Yue Mei

Xuan Feng

Yun Jin

Rongyao Kang

Xinyu Wang

Dongmei Zhao

Soham Ghosh

Corey P Neu

Stephane Avril
email: avril@emse.fr

Cell nucleus elastography with the adjoint-based inverse solver

Keywords: Cell mechanics, nuclear elastography, deformation microscopy, inverse problem, nonhomogeneous elastic distribution, mechanogenetics

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1.Introduction

The study of cellular and nuclear mechanics has gained increasing attention in recent years due to its essential role in the regulation of gene expressions [START_REF] Jacobs | Introduction to cell mechanics and mechanobiology[END_REF][START_REF] Ramérez-Mena | Explainable artificial intelligence to predict and identify prostate cancer tissue by gene expression[END_REF] affecting cellular function such as cell migration [START_REF] Weijer | Collective cell migration in development[END_REF][START_REF] Jain | The role of single-cell mechanical behaviour and polarity in driving collective cell migration[END_REF][START_REF] Lautenschläger | The regulatory role of cell mechanics for migration of differentiating myeloid cells[END_REF], proliferation or differentiation [START_REF] Titushkin | Modulation of cellular mechanics during osteogenic differentiation of human mesenchymal stem cells[END_REF][START_REF] Bongiorno | Mechanical stiffness as an improved single-cell indicator of osteoblastic human mesenchymal stem cell differentiation[END_REF]. A number of diseases have been associated to altered cellular mechanics, as for instance, atherosclerosis and other cardiovascular diseases [START_REF] Kliche | Role of cellular mechanics in the function and life span of vascular endothelium[END_REF]. Analysis of full-field strain maps depicting the behavior of cardiomyocyte nuclei during in vitro contractions has revealed a discernible relationship between nuclear deformation, chromatin reorganization, and the subsequent reduction in the expression of genes associated with cardiac development [START_REF] Seelbinder | Nuclear deformation guides chromatin reorganization in cardiac development and disease[END_REF]. Consequently, there is a pressing need in elucidating the intricate mechanosensitive role of the cell nucleus in the pathophysiology of cardiovascular diseases. To name another example, osteoarthritis [START_REF] Athanasiou | Biomechanics of single chondrocytes and osteoarthritis[END_REF][START_REF] Huang | Three-dimensional surgical planning and clinical evaluation of the efficacy of distal tibial tuberosity high tibial osteotomy in obese patients with varus knee osteoarthritis[END_REF], which is a common joint disorder affecting millions of individuals worldwide, it is now well understood that changes in the extracellular matrix, cell and nuclear mechanics are associated with the cartilage degeneration [START_REF] Mccreery | Nuclear stiffness decreases with disruption of the extracellular matrix in living tissues[END_REF]. Many other pathological conditions can be attributed to compromised mechanobiological pathways that are triggered by or result into compromised cell and nuclear mechanics [START_REF] Carrasco-Mantis | An in silico study on the influence of extracellular matrix mechanics on vasculogenesis[END_REF][START_REF] Mora | Insights from an electro-mechanical heart failure cell model: Role of SERCA enhancement on arrhythmogenesis and myocyte contraction[END_REF][START_REF] Rahman | A chemo-mechano-biological modeling framework for cartilage evolving in health, disease, injury, and treatment[END_REF].

To investigate cell mechanics, a variety of methods have been developed to measure their elastic and viscoelastic properties [START_REF] Moeendarbary | Cell mechanics: principles, practices, and prospects[END_REF]. Among these methods, atomic force microscopy (AFM) is commonly used to estimate the nano-indentation response of cells and relate it to their viscoelastic properties [START_REF] Haase | Investigating cell mechanics with atomic force microscopy[END_REF][START_REF] Li | Atomic force microscopy in characterizing cell mechanics for biomedical applications: A review[END_REF]. However, the indentation response combines mechanical contributions of subcellular structures including the cell membrane and nucleus which are difficult to separate [START_REF] Moeendarbary | Cell mechanics: principles, practices, and prospects[END_REF]. To better measure the separate mechanical contributions of each cellular compartment, particle tracking microrheology (PTM) was also developed [START_REF] Wirtz | Particle-tracking microrheology of living cells: principles and applications[END_REF][START_REF] Wu | A comparison of methods to assess cell mechanical properties[END_REF], although applications to identify local elastic properties are rare.

A recent progress in the field is the introduction of optical elastography to measure full-field deformation fields induced by subjecting the cell to adapted mechanical loading [START_REF] Grasland-Mongrain | Ultrafast imaging of cell elasticity with optical microelastography[END_REF]. By leveraging the theory of elastic waves [START_REF] Graff | Wave motion in elastic solids[END_REF] and assuming local homogeneity [START_REF] Sinkus | High-resolution tensor MR elastography for breast tumour detection[END_REF][START_REF] Papazoglou | Algebraic Helmholtz inversion in planar magnetic resonance elastography[END_REF], the spatial variation of cell elasticity could be obtained. However, these elastic maps did not resolve spatial heterogeneities within the nucleus, whereas internal architecture of the cell nucleus show heterogeneous distribution of dense heterochromatin and loose euchromatin domains [START_REF] Baumann | A vision of 3D chromatin organization[END_REF][START_REF] Maloney | Mesenchymal stem cell mechanics from the attached to the suspended state[END_REF][START_REF] Grewal | Heterochromatin revisited[END_REF],

normally giving rise to spatially variable stiffness properties [START_REF] Ghosh | Image-Based Elastography of Heterochromatin and Euchromatin Domains in the Deforming Cell Nucleus[END_REF]. Therefore, continued efforts to refine these techniques and develop new ones to measure the nonhomogeneous nucleus elasticity are essential.

Recent research has focused on characterizing the mechanical properties of euchromatin and heterochromatin due to their significant role in gene expressions [START_REF] Alisafaei | Regulation of nuclear architecture, mechanics, and nucleocytoplasmic shuttling of epigenetic factors by cell geometric constraints[END_REF]. Deformation microscopy, a non-invasive method based on high-resolution optical microscopy, has been proposed to obtain mechanical properties of the cell nucleus. This technique overcomes the challenges of AFM indentation as it does not require to invasively probe inside the cell nucleus [START_REF] Ghosh | Deformation microscopy for dynamic intracellular and intranuclear mapping of mechanics with high spatiotemporal resolution[END_REF]. While displacement and strain fields could be acquired successfully by this method, attempts to solve the inverse problem for relating them to local elastic properties has been limited by technical challenges. Only a recent study used an optimization scheme to estimate the effective elastic moduli of the heterochromatin and euchromatin regions based on pixel grayscale intensity [START_REF] Ghosh | Image-Based Elastography of Heterochromatin and Euchromatin Domains in the Deforming Cell Nucleus[END_REF], but this method reduced the inverse problem to the identification of only two stiffness parameters and neglected the possible heterogeneity within heterochromatin domains. There still remains a need to develop a complete description of intranuclear mechanics, including detailed spatial distributions of material properties, that may give insights into cellular structure, architecture, and biological processes.

In this paper, we address this technically challenging problem with an adjoint-based inverse solver without any a priori assumption about the distribution of stiffness properties within the nucleus. The paper is structured as follows: In the Methods Section, we describe the experimental setup of deformation microscopy and the inverse solver algorithm. In the Numerical Example Section, we demonstrate the feasibility of the proposed method using simulated datasets generated by finite-element simulations. In the Experimental Example Section, we identify the nonhomogeneous distribution of nuclear stiffness using the full-field displacement fields measured by deformation microscopy in beating cardiomyocytes. Then we discuss and analyze the numerical and experimental results in the Discussion Section. Finally, we conclude the paper with a summary of our findings and suggestions for future work.

