# Asymptotic strain-gradient theory for one-dimensional continua

Manon Thbaut, Basile Audoly, Claire Lestringant

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Manon Thbaut, Basile Audoly, Claire Lestringant. Asymptotic strain-gradient theory for one-dimensional continua. Journal of the Mechanics and Physics of Solids, 2025, 206, pp.106392. 10.1016/j.jmps.2025.106392. hal-05129643v2

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# Asymptotic strain-gradient theory
for one-dimensional continua

MANON THBAUT

Laboratoire de Mécanique des Solides
CNRS, Institut Polytechnique de Paris
91120 Palaiseau, France

BASILE AUDOLY

Laboratoire de Mécanique des Solides
CNRS, Institut Polytechnique de Paris
91120 Palaiseau, France

CLAIRE LESTRINGANT

Institut Jean Le Rond d'Alembert,
Sorbonne Université, CNRS,
75005 Paris, France

The goal of periodic homogenization is to identify an effective model specified by an energy functional $\Phi_{\varepsilon}[u]$ depending on the macroscopic displacement $u$. We consider second-order homogenization, a case where the effective energy depends not only on the strain $u'$ but also on its gradients $u''$ and $u'''$. Functionals $\Phi_{\varepsilon}[u]$ obtained in prior work are typically made stationary *order by order* in the expansion parameter, and are not positive when truncated: they are not proper strain-gradient theories. Starting from a functional $\Phi_{\varepsilon}[u]$ produced by linear, second-order homogenization of a periodic elastic lattice in dimension 1, we propose a systematic method to upgrade it to a positive strain-gradient energy $\Psi_{\varepsilon}[u]$. This enables us to formulate second-order homogenization as a variational problem. Boundary layers are represented in an effective and asymptotically correct way by boundary terms in the energy $\Psi_{\varepsilon}[u]$.

**Keywords** Strain-gradient elasticity, higher-order homogenization, variational methods.

## 1. INTRODUCTION

Strain-gradient theories have been developed to explain size effects in heterogeneous materials [Ger73, Min65, Tou62, Min64]. Their starting point is a postulated strain energy depending not only on the strain but also on its successive gradients. Due to the higher-order gradients in the energy, the equilibrium equations that are derived variationally are higher order than in classical elasticity, and there are more boundary conditions. The main limitation of strain-gradient models is their phenomenological character. Typically, the strain gradient terms cannot be related in a quantitative way to the properties of the microstructure.

Asymptotic homogenization, on the other hand, is a systematic tool that delivers equivalent properties of a periodic microstructure, by averaging out microscopic fluctuations. Homogenization is in essence an expansion with respect to the small size $\varepsilon$ of the unit cell. It delivers an expansion of the strain energy [Lio81, BKS11, S18, LM18, AL23] in terms of the strain and its successive gradients. By accounting for the strain-gradient effect, higher-order homogenization improves the accuracy of the solution *in the bulk* when the scale separation parameter is moderately small [GK89, BP89, SC00, YAL24]. The additional size effect arising from *boundary layers* can be captured by effective boundary conditions obtained by combining homogenization with boundary layer analysis [AA99, Pan09, TAL25a].

Strain-gradient theories and asymptotic homogenization thus address the same fundamental problem. Ideally, asymptotic homogenization should offer a rigorous justification for the effective energy functionals postulated in strain-gradient theories. Unfortunately, this is not the case and the two approaches have remained largely disconnected, except for the special case of infinite continua [TMB12]. Indeed, the functional $\Phi_{\varepsilon}[u]$ of the macroscopic displacement $u$ obtained by homogenization does not satisfy the standard properties associated with the notion of energy — we call $\Phi_{\varepsilon}[u]$ a *pseudo-energy* for this reason. It is a formal series expansion in powers of $\varepsilon$ whose truncation at a particular order, say $\varepsilon^2$:

* is typically not positive in the sense of quadratic forms [ABV16, AL23, LM18, SC00] and can be made arbitrarily large and negative by adding short-wavelength, small-amplitude oscillations to $u$;

* can be made stationary *order by order in* $\varepsilon$ (perturbatively), but not in the usual sense for fixed $\varepsilon > 0$.

By contrast, energy functionals postulated in strain-gradient theories are positive. They can thus be minimized for fixed $\varepsilon > 0$ and not just be made stationary in a perturbative sense. The essential difference between the two approaches is clearly highlighted in [Bou20].

The goal of this paper is to reconcile strain-gradient theory and asymptotic homogenization. We work in the one-dimensional case. Specifically, we propose a step-by-step recipe that can be applied to any pseudo-energy $\Phi_{\varepsilon}[u]$ produced by second-order asymptotic homogenization. The recipe delivers a proper strain-gradient energy functional $\Psi_{\varepsilon}[u]$. The minimizer of $\Psi_{\varepsilon}[u]$ agrees to second order in $\varepsilon$ with the prediction of asymptotic homogenization, i.e., with the formal expansion $u_{\varepsilon}$ in powers of $\varepsilon$ that makes $\Phi_{\varepsilon}[u]$ stationary. Our main contribution is thus a variational reformulation of second-order homogenization that upgrades the perturbative stationarity condition for the pseudo-energy $\Phi_{\varepsilon}[u]$ into a proper minimum-energy principle.

In a recent work [TAL25b], we proposed a variational reformulation of second-order homogenization applicable to *infinite* periodic structures. We propose here an extension to continua with *finite extent*, a case which is significantly more challenging due to the existence of boundary layers.
The first minimum-energy principle for second-order homogenization goes back to the seminal work of [SC00], where the authors derive a strain-gradient energy $\Psi_{\varepsilon}[u]$ by (i) truncating the formal expansion of the microscopic strain beyond order $\varepsilon^2$, and (ii) inserting it into the microscopic energy without any further truncation. In this truncation-limited-to-strain approach, the positivity of the strain-gradient energy $\Psi_{\varepsilon}[u]$ is directly inherited from the microscopic energy. The main drawback is that the pseudo-energy contains terms of order $\varepsilon^3$ and $\varepsilon^4$ whose expressions are cumbersome, even for the simplest periodic structures. While these terms are beyond what second-order homogenization can resolve, they had to be retained in order to avoid breaking positivity. The approach we proposed in our recent work [TAL25b] proceeds instead by (i) truncating the energy beyond order $\varepsilon^2$, giving up temporarily on positivity, and (ii) restoring positivity by applying a Cholesky decomposition to the truncated energy. It produces much simpler strain-gradient functionals and involves effective constants having a clear interpretation.

The truncation-limited-to-strain approach of [SC00] was extended to finite continua by [Pan09], who included boundary-layer correctors in the expression of the strain before inserting it into the microscopic energy. The outcome was a strain-gradient theory whose boundary terms capture boundary layers effectively. The approach of Panasenko, however, is only applicable in the special case where the microstructure is invariant in the direction perpendicular to the boundary, as we show in Section 8. We thus follow and extend the alternative strategy which we introduced in [TAL25b]. It delivers a tractable strain-gradient model, see Equation (6.1), and a simple equilibrium problem whose interpretation is transparent, see Equation (6.2).

We limit attention to one-dimensional continua. The benefit is that the energy, the equilibrium problems and the solutions can all be obtained in closed analytical form. This eases considerably the identification of the key mathematical properties. To ease future extensions to higher dimensions, we have refrained as much as possible from using properties that are specific to dimension 1. For instance, we did not use the fact that the macroscopic strain $u'$ is always geometrically compatible in 1D (i.e., it is an exact derivative) or that a strain-gradient energy of the form $\int u'u''' dx$ can be integrated by parts.

Our starting point is the expression of the pseudo-energy $\Phi_{\varepsilon}[u]$ covering a broad family of periodic one-dimensional lattices, which we established recently using formal two-scale analysis [TAL25a]—these formal derivation could likely be justified rigorously using $\Gamma$-developments or expansions [AB93, BT08, SSZ11]. In [TAL25a], we derived the pseudo-energy $\Phi_{\varepsilon}[u]$ in symbolic form in terms of the microscopic properties of the lattice. This revealed two key properties of $\Phi_{\varepsilon}[u]$, which we leverage in the present work:

* in the integral term of $\Phi_{\varepsilon}[u]$, the effective elasticity coefficients satisfy two inequalities arising from the positivity of the microscopic spring constants, see Equation (3.2);

* in the boundary terms of $\Phi_{\varepsilon}[u]$, the effective higher-order tractions can be expressed in terms of other effective properties, namely the bulk elasticity constants and the boundary loading, see Equations (3.3)–(3.4).

At the core of the present work is the observation that a naïve truncation of the pseudo-energy $\Phi_{\varepsilon}[u]$ fails to deliver a strain-gradient theory due to positivity issues, see Section 2.4. Our general idea is thus to modify negligible terms in the pseudo-energy $\Phi_{\varepsilon}[u]$ in a way that restores its positivity, before truncating it. The two key properties noted above provide a natural way to do so:

* in Section 5.1, we eliminate the non-positive boundary elasticity by taking advantage of the special form of the higher-order tractions,

* in Section 5.2, we apply a Cholesky factorization to the strain energy in the bulk [TAL25b], a natural strategy in view of the inequalities satisfied by the effective elasticity coefficients in the bulk.

The paper is organized as follows. In Section 2, we present an overview of our results for a particular pseudo-energy $\Phi_{\varepsilon}[u]$. In Section 3, we recall the general form of the pseudo-energy $\Phi_{\varepsilon}[u]$ established in our previous work [TAL25a] for a family of one-dimensional lattices, along with its main properties. We also introduce the example of a chain of identical springs connecting nearest neighbors, which we use as a simple illustration throughout the paper. In Section 4, we make $\Phi_{\varepsilon}[u]$ stationary in a perturbative sense, and obtain an equilibrium solution as a formal expansion in powers of $\varepsilon$. The standard results from second-order homogenization are recovered, which we use later as a reference. In Equation (4.4), we make the key observation that the equilibrium problem takes a simple form when the special form of the higher-order tractions is used. In Section 5, we present our core contribution: we rewrite the pseudo-energy by modifying negligible terms in a way that restores its positivity. In Section 6, we quit formal expansions and introduce a positive truncation $\Psi_{\varepsilon}[u]$ of the pseudo-energy. We minimize $\Psi_{\varepsilon}[u]$ for fixed $\varepsilon > 0$, and show that the minimizer agrees with the perturbative solution, to order $\varepsilon^2$ in the limit $\varepsilon \to 0$. The strain-gradient theory $\Psi_{\varepsilon}[u]$ thus provides a variational reformulation of second-order homogenization.

## 2. OVERVIEW OF THE MAIN RESULTS BASED ON AN EXAMPLE

In this section, we present an overview of our main results based on a particular one-dimensional continuum. The general case and the detailed proofs will be presented in Section 3 onwards.
In Sections 2.1 we provide some background for the expression (2.1) of the particular pseudo-energy used as an illustration. In Section 2.2 we derive the associated equilibrium problem order by order. These results summarize the results of our previous paper [TAL25a]. In Section 2.3 we produce the perturbative solution that will be used later for verification. In Section 2.4, we motivate the present work by showing that an uninformed truncation of the pseudo-energy fails to deliver a variational principle. In Section 2.5–2.8, we present the main results of this paper, namely a variational principle delivering the same equilibrium solution as the perturbative method up to negligible terms.

