# On imperfect KUBC with view toward experimental homogenisation

Robin Valmalette, Cristobal Romo-Alvarez, Martin Poncelet, Nicolas Auffray

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## To cite this version:

Robin Valmalette, Cristobal Romo-Alvarez, Martin Poncelet, Nicolas Auffray. On imperfect KUBC with view toward experimental homogenisation. 2025. hal-05189056

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# On imperfect KUBC with view toward experimental homogenisation

Robin Valmalette<sup>a,c,*</sup>, Cristobal Romo-Alvarez<sup>a,d</sup>, Maxence Wangermez<sup>a,e</sup>, Martin Poncelet<sup>a</sup>,
Nicolas Auffray<sup>b</sup>

<sup>a</sup> Université Paris-Saclay, CentraleSupélec, ENS Paris-Saclay, CNRS UMR 9026

LMPS – Laboratoire de Mécanique Paris-Saclay, Gif-sur-Yvette, 91190, France

<sup>b</sup>Sorbonne-Université, CNRS UMR 7190

Institut Jean le Rond d'Alembert, Paris, 75005, France

<sup>c</sup>Now at: Aix Marseille Univ, Centrale Marseille, CNRS UMR 7031

Laboratoire de Mécanique et d'Acoustique, Marseille, 13013, France

<sup>d</sup>Now at: Astemo France, Drancy, 93700, France

<sup>e</sup>Now at: Dassault Systemes Provence, Aix-en-Provence, 13082, France

## Abstract

This study investigates the use of imperfect Kinematic Uniform Boundary Conditions (KUBC) in experimental setups to measure the effective stiffness of architECTured materials. To this aim, a numerical model simulates various experimental configurations: the number of grips applying the discretized KUBC, their compliance and clearance, etc. Simulations are conducted on bulk, periodic, and quasi-periodic specimens to understand the interactions between the grips and the mesostructures. The paper compares results from this Experimental KUBC (E-KUBC) to classical numerical KUBC to assess accuracy. Errors from E-KUBC are typically within 2–5%, but can significantly rise for some specimen design choices. The study identifies design guidelines to mitigate these errors, including adding notches to lower the stiffness of the specimen boundary and avoiding matching the specimen pattern with machine one. Quasi-periodic structures tend to be less sensitive to these issues. Overall, E-KUBC setups can yield accurate stiffness measurements if carefully implemented.

**Keywords:** homogenisation, experiments, identification, architECTured materials, imperfect boundary conditions

## 1. Introduction

ArchitECTured materials (AMs) have become an important trend in industry research in recent years. Indeed, AMs allow the design of interesting mechanical properties directly from mesostructures, thus reducing, for example, the mass of components while increasing their performances (Ashby, 2013). Until a few years ago, these mesostructures were rarely possible to produce and were mainly limited to toy models for homogenisation methods (Milton and Cherkaev, 1995; Prall and Lakes, 1997), topological optimisation (Wang et al., 2003), or illustrations of generalisedcontinua (Alibert et al., 2003). The development of 3D printing techniques has been a game-changer

<footnote>*Corresponding author</footnote>

<footnote>Email addresses: robin.valmalette@ens-paris-saclay.fr (Robin Valmalette), cristobal-romo.alvarez@hitachiastemo.com (Cristobal Romo-Alvarez), maxence.wangermez@3ds.com (Maxence Wangermez), martin.poncelet@ens-paris-saclay.fr (Martin Poncelet), nicolas.auffray@sorbonne-universite.fr (Nicolas Auffray)</footnote>
since it enables more and more mesostructures to be accurately manufactured with various constitutive materials (Sai Saran et al., 2022). Now that industrial applications are possible, one of the significant research directions in AMs lies in the predictive modelling of their mechanical behaviour without and with manufacturing defects.

Because of the multi-scale nature of these materials, the associated numerical models are computationally intensive. Thus, direct numerical simulations of complete structures at the mesoscopic scale are still challenging (Patel et al., 2018). Moreover, they are ill-suited for providing information on the effective behaviour of the AM and, thus, for predicting the consequences of any mesostructure modification at the macroscopic scale. Relevant approaches are to use homogenisation schemes to assess the effective behaviour of the AM. In most cases, the models used in these simulations are based on perfect Computer-Aided Design (CAD) geometries and presumed constitutive material behaviour. Due to the significant discrepancies between the theoretical and actual mesostructure (Wang and McDowell, 2020), it is important to question the relevance of these homogenisation results. To this aim, recent developments have focused on the digitalisation of actual mesostructures in order to introduce them into multi-scale computations (Bénézech and Couégnat, 2019; Mendoza et al., 2019; Wangermez et al., 2020). These methods require expensive equipment since they are based on Computed Tomography (CT) scans. Moreover, the real behaviour of the parent material — potentially heterogeneous — is only assumed, which is another source of modelisation error.

In order to circumvent these limitations, one solution is to measure the effective behaviour of the mesostructure experimentally. Based on the numerical homogenisation techniques (Kanit et al., 2003), the effective behaviour might be measured on a Representative Volume Element (RVE) specimen with testing machines that would mimic the theoretical Kinematic Uniform Boundary Conditions (KUBC), Static Uniform Boundary Conditions (SUBC), or Periodic Boundary Conditions (PBC). Until now, SUBC and PBC experimental setups have not yet been designed, despite they could in principle be realised. Several machines applying in-plane Experimental KUBC (E-KUBC) already exist: Meng et al. (2013) for uniaxial tension, C.Jones (1992); Hitt and Gilbert (1994) for biaxial tension and Penzien and Didriksson (1964); Bjerrum and Landva (1966); McNatt (1969) for in-plane shear. Some mechanisms are even able to impose the three elementary KUBC at the same time: uniaxial tension, biaxial tension, and in-plane shear (Joer, 1991; Eberhardsteiner, 1995; Dassonville et al., 2020; François, 2024). Except Dassonville et al. (2020) machine, these multiaxial setups were never designed with homogenisation in mind. To date, all such machines apply imperfect E-KUBC, i.e. a discretised version of KUBC where the boundary conditions are imposed at a finite number of grips along the specimen boundary. However, to the best of the authors' knowledge, there have been no comprehensive studies investigating the consequences of highly degraded KUBC conditions on the measurement of effective properties.

