-to name a few examples. This cell-centric approach of tissue mechanics has stressed the importance of actomyosin cortex contractility in tissue mechanics, as well as cell-cell adhesion. Studies were often conducted on simple monolayered epithelia and, most of the time, emphasized the role of cortex mechanics at apical junctions-although some recent studies now stress the role of basal mechanics and ECM in shaping three-dimensional tissues [START_REF] Daley | ECM-modulated cellular dynamics as a driving force for tissue morphogenesis[END_REF][START_REF] Chlasta | Variations in basement membrane mechanics are linked to epithelial morphogenesis[END_REF][START_REF] Suzanne | Physical and functional cell-matrix uncoupling in a developing tissue under tension[END_REF].

In line with this cell-centred approach, cell-based simulations were developed to explore tissue mechanics (reviewed in [START_REF] Davidson | Emergent morphogenesis: elastic mechanics of a self-deforming tissue[END_REF]). The goal is to make simple assumptions on cell behaviour (e.g. spatial anisotropies of cell interface tension) based on experimental observations (e.g. spatial distribution of molecular motors) and explore how these hypotheses translate to the tissue scale with computer simulations. Widely endorsed models are the vertex model [START_REF] Alt | Vertex models: from cell mechanics to tissue morphogenesis[END_REF], in which cells shapes are described by the vertices that join three neighbouring cells (in two dimensions), or the cellular Potts model, in which cells are represented by domains of nodes on a lattice with like identity (akin to magnetic domains in an Ising model) [START_REF] Glazier | Simulation of the differential adhesion driven rearrangement of biological cells[END_REF]. Finite element (FE) methods can also be used to represent cells, which are then divided into small polygons. In vertex, cellular Potts and FE methods, the configuration of the tissue is set by an energy minimization. In centre-based models (reviewed in [START_REF] Van Liedekerke | Simulating tissue mechanics with agent-based models: concepts, perspectives and some novel results[END_REF]), cells can be represented by simple shapes (typically spheres) with slight deformability, for which trajectories are computed based on Newtonian dynamics. More realistic cell shapes can be achieved by discretizing cells in a number of viscoelastic elements.

In a seminal study on the link between mechanics and proliferation, Farhadifar and co-wokers used vertex models to simulate the cell assembly of developing Drosophila wings [START_REF] Farhadifar | The influence of cell mechanics, cell-cell interactions, and proliferation on epithelial packing[END_REF]. In this work, mechanical stresses can be relaxed through topological rearrangements: whenever energy minimization reduces a cell-cell interface below a certain cut-off length, the interface disappears through a T1 process when energitically favoured. Likewise, Ranft et al. used a centre-based model (dissipative particle dynamics) to model proliferating tissues [START_REF] Ranft | Fluidization of tissues by cell division and apoptosis[END_REF]. Whenever cell distorsion reaches a certain threshold through the combined effect of mechanical stress and growth, a cell division is triggered. In their context, cell divisions fluidize the tissue leading to a stress free state. These two influential papers promoted the viewpoint that topological transitions such as T1 and cell divisions relax stresses in growing tissues, which emphasized the fluid behaviour of the tissues at long time scales.

(b) Organ biomechanics: the macroscopic view

Changes in the shape of tissues proceed at spatial scales much larger than the cell. In this context, continuum mechanics seems appropriate to describe processes with large-scale fields. This formalism gives a rigorous formulation of morphogenesis, providing explicit results as a function of macroscopic quantities, which can be evaluated experimentally. Comparison of the resulting analytical laws with experiments provides a strong predictive power and the sense that the problem under study is understood. Continuous models are also appropriate when mechanics interplays with processes that translate well into continuous equations, such as diffusion and transport in the context of chemotaxis [START_REF] Ouakin | Stochastic collective movement of cells and fingering morphology: no maverick cells[END_REF][START_REF] Amar | Onset of nonlinearity in a stochastic model for auto-chemotactic advancing epithelia[END_REF].