## 2.1. Pseudo-energy used as an illustration

In this section, we consider a particular pseudo-energy,

$$ \Phi_{\varepsilon}[u] = \mathcal{Q} \int_{0}^{\ell} \left( \frac{u'^{2}(x)}{2} + \varepsilon^{2} \left( \frac{u''^{2}(x)}{2} + \frac{5u'(x)u'''(x)}{8} \right) - \mathcal{F}(x)u(x) \right) dx \\ - \mathcal{Q} \varepsilon^{2} u'(0) u''(0) - \rho^{0} u(0) - \varepsilon \left( 0^{*} + \frac{11^{*} \varepsilon}{8} \mathcal{F}(0) \right) u'(0) - \frac{13^{*} \varepsilon^{2}}{8} \rho^{0} u''(0) \\ + \mathcal{Q} \varepsilon^{2} u'(\ell) u''(\ell) - \rho^{\ell} u(\ell) - \varepsilon \left( 0^{*} - \frac{11^{*} \varepsilon}{8} \mathcal{F}(\ell) \right) u'(\ell) - \frac{13^{*} \varepsilon^{2}}{8} \rho^{\ell} u''(\ell) + \mathcal{O}(\varepsilon^{3}). \quad (2.1) $$

Although it is not derived from any particular microscopic lattice, this functional is typical of the functionals produced by second-order homogenization of periodic lattices in dimension 1. Indeed, it is of the general form obtained in Equation [7.7] from our previous work [TAL25a] and recalled in Equation (3.1).

The pseudo-energy (2.1) is not a strain-gradient theory, as we start by recalling briefly. Our point is that it can, however, be reformulated as a strain-gradient theory.

The parameters appearing in Equation (2.1) are the microscopic length $\varepsilon$ (typically, the spatial period of the lattice), the overall dimension of the continuum $\ell \gg \varepsilon$, the distributed applied load $\mathcal{F}(x)$, the endpoint forces $\rho^0$ and $\rho^\ell$, and a microscopic stiffness parameter $\mathcal{Q}$, which we require to be positive,

$$ \mathcal{Q} > 0. \quad (2.2) $$

In view of the first term in the integral of (2.1), $\mathcal{Q}$ can be identified with the effective traction modulus (at leading order) which we denote as $\mathcal{Q}$ in Section 3. Its positivity is a direct consequence of the positivity of the elastic constants in the lattice, see [TAL25a] for details. The distributed load $\mathcal{F}$ is assumed to be a smooth function of $x$. Pseudo-energies of the kind (2.1) can be established in the limit of well-separated scales, $\varepsilon/\ell \ll 1$, by formal two-scale expansions, see [TAL25a].

We systematically assume that the length parameters $\ell$ and $\varepsilon$ and variable $x$ have all been made dimensionless using an appropriate rescaling. There is no drawback in setting $\ell = 1$ throughout, but we keep the symbol $\ell$ for the sake of legibility.

The argument $u$ to the functional $\Phi_{\varepsilon}[u]$ is the macroscopic displacement. Along with the energy (2.1), the two-scale expansion method provides a way to reconstruct the nodal displacement in terms of $u$, see Section [7.4] in our previous work [TAL25a]. We will however ignore this relocalization step here, along with other microscopic aspects.

**Remark 2.1.** In the two-scale analysis of [TAL25a], any boundary layer arising in the microscopic solution is subtracted from the macroscopic displacement $u$, and represented in an effective way by the boundary terms of the pseudo-energy (2.1). As a result, $u$ is by construction a smooth function of $x$ that is free of boundary layers, even when some are present in the lattice.

Load control is used at the endpoints in (2.1), and no kinematic condition is applied on the unknown $u$. We will always assume that the necessary equilibrium condition of global load balance is satisfied,

$$ \rho^0 + \int_0^\ell \mathcal{F}(x) + \rho^\ell = 0. \quad (2.3) $$

We proceed to discuss the main properties of $\Phi_{\varepsilon}$ in (2.1). In the integral, the elasticity is of the strain-gradient type, with a dependence on both the macroscopic strain $u'$ and its gradients. In the density of strain energy, the $i$-th derivative of $u'$ is systematically multiplied by $\varepsilon^i$. In the integral, the potential energy $-\mathcal{F}u$ arising from the distributed force $\mathcal{F}$ involves $u$ only as the other potential energy terms of the form $-\mathcal{F}\varepsilon u''$ and $-\mathcal{F}\varepsilon^2 u'''$ have been integrated by parts.

The first boundary terms on the second and third lines of (2.1) are quadratic. This effective boundary elasticity is of the strain-gradient type as it depends not only on $u'$ but also on $\varepsilon u''$. Effective tractions $\rho^b$ appear in the potential energy terms $-\rho^b u(b)$ at each of the boundaries $b \in \{0, \ell\}$. The remaining boundary terms are linear terms in $u'$ and $u''$: they describe effective higher-order tractions. It is a key finding of our previous work [TAL25a] that these higher-order tractions can be parameterized in terms of the lower-order loads $(\rho^b, \varepsilon \mathcal{F}(b))$. The numerical constants $0^*, 11^*/8$ and $13^*/8$ that enter in the expressions of these higher-order tractions have been marked with an asterisk, so we can track them in the forthcoming calculations. These values have been obtained using the general formula for the higher-order tractions established in Equation [11.1] of our previous work [TAL25a], which we recall in Equation (3.3) below. This formula is a regularity condition expressing the fact that the pseudo-energy delivers a smooth solution $u$ by design, see Remark 2.1 above.

We call $\Phi_{\varepsilon}$ in (2.1) a pseudo-energy to emphasize that

* it can only be made stationary order by order in $\varepsilon$, by the perturbative approach discussed in Section 2.2;
* it is not required to satisfy any positivity property except at the leading order—note that $Q \int_0^\ell \frac{u'^2(x)}{2} dx \ge 0$ by (2.2).

The goal of the present work is to derive a proper energy $\Psi_\epsilon$ in explicit form that is free of these two important limitations, and still delivers the same solution up to negligible terms.

## 2.2. Perturbative approach

The standard approach to higher-order homogenization is perturbative and seeks the macroscopic displacement $u(x)$ as a formal asymptotic expansion,

$$u_{\epsilon}(x) = u_{(0)}(x) + \epsilon u_{(1)}(x) + \epsilon^2 u_{(2)}(x) + \mathcal{O}(\epsilon^3). \quad (2.4)$$

In previous work [TAL25a], we have identified a stationarity principle for second-order homogenization, whereby the macroscopic displacement is obtained by solving the following variational equality order by order in $\epsilon$,

$$\forall \delta u, \quad D_u \Phi_{\epsilon}[u_{\epsilon}, \delta u] = 0. \quad (2.5)$$

This stationarity principle makes use of the pseudo-energy $\Phi_\epsilon$ obtained by coarse-graining the energy of the discrete lattice.

Here, we use the pseudo-energy (2.1) as an illustration. At orders $\epsilon^0, \epsilon^1$ and $\epsilon^2$, the perturbative approach (2.4)–(2.5) yields the variational problem for $u_{(0)}$, $u_{(1)}$ and $u_{(2)}$ in the form:

$$\begin{align}
\forall \delta u, \quad \int_0^\ell \mathcal{Q} u'(0) \delta u' dx &= \rho^0 \delta u(0) + \rho^\ell \delta u(\ell) + \int_0^\ell \mathcal{F}(x) \delta u dx \nonumber \\
\forall \delta u, \quad \int_0^\ell \mathcal{Q} u'(1) \delta u' dx &= 0^* \delta u'(0) + 0^* \delta u'(\ell) \nonumber \\
\forall \delta u, \quad \int_0^\ell \mathcal{Q} u'(2) \delta u' dx &= +\frac{11^*}{8} \mathcal{F}(0) \delta u'(0) + \frac{13^*}{8} \rho^0 \delta u''(0) - \frac{11^*}{8} \mathcal{F}(\ell) \delta u'(\ell) + \frac{13^*}{8} \rho^\ell \delta u''(\ell) \nonumber \\
&\quad - \mathcal{Q} \int_0^\ell \left( \frac{5 u'''(0)}{8} \delta u' + u''(0) \delta u'' + \frac{5 u'(0)}{8} \delta u''' \right) dx \nonumber \\
&\quad + \mathcal{Q} (u'(0)(0) \delta u''(0) + u''(0)(0) \delta u'(0)) - \mathcal{Q} (u'(0)(\ell) \delta u''(\ell) + u''(0)(\ell) \delta u'(\ell)). \tag{2.6}
\end{align}$$

The boundary-value problems for $u_{(0)}$, $u_{(1)}$ and $u_{(2)}$ are derived from these variational problems as

$$\begin{array}{l|ccc}
\text{domain} & \text{coef.} & \varepsilon^0 & \varepsilon^1 & \varepsilon^2 \\
\forall x \in [0, \ell], & \delta u(x) & \mathcal{Q} u''_{(0)}(x) + \mathcal{F}(x) = 0 & \mathcal{Q} u''_{(1)}(x) = 0 & \mathcal{Q} \left( u''_{(2)}(x) + \frac{u'''_{(0)}(x)}{4} \right) = 0 \\
\hline
\delta u(b) & \mathcal{Q} u'_{(0)}(b) = \frac{\rho^b}{s^b} & \mathcal{Q} u'_{(1)}(b) = 0 & \mathcal{Q} \left( u'_{(2)}(b) + \frac{u'''_{(0)}(b)}{4} \right) = 0 & (2.7) \\
\forall b \in \{0, \ell\}, & \delta u'(b) & \times & 0 = 0^* & \frac{11 \mathcal{Q}}{8} u''_{(0)}(b) = -\frac{11^*}{8} \mathcal{F}(b) \\
& \delta u''(b) & \times & \times & \frac{13 \mathcal{Q}}{8} u'_{(0)}(b) = \frac{13^*}{8 s^b} \rho^b
\end{array}$$

where $s^b$ is the outward unit normal at each endpoint,

$$s^b = \begin{cases} -1 & \text{if } b=0, \\ +1 & \text{if } b=\ell. \end{cases} \quad (2.8)$$

Limiting attention to the first two rows in Equation (2.7) (labelled respectively as $\delta u(x)$ and $\delta u(b)$), we get a series of problems for classical elastic bars, with an effective distributed force, and effective tractions at both endpoints. As usual in perturbation methods, the effective loading relevant to $u_{(i)}$ at order $\epsilon^i$ is known in terms of the solutions $u_{(0)}$, ..., $u_{(i-1)}$ that have been found at lower orders. These problems are all classical elastic bar problems, of order 2, for the successive unknowns $u_{(0)}$, $u_{(1)}$ and $u_{(2)}$. It can easily be shown that the loading is globally balanced for each of these problems, see (2.3), implying that each has a unique solution $u_{(i)}$ for $0 \le i \le 2$, up to a rigid translation.