This paper aims to investigate the accuracy of the measurement of the effective in-plane linear stiffness tensor when an imperfect KUBC testing machine is used. Given that multiple sources of defects can influence measurements, quantifying the expected error level is crucial. Additionally, the paper aims to propose specimen design guidelines (pattern scale, orientation ...) that ensure the accurate measurement of the elastic properties. It will highlight key features that should be considered during this homogeneization operation. In other words, the discussion will refrain from delving into particular technological choices and will instead address common sources of defects present in mechanical testing machines, including the discretisation level of the KUBC frame and mechanical defects: compliance, friction, and clearances. To this end, a digital twin of a generic E-KUBC setup is developed for numerical simulations of experiments of various specimens. It is inspired by the design proposed by Dassonville et al. (2020), where the square contour of the KUBC
<figure><img src="image_3.png" /><figcaption>Figure 1: Differences between Theoretical (T-KUBC), Numerical (N-KUBC), and Experimental (E-KUBC)(c). The grey areas represent the specimen, and the green lines correspond to the regions of imposed displacement. In (a) these regions are continuous and correspond to a linear displacement, in (b) these regions are a series of discrete points with imposed translation. In (c) they are grips with imposed ridi-body-motion translation, while their rotation can be chosen free or not depending on the design.</figcaption></figure>

frame is discretised into ten points per side. The specimens under consideration include bulk elastic materials, periodic and quasi-periodic lattice structures with different pattern geometries and size, as well as position and orientation within the specimen. The measurement involves determining the effective stiffness tensor through three successive loadings. Results are compared with numerical PBC homogenisation results as well as N-KUBC results.

The paper is structured as follows: Section 2 addresses the defects associated with E-KUBC and explores the various options available when designing specimens. Section 3 provides a detailed description of the methodology, including the numerical model of E-KUBC and the tested mesostructures. Section 4 and Section 5 present the results and discuss the different E-KUBC configurations and the effect of specimen selections.

## 2. Theoretical vs. experimental KUBC

The primary question explored in this work is: what is the extent of the error incurred when evaluating the effective stiffness of a material through an experimental KUBC approach compared to numerical and theoretical KUBC methods? Both numerical and experimental implementations inevitably approximate the theoretical KUBC. However, while refining the mesh in numerical simulations can minimize the associated error to negligible levels, the inherently coarser nature of experimental boundary conditions may lead to a more significant error.

Throughout this article, Theoretical, Numerical, and Experimental methods are distinguished using the prefixes T-, N-, and E-, respectively (e.g., N-KUBC). They can also be interpreted as T = theoretical, N = node-controlled (displacement imposed at mesh nodes), and E = edge-controlled (displacement imposed via rigid segments later called ‘grips’).

The first limitation arising from the T-KUBC approximation is kinematic. As shown in Fig. 1(a) and (b), the discrete nature of numerical models (N-KUBC) restricts the imposition of a continuous displacement field along the boundary; instead, displacements are prescribed solely at discrete mesh nodes. For the aforementioned E-KUBC experiments (Fig. 1(c)), the discretization is coarser than in the numerical scenario. Here, boundary displacement is approximated in a piecewise manner, with a set of $4 \times N_G$ grips (represented as green elements in Fig. 1(c)) imposing rigid-body motions at specific locations. In contrast, the boundary segments between these grips remain traction-free. For each grip $i \in [1, 4 \times N_G]$, a mechanical coupling, illustrated by the blue elements in Fig. 2, relates the grip center $G_i$ to its control point $T_i$. All the $T_i$ points follow the same KUBC motion, i.e. a linear function of the position (shown as red line in Fig. 2). This arrangement introduces
<figure><img src="image_4.png" /><figcaption>Figure 2: E-KUBC actual setup: a blue coupling is between each green grip G and its red point of control T. For the sake of clarity, the number of grips per side $N_G$ is here equal to 2.</figcaption></figure>

additional rotational degrees of freedom (DoF) for each grip, which can either be constrained or remain free, depending on the experimental design. In practice, achieving perfectly free rotation is unrealistic due to friction. Moreover, the couplings introduce further sources of error, such as friction, clearances, and compliance, which are absent in the numerical context.

A second limitation specific to E-KUBC pertains to measuring the average stress tensor. In contrast to T- or N-KUBC, where the average stress can be determined by integrating local stresses throughout the specimen, the experimental approach for E-KUBC relies on measuring resultant forces at the specimen boundary, as proposed by Dassonville et al. (2020). This method, however, is based on an idealized setup, and in practice, it is crucial to consider deviations caused by imperfections in the E-KUBC apparatus. Therefore, accounting for how these experimental defects may impact the measured effective stiffness is essential. Here follows a detailed description of the considered defects.

## 2.1. Imperfect boundary conditions

Whatever the E-KUBC machine design, they share several type of generic imperfections which may significantly contribute to an error in the effective stiffness measurement compared to T-KUBC. The main ones that must be investigated are:

* **Discretization of the T-KUBC frame.** The discretisation in E-KUBC is coarser than that of N-KUBC because of technical limitations. One may reasonably expect around 2 to 30 control points $T_i$ per side of the specimen. While this aspect is often neglected in N-KUBC, the low spatial discretisation frequency of E-KUBC may severely affect the measurements.

* **Relative length of the rigid edges.** If this ratio tends to 100%, the free boundary edges of the specimen between the rigid edges (green lines on Fig. 2) virtually do not exist. This case is to be avoided because the strain between rigid edges would tend to have infinite value.

* **The shape of the contact surface between the sample and the grips.** It was a straight line on Fig. 1 (c) and Fig. 2, but might be chosen otherwise, as a curved shape.

* **Translation offset.** In the case of a perfect setup, the KUBC frame point $T_i$ and the grip point $G_i$ would be the same. However, it might be technologically difficult to achieve this
<figure><img src="image_5.png" /><figcaption>Figure 3: Rheological model of a coupling. Horizontal axis corresponds to translation/force variables and vertical axis to rotation/torque variables. Torque *M* is dependent of force *F* via the friction coefficient *f*<sub>θ</sub> and the bearing preload *p*<sub>r</sub>.</figcaption></figure>

perfect match, and a small shift between the two points may remains. In this case, the grips
are not rigorously 'superimposed' on the KUBC frame as shown in Fig. 2.

*   **The coupling defects.** The discretised KUBC points $T_i$ are connected to the grips through the couplings that are subject to defects. A general rheological model of imperfect coupling is shown Fig. 3. For clarity, translations normal and tangential to the specimen side are combined into a single radial translation, which is illustrated along the horizontal axis of Fig. 3. The corresponding radial DoF $r_t$ is the imposed theoretical displacement (input of the coupling), $r_s$ is the displacement of the grip (output of the coupling), and $F_r$ is the transmitted force. The vertical axis corresponds to the rotation DoF, namely $\theta_t$ the imposed theoretical rotation, $\theta_s$ the rotation of the grip, and $M$ the transmitted torque. This representation shows the relation between rotation and translation due to the bearing friction. This model is characterised by:

- *The radial clearance* $\delta_r$ in the coupling. A perfect mechanism would have zero clearance;

- *The radial compliance* $C_r$ of the coupling. A perfect mechanism would have zero compliance;

- *The rotation of the grips.* It can be imposed as equal to a specific value or let free as shown in Fig. 1. The friction occurring during the rotation is governed by the friction coefficient $f_\theta$ and the radial force $F_r$ plus the internal preload $M_p = p_r \times f_\theta$. A perfect mechanism would feature a free rotation $M = 0$ or a perfect rotation transmission $\theta_s = \theta_t$. These two limit cases are obtained by choosing $f_\theta = 0$ and $f_\theta = \infty$.