There is a large body of work on the continuum mechanics of tissues from the bioengineering community [START_REF] Taber | Biomechanics of growth, remodeling, and morphogenesis[END_REF][START_REF] Fung | Biomechanics: mechanical properties of living tissues[END_REF]. Interestingly, contrary to cell-based approaches, the elastic nature of tissues is often emphasized. Studies on arteries [START_REF] Chuong | Residual stress in arteries[END_REF], the mechanical properties of skin [START_REF] Langer | On the anatomy and physiology of the skin: I. The cleavability of the cutis[END_REF][START_REF] Liang | Biomechanical properties of in vivo human skin from dynamic optical coherence elastography[END_REF], gut lumen ridges [START_REF] Amar | Anisotropic growth shapes intestinal tissues during embryogenesis[END_REF][START_REF] Shyer | Villification: how the gut gets its villi[END_REF] and gut looping [START_REF] Savin | On the growth and form of the gut[END_REF], as well as cortical brain convolution [START_REF] Tallinen | On the growth and form of cortical convolutions[END_REF][START_REF] Amar | Mimicking cortex convolutions through the wrinkling of growing soft bilayers[END_REF], are examples of whole organ mechanics well described by elasticity theory. In line with elastic material properties, cutting experiments reveal residual stresses that perdure long after the growth process is finished. How can we reconcile the fluid perspective of the cell-based approach (see above) and the elastic perspective of the the bioengineering approach? As discussed below (see §1c(ii)), one possibility is that whole organ biomechanics encompasses the inhomogeneous structure of organs made of connected layers, which may confer elasticity to tissues. On theoretical grounds, a long-standing challenge in the continuum description of tissues has been to find the correct description for growing tissues. The mechanical description of growth is a quite recent addition to the theory of elasticity. Precursor works include those of Cowin & Hegedus [START_REF] Cowin | Bone remodeling I: theory of adaptive elasticity[END_REF], Skalak et al. [START_REF] Skalak | Analytical description of growth[END_REF] and Fung [START_REF] Fung | Biomechanics: mechanical properties of living tissues[END_REF]. In their seminal work, which recapitulated and extended these initial efforts, Rodriguez et al. [START_REF] Rodriguez | Stress-dependent finite growth in soft elastic tissues[END_REF] developed the formalism of finite elasticity, which has now become the method of choice to account for the elasticity of growing bodies. Finite elasticity describes the final shape of a growing body as resulting from the elastic accommodation of a prescribed volumetric strain due to growth. Taking its roots from plasticity theory, this model turns out to describe a wide range of volume variation processes, for both inanimate matter (gels) and biological systems. A lot of works have been done with this formalism with the aim to extend its ability to treat biological features more efficiently as dissipation, pre-strained or pre-stressed tissues, and adequate constitutive laws, especially for fibre networks.

(c) Current challenges for continuum theory (i) Bridging scales

It is certainly relevant to merge the best of the two worlds: the analytical capability of continuum theory to encapsulate tissue mechanics in a few equations, and the capability of cell-based models to take into account the wealth of information that modern biology provides. Some attention has already been given to mapping vertex to continuum models [START_REF] Murisic | From discrete to continuum models of three-dimensional deformations in epithelial sheets[END_REF]. Methods to integrate experimental cell descriptors at the mesoscopic level were also recently published [START_REF] Etournay | TissueMiner: a multiscale analysis toolkit to quantify how cellular processes create tissue dynamics[END_REF][START_REF] Guirao | Unified quantitative characterization of epithelial tissue development[END_REF][START_REF] Heemskerk | Dynamic clonal analysis based on chronic in vivo imaging allows multiscale quantification of growth in the Drosophila wing disc[END_REF]. In parallel, as mentioned above, theoretical efforts should try to extend current formalisms to account for cell behaviour at a mesoscopic scale-trying to address for example questions of dissipation through the exchange of neighbours. Alan Turing, in his classic work on morphogenesis, has analysed this alternative between discrete and continuous modelling [START_REF] Turing | The chemical basis of morphogenesis[END_REF]. His work focused on reaction-diffusion of chemical species-not mechanics. However, the core of his pioneering paper concerns mostly a finite number of cells reacting with each other through morphogens. His effort to link the cellular scale with a process usually formalized in a continuum description (here diffusion) could be inspiring for tissue mechanics.

(ii) Which constitutive equation?

Different constitutive equations that range from elastic, visco-plastic to liquid-like behaviour have been used to model tissue mechanics [START_REF] Gonzalez-Rodriguez | Soft matter models of developing tissues and tumors[END_REF][START_REF] Liu | Matrix viscoplasticity and its shielding by active mechanics in microtissue models: experiment s and mathematical modeling[END_REF], suggesting that the general behaviour of tissues is still controversial. Some cases are quite intuitive. For instance, plants are appropriately described as elastic solids, consistent with the solid wall of plant cells and limited remodelling [START_REF] Amar | Petal shapes of sympetalous flowers: the interplay between growth, geometry and elasticity[END_REF][START_REF] Milani | Shrinking the hammer: micromechanical approaches to morphogenesis[END_REF]. This is also true for algae, although their cell walls are softer than the plant ones [START_REF] Dervaux | Morphogenesis of growing soft tissues[END_REF][START_REF] Jia | Morphoelasticity in the development of brown alga Ectocarpus siliculosus: from cell rounding to branching[END_REF]. For tumour models, the situation is less clear, since tumour multicellular spheroids have been shown to be characterized by a surface tension reminiscent of a liquid at long times [START_REF] Forgacs | Viscoelastic properties of living embryonic tissues: a quantitative study[END_REF], while real tumours, whose microstructures are reinforced with stiff ECM [START_REF] Stylianopoulos | Causes, consequences and remedies for growth-induced solid stress in murine and human tumors[END_REF][START_REF] Jain | The role of mechanical forces in tumor growth and therapy[END_REF] or pathological fibrosis are well described on elasticity basis [START_REF] Amar | Morpho-elasticity of inflammatory fibrosis: the case of capsular contracture[END_REF]. Indeed, in vivo tumours (by opposition to in vitro cancer cell spheroids) recapitulate the organ complexity, with dynamic regulation and constituent cellular populations, during initiation, maintenance and progression [START_REF] Hanahan | Hallmarks of cancer: the next generation[END_REF]. Likewise, mechanical instabilities that seem to drive gut looping [START_REF] Savin | On the growth and form of the gut[END_REF], vilification [START_REF] Amar | Anisotropic growth shapes intestinal tissues during embryogenesis[END_REF][START_REF] Shyer | Villification: how the gut gets its villi[END_REF] and cortical folds [START_REF] Tallinen | On the growth and form of cortical convolutions[END_REF][START_REF] Amar | Mimicking cortex convolutions through the wrinkling of growing soft bilayers[END_REF][START_REF] Rashid | Mechanical characterization of brain tissue in compression at dynamic strain rates[END_REF][START_REF] Mihai | A comparison of hyperelastic constitutive models applicable to brain and fat tissues[END_REF][START_REF] Mihai | A family of hyperelastic models for human brain tissue[END_REF][START_REF] Tallinen | Gyrification from constrained cortical expansion[END_REF] point to the elastic behaviour of these structures. An argument in favour of elastic behaviour is that cells usually tightly adhere to each other and most importantly to the underlying ECM. An argument in favour of fluid behaviour at long times is the presence of topological rearrangements associated with proliferation [START_REF] Ranft | Fluidization of tissues by cell division and apoptosis[END_REF] leading to stress relaxation. Whether elastic energy is stored and leads to residual stress, or whether cell proliferation induces fluidization of the tissue, is still debated to the best of our knowledge. The fluidization of tissues finds its root in structural re-organization of the living samples, which is not always possible when layers are inter-connected, a fortiori when fibres are present. As a consequence, it mainly concerns epithelia without interactions. Inside an organ, cell layers are attached to connective tissues via collagenous membranes such as in the epidermis of our skin, which is separated from the dermis via the basement membrane. Connective tissues provide elasticity. These simple arguments may explain the difference between the fluid emphasis of the cell-based approaches, which focuses on model tissues and young embryos, and the elastic emphasis of bio-mechanicians, which focuses on whole organs.