2.Methods

Deformation microscopy within the cell nucleus

To characterize the nonhomogeneous elastic property distribution of the nucleus, the first step is to measure the displacement field inside a deforming nucleus using deformation microscopy. As depicted in Fig. 1, cardiomyocytes were cultured on a PDMS substrate with two different stiffness values resembling a normal (soft, E~12kPa) vs fibrotic (stiff, E~434kPa) heart. Highresolution images of the beating cardiomyocyte nucleus at the medial plane was captured using a epifluorescence microscope. The two-dimensional displacement fields in the imaging plane was quantified by deformation microscopy, a technique based upon digital image correlation and hyperelastic warping, that utilizes the undeformed and deformed images. Further details regarding the experimental arrangements and displacement calculation can be found in [START_REF] Ghosh | Deformation microscopy for dynamic intracellular and intranuclear mapping of mechanics with high spatiotemporal resolution[END_REF].

Inverse problem solution algorithm

As the image of the cell nucleus is two-dimensional, the cell nucleus is assumed to be in a state of plane stress in its medial plane. It is also assumed that the nucleus behaves like an incompressible [START_REF] Lomakin | The nucleus acts as a ruler tailoring cell responses to spatial constraints[END_REF][START_REF] Venturini | The nucleus measures shape changes for cellular proprioception to control dynamic cell behavior[END_REF] and neo-Hookean material such as:

( )

11 pJ  -- -+ S = I C C (1)
where S is the second Piola-Kirchhoff stress tensor,  is the shear modulus, p denotes the hydrostatic pressure,

T = C F F , ( ) det J =
F , where F is the deformation gradient. F derives from the displacement field u such as: F = I + Grad u. For a compressible and neo-Hookean material, the constitutive equation is given by: ( ) ( )

11 2 ln 12 J    -- -+ - S = I C C (2)
where  is the Poisson's ratio.

The inverse problem is stated as: with the measured displacement field m u obtained by deformation microscopy, we seek the spatial distribution of shear modulus µ(x) within the nucleus that minimizes the following objective function:

( )

2 2 2 1 , d c d 2 m      = -  +  +   u u u (3)
where  represents the entire domain of the nucleus and u is the displacement field satisfying quasi-static equilibrium such as div(FS)=0 in  and u=g in  ,  being the boundary of  . The shear modulus µ(x) is nodally defined and interpolated by the same shape function as the displacement fields. The second term in the objective function (regularization term) is used to avoid overfitting;  is the regularization factor that is used to control the significance of the regularization. For a very small  , the elasticity map of the nucleus has strong oscillations and distortions, whereas it is very smooth if  is very large. In this paper, the optimal regularization factor is chosen by the L curve method [START_REF] Hansen | Analysis of discrete ill-posed problems by means of the L-curve[END_REF]. c=10 -2 is a stabilizing factor mitigating potential singularity arising from the differentiation of the objective function with respect to the shear modulus.

We utilize the finite-element method to solve quasi-static equilibrium and compute u. The weak form is defined as: Find   , p  u M P such that:

( )   , 0, , q p q =    w,u, w P A (4)
where

( ) ( ) , : d 1 d q p q J  =   + -  w,u, w P A (5)
The function spaces M and are defined as:

( )   ( )   ( ) 1 1 2
; on ; 0 on

i i i u i i u u H u g w H w L    =     =   u w M P (6)
The domain of interest is discretized with quadrilateral elements and the corresponding discretized formulation is stated as:

Find ,

h h h h p    u M P such that ( ) , , 0, , h h h h h h h h q p q  =     w ,u w P A (7)
where h M , h and h P are the finite-element subspaces of M , and P , respectively. Here, we use the standard Galerkin finite-element method. To address volumetric locking induced by the incompressibility assumption, a stabilization term is introduced into the weak form ( 7) such as:

( ) ( ) ( ) ( ) ( ) NE 1 , , , , div , i T h h h h h h h h h h h h i q p q p q  -   = = -   w ,u w ,u F S F AA ( 8
)
where NE is the total number of finite elements across the discretized nucleus. Moreover,

2 2 h    =
represents the stabilization factor, where h is the characteristic element length. α is set to 1.0. More details on the stabilization method can be found in [START_REF] Hughes | A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuška-Brezzi condition: A stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations[END_REF].

To solve the inverse problem, we use the adjoint-based optimization method [START_REF] Oberai | Solution of inverse problems in elasticity imaging using the adjoint method[END_REF][START_REF] Mei | Quantifying the anisotropic linear elastic behavior of solids[END_REF][START_REF] Mei | Mapping the viscoelastic behavior of soft solids from time harmonic motion[END_REF]. In this case, the Lagrangian is:

( ) ( ) ( ) , , ; , , , h h h h h h h h h q p q p     =+ w ,u u w ,u L = A (9)
where h w and h q can be interpreted as Lagrange multipliers. Note that if ( )

, , 0 h h h h qp  = w ,u A ,
we obtain the following equality:

 = L= . The variation of L with respect to h  u , h p  , h  w ,
h q  and  may be written such as

h h h h h h h h pq D D p D D q D        = + + + + uw uw L = L L L L L ( 10
)
where D is the directional derivative operator. Setting 0

hh hh q D D q  += w w LL yields the state equation: ( ) , , 0

, ,

h h h h h h h h q p q       =     w ,u w A P (11)
Eq. 11 is the forward problem, whose resolution yields h u and h p .

Subsequentially, if we set 0

hh hh p D D p  += u u LL
, the adjoint equation can be obtained: where ( ) ( )

( ) ( ) , ; , , , h h h h h h h h m h h h h q p p p        = - -    
0 , ; , , h h h h h h h h h h h h d q p p q p p d        → = + + w , , u u w , ,u u BA is the linearization of  A .
Eq. ( 12) is the adjoint equation and h w can be obtained accordingly. With the acquired state variables h u , h p and the dual variables h w , h q , the gradient of the objective function with respect to the shear modulus is deduced such as:

( )

2 2 d ; , , c h h h h D q p             =  + +  = w , , u L = L C (13)
where ( ) ( ) 0 ; , , , ,

h h h h h h h h d q p q p d        → =+ w , , u w , ,u CA .
With the objective function and spatial variation of the objective function with respect to the shear modulus, we can update the shear modulus distribution using the L-BFGS method [START_REF] Byrd | A limited memory algorithm for bound constrained optimization[END_REF]. This process is repeated until the objective function, or the L2 norm of gradients of the objective function with respect to nodal shear moduli, become lower than the machine precision.

In summary, the algorithm of the inverse solver is shown in Fig. 2. Fig. 2: Inverse algorithm diagram

3.Results

Numerical Examples

The proposed inverse approach for the identification of nonhomogeneous shear modulus distribution in the cell nucleus was first tested with several simulated cases of deformation generated by finite element method (forward problem), as shown in Fig. 3. We built a finite-element model of cell nucleus with regions of heterochromatin (H) and euchromatin (E)

following the validation example introduced in [START_REF] Ghosh | Deformation microscopy for dynamic intracellular and intranuclear mapping of mechanics with high spatiotemporal resolution[END_REF], shown in Fig. 3(b). Each domain was assigned a different stiffness, the A and B regions having the same stiffness because they belong to the same chromatin domain. Subsequently, the uniform tension of 500kPa along the outward normal of the boundary was applied to the boundary of the cell nucleus, and finite-element simulations were performed with the commercial FE software ABAQUS, while assuming an incompressible behavior. The resulting simulated full-field displacement fields were used as inputs for testing the inverse solver (then using simulated data instead of actual data). To account for the unknown force information in actual nuclear deformation microscopy. we have to apply the displacements on the entire boundary in the inverse problem. Furthermore, various levels of Gaussian noise were added to the simulated displacement fields to simulate measurement errors.

The noise level was defined such as:

( ) ( ) NN 2 1 2 1 noise level 100% m ii i NN i i uu u = = - =   ( 14
)
where NN is the total number of nodes in the meshed nucleus, i u and m i u are the exact nodal displacement and the "noisy" nodal displacement at the i-th node, respectively. The ratio between the shear modulus of the H (heterochromatin) region and the shear modulus of the E (euchromatin) region is denoted as the H:E ratio. This ratio is assigned in the finite-element model to simulate the nuclear displacement fields and the goal of the inverse solver is to recover the correct ratio, namely the target H:E ratio.