## 2.2. *Choices of specimen design*

In addition to the machine defects, a second source of error is related to the specimen design.
Because of the presence of grips and the nature of the tested materials (AMs), many choices of
specimen design must be made, as shown in Fig. 4:

* **The thickness:** an increased thickness in the out-of-plane direction likely creates a more significant discrepancy between the experimental results and the plane stress assumption. On the other hand, an overly thin specimen may experience out-of-plane global buckling under compressive load. Therefore, establishing a minimum thickness limit is essential.
* **Scale separation:** when examining heterogeneous materials, the scale separation is a fundamental consideration (Huet, 1990). This problem is generally investigated case by case (Michel et al., 1999), determining the minimum RVE size (or statistical representative element) using the integral range method (Kanit et al., 2003). For KUBC, a reasonable scale separation (RVE) is expected with a minimum grid size of approximately 10 × 10 unit cells.

* **Placing AM pattern within the specimen (shift and rotation):** this third option concerning the E-KUBC is unconventional and arises from the coarser discretisation and grips not found in the traditional N-KUBC. One denotes *p* as the distance between two grips (i.e. between the $G_i$) which is also the distance between the discretisation point of the KUBC (i.e. between the $T_i$). Concerning the imperfect boundary conditions, several questions need to be addressed:

- Ratio between specimen internal length and period *p*:

* At first glance, a cell size much smaller than the parameter *p* appears to ensure better scale separation. However, this choice may result in an undesirable loading heterogeneity at the boundary: the cells in contact with the grips would be loaded with a uniform displacement while the cells at the boundary between the grips are not (free boundary condition), see Fig. 4 (a);

* Conversely, a cell size much larger than *p* will lead to low scale separation and some beams might be 'stuck' between two grips, see Fig. 4 (b). All the cells at the boundary would however be loaded by the grips, hence a more uniform boundary condition;

* A cell size close to *p* might thus be a good compromise. However the lattice structures often exhibit high void density, which causes nefast effects like the absence of contact with one or several grips, see Fig. 4 (c). In addition, bridges between grips may occur, i.e. lattice beams connecting grips two or more grips in a narrow region, leading to overestimating the effective stiffness.

Is one phenomenon more significant than the other? How a balance between these factors can be achieved?

- Matching of specimen mesostructure period with machine period *p*, or being a multiple of *p*: since the KUBC was discretised with a specific period *p*, periodic materials have a potential risk due to a 'interference'/'resonance' effect that could impact measurements. How this issue can be addressed and an appropriate specimen design can be selected for periodic materials?

Each of these specimen design choices might contribute to increased error between the T- and the E-KUBC. Like for the machine defects, it is crucial to understand how to design specimens regarding imperfect boundary conditions. These choices will be studied in the following section.

### 3. Method

In order to assess the aforementioned effects of defects of the E-KUBC and investigate the proper design for the specimens, a numerical model of the E-KUBC protocole is developed. This model is used to measure the plane stress effective stiffness of specimens of different kinds with customisable defects of E-KUBC. For the periodic AMs, the measured stiffness tensor is then compared to N-PBC results. Numerical errors set aside, this N-PBC computation indeed delivers the exact
<figure><img src="image_6.png" /><figcaption>Figure 4: Various possibilities of specimen design with the same mesostructure. The designs are overexagerated for the sake of clarity. (a) a high scale separation but with heterogeneous boundary conditions mixing free edge (fe) and imposed displacement (id); (b) low scale separation but with beams ‘stuck’ between adjacent grips (st) and (c) cell size close to grips distance leading to missing contact (mc) and bridges (br).</figcaption></figure>

<table><caption>Table 1: Mesostructures properties. ‘Anisotropy’ stands for the classical linear elastic anisotropy. Symmetry notations follow the notation given by Auffray and Ropars (2016). $Z_n$ notation corresponds to n-fold $(2\pi/n)$ rotational invariance, and $D_n$ to n-fold rotational invariance completed by n mirror symmetries. O(2) is the infinitesimal orthogonal group, the dihedral limit group for $n \to \infty$. For Kite-and-Dart Penrose, see (Somera et al., 2022).</caption><thead><tr><th>Mesostructure</th><th>Periodicity</th><th>Symmetry</th><th>Anisotropy</th><th>Porosity [%]</th><th>Other specificity</th></tr></thead><tbody><tr><td>Bulk</td><td>-</td><td>O(2)</td><td>Iso.</td><td>0</td><td></td></tr><tr><td>Square</td><td>P.</td><td>D<sub>2</sub></td><td>Ortho.</td><td>61.2</td><td>Bending-dominated</td></tr><tr><td>Hexachiral</td><td>P.</td><td>Z<sub>6</sub></td><td>Iso.</td><td>50.4</td><td>Rotation-induced auxetism</td></tr><tr><td>Hexagonal</td><td>P.</td><td>D<sub>6</sub></td><td>Iso.</td><td>73.5</td><td>Bending-dominated</td></tr><tr><td>Triangle</td><td>P.</td><td>D<sub>6</sub></td><td>Iso.</td><td>61.7</td><td>Traction/compression-dominated</td></tr><tr><td>Kite-and-Dart Penrose</td><td>Q.P.</td><td>D<sub>10</sub></td><td>Iso.</td><td>70.6</td><td>Mixed Traction-Bending</td></tr></tbody></table>

equivalent stiffness tensors, while the N-KUBC results depend on the RVE as the E-KUBC. It thus allows to assess the KUBC error that is solely due to the finite size of the RVE. For the quasiperiodic AMs, the measured stiffness tensor is then compared to N-KUBC computations, performed with the same RVE than in the case of the E-KUBC.

The description of the used specimens, the numerical twin of the setup and the error calculation are presented in the following.

### 3.1. AM specimens

The E-KUBC measurement error has been numerically assessed using various types of specimens. Their different parameters are:

* **The mesostructure type.** For our study, five distinct mesostructures were selected (Fig. 5), chosen for their prevalence in the literature as well as their varying degrees of anisotropy, internal mechanisms, and void spatial organisation, as detailed in Tab. 1. Consequently, each mesostructure may reveal different unexpected phenomena resulting from the imperfections of the E-KUBC.

* **The mesostructure scale.** There is a reference scale for all mesostructure geometries. At this scale of 1/1, the length of the beams, designated as *L*, is set equal to the distance *p*.
<figure><img src="image_7.png" /><figcaption>Figure 5: Mesostructures: (a) square, (b) hexachiral, (c) hexagonal, (d) triangle, (e) Kite-and-Dart Penrose.</figcaption></figure>

between specimen grips, and the thickness *t* of the beams is established at 4 mm. For the hexachiral mesostructure, the parameter *r* is defined as 2*t* to achieve a negative yet near-zero Poisson's ratio of -0.1 (Spadoni and Ruzzene, 2012).