(iii) Heterogeneities

Connective tissues are constructed from an interconnecting network of collagen filaments, fibrils and fibres with a high tensile strength, fine and delicate reticular fibres and elastic fibres with a greater extensibility. They constitute the most predominant tissue in our body (20%). All these fibres are organized in a hierarchical multi-scale structure (see figure 1) inside a ground substance. How soft tissues adapt hierarchically to shear and compression is still little explored, which leaves unsolved the question of how, in soft connective tissues, cancer cells or fibroblasts change their mechano-phenotype to become more aggressive and invasive in pathological conditions. A unique modelling will never cover the whole diversity of growing living systems. Thus, different nonlinear modellings exist to represent the strain-stress laws under specific loadings. This is especially the case for fibre networks for which the link between the microscopic structure and the macroscopic constitutive laws have been investigated from different perspectives [START_REF] Holzapfel | An affine continuum mechanical model for cross-linked F-actin networks with compliant linker proteins[END_REF][START_REF] Shearer | A new strain energy function for the hyperelastic modelling of ligaments and tendons based on fascicle microstructure[END_REF][START_REF] Licup | Stress controls the mechanics of collagen networks[END_REF]. No attempt was made, however, to take into account the regulative function of fibroblasts. These isolated but very active cells are targeted to wound and change local stiffness there by making a fibrotic network. The consequent changes in compressive stresses affect in return fibroblast physiology. Another example of heterogeneous structures is sarcoma. Sarcomas are rare malignant tumours originating from mesenchymal tissue; they show an extremely high morphological and genotypic heterogeneity [START_REF] Chibon | Validated prediction of clinical outcome in sarcomas and multiple types of cancer on the basis of a gene expression signature related to genome complexity[END_REF][START_REF] Italiano | Genetic profiling identifies two classes of soft-tissue leiomyo-sarcomas with distinct clinical characteristics[END_REF]. Their mesenchymal origin gives them an extremely high differentiation capacity, in various soft tissues. For example, and as shown in figure 2, leiomyosarcoma cells adopt a fibrillar structure. Their cytoplasm is finely fibrillara signature of a deep mechanical transformation of the embedded cells, both locally and globally. There is also a long-range ordering of cells (alignment) which is the result of a higher deformability of their cytoplasm, suggesting a higher unidirectional actomyosin tension. This ordering leads to the alignment of two cells which then fuse to give a very aggressive entity suspected to be at the origin of metastasis. Here again, this process is driven by mechanical cues since it seems to be induced by a local shear inside the tissue. and myofibroblasts [START_REF] Martin | Wound healing-aiming for perfect skin regeneration[END_REF][START_REF] Gurtner | Wound repair and regeneration[END_REF], other examples of singular behaviour that may be hard to grasp in a continuum model are cancer stem cells [START_REF] Lathia | Cancer stem cells in glioblastoma[END_REF][START_REF] Porta | Senescent cells in growing tumors: population dynamics and cancer stem cells[END_REF][START_REF] Battle | Cancer stem cells revisited[END_REF][START_REF] Nassar | Cancer stem cells: basic concepts and therapeutic implications[END_REF], which are also isolated cells within a tumour that respond to environmental changes through proliferation. A single undifferentiated cancer cell can generate a complete tumour. They involve niches where stem cells are located and perhaps the cancer stem cells also. Cancer stem cells are then localized sources of proliferation. To conclude, the variability of living species (bacteria, plants, algae and fungi) and additionally the variability of our organ properties at different ages (fetus, childhood, adult or ageing) is too vast, quite infinite, to be relevant to a unique formalism. The formalism of finite elasticity alone cannot describe all the material properties of complex soft tissues at an intermediate scale above the cell size, with refinements including dissipative effects, feedback mechanism between growth and stress, localization and spatial disorders. There is also a price to pay to extend a formalism: first, the added complexity; second, the increase of unknown parameters.