The three boundary conditions appearing on the last two rows, $0 = 0^*$, $\frac{11\mathcal{Q}}{8}u''_{(0)}(b) = -\frac{11^*}{8}\mathcal{F}(b)$ and $\frac{13\mathcal{Q}}{8}u'_{(0)}(b) = \frac{13^*}{8s^b}\rho^b$ are considered next. They are a potential source of concern since the solution has already been obtained and there is no degree of freedom left to adjust them. ‘Luckily’, these extra boundary conditions are all automatically satisfied:

* the first equation $0 = 0^*$ is satisfied thanks to the leading-order coefficient $0^*$ in the first-order traction $\epsilon \left(0^* - \frac{11^* s^b \epsilon}{8} \mathcal{F}(b)\right)$ in (2.1);

* the second equation $\frac{11\mathcal{Q}}{8}u''_{(0)}(b) = -\frac{11^*}{8}\mathcal{F}(b)$ duplicates the leading-order equilibrium $\mathcal{Q}u''_{(0)}(x) + \mathcal{F}(x) = 0$ with an overall coefficient $11^*/8$, thanks to the choice of the coefficient $11^*/8$ in the first-order traction $\epsilon \left(0^* - \frac{11^* s^b \epsilon}{8}\mathcal{F}(b)\right)$;

* the third equation $\frac{13\mathcal{Q}}{8}u'_{(0)}(b) = \frac{13^*}{8s^b}\rho^b$ duplicates the boundary condition $\mathcal{Q}u'_{(0)}(b) = \frac{\rho^b}{s^b}$ used to obtain $u_{(0)}$ with an overall coefficient $13^*/8$, thanks to the choice of the coefficient $13^*/8$ in the second-order traction $\frac{13^* \epsilon^2}{8}\rho^\ell$ in (2.1).
These cancellations therefore follow directly from the special form of higher-order tractions in the pseudo-energy (2.1). In previous work, we have proposed a general argument showing that these cancellations have to happen in the context of second-order homogenization, see Section [8.3] in [TAL25a]. The essence of the argument is that the perturbative method would fail if these cancellations did not take place, whereas the coarse-graining process by which $\Phi_{\varepsilon}$ has been obtained warrants the existence of a perturbative solution $u(x)$—this solution being an avatar of the microscopic solution to the underlying lattice problem.

To sum up, the equations appearing on the first two rows in (2.7) are the only ones that matter, and those appearing on the last two rows are automatically satisfied. Rewriting the former in terms of the series $u=u_{\varepsilon}$ in (2.4), we get the equilibrium problem in re-summed form as

$$ \begin{array}{c|l} \text{equilibrium} & \begin{array}{l} \forall x \in [0, \ell], \quad n'(x) + \mathcal{F}(x) = \mathcal{O}(\varepsilon^3) \\ \forall b \in \{0, \ell\}, \quad n(b) = \frac{\ell^b}{s^b} + \mathcal{O}(\varepsilon^3) \end{array} \\ \text{constitutive law} & \forall x \in [0, \ell], \quad n(x) = \mathcal{Q}\left(u'(x) + \frac{\varepsilon^2}{4}u'''(x)\right) + \mathcal{O}(\varepsilon^3). \end{array} \qquad (2.9) $$

Here, we have introduced the internal force $n(x)$. It is defined by the constitutive law (2.9)$_3$ which extends to order $\varepsilon^2$ that of a *classical* bar, $n(x) = \mathcal{Q}u'(x)$, when the gradient effect is neglected.

**Remark 2.2.** We have introduced the series expansion (2.4) into the variational equality (2.5), worked out the equilibrium order by order, and finally re-summed it in (2.9). Alternatively, Equation (2.9) can be derived directly from the variational equality $\forall \delta u, D_u \Phi_{\varepsilon}[u, \delta u] = 0$ in (2.5) without ever using the expanded form (2.4) of the series $u$.

## 2.3. Perturbative solution

We proceed to derive the solution to the equilibrium problem (2.9). For the sake of conciseness, we present this solution at a high level, by equating series as a whole rather than their coefficients. This concise presentation does not change the fact that equilibrium problem (2.9) is perturbative, and that the solution we derive is nothing more than a *formal series* solution.

We start with the equilibrium equations in (2.9)$_{1,2}$, which are solved by the standard method. Integrating the equilibrium (2.9)$_1$ in the bulk and setting the constant of integration using the equilibrium (2.9)$_2$ of the first boundary ($b=0$), we get

$$ n_{\varepsilon}(x) = -\ell^0 - \int_0^x \mathcal{F} \, dx + \mathcal{O}(\varepsilon^3). \qquad (2.10) $$

The equilibrium (2.9)$_2$ of the second boundary ($b=\ell$) is automatically satisfied in view of the global load balance (2.3).

Next, we use the ansatz (2.4) to derive a perturbative solution $u_{\varepsilon}(x)$ to the constitutive law (2.9), either by solving (2.9)$_3$ order by order or by solving the sequence of problems in (2.7)$_{1,2}$. The result is

$$ u_{\varepsilon}(x) = u_{\varepsilon}(0) + \int_{0}^{x} \frac{1}{\mathcal{Q}} \left( n_{\varepsilon} - \frac{\varepsilon^2}{4} n_{\varepsilon}'' \right) dx + \mathcal{O}(\varepsilon^3). \qquad (2.11) $$

For the purpose of illustration, we use a harmonic distributed force profile balanced by a point-like force $\ell^0$ at the left endpoint,

$$ \begin{array}{c|l} \text{wavenumber} & m = \frac{\pi}{2\ell} \\ \text{tractions} & \ell^0 = -\frac{\mathcal{F}_0}{m} \quad \ell^\ell = 0 \\ \text{distributed force} & \mathcal{F}(x) = \mathcal{F}_0 \cos(mx) \end{array} \qquad (2.12) $$

The argument $mx$ of the cosine varies in the range $0 \le mx \le \pi/2$ over the domain $0 \le x \le \ell$. This loading satisfies the global balance equation (2.3). With this particular loading, the perturbative solution in (2.10)–(2.11) becomes

$$ \begin{aligned} n_{\varepsilon}(x) &= \frac{\mathcal{F}_0}{m} (1 - \sin(mx)) + \mathcal{O}(\varepsilon^3), \\ u_{\varepsilon}(x) &= u_{\varepsilon}(0) + \frac{\mathcal{F}_0}{\mathcal{Q}} \left( m^{-1}x - \frac{1 + \frac{1}{4}m^2\varepsilon^2}{m^2} (1 - \cos(mx)) \right) + \mathcal{O}(\varepsilon^3). \end{aligned} \qquad (2.13) $$

We will use this solution as a reference, i.e., we will ensure that the variational principle which we set up next delivers the same solution, up to negligible terms.

## 2.4. Difficulties in identifying a variational principle

In the remainder of this section, our goal is to recover the reference solution (2.13) by minimizing an energy functional $\Psi_{\varepsilon}[u]$. We are therefore switching from a *perturbative* to a *variational* approach. In the variational approach, a positive functional $\Psi_{\varepsilon}[u]$ delivers a solution $u$ of the variational problem (stationarity condition) for fixed $\varepsilon > 0$,

$$ \forall \delta u, \quad D_u \Psi_{\varepsilon}[u, \delta u] = 0. \qquad (2.14) $$

The key difference with the perturbative approach in (2.4)–(2.5) is that we consider the limit $\varepsilon \to 0$ after solving the variational equality (2.14), and not concurrently.
We start in this section with an obvious energy candidate $\tilde{\Psi}_{\varepsilon}[u]$ that will turn out to be unsuccessful, namely the naïve truncation of the pseudo-energy $\Phi_{\varepsilon}[u]$ in (2.1) beyond order $\varepsilon^2$,

$$ 
\begin{aligned}
\tilde{\Psi}_{\varepsilon}[u] = & \mathcal{Q} \int_{0}^{\ell} \left( \frac{u'^{2}}{2} + \varepsilon^2 \left( \frac{u''^{2}}{2} + \frac{5u'u'''}{8} \right) - \mathcal{F}(x)u \right) dx \\
& - \mathcal{Q} \varepsilon^2 u'(0) u''(0) - \rho^0 u(0) - \frac{11 \varepsilon^2}{8} \mathcal{F}(0) u'(0) - \frac{13 \varepsilon^2}{8} \rho^0 u''(0) \\
& + \mathcal{Q} \varepsilon^2 u'(\ell) u''(\ell) - \rho^\ell u(\ell) + \frac{11 \varepsilon^2}{8} \mathcal{F}(\ell) u'(\ell) - \frac{13 \varepsilon^2}{8} \rho^\ell u''(\ell).
\end{aligned} 
\quad (2.15) $$

Anticipating that this will fail, we mark this functional with a tilde. The absence of a remainder $\mathcal{O}(\varepsilon^3)$ in the right-hand side of (2.15) indicates that we are trying to set up a genuine variational principle of the form (2.14). The boundary-value problem expressing the stationarity condition (2.14) takes the form

$$ 
\left. 
\begin{array}{l|l}
\text{constitutive law} & \forall x \in [0, \ell], \qquad n(x) = \mathcal{Q} \left( u'(x) + \frac{\varepsilon^2}{4} u'''(x) \right) \\
\text{equilibrium in bulk} & \forall x \in [0, \ell], \qquad n'(x) + \mathcal{F}(x) = 0 \\
\text{boundary equilibrium} & \forall b \in \{0, \ell\}, \qquad \begin{aligned}[t]
n(b) &= \frac{\rho^b}{s^b} \\
&\quad \frac{11s^b \varepsilon}{8} (\mathcal{Q} u''(b) + \mathcal{F}(b)) = 0 \\
&\quad \frac{13s^b \varepsilon^2}{8} \left( \mathcal{Q} u'(b) - \frac{\rho^b}{s^b} \right) = 0.
\end{aligned}
\end{array}
\right\} (2.16) $$

If the load is globally balanced, the internal force $n(x)$ can be found by integrating (2.16)$_2$ using the boundary condition (2.16)$_3$ at any of the endpoints. We view the remaining equations in (2.16) as a boundary-value problem for $u'$—this eliminates the indeterminate constant in $u$ representing a free rigid-body translation. The differential equation for $u'$ in (2.16)$_1$ is of order 2, and we have two boundary conditions per endpoint in (2.16)$_{4,5}$, hence 4 boundary conditions overall. There are two boundary conditions in excess, hence no solution. The problem (2.16) is ill posed.

The reason is simple: the energy functional $\tilde{\Psi}_{\varepsilon}[u]$ in (2.15) is not bounded from below, hence not positive. Indeed, evaluating $\tilde{\Psi}_{\varepsilon}$ with the trial function $u(x) = \cos^x u''_{\varepsilon} \ell$, the bulk term $\varepsilon^2 \left( \frac{u''^2}{2} + \frac{5u'u'''}{8} \right)$ in (2.15) is negative and dominates the other terms, yielding the *large and negative* equivalent $\tilde{\Psi}_{\varepsilon}[u] \sim -\frac{1}{16 \ell^3 \varepsilon^6} \to -\infty$ in the limit $\varepsilon \to 0$. The function $u(x)$ above that defeats positivity is not of the smooth form (2.4): indeed, the elimination of high-frequency content, e.g., by a filtering technique [ALR22], provides a common work-around to non-positivity issues.

The non-positivity could also have been anticipated by observing that the highest derivative of $u$, namely $u'''$, appears in the integral term of (2.15) within the *coupled term* $\varepsilon^2 u' u'''$ only, and no *quadratic term* of the form $u'''^2$ being present. This is what causes the discrepancy between the order of the differential equation (four) and the number of boundary conditions (six overall).

These difficulties explain why pseudo-energies produced by asymptotic analyses have remained limited to the perturbative setting so far.