Specimens were generated and tested for each mesostructure at seven scales from 7/10 to 13/10. Applying homothetic transformations allows the scale to be adjusted while maintaining consistent lengths, ratios, and porosities among specimens of the same mesostructure. It was ensured that the tested scales matched realistic scales that align with the accuracy and resolution of professional FDM 3D printers using ABS and laser cutters with PMMA.

* **The mesostructure placement.** The AM specimen's design also considers a rigid-body-motion of the pattern: the in-plane rotation $\theta_z$ and the $T_x$ and $T_y$ translations. Various pattern placement of the unit cell within the specimen were examined at each scale, involving 6 $T_x$-translations, 6 $T_y$-translations, and 12 $\theta_z$-rotations.

The rotation step is equal to 60° for the [Z₆] hexachiral mesostructure, 30° for the [D₆] hexagonal and triangle mesostructures, and 90° for the [D₂] square mesostructure.

The translation step is equal to a sixth of the cell size in the *x* and *y* directions.

For all the specimens, given the low strain loading, the constitutive material is supposed to be purely linear, elastic, and isotropic. The Young modulus and the Poisson ratio of the ABS are chosen, respectively equal to 2.5 GPa and 0.3.

Last, in the case of a real specimen, it is challenging, if not impossible, to fix each beam directly to its grip: the beam end is thin, the incident angle between the beam and the grip may vary, multiple beams may be related to the same grips, etc. Therefore, an extra layer of matter covers
the grip to ensure optimal kinematic contact (green parts on Fig. 4). It is made out of the same material as the beams. The beams are fused with this layer, whose thickness equals the beam one. These layers are also part of the mesh of the specimen.

## 3.2. Numerical twin

### 3.2.1. Machine modelisation

The numerical twin of the experimental setup corresponds to Fig. 2, where only the specimen is meshed while the coupling and the KUBC frame displacement are modeled by the definition of adequate boundary conditions applied to the grips. Each coupling connecting a grip at its point $G_i$ and its corresponding control point $T_i$ is characterised by its internal behaviour. A general rheological behaviour model was chosen to consider compliance, clearances and angular friction; see Fig. 3. The specimen is modelled as 2D with plane stress assumption. For completeness, Appendix A presents a validation example of this assumption with a comparison to several 3D cases.

Model parameters are mostly inspired by Dassonville et al. (2020) but remain customisable. Among the multiple investigated parameters, the most relevant ones are:

* **Amplitude of the macroscopic strains.** The choice of this amplitude has no consequences on the stiffness evaluation as long as the problem is linear. When the defects are considered, the problem is nonetheless non-linear because of the clearances and friction; thus, the realistic amplitudes of strain must be used. It is chosen equal to 2%.

* **Number of grips per side $N_G$.* For a square specimen, it is within the [2, 30] range. For $N_G = 10$, the grip period is $p = 28$ mm. The lower value of $N_G$ is the minimum allowing E-KUBC, and the upper value given by a realistic technological limitation.

* **Relative length of the grips.** It is set to 2/3.

* **Shape of the grips.** A circular arc with a radius $R$ equal to 13 mm and a depth equal to 3 mm is chosen. The radius is much larger than the depth, so the grip edge is relatively close to a straight line.

* **Radial clearance in the coupling.** The radial clearances are picked randomly for each coupling in a truncated Gaussian distribution with a 28 µm mean and 11.5 µm standard deviation.

* **Radial compliance.** Different compliances along the horizontal and vertical sides were setup, which are 18 µm/kN and 144 µm/kN, respectively.

* **Rotation of the grips.** It is let free by default. When the bearing friction is not neglected, the maximal transmissible moment $M$ is given by the Coulomb law $|M| \le f_\theta |F_r| + M_p$ where $f_\theta$ is the bearing friction coefficient and $M_p$ the torque preload (see Fig. 3). In the following, the bearing friction is set to $f_\theta = 0.77$ mm. While a preload of $M_p \approx 10^2$ N.mm was measured on Dassonville's machine, the preload is here set randomly for each rigid part between 0 and $10^5$ N.mm. These large conservative values are used to investigate the impact of severe defects on the measured properties.
<figure><img src="image_8.png" /><figcaption>Figure 6: The three elementary loadings.</figcaption></figure>

### 3.2.2. Specimen FEM

The FEM computations are conducted with Abaqus under infinitesimal strain assumption. Bulk-type specimens are meshed with quadratic triangular elements with a size range of 0.8 – 7.0 mm, resulting in approximately $0.5 \times 10^6$ DoF. The lattice-type mesostructures were meshed with the same elements type with a rule of five elements in the beam thickness, producing around $2 \times 10^6$ DoF depending on the specimen.

### 3.2.3. Effective stiffness measurement

A 2D homogenizing machine measures the plane stress effective stiffness tensor $K$. This is usually done with three successive orthogonal loadings for which the mean strain $E$ is imposed:

$$ \mathbf{E}_1 = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}_{(x,y)} \quad \mathbf{E}_2 = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}_{(x,y)} \quad \mathbf{E}_3 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}_{(x,y)} $$

i.e. the ones depicted Fig. 6. Additional loadings could be chosen to reduce the experimental error, but in the context of this numerical assessment, only this minimal set of three loadings is considered.

In the case of E-KUBC, one must ensure the experimental feasibility of the whole process, particularly for the accessible measurements:

* The mean strain is obtained for each loading either by deducing it from the position of the frame (which is not measured but driven), or better by measuring the kinematic full field on the specimen surface (for example with a Digital Image Correlation technique). Since this second method is more accurate and easy to perform on a real E-KUBC test, the mean strain of the simulated test is obtained by averaging the strain field on the specimen surface.

* On the contrary, the mean stress is determined from the boundary nodes' reaction forces to calculate the resultant force on each side. The relationship between the average stress tensor and the boundary forces was demonstrated by Dassonville et al. (2020) in the case of rectangular specimens, and its extension to quadrilaterals almost rectangles is given in Appendix B.

Once the mean strain $E_j$ and the mean stress $\Sigma_j$ are obtained for each loading $j$ the effective stiffness tensor is calculated as:

$$ \mathbf{E}_i = \mathbf{K} : \mathbf{\Sigma}_i. \qquad (1) $$

Note that only the symmetrical part of the tensor is considered.
### 3.3. Comparison metric

In order to assess the accuracy of the stiffness tensor measured with the E-KUBC protocole, denoted $K_E$, reference effective stiffness tensors $K_P$ and $K_N$ are computed with N-PBC or N-KUBC homogenisation, respectively:

* For the periodic AMs, N-PBC are performed with a specimen limited to one unit cell. The N-KUBC are computed with the imposed displacements directly applied to the mesostructure ends defined by the square boundary of the specimen. It is thus slightly different from the E-KUBC geometry for which there is the fin extra layer of matter over the grips.