(iv) Singularities

A cross-fertilization of the two approaches we discussed-the macroscopic biomechanics and the cell-based biophysics-may allow us to revisit the material properties of growing tissues and putatively gain a more realistic insight into organ growth, shapes and pathologies. To explain this in a more concrete way, a selection of biological models is proposed hereafter where experimental data are available or will be available in a close future. It is important to underline that most debates in the last two decades originate from the lack of experimental data, suggesting that it is important to develop tools to access the mechanical properties of tissues in development. Both connective and epithelial tissues in growth will be considered with emphasis on the role of cells.

Active cells in connective tissues

Very active cells may be embedded in connective tissues generating mechanical stresses. To model this requires a good representation of the fibre network at the macro-scale modified by the presence of active, eventually aggressive cells which, in addition, may have the ability to change their phenotype. Classical models of fibres [START_REF] Humphrey | Determination of a constitutive relation for passive myocardium: I. A new functional form[END_REF][START_REF] Gasser | Hyperelastic modelling of arterial layers with distributed collagen fiber orientation[END_REF][START_REF] Ciarletta | Pattern formation in fiber-reinforced tubular tissues: folding and segmentation during epithelial growth[END_REF][START_REF] Melnik | Dynamic fiber reorientation in a fiber-reinforced hyperelastic material[END_REF][START_REF] Destrade | At least three invariants are necessary to model the mechanical response of incompressible, transversely isotropic materials[END_REF][START_REF] Pucci | On the use of universal relations in the modeling of transversely isotropic materials[END_REF] in biomechanics are based on symmetries and represent the constitutive laws in term of the first invariants constructed with the deformation gradients. These macroscopic models have the advantages to treat simultaneously both the anisotropy of growth and also the structural anisotropy of tissues [START_REF] Amar | Growth and instability in elastic tissues[END_REF]. They also have the advantage of simplicity even if they were a matter of debate in the recent past, concerning mainly their adequacy to treat correctly all kinds of deformations. Indeed, they present some difficulty in mimicking compression which always exists in three-dimensional incompressible samples: in three dimensions, a tensional state in one direction implies quite automatically a compression in another direction. Recently, microscopic approaches have been proposed [START_REF] Holzapfel | An affine continuum mechanical model for cross-linked F-actin networks with compliant linker proteins[END_REF] to establish more realistic constitutive laws for the fibre network (without solving the compression problem), introducing the compliance of the cross-linkers and viscoelasticity. These approaches are based on affine kinematic assumptions [START_REF] Holzapfel | An affine continuum mechanical model for cross-linked F-actin networks with compliant linker proteins[END_REF], a hypothesis which has not yet been validated experimentally for fibrous organs [START_REF] Jayyosi | Affine kinematics in planar fibrous connective tissues: an experimental investigation[END_REF], which have a more complex structure than the actin network proposed in [START_REF] Holzapfel | An affine continuum mechanical model for cross-linked F-actin networks with compliant linker proteins[END_REF].

More importantly, these biomechanical models of fibrous tissues do not include the existence of active cells, which generate local proliferation and global growth (fibroblasts, cancerous cells) but also active strain or stress (myofibroblasts), apart for the work of De Vita and colleagues [START_REF] Tan | A structural constitutive model for smooth muscle contraction in biological tissues[END_REF][START_REF] Vita | Mathematical model for muscle isometric and isotonic contractions[END_REF] on smooth muscles. Clearly, this biological situation, where small-scale individuals may perturb a complex environment, open new challenges in finite elasticity. One may naturally think of the theory of defects [START_REF] Maugin | Defects, dislocations and the general theory of material inhomogeneity[END_REF] as a starting point. In addition to the complexity of tissue modelling, the interpretation of the experimental measurements is difficult as cells tend to react to any imposed perturbation, and thus to the imposed measurements. This is what happens to cells when the cytoskeleton is probed by atomic force microscopy (AFM). Two time scales have been detected [START_REF] Digiuni | Single cell wall nonlinear mechanics revealed by a multi-scale analysis of AFM forceindentation curves[END_REF]: the typical time scale of a passive elastic response, then a typical time scale of reactivity of the molecular motors. Even if these two time scales are different, this indicates a competition, at the cellular level of the cell of both passive and active processes.

Many classical biophysical experiments (such as micro-pipette aspiration) have been interpreted in the past by surface tension. It requires, now, to be revisited in the context of nonlinear elasticity as done by Holzapfel et al. [START_REF] Holzapfel | An affine continuum mechanical model for cross-linked F-actin networks with compliant linker proteins[END_REF] for micro-pipette aspiration and by [START_REF] Digiuni | Single cell wall nonlinear mechanics revealed by a multi-scale analysis of AFM forceindentation curves[END_REF][START_REF] Martinez-Torres | Passive microrheology of soft materials with atomic force microscopy: a wavelet based spectral analysis[END_REF] for AFM indentation. This is necessary for a proper interpretation of the data and this is the only way to provide some portability to the models when applied to in vivo situations. It is especially true for frequency-dependent stress-strain experiments done on three-dimensional fibrous samples. A recent study on fibrous materials submitted to cyclic extension has shown a complex response of these tissues under time-dependent loading, which is not trivial to interpret in the context of established formalisms [START_REF] Natali | Constitutive modeling of the non-linear visco-elasticity of the periodontal ligament[END_REF][START_REF] Amar | Nonlinear visco-elasticity of soft tissues under cyclic deformations[END_REF].