**Remark 2.3.** The second and third boundary conditions (2.16)$_{3,4}$ are redundant *up to terms of order* $\varepsilon^3$ with the bulk equilibrium (2.16)$_1$ and with the first boundary condition (2.16)$_1$, respectively. This is consistent with the fact that we obtained automatic cancellations in the perturbative setting, when we discarded those terms of order $\varepsilon^3$ in Section 2.2. These boundary conditions however resurface in the variational setting as the cancellations do not take place beyond order $\varepsilon^2$.

**Remark 2.4.** The perturbative and variational approaches require a different number of boundary conditions. The perturbative one, presented in Sections 2.2 and 2.3, requires a *single boundary condition per endpoint* as the problems to be solved at the successive orders are all of order 2, see (2.7). By contrast, the counting argument following Equation (2.16) above shows that *the variational approach requires two boundary conditions per endpoint*. The reason for this discrepancy is that the differential equations produce a smaller space of solutions when interpreted in the perturbative sense. The homogeneous solution $u = \cos(2x/\varepsilon)$ of the differential equation (2.16)$_1$, for instance, is *not* of the regular form (2.4) postulated in the perturbative approach.

## 2.5. Variational principle

Our main result, fully justified later in Section 6, is a recipe that upgrades the pseudo-energy $\Phi_{\varepsilon}[u]$ into a *positive* energy functional $\Psi_{\varepsilon}[u, e]$. When applied to the particular pseudo-energy in (2.1), it delivers—see Equation (6.1)—,

$$ 
\left\{ 
\begin{array}{l}
\forall x, \qquad u'(x) = e(x) - \frac{5\varepsilon^2}{8} e''(x), \\
\\
\Psi_{\varepsilon}[u, e] = \int_0^{\ell} \left( \mathcal{Q} \left( \frac{1}{2}e^2(x) + \frac{\varepsilon^2}{2}e'^2(x) \right) - \mathcal{F}(x)u(x) \right) dx \qquad (2.17) \\
\qquad + \sum_{b \in \{0, \ell\}} \left[ -\rho^b \left( u(b) + \frac{5\varepsilon^2}{8} e'(b) \right) + \frac{3\varepsilon^2 s^b}{8} \mathcal{F}(b)e(b) + \frac{\varepsilon^2}{\mathcal{Q}} \rho^b \mathcal{F}(b) \right].
\end{array}
\right. $$
The equality on the first line (2.17)₁ is a kinematical constraint defining the generalized strain $e(x)$ through a differential equation in terms of the displacement gradient $u'$: the dependence of $e(x)$ on $u'(x)$ is thus non-local, as in the work of [RA93]. The classical theory of linear elastic bars uses the displacement gradient $u'(x)$ as the strain measure, and the generalized strain $e(x)$ introduced above differs by a correction of order $\varepsilon^2$.

The energy functional $\Psi_{\varepsilon}[u, e]$ in (2.17)₂ depends on both $u$ and on the generalized strain $e$. It is positive given that $Q > 0$ by (2.2).

The positive energy functional $\Psi_{\varepsilon}[u, e]$ differs from the original pseudo-energy $\Phi_{\varepsilon}[u]$ in (2.1) not only by negligible terms, of order $\mathcal{O}(\varepsilon^3)$, but also by a 'non-negligible' ghost term, of order $\varepsilon^2$. In Section 2.6, we specify the expression of the ghost terms and justify why they can be ignored, even though they are not negligible.

We proceed to minimize $\Psi_{\varepsilon}$ by solving the following variational equality for $(u_{\varepsilon}, e_{\varepsilon}, n_{\varepsilon})$

$$ \forall (\delta u, \delta e), \quad D_u \Psi_{\varepsilon}[u_{\varepsilon}, e_{\varepsilon}; \delta u, \delta e] + \int_0^{\ell} n_{\varepsilon}(x) \left( \delta u'(x) - \left( \delta e(x) - \frac{5 \varepsilon^2}{8} \delta e''(x) \right) \right) dx = 0. \quad (2.18) $$

The kinematic constraint (2.17)₁ defining the generalized strain $e_{\varepsilon}(x)$ has been treated using the method of Lagrange multipliers. The Lagrange multiplier, denoted as $n_{\varepsilon}$, can be interpreted as the internal force.

The corresponding boundary-value problem is derived later in (6.2) in the form

$$ \begin{array}{ll} \text{equilibrium} & \begin{array}{l} \forall x \in [0, \ell], \quad n'(x) + \mathcal{F}(x) = 0 \\ n(0) = -n' \quad 0 \\ n(\ell) = +n' \quad \ell \end{array} \\ \text{constitutive law} & \begin{array}{l} \forall x \in [0, \ell], \quad \mathcal{Q}(e(x) - \varepsilon^2 e''(x)) = n(x) - \frac{5\varepsilon^2}{8} n''(x) \\ \forall b \in \{0, \ell\}, \quad \mathcal{Q}e'(b) + \mathcal{F}(b) = 0 \end{array} \\ \text{generalized strain} & \forall x \in [0, \ell], \quad u'(x) = e(x) - \frac{5\varepsilon^2}{8} e''(x), \end{array} \quad (2.19) $$

This variational boundary-value problem can be compared to that obtained perturbatively in (2.9):

* The equilibrium equations (2.19)₁,₂ are identical to (2.9)₁,₂ but with the $\mathcal{O}(\varepsilon^3)$ remainders dropped.

* As noted earlier, the constitutive law (2.19)₄ is now a second-order differential equation for the generalized strain $e(x)$. Its right-hand side depends on the internal force $n$ and its second derivative. This differential equation comes with one boundary condition at each endpoint, see (2.19)₄, which makes it well posed.

* Once the generalized strain $e_{\varepsilon}$ has been found, an additional integration of the kinematical relation (2.19)₆ yields the displacement $u_{\varepsilon}$ up to a rigid translation, much like in the perturbative case, see (2.11).

The boundary-value problem in (2.19) has two remarkable features:

* It has the right number of boundary conditions: by relying on a positive energy functional, we have generated a well-posed boundary-value problem.

* The boundary-value problem is solved into 3 simple steps, corresponding to each of the blocks separated by horizontal lines in Equation (2.19):

1. the internal force $n(x)$ is found by solving the equilibrium equations,

2. the generalized strain $e(x)$ is found by solving a second-order boundary-value problem,

3. the displacement $u(x)$ is obtained by a final integration.

The second step is the only part where the gradient effect comes into play: the constitutive law is a differential equation of order 2 for $e$ (and not a proportionality relation as in classical bar theory). The differential equation comes with one boundary at each endpoint, which makes it well posed. Overall, the formulation of the equilibrium problem is not very different from that of a classical elastic bar, and its solution is moderately more difficult. This cannot be said of the variational models produced by the truncation-limited-to-strain approach [Pan09].

The solution to (2.19) is derived in the general case in Section 6.3, following the three-steps procedure outlined above. For the problem at hand and using the cosine profile (2.12) for the distributed force by way of illustration, the minimizer of $\Psi_{\varepsilon}$ takes the form

$$ \begin{aligned} n_{\varepsilon}(x) &= \frac{\mathcal{F}_0}{m} (1 - \sin m x) \\ e_{\varepsilon}(x) &= \frac{\mathcal{F}_0}{mQ} \left( 1 - \frac{1 + \frac{5}{8} m^2 \varepsilon^2}{1 + m^2 \varepsilon^2} \sin m x \right) + \frac{3 \varepsilon^3 m^2 \mathcal{F}_0}{8(1 + m^2 \varepsilon^2)Q} \frac{\exp(-\frac{x}{\varepsilon}) - \exp(-\frac{\ell - x}{\varepsilon})}{1 - \exp(-2\ell/\varepsilon)} \\ u_{\varepsilon}(x) &= C_{\varepsilon} + \frac{\mathcal{F}_0}{Q} \left( m^{-1} x - \frac{(1 + \frac{5}{8} m^2 \varepsilon^2)^2}{m^2(1 + m^2 \varepsilon^2)} (1 - \cos m x) \right) - \frac{3^2 \varepsilon^4 m^2 \mathcal{F}_0}{8^2(1 + m^2 \varepsilon^2)Q} \frac{\exp(-\frac{x}{\varepsilon}) + \exp(-\ell/\varepsilon)}{1 - \exp(-2\ell/\varepsilon)}, \end{aligned} \quad (2.20) $$
where the constant $C_\varepsilon$ represents a rigid-body translation. The generalized strain $e_\varepsilon(x)$ and the displacement $u_\varepsilon(x)$ are both made up of a regular part that is a smooth function of $x$, see the first column in (2.20), and of boundary layers present in a neighborhood of size $\sim \varepsilon$ of the endpoints, see the second column. The boundary layers are pure mathematical artifacts that are unrelated to the boundary layers present in the real elastic continuum—the latter have been effectively hidden in this coarse-grained formulation, as noted in Remark 2.1 above. The boundary layers seen in the second column of (2.20) are an undesirable side effect of the truncation. Their amplitudes is so small, however, that they have no consequence. Indeed, they affect the generalized strain $e_\varepsilon$, the scaled strain gradient $\varepsilon e'_\varepsilon$ and the displacement $u_\varepsilon$ at order $\mathcal{O}(\varepsilon^3)$, $\mathcal{O}(\varepsilon^3)$ and $\mathcal{O}(\varepsilon^4)$, respectively, in two layers of size $\sim \varepsilon$ near the endpoints. In second-order methods, these quantities are respectively resolved at order $\varepsilon^2$, $\varepsilon^2$ and $\varepsilon^3$, respectively. The error associated with the boundary-layer artifacts is thus negligible.

**Remark 2.5.** The boundary layers in (2.20) produced by the particular pseudo-energy (2.1) happen to be an order of magnitude smaller than with a generic pseudo-energy, see (6.11). The conclusion will however remain the same in the general case, as discussed in Section 6.4: they are pure mathematical artifacts and are beyond what can be resolved in second-order homogenization.

A key feature of the variational formulation is that its solution (2.20) is consistent with the perturbative one derived earlier in (2.13). Indeed, the second-order Taylor expansion of the regular part in the first column agrees with the perturbative solution (2.13), and the boundary-layer artifacts in the second column are negligible. The strain-gradient model in (2.17) thus provides a variational reformulation of the perturbative approach based on the pseudo-energy (2.1) which we started from.

## 2.6. Pseudo-energy and energy differ by ghost terms

The original pseudo-energy $\Phi_\varepsilon$ in (2.1) and the positive energy $\Psi_\varepsilon[u, e]$ in (2.17)$_2$ differ not only by negligible terms of order $\mathcal{O}(\varepsilon^3)$ but also by larger ‘ghost’ terms, of order $\mathcal{O}(\varepsilon)$ in general. In Section 6.5, see (6.16) in particular, we capture the ghost terms $\Gamma_\varepsilon[e]$ via a decomposition

$$ \Phi_\varepsilon[u] = \Psi_\varepsilon[u, e] + \Gamma_\varepsilon[e] + \mathcal{O}(\varepsilon^3). \quad (2.21) $$

By design, the ghost terms $\Gamma_\varepsilon[e]$ are boundary terms that can be written as a sum of products,

$$ \Gamma_\varepsilon[e] = \sum_{b \in \{0, \ell\}} \frac{\varepsilon^2 s^b}{\mathcal{Q}} \left\{ \mathcal{Q} \left( 1 - \varepsilon^2 \frac{d^2}{dx^2} \right) e(b) - \left( \frac{n^b}{s^b} + \frac{5}{8} \varepsilon^2 \mathcal{F}'(b) \right) \right\} \left\{ \mathcal{Q} e'(b) + \mathcal{F}(b) \right\}. \quad (2.22) $$

Still by design, each one of the factors denoted by curly braces above vanishes when evaluated on a solution of the boundary-value problem (2.19),

$$ (u, e, n) \text{ solution of (2.18)} \Rightarrow \{\cdots\} = 0. \quad (2.23) $$

This can be shown as follows:

* for the first curly bracket in (2.22), this follows by inserting the boundary equilibrium (2.19)$_{2,3}$ and the differentiated form of the bulk equilibrium (2.19)$_1$ into the constitutive law (2.19)$_4$ evaluated at the boundary $x=b$;

* for the second curly bracket, this follows directly from the boundary condition (2.19)$_5$.