When the H:E ratio is 2:1, as depicted in Fig. 4, the inverse solver is able to correctly recover the target shear modulus distribution even when the noise level is at 10%. However, as the noise level increases, errors increase at the boundaries of the H domain. In Fig. 5, the displacement and strain fields from the forward simulation with different noise levels and the inverse results were thoroughly compared. We observed that with the increase of the noise level, the strain fields from the forward simulation become seriously noisy compared to the associated displacement fields. A similar trend can be observed in Figs. 6 and7, where the target H:E ratio is 10:1. However, the shape of the H domain deteriorates when the noise level exceeds 4%. As shown in Table 1, the average shear moduli in the two regions (A and B) of the H domain are very close to the target values. The maximum error is less than 5%, indicating a strong reliability of the inverse solver. These results demonstrate the feasibility and robustness of the inversion method for characterizing nonhomogeneous shear modulus distributions of the cell nucleus in the presence of a Gaussian noise. Since the cell nucleus might be compressible [START_REF] Rowat | Mechanical properties of the cell nucleus and the effect of emerin deficiency[END_REF], the influence of Poisson's ratio on the reconstructed results was considered. To this end, we simulated nuclear displacement fields with Poisson's ratio v=0. [START_REF] Weijer | Collective cell migration in development[END_REF] and checked what would be the impact on the inverse solver if we assume that the cell nucleus is incompressible in the inverse problem (fix the Poisson's ratio v=0.5 in solving the inverse problem). The obtained results demonstrate that the regions of the H domain could still be well recovered even when a wrong Poisson's ratio is assumed (see Figs. 8 and10).

We also compared the displacement and strain fields from the forward simulation with different noise levels and the inverse results are shown in Figs. 9 and 11. Although relative error for H:E ratio is larger as shown from the calculated values, the maximum relative error in this case remains below 15% even in the presence of 10% noise (see Table 2). This demonstrates that the sensitivity of stiffness reconstruction to the Poisson's ratio is relatively small. The reliability of the plane stress assumption necessitates validation. Therefore, we modeled a homogeneous, incompressible 3D spherical nucleus subjected to a uniform tension of 500kPa applied on its equatorial plane, as illustrated in Fig. 12(a). There are two ellipsoids with the same dimension embedded in the sphere. The sphere was discretized into 62084 tetrahedral elements.

Using the displacement fields in the equatorial plane, we identified the elastic map with our inverse solver, as shown in Fig. 12(b). We assumed that the plane was under a state of plane stress. The reconstructed stiffness, as depicted in Figs. 12(c) and 12(d), reveal that the H:E distributions align closely with the target. The relative error for H:E ratios is kept under 1% in both scenarios. In the strain fields shown in Fig. 13, the two inclusions can be clearly observed from the strain image. These findings suggest that the bias due to the plane stress assumption in the identification process is not significant for a spherical nucleus. However, it is worth noting that the previous 3D model exhibited a significant degree of symmetry. We also explored an alternate 3D model configuration with two irregular inclusions embedded within the upper portion of the spherical nucleus, as shown in Fig. 14. For the mechanical loading conditions, we applied compression to the upper surface while constraining the lower region. To solve the inverse problem, we selected the plane corresponding to the upper surfaces of these two inclusions for analysis. When dealing with a lower target H:E (Hardnessto-Elasticity) ratio of 2:1, the quality of the reconstruction results was notably suboptimal. For a more substantial target H:E ratio of 10:1, although the two inclusions can be detected (as shown in the strain maps of Fig. 15), we observed that the reconstructed H:E ratio deviated significantly towards 4.0, a value substantially lower than the target. This discrepancy highlights that the plane stress assumption is a very strong assumption in our study which is not satisfied in case on non-symmetric internal morphology of the cell nucleus.

Examples with experimental data

After validating the inverse solver using different simulated cases, we applied the technique on displacement fields measured in the nuclei of two sample cardiomyocytesone in physiological and another in pathological condition. In solving the inverse problem, the bound of the shear modulus ratio was set to [START_REF] Jacobs | Introduction to cell mechanics and mechanobiology[END_REF][START_REF] Ghosh | Deformation microscopy for dynamic intracellular and intranuclear mapping of mechanics with high spatiotemporal resolution[END_REF]. In Fig. 16, the raw image, the displacement fields, and the strain fields are shown. We observed that the resulting strain components can be higher than 10%. This demonstrates that the cell nucleus undergoes a large deformation (see Fig. 16(d)-(f)).

Furthermore, in Fig. 16(m), the heterochromatin is primarily gathered on the left side of the cardiomyocyte nucleus, consistently with the H domain shown through pixel grayscale intensity (see Fig. 16(l)). We also observed that the H:E distributions acquired by the pixel grayscale intensity or the adjoint-based inverse solver are completely different from the strain fields. Thus, it is impossible to determine the H:E distributions qualitatively based on the associated strain fields. Note that our approach is not only able to recover the H domain as this can also be done based on pixel grayscale intensity, but our approach can also identify stiffness heterogeneities within the H domain itself. Table 3 reports the H:E values at three different regions. They remain within the same range order of magnitude as the average H:E value obtained from the pixel grayscale intensity. Additionally, we conducted a comparison of the reconstruction outcomes under varying regularization factors, as illustrated in Fig. 17. Our observations revealed a noteworthy trend: for smaller regularization factors, the boundary delineating the heterochromatin region exhibits a pronounced sharpening effect. Conversely, as the regularization factor increases, the resultant elasticity map of the nucleus tends to manifest an undesirable degree of smoothing, thus potentially compromising the accuracy of the reconstructed model.

For the other cardiomyocyte sample shown in Fig. 18, the strain level is higher than what we observed in Fig. 16. Moreover, the H:E distribution identified by the adjoint-based inverse solver (see Fig. 18(m)) is different from the one acquired by the pixel grayscale intensity (see Fig. 18 (l)).

More specifically, there is a region of very low stiffness in the center of the nucleus in both H:E distributions but its location and dimension are different in Figs.18(l) and 18(m). Nevertheless, the H:E ratios were in the same range order as the average H:E ratio of [START_REF] Ghosh | Image-Based Elastography of Heterochromatin and Euchromatin Domains in the Deforming Cell Nucleus[END_REF] (see Table 3). The difference between the H:E ratio in region A and the average H:E ratio of ref [START_REF] Ghosh | Image-Based Elastography of Heterochromatin and Euchromatin Domains in the Deforming Cell Nucleus[END_REF] is only 4%. Table 3: Estimated H:E ratios at regions A, B and C for the two nuclei shown in Figs. [START_REF] Moeendarbary | Cell mechanics: principles, practices, and prospects[END_REF] and 18(Note that H:E ratio predicted in [START_REF] Ghosh | Image-Based Elastography of Heterochromatin and Euchromatin Domains in the Deforming Cell Nucleus[END_REF] was solved by an optimization scheme assuming H and E domains are homogeneous) H:E=5.03 [28] H:E=2.51 [28

4.Discussion

This paper presents a novel approach to identify the stiffness distribution within the cell nucleus from high-resolution displacement map. The approach relies on an in-house adjoint-based inverse solver. Since the cell nucleus undergoes deformations with strains larger than 5%, the inverse solver is based on the finite deformation theory. The major achievement is to reconstruct heterogeneous stiffness distributions without any a priori assumption about the distribution of heterochromatin and euchromatin. Results show that the stiffness distribution in the heterochromatin domain is not homogenous, which can be attributed to the variations in density across the cell nucleus. Another important observation is that the boundaries of the heterochromatin domain cannot be detected from the strain fields due to the amplification of the noise level when computing strain by differentiating the measured displacement fields. The inverse solver addresses these effects by the regularization term. Another important point is that the shear modulus distribution can only be mapped up to a multiplicative factor as only displacements are measured but no force value is available. To obtain quantitative information on the shear moduli of the cell nucleus, tractions at the boundary would need to be measured, or alternatively, if the density of the cell nucleus is known, its spatial variation of shear moduli could be quantified by measuring dynamic deformations [START_REF] Wijesinghe | Ultrahigh-resolution optical coherence elastography images cellular-scale stiffness of mouse aorta[END_REF][START_REF] Leartprapun | Photonic force optical coherence elastography for three-dimensional mechanical microscopy[END_REF].