* For quasiperiodic AMs, N-PBC is meaningless and only the N-KUBC is computed.

Both N-PBC and N-KUBC computations are performed in Abaqus with the homogenisation toolbox developed by Lejeunes and Bourgeois (2011).

In order to compare the stiffness tensors obtained through the different methods, the relative error $r$ of stiffness tensor $K_1$ w.r.t to the stiffness tensor $K_2$ is defined as:

$$r(K_1, K_2) = \frac{\|K_1 - K_2\|_F}{\|K_2\|_F} \qquad (2)$$

with $\|K\|_F = \sqrt{K} :: K$ the Frobenius norm. The conservative feature of this error (it stands for the sum of the error on each component and not the mean or maximum one) must be noted.

## 4. Results

To evaluate the mechanical performances and limitations of the E-KUBC, different categories of specimens are examined: (i) a homogeneous bulk material, (ii) periodic and (iii) quasi-periodic 2D lattice structures. The bulk specimen is a reference due to its clearly defined elastic properties and lack of internal length scales. This allows us to isolate and understand the effects of machine-induced artefacts. Conversely, the lattice specimens reflect more realistic architected materials found in additive manufacturing, enabling an assessment of how the interplay between geometry, boundary conditions, and machine design influences the homogenized response. Our approach involves varying the specimen configurations and testing conditions to quantify errors about known reference solutions.

### 4.1. Bulk specimen

The interest in the 2D bulk specimen is twofold. First, its stiffness is well-established and clearly defined by the 2D plane stress elasticity matrix, denoted as $K_{PS}$. Second, it lacks internal length, periodicity, or scale separation, eliminating the risk of 'interference' with the discretised KUBC period. This characteristic allows for a more apparent distinction between the effects of machine defects and the choices of specimens, which will only be relevant for lattice mesostructures.

#### 4.1.1. Effect of the number and position of the grips

The minimal number of grips and their free rotation must first be investigated. A series of 2D plane stress numerical simulations is performed to this aim. The number of grips ranges from 2 to 30 per side, and their rotations are free or fixed. Fig. 7 shows the heterogeneity of the strain field over a centred square corresponding to 90% of the surface of the specimen. The curves represent the evolution of the relative standard deviation of the strain components as a function of the
<figure><img src="image_9.png" /><figcaption>Figure 7: Evolution of the relative standard deviation of the strain components $\tilde{\epsilon}_{ij}$ as a function of the number of grips per side $N_G$ and the possibility for them to rotate ('free') or not ('fixed'). Values are given for a centred square area representing 90% of the specimen's total area. The upper plot corresponds to tension loading, and the lower corresponds to shear.</figcaption></figure>

number of grips. The relative standard deviation of the strain component is defined as $\tilde{\epsilon}_{ij} = \frac{\text{STD}(\epsilon_{ij})}{||E||}$ where STD(·) is the standard deviation of (·) and $|| \cdot ||$ is the L1 norm of (·). As expected, the strain field's heterogeneity decreases as more grips are used. Their free rotation slightly increases the heterogeneity during a tensile test but improves homogeneity in the case of a shear test. For a setup with ten freely rotating grips per side, the expected heterogeneity is below 4% during a tensile test and 5% during a shear test. The authors consider this configuration a realistic compromise, and ten grips per side is thereafter taken as a reference.

Regardless of the number of grips per side, heterogeneities will always be present near the boundary conditions within the specimen's boundary layer. Figure 8 depicts the strain distribution when ten rigid parts are employed on each side. Measurements are taken across various centred square zones of the specimen under a 2% tensile loading. As expected, the larger the area, the greater the strain variability. This is obvious that near the boundaries the strain is heterogeneous (see the 1st and 3rd quartils in the 10p × 10p case on Figure 8). Nonetheless the field becomes very homogeneous as soon as a thin peripheral layer is not considered (the 1st and 3rd quartils in the 8p × 8p case is not even visible on Figure 8). Meanwhile, the mean strain remains very stable, with a relative variation limited to ±0.2% regardless of the selected region. In the context of AMs E-KUBC homogeneisation, the results are of course improved if the area is the largest possible. Unfortunately, because of the aforementioned small heterogeneities at the boundary, and the fact that the AMs internal length is expected to be of the same order of magnitude of the cell size L and p, and, last, the geometry of the AM is necessarily slightly modified close to the boundary (addition and removal materials), it is not wise to measure the strain over the entire specimen. A thin peripheral layer is excluded to mitigate boundary effects. In the following, the layer of thickness is chosen equal to p, though in-depth studies of particular mesostructures might show that even lower values might be relevant.
<figure><img src="image_10.png" /><figcaption>Figure 8: Spread of strain value in the bulk specimen in different areas for a $\epsilon_{xx}$ = 2% loading. Minimum - 1st quartil - median - 3rd quartil - maximum.</figcaption></figure>

A systematic bias of the mean strain is also observed. While the theoretical mean strain applied is 2%, the measured mean value is 2.12% in Fig. 8. This 5.9% deviation can be explained by the translation shift between the KUBC frame, the centre of the grip and the interface between the grips and the specimen, i.e., points $T_i$ and $G_i$ are not coincident (see Fig. 2). Given the dimensions, the deviation was expected and the detailed of the calculus is given in Appendix C. This bias vanishes if $T_i$ and $G_i$ are superimposed, which is technically more complex but still possible.

### 4.1.2. Effect of the remote measurement of the stress

The simulations on a 2D bulk specimen resulted in an error of 2.7% for $r(K_E, K_{PS})$. This discrepancy can be attributed to the relative length of the grips and the significant bridging effect between them caused by the bulk geometry. Figure 9 (left) illustrates the maximum in-plane principal stress between the grips. The elevated stress in this region arises from the specimen being 'pinched' or 'stretched' between the surrounding grips. The shape of the grips does not contribute to this issue; rather, the area between them. This spurious phenomenon of 'pinching/stretching' generates higher local stress, leading to an increase in the locally measured force (beyond the global mean stress) and, consequently, an overestimation of the specimen's stiffness. The cause of this discrepancy makes it independent of the size of the centred square zones where the strain is measured: it is due to the forces at the boundary.

Figure 9 (right) shows a specimen with notches between the grips. The stress field amplitude is drastically reduced, and the error $r(K_E, K_{PS})$ decreases to 0.9%. A direct effect of this design is the better load balance in the grips, especially in the ones located in the corners, which used to be more heavily loaded.

### 4.1.3. Effect of clearances and compliances

Considering the bulk specimen without notches, adding the defects — namely the aforementioned compliance, clearances, and friction — reduces the error $r(K_E, K_{PS})$.