Now, consider the first variation of the ghost term,

$$ D\Gamma_\varepsilon[e, \delta e] = \sum_{b \in \{0, \ell\}} \varepsilon^2 s^b \left\{ \begin{aligned} & \mathcal{Q} \left( 1 - \varepsilon^2 \frac{d^2}{dx^2} \right) e(b) - \left( \frac{n^b}{s^b} + \frac{5}{8} \varepsilon^2 \mathcal{F}'(b) \right) \delta e'(b) \\ & + \left\{ \mathcal{Q} e'(b) + \mathcal{F}(b) \right\} (\delta e(b) - \varepsilon^2 \delta e''(b)) \end{aligned} \right\}. \quad (2.24) $$

Using (2.23), this yields

$$ (u, e, n) \text{ solution of (2.18)} \Rightarrow D\Gamma_\varepsilon[e, \delta e] = 0. \quad (2.25) $$

In view of (2.18), we have

$$ (u, e, n) \text{ solution of (2.18)} \Rightarrow \forall \delta u, D_u(\Psi_\varepsilon + \Gamma_\varepsilon)[u_\varepsilon, e_\varepsilon; \delta u, \delta e] + \int_0^\ell n(x) \left( \delta u'(x) - \left( \delta e(x) - \frac{5\varepsilon^2}{8} \delta e''(x) \right) \right) dx = 0. $$

Using (2.21) and slaving the increment of generalized strain $\delta e$ to the increment of displacement in such a way that $\delta u' - (\delta e - \frac{5\varepsilon^2}{8} \delta e'') = 0$, we get

$$ (u, e, n) \text{ solution of (2.18)} \Rightarrow \forall \delta u, D_u(\Psi_\varepsilon + \Gamma_\varepsilon)[u; \delta u] = 0. \quad (2.26) $$

Even though $\Psi_\varepsilon[u]$ has been obtained from $\Phi_\varepsilon[u]$ by subtracting the non-negligible ghost term $\Gamma_\varepsilon[e]$, Equation (2.26) reveals that the minimizer $u$ of $\Psi_\varepsilon$ satisfies a similar variational equality as the perturbative solution produced by $\Phi_\varepsilon[u]$, compare with (2.5). This explains qualitatively why the minimizer of $\Psi_\varepsilon[u]$ in (2.20) agrees with the perturbative solution up to small (exponential) terms.
The boundary elasticity $-Q\epsilon^2 u'(0)u''(0) + Q\epsilon^2 u'(\ell)u''(\ell)$ present in the pseudo-energy (2.1) is not positive in the sense of quadratic forms. The ghost terms have been designed to absorb this boundary elasticity, which has indeed disappeared in (2.17). By removing a source of non-positivity at the boundaries, the ghost terms are instrumental in producing a positive energy $\Psi_{\epsilon}$.

**Remark 2.6.** We have shown that the terms in curly brackets are zero when evaluated with the minimizer of $\Psi_{\epsilon}$, see (2.23). As a result, $\Gamma_{\epsilon}$ vanishes on the minimizer, and the values of the pseudo-energy $\Phi_{\epsilon}$ and of the energy $\Psi_{\epsilon}$ agree up to $\mathcal{O}(\epsilon^3)$ at this minimizer.

### 3. GENERIC PSEUDO-ENERGY USED AS A STARTING POINT

In this section, we return to a general setting and start the proof of the main claim of the paper. The proof extends up to Section 6, where we obtain the variational principle. In this section, we introduce the generic form of the pseudo-energy obtained by coarse-graining a periodic lattice of finite length $\ell$, having cell size $\epsilon \ll \ell$. This section summarizes the main findings obtained in our previous work [TAL25a].

#### 3.1. Pseudo-energy

The pseudo-energy of a broad family of one-dimensional elastic lattice featuring both nearest and next-nearest interactions has been obtained in Equation [7.7] of [TAL25a] in the form

$$ \Phi_{\epsilon}[u] = \int_{0}^{\ell} \left( \frac{1}{2} \begin{pmatrix} u' \\ \epsilon u'' \\ \epsilon^2 u''' \end{pmatrix} \cdot \begin{pmatrix} \bar{\mathbf{K}} \\ \overline{\begin{pmatrix} \bar{Q} & \bar{A} & \bar{C} \\ \bar{A} & \bar{B} & 0 \\ \bar{C} & 0 & 0 \end{pmatrix}} \\ \begin{pmatrix} u' \\ \epsilon u'' \\ \epsilon^2 u''' \end{pmatrix} \cdot \bar{F}_{\epsilon} u \right) dx \\ + \sum_{b \in (0, \ell)} \left[ \frac{\epsilon}{2} \begin{pmatrix} u' \\ \epsilon u'' \end{pmatrix} \cdot \underbrace{\begin{pmatrix} \bar{k}_{11}^b & \bar{k}_{12}^b \\ \bar{k}_{12}^b & 0 \end{pmatrix}}_{\bar{k}^b} \cdot \begin{pmatrix} u' \\ \epsilon u'' \end{pmatrix} - \bar{t}_{\epsilon}^b u - \epsilon \begin{pmatrix} \bar{q}_{\epsilon}^b \\ \bar{r}_{\epsilon}^b \end{pmatrix} \cdot \begin{pmatrix} u' \\ \epsilon u'' \end{pmatrix} - \epsilon \bar{c}_{\epsilon}^b \right]_{x=b} + \mathcal{O}(\epsilon^3). \quad (3.1) $$

The pseudo-energy (2.1) used in the introductory example of Section 2 is of this kind, as shown in Table 3.1. All the effective constants that are produced by the two-scale analysis are denoted with an over-bar. The quantities such as $\bar{F}_{\epsilon}(x)$ or $\bar{t}_{\epsilon}^b$ which bear a subscript may depend on $\epsilon$ as well: they are regular series of the form $\bar{F}_{\epsilon}(x) = \sum_{k=0}^{2} \epsilon^k \bar{F}_{(k)}(x)$ and $\bar{t}_{\epsilon}^b(x) = \sum_{k=0}^{2} \epsilon^k \bar{t}_{(k)}^b$.

The pseudo-energy (3.1) has been obtained in [TAL25a] by combining second-order homogenization in the bulk and boundary-layer analysis at the endpoints. As a result, the nodal displacements in the lattice could be parameterized in terms of the macroscopic displacement $u$. The energy of the lattice was then viewed as a functional of $u$. The pseudo-energy $\Phi_{\epsilon}[u]$ in (3.1) is nothing but a continuous approximation of the energy of the lattice, in which the discrete sums over nodes have been replaced by a continuous integral. As noted earlier in Remark 2.1, the boundary layers are accounted for in an effective way by the boundary terms in (3.1). The macroscopic displacement $u(x)$, however, is a smooth function of $x$ that is free of boundary layers.

The matrix of homogenized moduli $\bar{\mathbf{K}}$ is a $3 \times 3$ symmetric matrix satisfying the following positivity properties,

$$ \bar{Q} > 0, \quad \bar{Q}\bar{B} - \bar{A}^2 \ge 0. \quad (3.2) $$

These inequalities follow from the positivity of the microscopic elastic moduli, see Equation [11.2] in [TAL25a]. The matrices $\bar{\mathbf{K}}^0$ and $\bar{\mathbf{K}}^{\ell}$ are $2 \times 2$ symmetric matrices which are not positive in general, see Table 3.1, and account for an effective boundary elasticity arising from the boundary layers. Both the bulk elasticity $\bar{\mathbf{K}}$ and the boundary elasticity $\bar{\mathbf{K}}^b$ are of the strain-gradient type. In both of them, we have set to 0 the entries that produce negligible contributions to the energy, of order $\mathcal{O}(\epsilon^3)$.

The pseudo-energy (3.1) is relevant to the case of load control at the endpoints. Overall, the effective loading is made up of a distributed force $\bar{F}_{\epsilon}(x)$, boundary tractions $\bar{t}_{\epsilon}^b$ and higher-order boundary tractions $\bar{q}_{\epsilon}^b$ and $\bar{r}_{\epsilon}^b$. An important result of [TAL25a] is that the higher-order boundary tractions are not independent: they can be expressed in terms of the bulk elasticity $\bar{\mathbf{K}}$, of the boundary elasticity $\bar{\mathbf{K}}^b$ and of the boundary loading $(\bar{t}_{\epsilon}^b, \bar{F}_{\epsilon}(b))$ as

$$ \left( \begin{array}{c} \bar{q}_{\epsilon}^{b} \\ \bar{r}_{\epsilon}^{b} \end{array} \right) = \frac{\bar{\mathbf{K}}^{b} + s^{b} \bar{\mathbf{l}}(\bar{\mathbf{K}})}{\bar{Q}} \cdot \left( \begin{array}{c} 1/s^{b} \bar{t}_{\epsilon}^{b} \\ -\epsilon \bar{F}_{\epsilon}(b) \end{array} \right) + \left( \begin{array}{c} \mathcal{O}(\epsilon^2) \\ \mathcal{O}(\epsilon) \end{array} \right), \quad (3.3) $$

where $s^b = \pm 1$ is the unit normal, see (2.8), and $\bar{\mathbf{l}}(\bar{\mathbf{K}})$ is the $2 \times 2$ matrix depending on the effective elasticity $\bar{\mathbf{K}}$,

$$ \bar{\mathbf{l}}(\bar{\mathbf{K}}) = \begin{pmatrix} \bar{A} & \bar{B} - \bar{C} \\ \bar{C} & 0 \end{pmatrix}. \quad (3.4) $$

The special form (3.3) of the higher-order tractions ensures that the pseudo-energy $\Phi_{\epsilon}[u]$ can be made stationary order by order up to order $\epsilon^2$ included, as has been illustrated in Section 2.2. In Section [8.3] of [TAL25a], we have proven that pseudo-energies $\Phi_{\epsilon}[u]$ obtained by coarse-graining a lattice model always produce higher-order tractions of the form (3.3). In the pseudo-energy (2.1) used in the introductory example, the higher-order boundary tractions (which are the terms marked with asterisks) have been produced using Equation (3.3).
The pseudo-energy in (3.1) includes a boundary term $-\epsilon \bar{c}_\epsilon^b$ representing the energy stored in boundary layers, arising from the applied forces. This boundary term is uncoupled to $u$ and will therefore not appear in the equilibrium problem obtained by making $\Phi_\epsilon[u]$ stationary—we have set it to zero in both our examples, see Table 3.1. We keep this term, however, as it plays a role when we consider fixed-endpoint conditions later in Appendix D.