The stiffness map of the cell nucleus exhibits significant sensitivity to image quality, thereby presenting an opportunity to enhance the resolution of the stiffness map through improvements in deformation microscopy resolution. Figs. [START_REF] Moeendarbary | Cell mechanics: principles, practices, and prospects[END_REF] and 18 demonstrate that the adjoint-based inverse solver effectively captures the topology of the heterochromatin region, akin to the information obtained from pixel grayscale intensity. Consequently, employing the pixel grayscale intensity distribution as the initial estimation in the optimization problem holds potential for enhancing both convergence performance and accuracy.

Although the cell nucleus may typically be assumed to be incompressible or nearly incompressible as most living matter [START_REF] Chan | Volume transitions of isolated cell nuclei induced by rapid temperature increase[END_REF], experiments using micropipette aspiration have shown significant volumetric strain in the nucleus [START_REF] Rowat | Mechanical properties of the cell nucleus and the effect of emerin deficiency[END_REF]. Therefore, it was important to investigate the impact of the Poisson's ratio on the identification of nucleus stiffness. Figs. [START_REF] Kliche | Role of cellular mechanics in the function and life span of vascular endothelium[END_REF] and 10 show that the identified stiffness maps are in agreement with the shape of the heterochromatin domain independently of the Poisson's ratio. Moreover, the estimated H:E stiffness ratio is only slightly influenced by the Poisson's ratio. This observation is consistent with a previous study at the tissue scale [START_REF] Mei | Introducing regularization into the virtual fields method (VFM) to identify nonhomogeneous elastic property distributions[END_REF].

In this study, we employ the assumption of plane stress. Given the typically spherical or oval shape of the nucleus, as referenced in [START_REF] Webster | Sizing up the nucleus: nuclear shape, size and nuclear-envelope assembly[END_REF], the application of either 2D plane stress or plane strain assumptions may be inappropriate. This could potentially induce a bias in the identification process. For a symmetric model as shown in Fig. 12, the error generated by the plane stress assumption is notably minor, the identification bias could escalate with variations in shape or loadings such as the case presented in Fig. 14. Consequently, to overcome this limitation, the implementation of three-dimensional full-field measurements becomes essential in the future.

Our future work will primarily focus on improving the resolution of deformation microscopy and the accuracy of the inverse scheme to increase the robustness and reliability of the proposed method. In this study, the nucleus was assumed to follow the simple neo-Hookean model, and further research is needed to investigate whether the nonlinear elastic behavior of the cell nucleus follows a more complex hyperelastic model under very large deformations. Furthermore, understanding the relationship between the nuclear stiffness distribution and distribution of epigenetic modifiers [START_REF] Nims | Mechanogenetics: harnessing mechanobiology for cellular engineering[END_REF] is crucial because a denser chromatin region is most likely enriched with gene repression specific epigenetic modification. More investigation is needed to relate nuclear stiffness and gene expressions to potentially uncover new insights into the role of nuclear mechanics in cellular biology.

5.Conclusion

In this paper, we combined deformation microscopy-based displacement map and an adjointbased inverse solver to nondestructively identify the nonhomogeneous stiffness distribution of the cell nucleus. The feasibility of the proposed method was successfully demonstrated using simulated data generated by finite-element simulations. Our results indicate that the distribution of the heterochromatin domain can be characterized by the inverse solver, with a maximum relative error of less than 5% for the H:E ratio. We also found that the reconstructed heterochromatin domain is not sensitive to the Poisson's ratio. Finally, we obtained the nuclear stiffness distribution in beating cardiomyocytes using deformation microscopy-based displacement map and previously found two-domain elastography map. Overall, the proposed approach shows great potential for nuclear elastography, with promising directions to investigate mechano-genetics and mechano-epigenetics.

Fig. 1 :

 1 Fig.1: Schematic representation of the deformation microscopy setup to measure the

Fig. 3 :

 3 Fig.3: (a) Microscopy image of the nucleus in the medial plane; (b) Finite-element mesh and

Fig. 4 : 1 Fig. 5 :

 415 Fig.4: The reconstructed shear modulus distribution of the nucleus for varying noise levels when

Fig. 6 : 1 Fig. 7 : 6 .

 6176 Fig.6: The reconstructed shear modulus distribution of the nucleus for varying noise levels when

Fig. 8 :Fig. 9 : 8 .

 898 Fig.8: Reconstructed shear modulus distribution of the nucleus for varying noise levels when the

Fig. 10 :Fig. 11 : 10 .

 101110 Fig.10: Reconstructed shear modulus distribution of the nucleus for varying noise levels when

Fig. 12 :

 12 Fig.12: (a) 3D spherical nucleus subjected to a uniform tension in its equatorial plane; (b) Target

Fig. 13 :

 13 Fig. 13: Displacement and strain fields obtained with the forward simulation and with the inverse results for all cases shown in Fig. 12. The 1 st and 2 nd columns correspond to the x and y displacement components, respectively. The 3 rd , 4 th , 5 th columns correspond to three in-plane

Fig. 14 :

 14 Fig.14: (a) 3D spherical nucleus subjected to a uniform compression on the upper surface with a

Fig. 15 :

 15 Fig. 15: Displacement and strain fields obtained with the forward simulation and with the inverse results for all cases shown in Fig. 14. The 1 st and 2 nd columns correspond to the x and y displacement components, respectively. The 3 rd , 4 th , 5 th columns correspond to three in-plane

Furthermore, it isFig. 16 :

 16 Fig.16: Spatial maps of stiffness were computed in the deforming nucleus using inverse methods.

Fig. 17 :

 17 Fig.17: Spatial maps of stiffness for varying regularization factors.

Fig. 18 :

 18 Fig.18: Spatial maps of stiffness reveal heterogeneous material properties in the deforming

Fig. 19 :

 19 Fig.19: Spatial maps of stiffness for varying regularization factors.

Table 1 :

 1 Identified H:E ratios in regions A and B for the reconstruction shown in Figs.4 and 6

	Noise	H:E=2:1		H:E=10:1
	level/%	A	B	A	B
	0	2.03	2.05	10.06	10.29
	2	2.02	2.00	10.06	9.99
	4	2.00	1.98	9.99	10.03
	6	2.01	1.90	10.13	10.08
	8	2.04	1.94	9.84	9.92
	10	2.09	2.00	10.34	9.81

Table 2 :

 2 Identified H:E ratios in regions A and B for the reconstruction shown in Figs.8 and 10

	Noise	H:E=2:1		H:E=10:1
	level/%	A	B	A	B
	0	1.91	1.95	9.47	10.29
	2	1.95	1.93	9.54	10.32
	4	1.95	1.85	9.77	10.15
	6	1.94	1.80	9.76	9.68
	8	1.86	1.75	9.91	9.46
	10	1.94	1.72	10.66	8.74

Acknowledgments

The authors acknowledge the support from the National Natural Science Foundation of China (12002075) and the National Natural Science Foundation of Liaoning Province in China (2021-MS-128). The authors additionally acknowledge funding from the National Science Foundation (CMMI 2212121 and CMMI/CAREER 1349735).

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


Maxime Wetta

Jean-Benoit Kopp

Louise Le Barbenchon

Philippe Viot

A thick cellular structural adhesive : Identification of its behavior under shear loading

Keywords: Foam, Epoxy adhesive, Arcan shear test, X-Ray Tomography, Microstructure

This study focuses on the link between the microstructure and the mechanical behavior under shear loading of a thick cellular structural adhesive (TCSA).