The modelling of tumour growth under mechanical stress must decipher local and global (multi-scale) aspects. Cancer cells inside a tumour are not identical but form different clones which can be defined as subgroups of cancer cells sharing a common genotype. The epigenetic tumour cell heterogeneity cannot be fully explained without considering that tumour cells are not isolated but surrounded by their microenvironment (extracellular matrix) which is also very heterogeneous in composition. Malignant transformation, although primarily driven by genetic mutations in cells, is also accompanied by specific changes in cellular and extracellular mechanical properties such as adhesion, contractility, mechano-sensing, rigidity, viscoelasticity and plasticity. As the transformed cells grow into tumours, they interact with their microenvironment via physical contacts and application of forces which in turn alter the mechanical regulation of cells. It is therefore important to understand the effect of these peculiar changes in cellular and extracellular mechanical properties on the transformation of benign cells into cancer cells to establish the basis of a model for tumour growth and metastasis. For that purpose, computing tools are fundamental for estimating to which extent and at which scale the cross-talk of cancer cells with their microenvironment is going to impact their proliferation. Most of the previous studies of the biomechanics of cancer cells [START_REF] Katira | How changes in cell mechanical properties induce cancerous behavior[END_REF] have focused on the mechanical properties of the cancerous cells themselves and more rarely on their surrounding niche. They have led to disparate results since cancer cells can become stiffer or softer depending on their embedding tissue or their progression in the disease. The study of these fibrous tissues is especially important in the context of tumour growth.

Assembly of cells seen as tissues: cortex anomalies and ageing

The circumvolutions of the brain, called gyri, appear approximately at four months post fertilization for human fetuses. A number called the gyrification index is defined to characterize this shape instability. It is a number measuring the ratio between the outer surface of the cortex versus the averaged middle surface of the cortex layer. It is closed to 1 for smooth brain (rodents), Recently, the bilayer model has encountered some success for the prediction of the gyrification index of the brain [START_REF] Tallinen | On the growth and form of cortical convolutions[END_REF][START_REF] Amar | Mimicking cortex convolutions through the wrinkling of growing soft bilayers[END_REF][START_REF] Khul | Biophysics: unfolding the brain[END_REF]. This model is based on simple ingredients. It assumes that the grey and white matter can be considered as two hyperelastic layers in growth and that the origin of cortex gyrification comes from the differential growth between these two layers. A growth anisotropy in the grey matter originates from the fact that stem cell differentiation and differentiated cell proliferation occur simultaneously, in fetal life. The main assumption is only a differential and anisotropic volumetric growth of two gel layers (grey and white matter) with the added hypothesis that white matter is slightly stiffer than grey matter. These assumptions lead to a super-critical bifurcation in growth modelling. Except for the case of brain pathologies, this two-layer model recapitulates the gyrification index in fetal life quantitatively [START_REF] Amar | Mimicking cortex convolutions through the wrinkling of growing soft bilayers[END_REF] by comparison with data compiled by [START_REF] Zilles | The human pattern in gyrification in the cerebral cortex[END_REF]100] and new magnetic resonance imaging (MRI) studies for fetuses and pre-term newborns [101]. The model explains standard global pathologies as polymicrogyry (excess of cortical folds) and lissencephaly (lack of cortical folds).

Another challenge to raise is how to explain the existence of local creases in the cortex and their possible link to pathologies. If local creases have been observed in experiments involving layers of swelling gels [102-104], they are not completely understood from a theoretical point of view. Indeed the theory of bifurcations, which is well adapted for buckling, fails for local and singular instabilities like creases. Recent investigations in compression involving local defects are promising [105] although possibly not applicable to the brain. The question is mathematical: why does a growing elastic system select a periodic array of cusps instead of a nonlinear buckling? What kind of growth will lead to these cusps instead of smooth surface instabilities? Do we need specific growth repartition to observe such pattern? It is a challenging problem in the context of nonlinear elasticity. Material properties of the two layers (white and grey matter) change in the course of ageing. Not only the relative size of both layers change but their relative stiffness changes as well. This has been shown recently by studies on cadavers [106] indicating that the stiffness ratio between layers varies with age and that the cortex becomes stiffer than the white matter, in contrast to what happens for babies. How changes in material properties affect gyrification and the morphology of the cortical layers can be studied through a combination of modelling and MRI. In this context, important points to address are local variations or inhomogeneities, which seem to affect cognition such as language functions, and the width of sulci, which increases with age and seem to correlate with cognitive decline in the elderly. Neuro-degenerative diseases have also been linked with gyrification through the appearance of neuro-fibrillar tangles in the cortex, which increases the stiffness in comparison with the white matter, again modifying the gyri and sulci.