<table><caption>**Table 3.1.** Parameters defining the two examples covered in this paper, namely the introductory example presented in (2.1) and the spring chain in (3.6). With the exception of the matrix $\bar{l}$ which is calculated using (3.4), all the parameters $\bar{Q}, \bar{A}, \bar{B}, \bar{C}$, etc. have been obtained by identification of either (2.1) or (3.6) with the generic pseudo-energy in (3.1). Note that both examples satisfy the positivity requirements in (3.2), and have higher-order tractions of the particular form (3.3). We use calligraphic letters for constants and functions relating to the illustrations.</caption><thead><tr><th rowspan="3">example</th><th colspan="4">bulk properties</th><th colspan="6">endpoint properties</th></tr><tr><th colspan="4">elasticity</th><th colspan="3">elasticity</th><th colspan="3">loading</th></tr><tr><th>Q&#773;</th><th>A&#773;</th><th>B&#773;</th><th>C&#773;</th><th>F&#773;<sub>&epsilon;</sub>(x)</th><th>k&#773;<sup>b</sup></th><th>l&#773;</th><th>r&#773;<sub>&epsilon;</sub><sup>b</sup></th><th>q&#773;<sub>&epsilon;</sub><sup>b</sup></th><th>r&#773;<sub>&epsilon;</sub><sup>b</sup></th><th>c&#773;<sub>&epsilon;</sub><sup>b</sup></th></tr></thead><tbody><tr><td>introductory</td><td>Q&#773; > 0</td><td>0</td><td>Q&#773; <sup>5</sup>&frasl;<sub>8</sub></td><td>F&#773;(x)</td><td>Q&#773; s<sup>b</sup> (<sup>0</sup>&frasl;<sub>1</sub>, <sup>0</sup>&frasl;<sub>0</sub>)</td><td><sup>Q</sup>&frasl;<sub>8</sub> (<sup>0</sup>&frasl;<sub>5</sub>, <sup>0</sup>&frasl;<sub>0</sub>)</td><td></td><td>11 &epsilon; s<sup>b</sup> / 8</td><td>F&#773;(b)</td><td><sup>13</sup>&frasl;<sub>8</sub> &rho;<sup>b</sup></td><td>0</td></tr><tr><td>spring chain</td><td>Q&#773; > 0</td><td>0</td><td>0</td><td><sup>Q</sup>&frasl;<sub>24</sub></td><td>-<sup>Q</sup>&frasl;<sub>24</sub> s<sup>b</sup> (<sup>0</sup>&frasl;<sub>1</sub>, <sup>0</sup>&frasl;<sub>0</sub>)</td><td><sup>Q</sup>&frasl;<sub>24</sub> (<sup>0</sup>&frasl;<sub>1</sub>, <sup>-1</sup>&frasl;<sub>0</sub>)</td><td></td><td><sup>&epsilon;s<sup>b</sup></sup>&frasl;<sub>12</sub></td><td>F&#773;(b)</td><td>0</td><td>0</td></tr></tbody></table>

**Remark 3.1.** We use calligraphic letters to denote the parameters entering in the definition of the illustrative examples, see Table 3.1. By contrast, we use an over-bar for those defining the generic model, see Equation (3.1). For convenience, all our examples are set up in such a way that $\bar{Q} = \mathcal{Q}$, see the first column in the table, and it is possible to identify $\bar{Q}$ and $\mathcal{Q}$. We prefer to use different symbols, however. The advantage is that the relation $\bar{B} = \mathcal{Q}$ applicable to the introductory example, for instance, cannot be mistaken for a general statement such as the ones found in Equations (3.2) or (3.3).

## 3.2. Spring chain illustration

In addition to the introductory example covered in Section 2, we illustrate our results with a chain of linearly elastic springs. The spring chain is sketched in Figure 3.1. The introductory example from Section 2 was based on the abstract pseudo-energy $\Phi_\epsilon[u]$ in (2.1), for which we provided no microscopic justification. By contrast, the spring chain is intended to illustrate how a continuous pseudo-energy $\Phi_\epsilon[u]$ can be derived from a discrete lattice model. More complex one-dimensional lattices featuring next-neighbor-interactions have been treated in our previous work [TAL25a]. A lattice of this kind is analyzed in Section 7 below.

<figure><img src="image_2.png" /><figcaption>**Figure 3.1.** A chain of identical springs is homogenized into a strain-gradient continuum.</figcaption></figure>

The chain has a variable number of springs, $N$. We denote its total length by $\ell$, and length of individual springs by $\epsilon = \ell/N$. We consider the limit $\epsilon \to 0$, and use the convention that $\ell = \mathcal{O}(1)$ remains fixed in this limit, whereas $N \to \infty$. To ease the discussion, we assume that the spring constants are given as $\epsilon^{-1}\mathcal{Q}$ in terms of a constant $\mathcal{Q} = \mathcal{O}(1) > 0$ which we will identify later as the equivalent traction modulus in the limit $\epsilon \to 0$. A generic node is labelled by $i$, with $0 \le i \le N$, and is subjected to a force $\epsilon \mathcal{F}(\epsilon i)$ in terms of a smooth, prescribed force profile $\mathcal{F}(x)$ per unit length. The terminal nodes, for $i \in \{0, N\}$, are not only subjected to this force $\epsilon \mathcal{F}(\epsilon i)$ but also to an additional ‘point-like’ force $\rho^b = \mathcal{O}(1)$. The energy of the chain of springs is thus given, in terms of the nodal displacements $(u_0, \dots, u_N)$, as

$$ \Phi_{\epsilon}^{\text{spr}}(u_0, \dots, u_N) = \frac{1}{2} \sum_{i=0}^{N-1} \epsilon^{-1} \mathcal{Q} (u_{i+1} - u_i)^2 - \rho^0 u_0 - \rho^\ell u_N - \sum_{i=0}^{N} \epsilon \mathcal{F}(\epsilon i) u_i, \quad (3.5) $$

where the first sum yields the strain energy, and the other terms are the potential of the external forces.

A continuous version of the energy of the spring chain is derived in Appendix A in the form

$$ \Phi_{\epsilon}^{\text{spr}}[u] = \int_{0}^{\ell} \left( \frac{\mathcal{Q}}{2} \left( u'^2 + \frac{\epsilon^2}{12} u' u''' \right) - \mathcal{F} u \right) dx \\ + \sum_{b \in \{0, \ell\}} \left[ -\rho^b u - \frac{\mathcal{Q} \epsilon^2 s^b}{24} u' u'' - \frac{\epsilon}{2} \mathcal{F}(b) u - \frac{\epsilon^2 s^b}{12} (\mathcal{F}'(b) u + \mathcal{F}(b) u') \right]_{x=b} + \mathcal{O}(\epsilon^3). \quad (3.6) $$

It is of the generic form in Equation (3.1), the set of effective constants being identified in Table 3.1. It also satisfies the positivity condition (3.2), and features higher-order tractions $(\bar{q}_\epsilon^b, \bar{r}_\epsilon^b) = (\frac{\epsilon s^b}{12} \mathcal{F}(b), 0)$ of the form announced in (3.3), as can be checked by using the values in Table 3.1.

As we will see later, the spring chain is somewhat peculiar, as it produces no boundary layers and features no gradient effect. Both these features are a consequence of the fact that it only includes nearest-neighbor interactions. The generic form (3.1) of the pseudo-energy does cover next-nearest-neighbor interactions, as demonstrated in Section 7.
## 4. PERTURBATIVE APPROACH

Along the lines of what was done earlier in Sections 2.2 and 2.3, we start by presenting the perturbative approach. A stationary point of the pseudo-energy is obtained order by order in $\epsilon$. We follow the method of formal asymptotic expansions until end of Section 5, postulating that the remainder $\mathcal{O}(\epsilon^3)$ remains unaffected, both when the pseudo-energy is made stationary or when it is integrated by parts. This formal approach is used in most of the existing work on higher-order homogenization. Its main benefit is to generate a reference solution, see Equation (4.9) below. Our own contribution is presented later, in Section 5 onwards.

### 4.1. Equilibrium problem to be solved order by order

The first variation of a pseudo-energy functional of the form $\Phi_{\epsilon}$ in (3.1) is obtained in Appendix [G] of our previous work [TAL25a] as

$$ D_u \Phi_{\epsilon}[u, \delta u] = - \int_0^{\ell} (n' + \bar{F}_{\epsilon}) \delta u \, dx \\ - \sum_{b \in \{0, \ell\}} \left( \begin{pmatrix} \bar{t}_{\epsilon}^b - s^b n(b) \\ \bar{q}_{\epsilon}^b - (\bar{k}^b + s^b \bar{l}) \end{pmatrix} \delta u(b) \right. \\ \left. + \epsilon \left( \begin{pmatrix} \bar{q}_{\epsilon}^b \\ \bar{r}_{\epsilon}^b \end{pmatrix} - (\bar{k}^b + s^b \bar{l}) \cdot \begin{pmatrix} u'(b) \\ \epsilon u''(b) \end{pmatrix} \right) \cdot \begin{pmatrix} \delta u'(b) \\ \epsilon \delta u''(b) \end{pmatrix} \right) + \mathcal{O}(\epsilon^3). \quad (4.1) $$

where $\bar{l} = \bar{l}(\bar{k})$ is the quantity introduced in (3.4), and $n(x)$ is the internal force, defined by the constitutive law

$$ n(x) = \bar{Q} u'(x) - \epsilon^2 (\bar{B} - 2\bar{C}) u'''(x) + \mathcal{O}(\epsilon^3). \quad (4.2) $$

The coefficient $\bar{Q} > 0$ is the effective traction modulus. The strain-gradient modulus $\bar{B} - 2\bar{C}$ is not positive in general, see Table 3.1, and we will call it an *apparent strain-gradient modulus*. The positive energy $\Psi_{\epsilon}[u]$ which we introduce later in Section 6 captures the gradient effect using a different, positive constant.

Using the special form of the higher-order boundary tractions in (3.3), the first variation can be rewritten as

$$ D_u \Phi_{\epsilon}[u, \delta u] = - \int_0^{\ell} (n' + \bar{F}_{\epsilon}) \delta u \, dx \\ - \sum_{b \in \{0, \ell\}} \left( \begin{pmatrix} \bar{t}_{\epsilon}^b - s^b n(b) \\ \bar{p}_{\epsilon}^b \end{pmatrix} \delta u(b) \right. \\ \left. + \epsilon \left( \bar{k}^b + s^b \bar{l} \right) \cdot \left( \begin{pmatrix} \underbrace{\bar{Q} \begin{pmatrix} 1/s^b \bar{r}_{\epsilon}^b \\ -\epsilon \bar{F}_{\epsilon}(b) \end{pmatrix}_{\bar{p}_{\epsilon}^b} \\ \underbrace{\bar{u}^b(b)}_{\bar{u}''(b)} \end{pmatrix} \right) \right) \cdot \left( \begin{pmatrix} \delta u'(b) \\ \epsilon \delta u''(b) \end{pmatrix} \right) + \mathcal{O}(\epsilon^3). \quad (4.3) $$

By the fundamental lemma of the calculus of variations, we solve the variational equality $\forall \delta u$, $D_u \Phi_{\epsilon}[u, \delta u] = 0$ by setting to zero the coefficients of $\delta u(x)$ in the integral and of $\delta u(b)$, $\delta u'(b)$ and $\delta u''(b)$ in the boundary terms. Ignoring the conditions associated with $\delta u'(b)$ and $\delta u''(b)$ for the moment and recalling the form of the constitutive law (4.2), we obtain a set of necessary equilibrium conditions,

$$ \begin{array}{c|c|l} \text{equilibrium} & \delta u(x) & \forall x \in [0, \ell], \quad n'(x) + \bar{F}_{\epsilon}(x) = 0 + \mathcal{O}(\epsilon^3) \\ & \delta u(b) & \forall b \in \{0, \ell\}, \quad n(b) = \frac{\bar{t}_{\epsilon}^b}{s^b} + \mathcal{O}(\epsilon^3) \\ \text{constitutive law} & \times & \forall x \in [0, \ell], \quad n(x) = \bar{Q}u'(x) - \epsilon^2(\bar{B}-2\bar{C})u'''(x) + \mathcal{O}(\epsilon^3). \end{array} \quad (4.4) $$

It is understood that this equilibrium problem must be solved order by order in $\epsilon$.