X-ray microtomography and image post-processing were first used to perform 3D-quantitative microstructure analysis. The cells morphometric parameters and their orientations were studied. The foaming process boundary conditions seems to create local density gradients changing the cells dimensions and shape. The cells are more spherical in the core of the material whereas being more ellipsoidal close to the upper and lower faces of the samples creating a skin layer. The effect on the strain field of this skin layer has then been highlighted . Secondly, a shear test method using an Arcan set-up coupled with digital image correlation was used and allowed to observe the mechanical behavior of the material under shear loadings. Instead of the material being cellular and heterogenous, it has been found that the strain field can be considered homogeneous at macroscopic scale to extract the properties on a homogeneous equivalent material. Shear test on samples with different

Introduction

Structural bonding is a stiffness reinforcement technique that has been developing over the last 50 years. There are other joining methods such as riveting, welding, and bolding but these techniques create stress concentrations as well as oxidation and weight problems. Therefore, bonding is very popular in a large number of industrial sectors that use metals and composites: the transportation industry (cars, boats, airplanes, trains), civil engineering (solar panels, bridges) or medical prosthesis. For structural adhesives, the base material is synthetic polymers like epoxy (thermosetting) or acrylic (thermoplastic) [START_REF] Cognard | Science et technologie du collage[END_REF] resins.

During their lifetime, structural bondings may endure all types of loadings. These materials are usually strongest in shear and compression but relatively weak under peel and cleavage. The design of the joints should take this into account to reach the highest resistance [START_REF] Adderley | Adhesive Bonding[END_REF]. Manufacturers and suppliers consider that the main role of the adhesive is to transfer load from one adherend to another through a shearing action [START_REF] Cognard | Science et technologie du collage[END_REF]. Thus, the mechanical characteristics of the material under shear loading are very important for design calculations.

Several shear tests are used to characterize adhesives. The most commonly used in the industry are standardized as Single Lap [START_REF]D14 Committee, Test Method for Thick-Adherend Metal Lap-Shear Joints for Determination of the Stress-Strain Behavior of Adhesives in Shear by Tension Loading[END_REF] or TAST for thicker adhesives [START_REF]Test Method for Apparent Shear Strength of Single-Lap-Joint Adhesively Bonded Metal Specimens by Tension Loading (Metal-to-Metal)[END_REF]. There is also the Scarf test [START_REF] Liao | Effect of adhesive thickness, adhesive type and scarf angle on the mechanical properties of scarf adhesive joints[END_REF], which is not standardized. However, the Arcan test, initially developed by the researcher of the same name [START_REF] Arcan | A method to produce uniform planestress states with applications to fiber-reinforced materials: A specially designed specimen yields material properties under pure shear or uniform plane-stress conditions[END_REF], has been shown to create uniform plane-stress states [START_REF] Jouan | A critical comparison of shear tests for adhesive joints[END_REF][START_REF] Cognard | Analysis of the nonlinear behavior of adhesives in bonded assemblies-Comparison of TAST and Arcan tests[END_REF] in adhesive bonds. Since then, this solution has been used with digital image correlation (DIC) to characterize the elastic shear moduli of orthotropic materials [START_REF] Xavier | Non-linear shear behaviour of bovine cortical bone by coupling the Arcan test with digital image correlation[END_REF][START_REF] Bilko | Determination of the Shear Modulus of Pine Wood with the Arcan Test and Digital Image Correlation[END_REF], to test adhesively-bonded assemblies for the automotive industry [START_REF] Alfonso | Use of the modified Arcan fixture to study the strength of bonded assemblies for automotive applications[END_REF].

Several studies have shown that the properties of conventional adhesive joints tend to decrease with increasing joint thickness. Sources agree that the thickness impacts the stress field due to parasitic moments and stress concentration related to the structure [START_REF] Burst | Investigating the Thin-Film Versus Bulk Material Properties of Structural Adhesives[END_REF][START_REF] Taib | Bonded joints with composite adherends. Part I. Effect of specimen configuration, adhesive thickness, spew fillet and adherend stiffness on fracture[END_REF][START_REF] Grant | Experimental and numerical analysis of single-lap joints for the automotive industry[END_REF]. It is then necessary to also take into account bulk properties rather than just joint properties [START_REF] Davies | Créac'hcadec, Influence of adhesive bond line thickness on joint strength[END_REF][START_REF] Zuo | Review of fatigue of bulk structural adhesives and thick adhesive joints[END_REF].

But most studies on structural adhesives involve bond lines that are between 0.1 and 2-3 mm thick. Studies on thicker adhesives (5 to 10 mm) are limited, recent and focused to specific fields such as the marine and wind industries [START_REF] Fernandes | From thin to extra-thick adhesive layer thicknesses: Fracture of bonded joints under mode I loading conditions[END_REF][START_REF] Zuo | Review of fatigue of bulk structural adhesives and thick adhesive joints[END_REF][START_REF] Delzendehrooy | A comprehensive review on structural joining techniques in the marine industry[END_REF].

To link part of structures with large gaps and complex geometry between parts while limiting the final weight, expanded structural adhesives have been developed in the industry. By using cellular-structured composite materials or high performance bonding resins like epoxy, the mechanical characteristics of the structure can be improved with a reduced weight [START_REF] Wilson | Vehicle weight is the key driver for automotive composites[END_REF][START_REF] Hoge | Epoxy resin infused boat hulls[END_REF][START_REF] Ding | A Review on Microcellular Injection Moulding[END_REF]. However, the knowledge on these cellular-structured adhesives in the literature is poor and the influence of the manufacturing process on their mechanical properties little understood. The state of industrial knowledge in the sector indicates that these materials have a cellular material behavior at the macroscopic level. However, in some cases, such as interface fractures, the behavior would seem to be closer to a classical adhesive. It is therefore necessary to explore knowledge from both adhesives and cellular materials fields of study.

Historically, many studies have focused on metallic foams which have very good mechanical properties [START_REF] Davies | Metallic foams: their production, properties and applications[END_REF][START_REF] Gibson | Mechanical Behavior of Metallic Foams[END_REF]. But in the last three decades the use of polymeric foams has increased significantly due to the improved strength of the materials coupled with their intrinsic low density.

Studies of polymeric cellular materials began really in the 1980s with, for example Siegmann et al. [START_REF] Siegmann | Mechanical behavior of reinforced polyurethane foams[END_REF] predicting elastic properties of polyurethane foams using the Kerner equations [START_REF] Kerner | The Elastic and Thermo-elastic Properties of Composite Media[END_REF]. Gibson and Ashby [START_REF] Gibson | Cellular Solids: Structure and Properties[END_REF] provided then description of the relationship between the cell structure morphology and the governing parameters of the mechanical behavior. Fabrice Saint-Michel et al. [START_REF] Saint-Michel | Mechanical properties of high density polyurethane foams: I. Effect of the density[END_REF] verified the effect of the density on the mechanical properties of high density polyurethane foams comparing Gibson and Ashby [START_REF] Gibson | Cellular Solids: Structure and Properties[END_REF] and Christensen and Lo [START_REF] Christensen | Mechanics of low density materials[END_REF] models. The macroscopic response of foams under high compression rates have also been shown to be linked to microstructure [START_REF] Viot | Multi-scale foam behaviour characterisation[END_REF][START_REF] Viot | Polymeric foam behavior under dynamic compressive loading[END_REF][START_REF] Bouix | Polypropylene foam behaviour under dynamic loadings: Strain rate, density and microstructure effects[END_REF] as well as its fracture properties [START_REF] Mars | Fracture toughness of rigid polymeric foams: A review[END_REF].

The foaming process for the cellular material creation is important to understand. Indeed, it is known in the literature, as for example for injection process, that anisotropy can appear in cellular structures due to the nucleation [START_REF] Ding | A Review on Microcellular Injection Moulding[END_REF][START_REF] Dong | The cell forming process of microcellular injection-molded parts[END_REF]. Gomez-Monterde et al. [START_REF] Gómez-Monterde | Effect of microcellular foaming on the fracture behavior of ABS polymer[END_REF] have shown that the mechanical properties can be modified because of this phenomenon. The anisotropic behavior of the polymer increased due to the cell elongation caused by the injection flow.