Active cells at work: the case of C. elegans embryonic elongation

C. elegans embryonic elongation asks specific questions of great interest for embryogenesis. It involves two time periods (figure 3) in which the lima-bean shape of the embryo is transformed into a quasi-cylinder with a fourfold increase in length. These two periods last approximately 2 h each. During the elongation period, no cell proliferation, cell migration, cell division or notable changes in embryonic volume are observed. The first period is controlled by the cytoskeleton of the epidermal cells, mostly the actomyosin couple, and provides an elongation by a factor of 1.8 [START_REF] Vuong-Brender | The interplay of stiffness and force anisotropies drives embryo elongation[END_REF]107,108]. To experimentally check this hypothesis, Vuong-Brender et al. [START_REF] Vuong-Brender | The interplay of stiffness and force anisotropies drives embryo elongation[END_REF] have measured the outer dimensions of each part of the epithelium during elongation which is composed of three different cell types. These cells located in four domains differ by the polarized orientation of the cytoskeleton. The cytoskeleton in the dorsal and ventral cells is strongly polarized in the orthoradial direction, while it is less polarized in the seam cells. By measuring the opening induced by laser ablation in different parts of the embryo, these authors assess the stresses which drive the elongation. Indeed, this technique allows them to evaluate quantitatively the local stresses in the living embryo and to assess the rheology and structure of its compartments [START_REF] Amar | Assessing the contribution of active and passive stresses in C. elegans elongation[END_REF]. To be quantitative the laser ablation technique requires to revisit the theory of fracture as the crack Inspired by [START_REF] Vuong-Brender | The interplay of stiffness and force anisotropies drives embryo elongation[END_REF]. In pink, the dorsal epithelial cells, in yellow the seam epithelial cells, in orange the ventral cells.

(Online version in colour.)

shape in orthotropic material under bi-axial stresses (doable in linear elasticity) or the full theory of fracture in finite elasticity. Contrary to a naïve intuition, in linear elasticity, not all the problems of fracture have been investigated. In [START_REF] Vuong-Brender | The interplay of stiffness and force anisotropies drives embryo elongation[END_REF], for example, the linear analysis of fracture achieved initially by [109,110] has been extended to orthotropic materials to interpret correctly the laser indentation on C. elegans embryos. However, for an elongation of 80%, probably, linear elasticity is not enough and a better model should involve nonlinear elasticity. Indeed only finite elasticity can involve pre-stress/pre-stretch, passive and active stresses in the same formalism.

The second period of elongation [111-113] which lasts 2 h is controlled by nascent muscles, located below the epidermis, in the gelatinous intestinal inner cylinder. To understand how contractile cells can finally lead to an elongation process is challenging. Here again, it is known that four lines of muscle cells located below the epidermis elongate the embryo by a factor 2. It will require coming back to microscopic models of muscle formation and to an evaluation of the strains, which can be obtained after successive periods of compression on one side and stretch on the other side of the body. Indeed, when two muscular chains are in compression on one side of the embryo, the opposite side turns out to be in extension. Understanding how this succession of compression/extension can lead to such elongation and what we will learn on muscles in embryogenesis is an ambitious goal that will bring new insight into the role of muscles in embryogenesis. Indeed a factor 2 is huge and one can wonder if only mechanics is involved in this process or if new muscle cells appear in the stretching period, according to the naïve idea that a tensional state induces proliferation. Another question concerns the relaxation of stresses which have been accumulated during the total elongation period. Indeed, such elongation will put mostly all embryonic tissues into tension, which can be followed by cell proliferation hereafter.

Proliferative epithelial cells on an elastic substrate

As already underlined, there are a number of model experimental systems in which the growth of epithelia has been investigated. Surprisingly, however, the paradigm of finite elasticity with growth has not been applied to this context. In most of the theoretical biophysics contributions, an elastic coupling with the substrate is discarded and the notion of fluidity is promoted. It may be relevant to revisit the question of the best rheological model between viscoelasticity, elasticity and fluidity by comparison of experimental data with a finite elasticity model involving the bilayer composed of the epithelium and the extracellular matrix. Potential experimental models to confront the theory are the Drosophila wing (in vivo) or in vitro epithelial cysts. For the first model system, quantifying cell deformations and stresses, it was found that the growing wing imaginal disc displays a highly patterned field of mechanical stress [114,115]. In particular, the cell at the periphery of the tissue sustains a strong mechanical stretch. These results are in disagreement with the viewpoint of a growing tissue in a fluid state through the combined effect of cell divisions and cell rearrangements [START_REF] Ranft | Fluidization of tissues by cell division and apoptosis[END_REF]. The observed pattern of mechanical stress impacts on tissue morphogenesis by polarizing the cytoskeleton of cells as well as the orientation of their divisions. To investigate more thoroughly the role of mechanics, localized perturbation (e.g. growth, or structure of the ECM) can be achieved, and the resulting tissues analysed under the microscope. This provides challenging objects of study for the modelling. Such experimental investigations allow us to assess the link between patterned growth and patterned stress. They should allow us to derive in the future the constitutive equations of the tissue, answering the question of their fluid or elastic nature. Figure 4 provides an example of a perturbation where the elastic ECM underlying the Drosophila imaginal wing epithelium has been degraded in a region at the centre of the tissue. The consequent distortions are typically reminiscent of an elastic buckling. Likewise, the tissue undergoes shape changes also analogous to the buckling of an elastic sheet in conditions of local hyper contractility of the apical surface of cells by expression of a constitutively activated form of Myosin (L. LeGoff 2018, unpublished).