We observe that the necessary equilibrium conditions in (4.4) yield the value of the strain $u'(b)$ and strain gradient $u''(b)$ at an endpoint $b \in \{0, \ell\}$ as

$$ \begin{aligned} \begin{pmatrix} u'(b) \\ \epsilon u''(b) \end{pmatrix} &= \frac{1}{\bar{Q}} \begin{pmatrix} \bar{Q} u'(b) \\ \epsilon \bar{Q} u''(b) \end{pmatrix} + \mathcal{O}(\epsilon^2) \\ &= \frac{1}{\bar{Q}} \begin{pmatrix} n(b) \\ \epsilon n'(b) \end{pmatrix} + \mathcal{O}(\epsilon^2) \\ &= \frac{1}{\bar{Q}} \begin{pmatrix} \bar{t}_{\epsilon}^b \\ -\epsilon \bar{F}_{\epsilon}(b) \end{pmatrix} + \mathcal{O}(\epsilon^2) \\ &= \tilde{p}_{\epsilon}^b + \mathcal{O}(\epsilon^2), \end{aligned} \quad (4.5) $$

where $\tilde{p}_{\epsilon}^b$ is the term identified in (4.3).

We proceed to write the stationarity condition associated with perturbations $\delta u'(b)$ and $\delta u''(b)$ in (4.3), which we have put on hold until now. In view of (4.3), they take the form

$$ \epsilon (\bar{k}^b + s^b \bar{l}) \cdot \left( \tilde{p}_{\epsilon}^b - \begin{pmatrix} u'(b) \\ \epsilon u''(b) \end{pmatrix} \right) \cdot \begin{pmatrix} 1 & 0 \\ 0 & \epsilon \end{pmatrix} = \mathcal{O}(\epsilon^3). \quad (4.6) $$

where the matrix diag $(1, \epsilon)$ collects the factors associated with the perturbation $(\delta u'(b) \quad \epsilon \delta u''(b))$. Equation (4.5) shows that these conditions are automatically satisfied. We conclude that the boundary-value problem (4.4) is not only a necessary but also a sufficient equilibrium condition. This extends the argument presented in Section 2.2 (introductory example): the special form (3.3) of the higher-order tractions is such that the higher-order boundary conditions are automatically satisfied.
## 4.2. Order-by-order solution

We proceed to solve the equilibrium problem (4.4) order by order. This is a straightforward extension of what we have done earlier in Section 2.3. The equilibrium (4.4)$_1$ in the bulk and at the first endpoint (4.4)$_2$ ($b=0$) yield the internal force as

$$n_{\varepsilon}(x) = -\bar{t}_{\varepsilon}^{0} - \int_{0}^{x} \bar{F}_{\varepsilon} dx + \mathcal{O}(\varepsilon^{3}). \qquad (4.7)$$

Like the external load $\bar{t}_{\varepsilon}^0$ and $\bar{F}_{\varepsilon}$ in the right-hand side, this solution $n_{\varepsilon}$ is a regular series in $\varepsilon$ which is known up to order $\varepsilon^2$ included. The dependence on $\varepsilon$ is reflected in the notation $n_{\varepsilon}$.

The equilibrium (2.9)$_2$ of the second boundary ($b=\ell$) yields the global force balance condition

$$\bar{t}_{\varepsilon}^{0} + \int_{0}^{\ell} \bar{F}_{\varepsilon} dx + \bar{t}_{\varepsilon}^{\ell} = \mathcal{O}(\varepsilon^{3}). \qquad (4.8)$$

Finally, we need to solve the constitutive law (4.4)$_3$ for $u'$ in perturbations. Rewriting it as $n(x) = \bar{Q}(1 - \varepsilon^2 \frac{\bar{B}-2\bar{C}}{\bar{Q}} \frac{d^2}{dx^2})u'(x) + \mathcal{O}(\varepsilon^3)$, and noting as earlier that the perturbative solution amounts to inverting the operator in the right-hand side, we get

$$u_{\varepsilon}(x) = u_{\varepsilon}(0) + \int_{0}^{x} \frac{1}{\bar{Q}} \left( 1 + \varepsilon^2 \frac{\bar{B}-2\bar{C}}{\bar{Q}} \frac{d^2}{dx^2} \right) n_{\varepsilon} dx + \mathcal{O}(\varepsilon^3). \qquad (4.9)$$

The constant of integration $u_{\varepsilon}(0)$ represents a free rigid-body translation.

## 4.3. Illustrations

In Table 4.1, these results are specialized to the two strain-gradient continua used as illustrations. For the introductory example (first line in the table), we recover the solution announced in (2.10) and (2.11) earlier.

<table><caption>**Table 4.1.** Perturbative solution (4.7) and (4.9) for the two strain-gradient continua used as illustrations. These solutions have been obtained by inserting the parameters identified in Table 3.1 into the generic solution in (4.7) and (4.9).</caption><thead><tr><th></th><th>n<sub>&epsilon;</sub>(x) = &hellip; + &Ocirc;(&epsilon;<sup>3</sup>)</th><th>u<sub>&epsilon;</sub>(x) = &hellip; + &Ocirc;(&epsilon;<sup>3</sup>)</th></tr></thead><tbody><tr><td>introductory example (2.1)</td><td>-&rho;<sup>0</sup> - &int;<sub>0</sub><sup>x</sup> &Fcirc; dx</td><td>u<sub>&epsilon;</sub>(0) + &int;<sub>0</sub><sup>x</sup> <sup>1</sup>&frasl;<sub>&Oslash;</sub> (n<sub>&epsilon;</sub> - <sup>&epsilon;<sup>2</sup></sup>&frasl;<sub>4</sub> n<sub>&epsilon;</sub>'' dx</td></tr><tr><td>chain of springs (3.6)</td><td>-(&rho;<sup>0</sup> + <sup>&epsilon;</sup>&frasl;<sub>2</sub> &Fcirc; (0) - <sup>&epsilon;<sup>2</sup></sup>&frasl;<sub>12</sub> &Fcirc; (0') - &int;<sub>0</sub><sup>x</sup> &Fcirc; dx</td><td>u<sub>&epsilon;</sub>(0) + &int;<sub>0</sub><sup>x</sup> <sup>1</sup>&frasl;<sub>&Oslash;</sub> (n<sub>&epsilon;</sub> - <sup>&epsilon;<sup>2</sup></sup>&frasl;<sub>12</sub> n<sub>&epsilon;</sub>'' dx</td></tr></tbody></table>

**Remark 4.1.** For the spring chain, the equivalence of the discrete equilibrium equations with the effective equilibrium problem (4.4) can be established as follows. First, we use the constants from Table 3.1 to rewrite the constitutive law (4.4)$_3$ as $n_{\varepsilon}(x) = \mathcal{Q}(u'_{\varepsilon} + \varepsilon^{2/12}u''_{\varepsilon''}) (x) + \mathcal{O}(\varepsilon^3)$, where an $\varepsilon$ in subscript denotes a macroscopic quantity. Combining with the relocalization formula (A.1) for the displacement used in Appendix A and using Taylor expansions, one obtains the relocalization formula for the stress as $n^j = \mathcal{Q}|_c(u_{j+1} - u_j) = \mathcal{Q}|_c(u_{\varepsilon}(\varepsilon(j+1)) - u_{\varepsilon}(\varepsilon j)) = \mathcal{Q}(u'_{\varepsilon} + \varepsilon^{2/24}u''_{\varepsilon''})_{x=\varepsilon(j+1/2)} + \mathcal{O}(\varepsilon^3) = (n_{\varepsilon} - \frac{\varepsilon^2}{24}n_{\varepsilon''})_{x=\varepsilon(j+1/2)} + \mathcal{O}(\varepsilon^3)$, where $x = \varepsilon(j+1/2)$ is the spring midpoint. Note that the naïve relocalization formula $n^j \approx n_{\varepsilon}(\varepsilon(j+1/2))$ is not correct at order $\varepsilon^2$. The discrete equilibrium in the bulk at an interior node $i$ can therefore be written as $0 = n^i - n^{i-1} + \varepsilon \mathcal{F}(\varepsilon i) = (n_{\varepsilon} - \frac{\varepsilon^2}{24} n_{\varepsilon''})_{x=\varepsilon(i+1/2)} - (n_{\varepsilon} - \frac{\varepsilon^2}{24} n_{\varepsilon''})_{x=\varepsilon(i-1/2)} + \varepsilon \mathcal{F}(\varepsilon i) + \mathcal{O}(\varepsilon^3) = \varepsilon(n' + \mathcal{F})_{x=\varepsilon i} + \mathcal{O}(\varepsilon^3)$, which is nothing but the coarse-grained equation of equilibrium (4.4)$_1$. The compatibility of the discrete and effective boundary condition follows by a similar argument.

# 5. REDUCTION OF THE PSEUDO-ENERGY

In Section 2.4, we have shown on an example that an 'uninformed' truncation of the pseudo-energy $\Phi_{\varepsilon}$ fails to deliver a positive energy. In the present section, we work on the pseudo-energy $\Phi_{\varepsilon}$ to reveal some positivity properties. To do so, we introduce and modify negligible terms comparable to the $\mathcal{O}(\varepsilon^3)$ remainder, and identify 'ghost terms' that do not contribute to the equilibrium problem. This leads to a more compact expression of the pseudo-energy, see (5.22), which we will use as a basis for truncation later in Section 6.

## 5.1. Turning boundary elasticity into ghost terms

The three following observations suggest that it is desirable and possible to remove the boundary elasticity $\bar{k}^b$ from the pseudo-energy:

* as noted in the introductory example, it is not positive in general,

* it does not enter into the perturbative boundary-value problem (4.4),

* there is some arbitrariness in positioning the endpoints of the equivalent continuum with respect to the terminal nodes of the lattice, and this convention affects the boundary elasticity $\bar{k}^b$.

The following argument effectively removes the boundary elasticity $\bar{k}^b$.