Few studies exist in the literature to investigate in detail the behavior under shear stress of cellular materials. The shear strain of cellular materials have recently been locally analyze with DIC and Arcan fixtures [START_REF] Huo | Mechanical characterization and numerical modeling on the yield and fracture behaviors of polymethacrylimide (PMI) foam materials[END_REF][START_REF] Taher | Determination of mechanical properties of PVC foam using a modified Arcan fixture[END_REF]. The study of the material remains however often external and not volumetric.

The objective of this paper is a description and an analysis of the link between cellular microstructure of a thick cellular structural adhesive (TCSA) and its behavior under shear loading. The morphology heterogeneity in the material due to the process is first studied. A 3D-quantitative microstructure analysis is made possible thanks to X-ray microtomography and image post-processing. Then, a method using the Arcan test coupled with DIC is developed to characterize the mechanical behavior of the TCSA material under shear loading. The study will focus on samples of two different densities and then two different microstructures.

Material and Methods

In this section the studied material will be described. Then the X-Ray tomography images acquisition and their treatment are presented before a mechanical testing method using the Arcan shear test and DIC.

Thick cellular-structured adhesive (TCSA)

Epoxy resin is a thermosetting polymer commonly used for strong adhesion. It expands when cured at high temperature via the incorporation of a blowing agent reacting while the cross-linking of the resin is occurring. The resulting material is a thick adhesive with a cellular microstructure.

Material samples of dimensions ≈ 50 × 30 × 4 mm 3 are cut from an A5sized injected plate before being expanded using the assembly presented in the Figure 1. This assembly consists of two rigid metallic plates spaced by hollow shims of known thickness. Release paper is added between the plate and the material to prevent the adhesion of the material to the metal. The objective is first to control the density of the material by controlling the height of expansion. Samples with two different thicknesses t were manufactured : t = 5 mm and t = 7 mm. To describe cellular materials, the notion of relative density ρ r = ρ * /ρ s is commonly introduced. The density ρ * is that of the cellular material while ρ s is that of the same fully dense material. Two samples of the two different densities ρ r were cut with a band saw to obtain parallelepipeds of dimensions ≈ 45 × 10 × t mm 3 . They were then observed by X-ray tomography as explained in the section titled X-Ray micro-tomography.

Parallelepipeds were cut out of the expanded material samples using a band saw. To achieve the desired dimensions of 40 × 21 × e mm 3 , a sander with a P120 grit disc was used (step 1 in Figure 2). They were then glued with a device to create the Arcan test samples. The steps (step 2 for gluing and 3 for verification) in Figure 2) are described in the section Arcan shear test -sample preparation.

X-Ray micro-tomography

It is difficult to properly analyze the cellular materials structure with only 2D methods like light or scanning electron microscopy. Indeed, there is a loss of information into volume such as the coalescence of the cells. To eliminate this problem and observe in 3D, X-ray microtomography with a GE v|tome|x s tomograph available at Placamat Bordeaux was used. The general principle of the X-ray microtomography technique applied to the material science has already been described in an article by Maire et al. [START_REF] Maire | On the Application of X-ray Microtomography in the Field of Materials Science[END_REF].

The resolution was 8.5 µm/voxel for the low density sample and 10 µm/voxel for the high density one. The first one having a smaller thickness, it was possible to obtain a slightly better resolution with the laboratory tomograph. However, this did not result in a better observation afterwards, the difference being weak. Whatever the density, this technique is assured to have sufficient resolution regarding the size of the cells.

The 3D reconstruction of the samples and the quantitative analysis of the structural parameters of the material are performed using the Amira-Avizo software.

The study volume, also called volume of interest (VOI), is a prismatic subvolume. It is in this VOI that the morphometric parameters are calculated. This sub-volume is chosen with dimensions slightly smaller than the sample to avoid edge effects due to sample preparation. For the low density sample, its dimensions are 9×6×10 mm 3 . To keep the same width/height ratio, with the high density sample the dimensions of the VOI are 8.1 × 4.3 × 10 mm 3 .

3D Image analysis

The images being originally in 16 bits were converted to 8 bits (see in Figure 3a). Due to the large difference between the linear attenuation coefficients of air and epoxy, the grey-level distribution of the images is almost bimodal. A threshold then allows the image to be binarized as needed (see in Figure 3b). The pixels have intensity values from 0 to 90 for the cells (the void) and from 90 to 255 for the polymeric phase.

The 3D segmentation method is divided into three main steps [START_REF] Dillard | 3D quantitative image analysis of open-cell nickel foams under tension and compression loading using X-ray microtomography[END_REF]. First, the distance map image of the cells is computed. The right markers are then selected by topographic conditions. Finally, the watershed method is constructed from these markers with the help of the distance function (see results in Figure 3c). Then, one can label and study the cells one by one (Figure 3d). It is possible to analyze separately the polymeric phase (in Figure 4a)) or the porosities (in Figure 4b)).

Studies on cork [START_REF] Le Barbenchon | Multi-scale foam : 3D structure/compressive behaviour relationship of agglomerated cork[END_REF] and aluminum [START_REF] Olurin | The investigation of morphometric parameters of aluminium foams using micro-computed tomography[END_REF] foams highlight the difficulty of image segmentation on heterogeneous cellular materials. There is not yet a technique to quantify the accuracy of this method. Recently, deep learning was tested to improve these algorithms [START_REF] Liu | Efficient image segmentation based on deep learning for mineral image classification[END_REF]. The considered parameters are the length, width, volume, orientation, anisotropy and sphericity of the cells.

The length and width of the cells are defined as the maximum and minimum values of the Feret diameters respectively. One recalls that the Feret diameter is a one-dimensional measurement to estimate the size of an object along a direction. For the analysis of a 3D particle, the object is projected in a 2D plane and the Feret diameter is the distance between two parallel tangent lines [START_REF] Merkus | Particle Size, Size Distributions and Shape[END_REF].

All cells with a volume smaller than 10 -6 mm 3 are removed. It is impossible with the available resolution to identify them. The classical spherical coordinate angles θ and ϕ are used for the cell orientations. The mathematical convention calling θ the azimuthal angle and ϕ the polar angle is used to to simplify software processing (see in Figure 5). Sphericity ψ measures how closely the shape of an object approaches a mathematically "perfect" sphere. It is calculated according to Eq. 1

ψ = π 1 3 (6V p ) 2 3 A p , (1)
with V p , the volume of the particle and A p , the external surface area of the particle. The nearer to 1, the closer to a sphere.

Arcan shear test -sample preparation

The Arcan device consists of a bonded specimen clamped in two half disk shaped pieces. The Arcan adherends are manufactured from Aluminium 7075

T6 (E = 72 GP a and ν = 0.33). The size of the bonding surface is 40×21 mm (840 mm 2 ). The metallic adherends were cleaned with alcohol and acetone before being air dried. After setting and clamping the adherends, Loctite EA 9466 epoxy adhesive was applied. The material specimens were then placed on the adherends. The assembly was then put in place as shown in the step 2 in Figure 2. Large shims are used to control the adhesive thickness at 0.5mm on each side. Excess adhesive was directly removed.. This step was performed a second time after 45 min, as the glue was not yet fully cured.

The assembly was then left for at least 24 hours at room temperature before disassembly to ensure the glue cross-linking. The final result is shown in step 3 in Figure 2.

The tests were performed on 10 high density and 15 low density samples.

The difference in macroscopic density between each sample for both sample types (high or low density) was equal or less than 0.01 g/cm 3

Test set-up and protocol

The Arcan tests were performed with the help of a Zwick Z010 tensile machine as shown in Figures 6a and6b. The force was measured with a 10 kN load cell. The crosshead was driven with a constant displacement speed of 0.5 mm/min. To limit the displacement uncertainty at the beginning of the test due to the gaps in the set-up, a preload of 50 N was systematically applied.