In vitro cysts were recently developed with morphometric, physico-chemical and mechanical properties which can be tuned easily (see figure 5) [116,117]. Here, epithelial cells are encapsulated in hydrogel hollow micro-spheres whose diameter can be varied from few tens of micrometres to millimetres. In the presence of ECM proteins anchored to the inner walls of the capsule, cells proliferate, readily spread onto it and self-assemble into a closed spherical epithelium. This set up gives a new opportunity to investigate epithelial growth with a prescribed curvature and with variable parameters. The fact that one can vary the parameters, such as the stiffness of the artificial ECM by tuning the composition of the hydrogel, is a unique opportunity for testing specific aspects of the theory. Indeed, morphogenesis is a complex highly regulated sequences of processes with a few degrees of freedom including the use of mutants. Using in vitro cysts allows one to dissect the different potential morphogenetic events and to investigate the impact of geometrical and morphological control parameters. Both systems are geometrically complex to handle analytically, but being made of thin layers a lubrication analysis of the equations of elasticity or viscoelasticity may be suitable [START_REF] Dervaux | Morphogenesis of growing soft tissues[END_REF][START_REF] Jia | Morphoelasticity in the development of brown alga Ectocarpus siliculosus: from cell rounding to branching[END_REF].

Undoubtedly, these two model systems will be helpful to test many aspects debated in the field of morphogenesis of tissues, such as dissipation, localization and stochasticity [118,119]. Even within the simplifying assumption that they behave elastically, one can decipher the role and the impact of growth, epithelium and ECM stiffness, and adhesion/friction with the substrate (ECM). In addition, determining the elastic strain and stress tensors as well as the current shape of the tissue (be it the wing disc or the encapsulated cyst) gives the possibility of evaluating the growth tensor which dominates the geometric strain and the shape of the tissue during growth. To the best of our knowledge, such a study has never been achieved. Finite elasticity with growth has been established for organs where usually the growth tensor is not known a priori. This lack of information has forced theorists to make reasonable assumptions mainly based on symmetry arguments, then allowing to carry off semi-quantitative results. These model systems offer a situation to test in detail the development with the nonlinear elasticity of soft tissues. Comparing experimental data with theoretical predictions will clarify the question of viscoelasticity in growth processes, thus eliminating endless debates in the community.

Methodology (a) Analytical treatments

Giving the best description of living tissues in morphogenesis is an ambitious objective. Considering mostly the scale of organs or embryos, the continuum description is preferred, at least in the biomechanics community. However, living tissues present a diversity of behaviours that inert physical systems do not exhibit. In physical systems, volumetric growth is not so common. Swelling or drying gels, although very useful tools for mimicking growth with adaptable geometries do not fully correspond to volumetric growth or resorption. In physical situations with inert matter, growth is very often the result of an out-of-equilibrium phase transition process such as dendritic growth. This process is governed by diffusion equations and elasticity plays a minor role. It is why the theory of elasticity with growth [START_REF] Goriely | The mathematics and mechanics of biological growth[END_REF] has been so much debated. In addition, the fact that all geometric deformations have the scale of the samples indicate that the theorist cannot use linear elasticity. After two decades of discussions about the foundations of the modelling, the time has come to test the model in the full complexity of living systems, addressing their dynamics and disorders. Four up-to-date biological systems have been presented here where cells play a key role. Cells are the sole sources in the systems through their proliferation and their ability to generate new connective tissues. They also power other deformations, such as stretching via their actomyosin network. Finite elasticity allows analytical treatments when the shape result from bifurcations. However, when the study concerns extremely active cells embedded into a fibrous tissue, microscopic methods or semi-microscopic methods may be preferred.

(b) Microscopic methods

A possible microscopic method is the kinetic theory of active particles (KTAP), which takes into account both the cellular and tissue scales. The principle of the method can be found in [120,121] for the case of fibrosis. It is based on a generalization of the Boltzmann equation for gas. The system is decomposed into different functional subsystems involving a large number of interacting entities, called active particles: their microscopic state, in addition to geometrical and mechanical variables, also includes an additional variable, called activity, which represents the ability to express a specific strategy such as mutation or proliferation. The overall state of each subsystem is described by a distribution function over the microscopic states of the cells. Interactions, which modify the state of the interacting cells, produce transitions of the activity variable, proliferation and mutations. An external field (imposed by shear for example, as observed in sarcomas) is represented by an action into the system, thus moving the system out of equilibrium. By balancing the inlet and outlet flows into the elementary volume of the possible microscopic states, the evolution equation of each functional entity can be derived. The description at the cellular scale brings important information on the dynamics at tissue scale. The macroscopic behaviour of tissues can be derived from the dynamics at the cellular level by letting intercellular distances tend to those of the tissue level. The link between the cellular and the tissue scales is thus obtained through usual hydrodynamic limit procedures [122,123]. Typically sarcoma tumour growth would be a good candidate for such treatment.