Taking into account the setup compliance alone decreases the error to 2.3%. This compliance indeed reduces for example the effect of the displacement mismatch (Section 4.1.1) or the bridge effect (Section 4.1.2).

The error reaches 2.0% when compliance, clearance, and friction are considered. Compliance and friction are critical factors, while the clearance effect is insignificant, at less than 0.1% change.
<figure><img src="image_11.png" /><figcaption>Figure 9: Orientation and intensity of normalized principal stress in a bulk specimen between two rigid parts during horizontal tension test (upper side) without notching (left) and with notching (right). The specimen is the grey part, the grips the green curves and the couplings the blue parts.</figcaption></figure>

Indeed, the considered clearances are here small compared to the maximal imposed displacement of the grips (the ratio clearance/displacement is below 2%).

This improvement in the results is coherent with the previous observation of the effect of notches: adding compliance to the machine or to the external layer via the notches is similar. Such E-KUBC machines indeed enable specimens with vast gauge zones compared to standard biaxial setups, but a thin accommodation layer is still necessary. This layer distributes the imperfect E-KUBC displacement (because of the grips or the aforementioned kinematic defect) to the gauge zone more smoothly.

One must nonetheless be careful with this result, since it is influenced by the balance between the bridging effect (see Fig. 4), the specimen bulk stiffness and the compliance of the machine, meaning it varies based on the type of specimen and the testing machine used.

## 4.2. Periodic lattice specimens

Thanks to its simplicity, the bulk specimen allowed us to highlight the effects resulting from the discrete approximation of the KUBC. However, this material does not represent the potential candidates for such a testing machine type. Considering AMs, outcomes may differ because of possible intricate phenomena, especially the interaction between notch, lattice, and grips.

As explained earlier, the hexagonal, square, triangular, and hexachiral mesostructures underwent tests with different specimen configurations. These different configurations are obtained by applying a series of arbitrary rigid body motions to the elementary cell: 6 translation steps along *x*, 6 along *y* and 12 in-plane rotation steps. The 6 × 6 × 12 configurations resulted in 432 specimens per scale and mesostructure or about 12000 specimens tested with the E- and N-KUBC. Only one PBC homogenisation computation was made per mesostructure.

The extensive number of simulations would be quite time-consuming if the clearances and friction — factors that transform linear simulations into non-linear ones — were considered. Since the previous section has shown that the coupling defects effect turns out to be minor if not beneficial in the case of the bulk specimen, they were disregarded for the lattice specimens, allowing for the exploration of a wide range of specimen geometries.

### 4.2.1. Evolution of errors with the number of cells in the specimen

The first investigation on periodic mesostructures concerns the influence of the pattern scale, i.e. the number of cells in the specimen. The general trends for each mesostructure are shown in
<figure><img src="image_12.png" /><figcaption>Figure 10: Mean error between PBC homogenisation $K_P$ and $K_E$ from the KUBC machine for different lattice structures.</figcaption></figure>

Fig. 10. It represents the median values of the experimental error $r(K_E, K_P)$ calculated over the $6 \times 6 \times 12$ specimens versus the number of cells. The median value of $r(K_N, K_P)$ is also plotted to show the evolution of the error obtained in the case of a N-KUBC homogenisation with the same number of cells. Two main points can be underlined:

* First, for most mesostructures, one notes that while $r(K_N, K_P)$ is about 2 to 4%, the error $r(K_E, K_P)$ is about 3 to 6%, i.e. higher as expected but of the same order of magnitude. The worst case is obtained with the hexachiral mesostructure for which $r(K_N, K_P)$ is between 5 and 8%, and $r(K_E, K_P)$ around 8%.

* Second, the $r(K_N, K_P)$ curves demonstrate the usual convergence of KUBC results toward PBC ones (Kanit et al., 2003). In contrast, the $r(K_E, K_P)$ curves highlight noteworthy differences between N- and E-KUBC. The convergence process for all mesostructures is disrupted, and the error increases before decreasing toward non-zero constant values (case of hexachiral and square mesostructures) or zero (case of triangle and hexagonal mesostructures).

There is a clear difference between the E- and N-KUBC, but the sole evolutions of the median errors are insufficient to interpret the results. An in-depth analysis details the results for each scale and rotation angle of the lattice pattern.

### 4.2.2. Analysis of the worst configurations

Fig. 11 (resp. 12) illustrates the distribution of the error $r(K_E, K_P)$ over the $6 \times 6$ shifts as a function of the 12 angles and 7 scales of the hexagonal (resp. square) pattern. Fig. 13 also presents various examples of specimen geometries with their associated $r(K_E, K_P)$ errors.

Low errors are found for nearly any combination of scale and angle, but the error variability due to the shifts is more or less important. Since the optimal configuration of shift, angle, and scale in terms of error can not be found a priori, finding the rules leading to the smallest variability is best.

With the many results, a statistical analysis was performed to determine criteria correlated to the error. Multiple criteria were tested, such as the number of bridges between grips (i.e. when a beam directly relates two adjacent grips), the incident angle of the beams with the grip, or the number of missing contacts with grips. Surprisingly, no clear correlation was found between them.
<figure><img src="image_13.png" /><figcaption>Figure 11: Example of error $r(K_E, K_P)$ as a function of scale and orientation with the hexagonal mesostructure.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Figure 12: Example of error $r(K_E, K_P)$ as a function of scale and orientation with the square mesostructure.</figcaption></figure>

However, the worst variability scenarios are obtained in an apparent case, for which the spatial period of the beams linking the grips is close to the spatial period of the grips $p$. As an example, the most significant errors shown in Fig. 13 are obtained for the square configuration (k) and (l) for which the two periods match on every side for the square mesostructure, as well as the hexachiral configuration (g) where periods match on the upper and lower sides. Another example is seen Fig. 12, where the combination of scale and angle equal to (7/10, 45°) leads to substantial errors. The scale 7/10 at 45° indeed leads to a spatial period of the links equal to $7/10 \times \sqrt{2}p \approx 0.989p$, i.e., very close to $p$. Significant errors are also obtained when the geometrical configuration leads to a continuous bridge linking all the grips all along a side (Fig. 13 (g) and (l)).

Among all the mesostructures, the hexachiral structure exhibits the most significant error (Fig. 10). This can be attributed to the internal rotation mechanisms that generate the typical auxetic behaviour of such mesostructure. In the case of the real specimen, these rotations are constrained at the edges of the E-KUBC frame due to the overall high connectivity of the lattice
<figure><img src="image_15.png" /><figcaption>Figure 13: Various configurations with their associated $r(K_E, K_P)$ errors.</figcaption></figure>

structure: several beams — often 2 or 3 in the studies geometries — are related to each grip (Fig. 13 (f) and (g)). On the contrary, for large-scale specimens, the error $r(K_E, K_P)$ appears to be close to $r(K_N, K_P)$. In contrast to the T-KUBC homogenisation, the machine allows free rotation near the boundary and thus does not restrict the internal mechanisms of the AM. A more general observation is that blocking the rotation of the grips increases the errors $r(K_E, K_P)$ by around 4% on average for all mesostructures and scales.