For image acquisition during the test, a Canon 800D camera with a macroscopic lens is used. Its resolution is 4000 × 6000 pixels. The camera is mounted on a tripod and the focus is adjusted manually. The LED light allows to adjust the brightness without increasing the temperature of the sample (checked with a laser thermometer). A signal is sent by the computer to acquire pictures at a sampling rate of 0.33 Hz. To be able to use the image correlation tools, a white and black paint speckle was sprayed on the material as well as on the metallic adherends. Both the adherends and the sample were placed in the field of view.

Digital image correlation (DIC)

It is therefore necessary to take into account local behavior of the materials under loadings. Digital image correlation (DIC) is a powerful tool to analyze locally small displacements of materials [START_REF] Yoneyama | Basic principle of digital image correlation for in-plane displacement and strain measurement[END_REF]. This method has already been used coupled with the Arcan test to characterize the shear be-havior of heterogeneous materials such as wood [START_REF] Bilko | Determination of the Shear Modulus of Pine Wood with the Arcan Test and Digital Image Correlation[END_REF] or bones [START_REF] Xavier | Non-linear shear behaviour of bovine cortical bone by coupling the Arcan test with digital image correlation[END_REF] under shear loading. Studies have also shown the usefulness of this technique to study the local behavior of foams under high speed compression [START_REF] Liu | Impact testing of poly-meric foam using Hopkinson bars and digital image analysis[END_REF], [START_REF] Roux | Three-dimensional image correlation from X-ray computed tomography of solid foam[END_REF].

In the Arcan test, the displacements can be described by a 2D analysis.

For the DIC processing, the Vic2D software from Correlated Solutions is used. The data are then exported and processed using GNU Octave and Matlab.

On each image a calculation area called "Area of interest" (AOI) is defined. These areas are divided into "subsets" which each has their own gray scale spectrum. The size of these subsets is chosen to obtain a good estimation of the displacement field. At least 20 subsets in length and 5 in height of the sample were chosen. The post-processing consists in calculating the Lagrange deformation tensors E from the displacement field. From these it extracts the engineer's deformations and more particularly the shear strain,

γ, γ = sin -1 2E xy (1 + 2E xx )(1 + 2E yy ) , (2)
knowing that x is defined according to the length of the sample and y according to its height (as seen in Figure 6). The evolution of the average γ angle during the test is then exported.

The shear modulus, the ultimate shear strength and the maximum shear strain of the material were calculated. To obtain the stresses the force acquired by the machine was divided by the initial sheared surface, S 0 . This surface was considered to be the cross-section of an equivalent homogeneous material of the same dimension. Displacements and shear strain are obtained by DIC measurement.

Results and discussion

Quantitative morphometric data extraction using 2D slices is not suitable. Indeed, a large part of the information is only accessible from volumetric data. The former will therefore allow a qualitative analysis of the phenomenon while all the quantitative data will be extracted from the volume reconstruction analysis explained in section 2.3. After sorting the data to remove the noise one studies about 20000 cells for the low density sample and 10000 for the high density sample. The volume ratio being only 1.55 it seems that there are less cells in the high density sample. For the sample with lower density, the expansion height and therefore the expansion time is higher. The small bubbles that are initially unobservable will continue to grow and thus be detectable.

Microtomography results

It is important to note that the majority of the cells are reconstructed interconnected. The resolution is sufficient to reveal that the structure is predominantly with open cells.

The Figure 8 shows the anisotropy of the cells related to their height in the sample. It can be observed that, for this parameter, there is a difference between the upper and lower parts of the sample and the core. In the upper and lower parts the distribution seems to be linear, being more normally distributed in the core. To study the differences of the cells in the thickness, three study areas in the sub-volume were defined. The base and the top will correspond to the lower and upper 20 % of the sub-volume. The core will correspond to the rest. In this way, the study of the lower and upper layers, also known as skins or skin layers in the literature [START_REF] Banhart | Deformation characteristics of metal foams[END_REF][START_REF] Barzegari | The effect of injection molding conditions on the morphology of polymer structural foams[END_REF][START_REF] Jawhar | Study of the Coalescence Mechanisms During Silicone Foaming[END_REF] , and their effects is studied. Thereafter the term "skin effect" will be used to describe the influence of the skin layer on the material properties. The difference between the base and the side where the expansion ends is also studied.

Cell size

The length and width median values and first (Q1) and third (Q3) quartiles of the cells are presented in Table 1. In the core of the samples the cell sizes are comparable between the two samples. However, at the extremities, the cells are wider and longer.

Q1 There are some large cells but a large majority (around 70 % of the cells) has a volume less than 0.01 mm 3 . Of these, the majority have a volume inferior to 0.002 mm 3 . There is no significant difference between core and skin.

Orientation of the cells

The orientations according to the angle θ of the cells is represented in the Figure 9. The cells are mostly oriented with an angle θ close to 0 • in both samples.

For the low density sample (Figure 9a), about 50 % of the cells are oriented with an angle lower than 20 • . Variations are less than 5 % for all angles and for all three study areas. That means that the orientations are equivalent regardless of the area and the cells are directed perpendicular to the expansion.

For the high density sample (Figure 9b), about 50 % of the cells are oriented with an angle lower than 20 • in the core area. In the extremities, about 50 % of the cells are oriented with an angle lower than 10 • . That means that the cells are directed quite perpendicular to the expansion in all the sample but a skin is visible with a more pronounced phenomenon in the upper and lower extremities.

The study of the orientation with the ϕ angle presents equivalent results for both samples. About 40 % of the cells are oriented with an angle higher than 70 • . There is a quite constant increase in the proportion of cells with increasing angle. For low angles the proportion is about 5 % up to about 20 % for the highest angles. This indicates that the majority of cells tends to be oriented in the XY plane.

Cell shape

The anisotropy of the cells for both samples is represented in Figures 10a and10b. Both samples show similar cell anisotropies. In the extremities one has less than 20 % of cells with anisotropy inferior to 0.5. There is then a rapid, almost linear increase in proportion as a function of anisotropy (see in Figure 10b). In the core, however, a more harmonious distribution of cells between anisotropies of 0.3 and 1 is observed (see in Figure 10a). The sphericity of the cells for both samples is represented in Figures 10c and10d . A difference of the cells sphericity between core and skin on the two samples is obvious.

It is remarkable that for both samples almost half (45 %) of the cells have a high sphericity superior to 0.9 in the core (see in Figure 10c) . The distribution is much more heterogeneous in the skins (see in Figure 10d). For the low density sample, 80 % of cells have a sphericity greater than 0.7 in the core while this proportion drops to 65 % in the skin. For the high density sample 75 % of cells have a sphericity greater than 0.7 in the core while 50

(base) to 60 (top) % in the skin.

It has been shown that there is a difference in morphometric parameters between the cores of the samples and the extremities in contact with the metal plates covered with anti-adhesion paper. This skin layer exists for both sample types. Core cells have a less elongated shape, greater sphericity, and an orientation more perpendicular to the expansion than at the extremities.

In all likelihood the boundary conditions during the foaming process influence the microstructure. Indeed, the gas is released into the soft material and disperses. It will be blocked by the metal plate on each side. Körner et al. [START_REF] Körner | Modeling of Metal Foaming with Lattice Boltzmann Automata[END_REF] illustrated the developing of metallic foams within the constraints of walls. Nadella et al. used platen to block and heat a polymeric material to control the shape of the cells through the thickness and create a skin [START_REF] Nadella | Constrained Solid-State Foaming of Microcellular Panels[END_REF].

These two papers thus present results similar to those obtained for these TCSA materials confirming the hypothesis.