(c) Semi-microscopic modelling

Semi-microscopic models will consist in treating the tissue as a mixture of different entities [124][125][126]. The formalism can be extended to finite elasticity for fibres and for cells positioned in average position. Indeed, for fibrotic tissues, the fibroblast cells are located at specific positions (see figure 1). It is likewise the case for cancer stem cells which, as healthy stem cells, occupy preferential niches. The advantage in these models is that growth can be easily introduced. The introduction of drugs or nutrients can also be easily modelled. Another semi-analytical method will consist in using discrete-continuous models, then treating cells as singularities in a continuum medium. This approach is well adapted to the low density of extremely active cells, which have a notable impact on the tissues through proliferation or exertion of forces. When it is necessary to decipher local and global aspects of tumour growth in fibrotic tissues, continuum models (at scales larger than the cell size) must be combined to microscopic units to describe their fascicular orientation, with local anisotropic subcellular mechanical moduli to include the intracellular organization of the cell cytoskeleton and its change with mechanical modulation. These units may involve distinct cell populations, differentiated cancer cells, cancer stem cells, fused cells. Mixture models [124][125][126] are the most flexible tools, since various elasticities for the different cell phases and cell-cell interactions can be introduced according to their phases and their rate of proliferation. As an example, see fig. 6 from [127], where the patterns of cell repartition are shown in the presence of solid obstacles, always present in tumours. Also of importance will be the mechanical constraints imposed by the stroma which opposes a resistance on the tumour. The response of substrate elasticity that has been largely neglected so far may be at the heart of future studies.

Conclusion

Across four different examples chosen in embryogenesis, morphogenesis and tumour genesis, a panorama of living tissues properties and modelling is presented. The examples are at an intermediate scale between cells and organs with a moderate complexity. Among the biological diversity, we focus on the case where cells are not only present but very active [128], making ordinary modelling uneasy, sometimes not appropriate. We have emphasized the need to describe tissues at a scale intermediate between the large scale of whole organs and the cellular scale. A controversy originates from tissue growth: what is the most appropriate modelling? It was proposed that in growth, tissues behave as a fluid at a long time to reorganize the cells and dissipate the stresses. However, it was also suggested that because growth is a very slow process, all transient states disappear very quickly and tissues are mostly elastic when they grow. To clarify this debate, Mongera el al. propose a transition described by a fluid-to-solid jamming transition in vertebrate body elongation [129], also in favour of a solid view point at long times. However, the answer is probably not universal and only predictions of models adapted to specific biological experiments or situations where data are available will have a real scientific impact in the future. Data accessibility. This article has no additional data. Authors' contributions. All authors contributed equally, read and approved the manuscript. Competing interests. We declare that we have no competing interests. Funding. The authors acknowledge the support of ANR (Agency Nationale de la Recherche) under the contract MecaTiss (ANR-17-CE30-0007). M.B.A. acknowledges also the support of ANR under the contract (ANR-18-CE13-008) EPIMORPH and L.L.G. the support of ANR under the contract (ANR-18-CE13-028) GuideFusion.

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  have allowed developmental biologists to better understand how the behaviour of individual cells contributes royalsocietypublishing.org/journal/rsta Phil. Trans. R. Soc. A 377: 20180070 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Figure 1 .

 1 Figure 1. From [68], (a) collagen network within a Baker II capsule (Baker grade indicates the degree of the pathology, Baker II being an intermediate level of pathology). The green birefringence shows the loose organization of collagen fibres wrapped around tighter organized collagen fibres in red. (b) Periodic spacing of collagen fibres making up the thick cables in Baker III capsules. (c) Cells aligned parallel to the collagen fibres in Baker II capsules. (Online version in colour.)

  Continuum models are particularly challenged when a very diluted and scattered population, such as the fibroblasts, plays an important role in the properties of the tissue. Besides fibroblasts royalsocietypublishing.org/journal/rsta Phil. Trans. R. Soc. A 377: 20180070 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 2 .

 2 Figure 2. (a) 'Cigar shaped' configurations of tumour cell nuclei of a leiomyosarcoma with nuclear atypia [74]. (b) Histological appearance of a leiomyosarcoma after resection of a tumour in the subclavicular region [75]. (Online version in colour.)

  royalsocietypublishing.org/journal/rsta Phil. Trans. R. Soc. A 377: 20180070 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  royalsocietypublishing.org/journal/rsta Phil. Trans. R. Soc. A 377: 20180070 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . around 2.6 for babies and 5 for dolphins.

Figure 3 .

 3 Figure 3. Embryonic elongation in C. elegans is driven in part by epidermal actomyosin contractility and in part by muscle contraction. Inspired by [14]. In pink, the dorsal epithelial cells, in yellow the seam epithelial cells, in orange the ventral cells. (Online version in colour.)

Figure 4 .

 4 Figure 4. Generation of buckling-like three-dimensional shapes through localized destabilization of ECM after expression of the protease Mmp2 in a central band of the tissue, leading to a deep fold in the tissue (exact genotype: ubi-cad:GFP Gal80ts; UAS-Mmp2/DppGal4).

Figure 5 .

 5 Figure 5. (a) Confocal section of an alginate capsule (green) coated with a layer of ECM (red); (b) Phase contrast videomicrograph of an encapsulated cyst of MCF10A (epithelial cell from the mammary gland); (c) Confocal equatorial section of an MDCK cyst (red) in an alginate capsule (green).

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Figure 6 .

 6 Figure 6. Distribution of cells computed with the two-dimensional mixture model including cancer stem cells, differentiated cells and inert cells (dead, of different type such as blood cells). The computation was seeded from a small number of cells. (a) Differentiated cell distribution. (b) Cancer stem cell distribution. The black dots represent obstacles (vessels, fibres). This numerical computation includes a chemical signal exchange in between cells (such as growth factors) and growth feedback control loops (unpublished preliminary data [127]). (Online version in colour.)