Following the idea of the notches that enhance the measurement accuracy of the elastic tensor in bulk specimens, one may add them to lattice specimens. While the effect of notches located between grips is unclear, notches placed in the corners tend to improve the measure of the shear modulus and thus reduce the overall error. Especially for the hexachiral mesostructure, this modification is particularly efficient because of the relief of the rotation within the material.

### 4.3. A quasi-periodic lattice specimen (Penrose)

So far, only periodic materials have been tested, which does not represent the complete range of potential specimens. Specimens with Penrose-type mesostructures were tested at three scales with a 7.5° tilt angle. Notches in the corner were included based on the previous observation with periodic mesostructures.

As already mentioned, the PBC reference solution cannot be computed due to the absence of periodicity in the case of quasi-periodic AMs. While N-PBC offers a nearly exact solution for periodic AMs, N-KUBC may necessitate extensive RVE for quasi-periodic AMs. The stiffness matrix $K_N$ was thus computed with the smallest scale specimen.

The results are shown in Fig. 14. The $r(K_E, K_N)$ error was 6.2% for the largest scale, 2.1% for the medium, and 1.7% for the smallest tested scale.

Here an additional comparison tool is introduced to facilitate comparisons among isotropic quasi-periodic specimens. This comparison is based on the invariants of the effective stiffness tensor: $I_1$, $J_1$, and $J_2$ (Auffray and Ropars, 2016). The invariant $I_1$ assesses the sphericity of the material, $J_1$ measures the isotropic part of the deviatoricity, and $J_2$ measures the square norm of the anisotropic part. In isotropy case, $I_1$, $J_1$, and $J_2$ are respectively equals to the bulk modulus $K$, the shear modulus $G$, and zeros. These invariants enable a comparison between the E-KUBC and N-KUBC results, as, in the case of isotropic AMs, $J_2$ should equal 0. For completeness, the calculation details
<figure><img src="image_16.png" /><figcaption>Figure 14: Result of Penrose-type specimens. (a) $K_E$ coefficient with scales 10:10 (A), 7:10 (B), and 5:10 (C) specimens with a 7.5° tilt angle. (b) Invariants results of $K_E$ and $K_N$.</figcaption></figure>

for obtaining the invariants through harmonic decomposition are outlined by Auffray and Ropars (2016).

The obtained invariants are presented in Fig. 14 (b). The invariants associated with the isotropic part of the tensors $I_1$ and $J_1$ do not demonstrate a notable improvement at lower scale than 7:10, and a residual deviation is observed compared to $K_N$. In contrast, the invariant $J_2$, which quantifies the anisotropic part of the stiffness matrix, is significantly closer to zero at scale 5:10, compared to the larger scale specimens.

## 5. Discussion on the specimen design

It is common practice to enlarge the Representative Volume Element (RVE) to improve accuracy when using the N-KUBC method; however, alternative approaches may be explored with the E-KUBC approximated boundary conditions (Huet, 1990). These boundary conditions are sensitive to the design of the sample, necessitating careful consideration. Two contrasting strategies may be pursued.

First, specimens can be meticulously designed, for example, by aligning the symmetry axes of the mesostructure with those of the boundary frame so that it becomes significantly easier for the user to control the material links with the rigid parts. As observed with periodic materials, this approach is particularly hazardous because the error can surge significantly with minor changes, like a small shift.

The second approach is drastically different. Only two criteria leading to significant error variability were found: the spatial period match and a continuous bridge along the specimen's side. A relevant way of designing a specimen is thus to choose a generic configuration and check for these two conditions. Although the criteria may be coarse by eliminating potentially good specimens, the results obtained with the filtered specimens are less sensitive to the RVE choice. In addition to placing the pattern within the sample, notches in the corner decrease the error. Their length and width remain up to now empirically determined.

Finally, as with any non-standard test, this study shows that performing an inexpensive coarse simulation before the actual test makes sense to check that the intended specimen will behave as expected. Moreover a preliminary numerical study helps investigating the relevance of multiple specimens using the ergodicity principle (Huet, 1990; Kanit et al., 2003). This simulation must be linear, i.e., without machine defects and with the theoretical mesostructure or an approximation
of it. Its results must be compared to PBCs (or KUBCs for quasi or aperiodic mesostructures). Additional simulations may be performed with slight linear shifts and angle shifts of the pattern to expose potential sensitivity to geometrical defects.

## 6. Conclusion

The present study investigates the impact of imperfect kinematic uniform boundary conditions (E-KUBC) on measuring the effective stiffness tensor. The boundary conditions were discretely applied to samples using grips distributed evenly around them. Bulk and architectured material samples were subjected to numerical testing and compared with results obtained with periodic boundary conditions. The following conclusions were drawn:

* It was determined that ten grips per side were sufficient to ensure uniformity in the strain field (heterogeneity under 4% in the case of a bulk material). However, the strain between the grips was higher than in the central region of the sample, leading to an overestimated measure of the stiffness tensor. This issue can be partially addressed by incorporating notches into the design of the sample.

* Among the potential defects in the couplings, the free rotation of the grips was found to have the most significant impact. Allowing these rotations was beneficial, particularly for mesostructures where internal rotations occur.

* In the case of architectured materials, a high degree of variability was observed, depending on the pattern type, orientation, and scale. In the best-case scenario, a 2 to 4% error was observed with hexagonal, triangle, and square mesostructure. The auxetic hexachiral mesostructure exhibits a higher error of around 8%, but high errors (5-8%) were also measured with perfect N-KUBC.

* The errors encountered in the sample material are primarily influenced by:

- A continuous bridge of material connecting the grips along one side. This poses a considerable risk of significant error, but is easily circumvented by adding notches.

- The period match of the links and the grips. When the spatial periods are too closed, the error variability increases significantly, making it advisable to avoid such configurations in sample design.

Based on these observations (10 grips per side, free rotation of the grips, random specimen design following the period and bridge criteria), the E-KUBC described in this study led to measurement errors around 2-4% for many periodic mesostructures. Some cases, like chiral mesostructures, might lead to higher errors, but this fact is already known in the case of N-KUBC (El Ouafa, 2023). For the tested quasi-periodic sample, the E-KUBC also leads to an error of around 2%. These results prove the capabilities of the proposed approximated version of KUBC and give guidelines for designing E-KUBC testing machines and specimens.

## 7. Acknowledgements

This work benefited from the research project ANR-19-CE08-0005 (project Max-Oasis) funded by the French National Research Agency (ANR).