Strain field observed with DIC

The Figure 11a shows the evolution of the shear strain field in the material during the test. The Figures b) to f) confirms that the field is quasihomogeneous in the core of the sample throughout the test. The image correlation shows that at maximum the tensile strain ϵ yy reaches a value of 0.0006 or 2.4% of the shear strain γ. The hypothesis of pure shear with small deformations is therefore assumed to be valid. Thus, the stress tensor T is,

T =      σ 1 τ 0 τ σ 2 0 0 0 0      small strain ≈      0 τ 0 τ 0 0 0 0 0      (3)
and the Green-Lagrange tensor E is,

E =      0 γ 0 γ γ 2 /2 0 0 0 0      small strain ≈      0 γ 0 γ 0 0 0 0 0      . ( 4
)
The Figure 11b shows the evolution of the shear strain in the thickness along the line shown in the Figure 11a a). For this example, the sample thickness is 7 mm and the position 0 is the middle of it. One observes a small asymmetry of the curve explainable by the movements of the adherends and has been seen with the DIC. One undergoes effort and movement and the other is fixed. The bonding with epoxy adhesive is stiffer than the epoxy foam (but well below the aluminum)and can explain the strain gradient near the interface.

When the glue was removed during the preparation of the samples, some of it may have remained on the substrate (as seen in Figure 2). On the tensile strain field in Figure 12 it seems that the red area fits the residual glue location. This can explain the small shear strain in the substrates area.

Sample failure occurs cohesively in the core of the sample, always on the side of the effort. These results validate the use of this method to characterize the mechanical properties of this type of material. It is known in the literature that thermosetting polymers like epoxy have low viscosity [START_REF] Mullins | 2 -Mechanical properties of thermosets[END_REF] allowing to put aside viscosity effects for the curve slope softening. Heterogeneous quasi-brittle materials such as concrete [START_REF] Li | Evaluation method of rock brittleness based on statistical constitutive relations for rock damage[END_REF] are known to be damaged with a stable propagation of micro cracks before an unstable propagation of cracks until macro failure. It is assumed that this variation is due to the presence of micro-damages of the material. More local observation like in-situ tests under tomography might confirm it. Fracture tests with compact tension samples for example could to confirm the quasibrittle behavior of this material.

One can therefore affirm the quasi-brittle behavior of this material. The average maximum shear strain and standard deviation are presented in the The elastic modulus is 26 % lower for the sample with the lowest density.

The average maximum shear strain is also different between the two types of samples. The denser samples have on average a 26 % larger maximum shear strain. However, the average maximum stress is comparable between both samples series. The difference between the two mean values is smaller than the standard deviation of each series. That means that the driving factor is the stress.

Foam-like behavior

In the Figure 13 one could observe a difference in shear behavior for both types of samples. There is however a difference in density between the two.

The tomography allowed to show that the structure looks like an open cell foam. Gibson and Ashby [START_REF] Gibson | Cellular Solids: Structure and Properties[END_REF] show with a micromechanical model that the shear modulus G * (and thus the stress) is related to (ρ * ) 2 by the relation :

G * E s = C 2 ρ * ρ s 2 (5)
For these samples E s and ρ s are constant and identical for both types of samples. The shear stress values from Figure 13 were thus divided by the density according to eq. 5 (see in Figure 14). Discrepancies between the two densities are largely reduced. Furthermore, the maximum shear strain is, on average, related to the density of the samples on the same order of magnitude as the shear modulus.

The increase in porosity will therefore make the material softer.

It would appear that material failure is not directly related to macroscopic density. There are probably micro-cracks in the material and the complex structure will determine how the material will fail.

The use of a model based on micromechanics at the cellular scale thus seems to be a relevant first approximation to describe the macroscopic behavior of the material. The material failure being however not explained by this model, a study of the behavior of the material at the cellular scale is necessary. In-situ testing under X-ray microtomography could provide answers to these questions.

Conclusion

In the present study, the behaviour of TCSA under shear loading was The Arcan shear test allowed to determine the mechanical properties of the material and its behavior under shear loading. The homogeneity of the strain field through the test was verified thanks to DIC method. The skin effect does not seem to be the most relevant parameter influencing the macroscopic elastic mechanical properties of the material. It was shown that using the relation linking the shear modulus to the density squared, proposed by Gibson and Ashby, is a first approximation, at this scale, to predict the macroscopic shear modulus and the maximum shear strain at failure of TCSA which is not the case for the maximum shear stress at failure.

More local measurements and observations would be needed in order to link the heterogeneous microstructure and the fracture properties. Moreover the adhesive properties of cellular structural adhesive could also be of interest.

  After expansion, samples with relative density ρ r,high = 0.49 (for t = 5 mm) and ρ r,low = 0.45 (t = 7 mm) are obtained. These macroscopic densities have been measured with laboratory density scales. By disregarding the adhesive function, one tries to determine only the volumetric properties of the material.

Figure 1 :

 1 Figure 1: Set-up controlling the expansion height.

Figure 2 :

 2 Figure 2: Gluing protocol for Arcan samples with a set-up allowing parallelism.

Figure 3 :

 3 Figure 3: Image processing steps (a)-(d) for the 3D reconstruction of the cells.

Figure 4 :

 4 Figure 4: 3D reconstruction of the polymeric phase (a) and the porosities only (b). The represented volume size is 9 × 6 × 10 mm 3 .

Figure 5 :

 5 Figure 5: Orientation of the cells in a spherical coordinate system.

  (a) Arcan set-up. (b) Arcan schematic drawing.

Figure 6 :

 6 Figure 6: Arcan set-up and schematic drawing.

Figure 7 :

 7 Figure 7: X-Ray Tomography slices for both high and small densities samples.

Figure 8 :

 8 Figure 8: Relationship between sample height and anisotropy.

  Angle θ high density.

Figure 9 :

 9 Figure 9: Comparison of the cells θ-orientation for the two samples.

  sample Base Low density sample Top High density sample Base High density sample (b) Anisotropy near the faces for both samples.

1 Core

 1 Low density sampleCore High density sample (c) Sphericity in the core for both samples. Sphericity near the faces for both samples.

Figure 10 :

 10 Figure 10: Anisotropy and sphericity comparison between core and extremities for both samples.

  (a) Sample size on picture (i), and strain field during the test at loading = 0 % Fmax (ii), loading = 25 % Fmax (iii), loading = 50 % Fmax (iv), loading = 75 % Fmax (v), and before fracture (vi). -0% F max (c) -25% F max (d) -50% F max (e) -75% F max (f) -95% F max Adhesive thickness 0.5mm (b) Evolution of the shear strain along the thickness.

Figure 11 :

 11 Figure 11: Analysis of the shear stress field during the test.

Figure 12 :

 12 Figure 12: Tensile strain field at the end of the test.

3. 6 .

 6 Figure 13 shows the results curves of the shear tests for the samples of the two different thicknesses and density.One can observe a first part of the curve approximately linear until 50% of the maximum loading. Then, a non-linearity implies a change in behavior.After reaching a maximum stress the sample breaks in a brittle way. The image acquisition frequency does not allow to study the behavior after the maximum effort, explaining the sudden end of the curve. The last point of the curve corresponds to the last moment before a fall to zero of the force, not represented in the figure for reasons of readability.

Figure 13 :

 13 Figure 13: Stress-strain Arcan curves for both types of samples.

Figure 14 :

 14 Figure 14: Stress-strain Arcan curves normalized by squared density.

  studied. It was shown the existence of a skin layer in the TCSA open-cell microstructure due to the foaming process. The cells were observed to be more spherical in the core of the material whereas they becomes elongated close to the upper and lower faces. Samples with different densities display similar microstructure.

Table 1 :

 1 length Median length Q3 length Q1 width Median width Q3 width Length and width statistic of the cells

	Low density	164 µm	294 µm	520 µm	111 µm	176 µm	271 µm
	High density	139 µm	262 µm	544 µm	88 µm	160 µm	278 µm

Table 2

 2 as well as the shear modulus (normalized by a fixed value) and its variation.

		Average max shear strain	Standard deviation	Average normalized shear modulus	Modulus variation
	low density	0.023	0.003	23.8	±10 %
	high density	0.029	0.007	32.2	±11 %

Table 2 :

 2 Mechanical properties comparison between low and high density samples.