The authors thank the PIMM laboratory (Institut des Arts et Métiers / CNRS / Cnam / Université HESAM, Paris, France) for hosting the KUBC machine and helping with its development.
<figure><img src="image_17.png" /><figcaption>Figure B.15: Non-local forces on each edges, generalized to any quadrilateral close to a square.</figcaption></figure>

## Appendix A. Relevance of plane stress assumption simplification

In order to investigate the plane stress hypothesis, four bulk 3D specimens were considered with thicknesses of 10 and 50 mm and Poisson ratios of 0.01 and 0.49. The number of rigid parts per side is set to 10, which is, as one will see after that, a realistic and relevant value. Clearance, compliance or friction are here not considered in the couplings behaviour.

The most incompressible and thickest specimen leads to an error $r(K_E, K_{PS})$ of 4.8% while the three others result in an error ranging from 2.5 to 2.6%. The error is small but not negligible for bulk specimens. Since bulk and thick specimens with nearly incompressible behaviour are the worst-case scenarios, this error is expected to be significantly lower than that of lattice AMs. Consequently, this error is neglected by employing the plane stress assumption, allowing 2D simulations and improving computational speed.

## Appendix B. Expression of the average stress tensor derived from boundary forces

As presented in Dassonville et al. (2020), a linear kinematic allows to recover the average stress tensor $\Sigma$ of the specimen through the measurement of a limited number of force values. These forces $F$ are the loads at each end of each of the four edges as illustrated Fig. B.15.

In a rectangular shape configuration of the KUBC machine, the expression reads:

$$ \Sigma = \frac{L}{V} \left[ \begin{matrix} F_{2'}^B + F_{2'}^C + F_3^B - F_4^C & F_1^D + F_{2'}^C - F_4^C + F_4^D \\ -F_2^B + F_2^C - F_{3'}^B - F_{4'}^C & -F_1^D + F_2^C - F_{4'}^C - F_{4'}^D \end{matrix} \right] \quad (B.1) $$

with $V$ the surface of the specimen.

In case of a non-rectangular shape configuration, a multivariable second-order Taylor series expansion around the rectangular configuration can be given:

$$ \Sigma = \begin{bmatrix} \Sigma_{11} & \Sigma_{12} \\ \Sigma_{21} & \Sigma_{22} \end{bmatrix} $$
Each component is expressed a a function of the side length $L_i$ and the angles $\alpha_i$ Fig. B.15:

$$V\Sigma_{11} = -F_{1'}^D L_1(\frac{\pi}{2} - \alpha_1)$$

$$-F_2^B L_3(\frac{\pi}{2} - \alpha_2)$$

$$+F_{2'}^B L_3$$

$$-F_2^C L_3(\frac{\pi}{2} - \alpha_2)$$

$$+F_{2'}^C(L_3 + L_2(\frac{\pi}{2} - \alpha_2))$$

$$-F_3^B L_3$$

$$+F_4^C(L_4 + L_1(\frac{\pi}{2} - \alpha_1))$$

$$-F_{4'}^C L_4 \alpha_4$$

$$+F_4^D L_1(\frac{\pi}{2} - \alpha_1)$$

$$V\Sigma_{21} = F_1^D L_1(\frac{\pi}{2} - \alpha_1)$$

$$-F_2^B L_3$$

$$+F_{2'}^B L_3(\frac{\pi}{2} - \alpha_2)$$

$$-F_2^C(L_3 + L_2(\frac{\pi}{2} - \alpha_2))$$

$$+F_{2'}^C L_3(\frac{\pi}{2} - \alpha_2)$$

$$-F_3^B L_3$$

$$+F_4^C L_4 \alpha_4$$

$$+F_{4'}^C(L_4 + L_1(\frac{\pi}{2} - \alpha_1))$$

$$+F_{4'}^D L_1(\frac{\pi}{2} - \alpha_1)$$

$$V\Sigma_{12} = F_1^D L_1(\frac{\pi}{2} - \alpha_1)$$

$$-F_{1'}^D L_1$$

$$-F_2^C L_2(\frac{\pi}{2} - \alpha_2)$$

$$+F_{2'}^C L_2$$

$$+F_4^C L_1 \alpha_4$$

$$+F_{4'}^C(L_1 + L_4 \alpha_4)$$

$$+F_4^D L_1$$

$$-F_{4'}^D L_1 \alpha_4$$

$$V\Sigma_{22} = F_1^D L_1$$

$$-F_2^C L_2$$

$$-F_{2'}^C L_2(\frac{\pi}{2} - \alpha_2)$$

$$+F_4^C L_1(\alpha_4 - \alpha_1 + \frac{\pi}{2})$$

$$+F_{4'}^C(L_1 + L_4 \alpha_4)$$

$$+F_4^D L_1(\alpha_4 - \alpha_1 + \frac{\pi}{2})$$

$$+F_{4'}^D L_1$$

## Appendix C. Deviation in applied strain

The 5.9% bias of the mean strain is a direct consequence of Dassonville's setup design. As shown in Fig. C.16, the specimen's grips do not share the exact same positions as the discretised KUBC displacement points. This difference makes the specimen a little smaller than the discretised KUBC frame, with the couplings interposed between the two. The actual mean strain in the specimen is thus inevitably slightly higher than the one of the discretised KUBC frame. In the case of a tensile test along x axis, the tangent displacement $U_x$ imposed on the upper and lower sides of the specimen corresponds to the theoretical one so that the average strain along the upper boundary $\tilde{\epsilon}_{xx}^{up}$ equals the expected one $\epsilon_{xx}$. The normal displacement imposed at the left and right sides nonetheless differs from the one of the KUBC frame because of the radius R. The apparent average strain in the core is consequently closer to $\tilde{\epsilon}_{xx}^{core} = (1 - \frac{2R}{np})^{-1}\epsilon_{xx}$. The lower (resp. upper) bound corresponds to a specimen only loaded by the upper and lower (resp. left and right) sides. This error decreases when grips number n or pitch p increases. It completely vanishes if the axis of rotation is at the interface ($R = 0$ in the case of circular rigid parts). This last condition is technically challenging to achieve, though not impossible. To represent a KUBC experimental setup realistically in the following study, the axis of rotation of the grips is kept away from the interface at a distance equal
<figure><img src="image_18.png" /></figure>

Figure C.16: Geometrical explanation of the average strain error. The imposed displacement $U_x$ is exactly equal to T-KUBC one on the upper side, but not on the right side, leading to an effective strain $\tilde{\epsilon}_{xx}^{core}$ higher than the theoretical one $\epsilon_{xx}$. The grey zone represents the upper right corner of the specimen. The green arcs (resp. red crosses) are the grips (resp. the frame control points). Blue disc are the couplings.

to 13 mm (i.e., the diameter of the cylindrical grips).

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