To start with, we insert the particular form (3.3) of the higher-order tractions into the pseudo-energy (3.1),

$$\Phi_{\varepsilon}[u] = \int_{0}^{\ell} \left( \frac{1}{2} \begin{pmatrix} u' \\ \varepsilon u'' \\ \varepsilon^2 u''' \end{pmatrix} \cdot \bar{\mathbf{K}} \cdot \begin{pmatrix} u' \\ \varepsilon u'' \\ \varepsilon^2 u''' \end{pmatrix} - \bar{F}_{\varepsilon} u \right) dx \\ + \sum_{b \in \{0, \ell\}} \left[ \varepsilon g_{\varepsilon}^b - \bar{t}_{\varepsilon}^b u - \varepsilon s^b (\bar{\mathbf{l}} \cdot \bar{\mathbf{p}}_{\varepsilon}^b) \cdot \begin{pmatrix} u' \\ \varepsilon u'' \end{pmatrix} - \varepsilon \bar{c}_{\varepsilon}^b \right]_{x=b} + \mathcal{O}(\varepsilon^3), \qquad (5.1)$$
where $\tilde{\boldsymbol{p}}_{\varepsilon}^{b}$ is the quantity already encountered in (4.3), and $g_{\varepsilon}^{b}$ collects the two boundary terms that depend on $\overline{\boldsymbol{k}}^{b}$,

$$ 
\begin{aligned}
\tilde{\boldsymbol{p}}_{\varepsilon}^{b} &= \frac{1}{\overline{Q}} \left( \begin{array}{c} \frac{1}{\overline{\varepsilon}} \overline{t}_{\varepsilon}^{b} \\ -\varepsilon \overline{F}_{\varepsilon}(b) \end{array} \right) + \mathcal{O}(\varepsilon^2) \\
g_{\varepsilon}^{b} &= \left[ \frac{1}{2} \left( \begin{array}{c} u' \\ \varepsilon u'' \end{array} \right) \cdot \overline{\boldsymbol{k}}^{b} \cdot \left( \begin{array}{c} u' \\ \varepsilon u'' \end{array} \right) - (\overline{\boldsymbol{k}}^{b} \cdot \tilde{\boldsymbol{p}}_{\varepsilon}^{b}) \cdot \left( \begin{array}{c} u' \\ \varepsilon u'' \end{array} \right) \right]_{x=b} + \mathcal{O}(\varepsilon^2).
\end{aligned} 
\quad (5.2) $$

We view $g_{\varepsilon}^{b}$ as a polynomial of degree 2 in $\boldsymbol{u} = (u', \varepsilon u'')$ and rewrite it in 'canonical form' as

$$ g_{\varepsilon}^{b} = \frac{1}{2} \left( \underbrace{\left( \begin{array}{c} u'(b) \\ \varepsilon u''(b) \end{array} \right) - \tilde{\boldsymbol{p}}_{\varepsilon}^{b}}_{\tilde{g}_{\varepsilon}^{b} \begin{bmatrix} u'(b) \\ u''(b) \end{bmatrix}} \cdot \overline{\boldsymbol{k}}^{b} \cdot \left( \begin{array}{c} u'(b) \\ \varepsilon u''(b) \end{array} \right) - \tilde{\boldsymbol{p}}_{\varepsilon}^{b} \right) - \frac{1}{2} \tilde{\boldsymbol{p}}_{\varepsilon}^{b} \cdot \overline{\boldsymbol{k}}^{b} \cdot \tilde{\boldsymbol{p}}_{\varepsilon}^{b} + \mathcal{O}(\varepsilon^2). \quad (5.3) $$

Here, the first term has been rewritten in terms of a quadratic form $\tilde{g}_{\varepsilon}^{b}(\boldsymbol{u})$ taking $\boldsymbol{u} = (u', \varepsilon u'')$ as an argument,

$$ \tilde{g}_{\varepsilon}^{b}(\boldsymbol{u}) = \frac{1}{2}(\boldsymbol{u} - \tilde{\boldsymbol{p}}_{\varepsilon}^{b}) \cdot \overline{\boldsymbol{k}}^{b} \cdot (\boldsymbol{u} - \tilde{\boldsymbol{p}}_{\varepsilon}^{b}) + \mathcal{O}(\varepsilon^2). \quad (5.4) $$

Inserting the identity (5.3) into (5.1), we have

$$ \begin{aligned} \Phi_{\varepsilon}[u] = & \int_{0}^{\ell} \left( \frac{1}{2} \begin{pmatrix} u' \\ \varepsilon u'' \\ \varepsilon^2 u''' \end{pmatrix} \cdot \overline{\boldsymbol{K}} \cdot \begin{pmatrix} u' \\ \varepsilon u'' \\ \varepsilon^2 u''' \end{pmatrix} - \overline{F}_{\varepsilon} u \right) dx \\ & + \sum_{b \in \{0, \ell\}} \left[ \varepsilon \tilde{g}_{\varepsilon}^{b} \begin{pmatrix} u' \\ \varepsilon u'' \end{pmatrix} - \overline{t}_{\varepsilon}^{b} u - \varepsilon s^{b} (\overline{\boldsymbol{I}} \cdot \tilde{\boldsymbol{p}}_{\varepsilon}^{b}) \cdot \begin{pmatrix} u' \\ \varepsilon u'' \end{pmatrix} - \varepsilon c_{\varepsilon}^{b} \right]_{x=b} + \mathcal{O}(\varepsilon^3), \end{aligned} \quad (5.5) $$

where we have introduced a new version of the pure-loading energy that includes the constant term produced by (5.3),

$$ c_{\varepsilon}^{b} = \overline{c}_{\varepsilon}^{b} + \frac{1}{2} \tilde{\boldsymbol{p}}_{\varepsilon}^{b} \cdot \overline{\boldsymbol{k}}^{b} \cdot \tilde{\boldsymbol{p}}_{\varepsilon}^{b} + \mathcal{O}(\varepsilon^2). \quad (5.6) $$

Now, the tractions in (5.5), i.e., the terms that are linear in $u$, $u'$ and $u''$, can be rewritten using the expressions (3.4) of $\overline{\boldsymbol{l}}$ and (5.2)$_1$ of $\tilde{\boldsymbol{p}}_{\varepsilon}^{b}$ as

$$ \begin{aligned} -\overline{t}_{\varepsilon}^{b} u(b) - \varepsilon s^{b} (\overline{\boldsymbol{I}} \cdot \tilde{\boldsymbol{p}}_{\varepsilon}^{b}) \cdot \begin{pmatrix} u'(b) \\ \varepsilon u''(b) \end{pmatrix} &= -\overline{t}_{\varepsilon}^{b} u(b) - \frac{\varepsilon s^{b}}{\overline{Q}} \left( \begin{pmatrix} \overline{A} & \overline{B} - \overline{C} \\ \overline{C} & 0 \end{pmatrix} \cdot \begin{pmatrix} \overline{t}_{\varepsilon}^{b} / s^{b} \\ -\varepsilon \overline{F}_{\varepsilon}(b) \end{pmatrix} \right) \cdot \begin{pmatrix} u'(b) \\ \varepsilon u''(b) \end{pmatrix} + \mathcal{O}(\varepsilon^3) \\ &= \left[ -\overline{t}_{\varepsilon}^{b} \left( u + \varepsilon \frac{\overline{A}}{\overline{Q}} u' + \varepsilon^2 \frac{\overline{C}}{\overline{Q}} u'' \right) + \varepsilon^2 s^{b} \frac{\overline{B} - \overline{C}}{\overline{Q}} \overline{F}_{\varepsilon}(b) u' \right]_{x=b} + \mathcal{O}(\varepsilon^3). \end{aligned} \quad (5.7) $$

This yields the pseudo-energy (5.5) in the form

$$ \Phi_{\varepsilon}[u] = \int_{0}^{\ell} \left( \frac{1}{2} \begin{pmatrix} u' \\ \varepsilon u'' \\ \varepsilon^2 u''' \end{pmatrix} \cdot \overline{\boldsymbol{K}} \cdot \begin{pmatrix} u' \\ \varepsilon u'' \\ \varepsilon^2 u''' \end{pmatrix} - \overline{F}_{\varepsilon} u \right) dx \\ + \sum_{b \in \{0, \ell\}} \left[ \varepsilon \tilde{g}_{\varepsilon}^{b} \begin{pmatrix} u' \\ \varepsilon u'' \end{pmatrix} - \overline{t}_{\varepsilon}^{b} \left( u + \varepsilon \frac{\overline{A}}{\overline{Q}} u' + \varepsilon^2 \frac{\overline{C}}{\overline{Q}} u'' \right) + \varepsilon^2 s^{b} \frac{\overline{B} - \overline{C}}{\overline{Q}} \overline{F}_{\varepsilon}(b) u' - \varepsilon c_{\varepsilon}^{b} \right]_{x=b} + \mathcal{O}(\varepsilon^3). \quad (5.8) $$

The boundary elasticity $\overline{\boldsymbol{k}}^{b}$ has been entirely absorbed into the term $\tilde{g}_{\varepsilon}^{b}(u', \varepsilon u'')$, which we call *ghost term*. We will show that this term does not contribute to the equilibrium problem.

We are done rearranging the boundary terms and proceed with the integral.

## 5.2. Cholesky reduction of the bulk elasticity

In recent work [TAL25b], we have shown that the positivity of the elastic energy in the bulk can be restored by replacing $\overline{\boldsymbol{K}}$ by its Cholesky LDLT factorization. For the symmetric $3 \times 3$ matrix $\overline{\boldsymbol{K}}$ describing a one-dimensional continuum, the Cholesky factorization is carried out by the following identity,

$$ \left( \begin{array}{cc} \overline{Q} & \overline{A} \\ \overline{A} & \overline{B} \\ \overline{C} & 0 \end{array} \right) = \left( \begin{array}{cc} 1 & \overline{A}/\overline{Q} \\ 0 & 1 \\ 0 & 0 \end{array} \right)^{\top} \cdot \left( \begin{array}{cc} \overline{Q} & 0 \\ 0 & G \\ 0 & 0 \end{array} \right) \cdot \left( \begin{array}{cc} 1 & \overline{A}/\overline{Q} \\ 0 & 1 \\ 0 & 0 \end{array} \right) + \left( \begin{array}{cc} 0 & 0 \\ 0 & 0 \\ 0 & * \end{array} \right), \quad (5.9) $$

which can be checked using elementary matrix algebra. The symbols $*$ denote entries which we do not need to calculate as they give rise to negligible energy contributions, of order $\mathcal{O}(\varepsilon^3)$: in view of (5.8), the $*$ in the second row, for instance, gets multiplied by $(\varepsilon u'') \times \varepsilon^2 u''' = \mathcal{O}(\varepsilon^3)$ in the energy—the values of the matrix entries themselves, which are denoted by the symbol $*$, are not necessarily small. In the identity above, we have introduced the gradient modulus $G$,

$$ G = \frac{\overline{Q} \overline{B} - \overline{A}^2}{\overline{Q}}, \quad (5.10) $$
which is positive by (3.2)$_2$,

$$G \ge 0. \qquad (5.11)$$

In the right-hand side of (5.9), the same upper-triangular matrix is applied on both sides of the central, positive diagonal matrix diag $(\overline{Q}, G, 0) \ge 0$, with an additional transpose on the left-hand side.

Following our previous work [TAL25b], we use the coefficients appearing in the upper-triangular matrix to define the generalized strain $e(x)$ as

$$ \begin{aligned} e(x) &= \left( 1 \frac{\overline{A}}{\overline{Q}} \frac{\overline{c}}{\overline{Q}} \right) \cdot \begin{pmatrix} u' \\ \varepsilon u'' \\ \varepsilon^2 u''' \end{pmatrix} + \mathcal{O}(\varepsilon^3) \\ &= u'(x) + \varepsilon \frac{\overline{A}}{\overline{Q}} u''(x) + \varepsilon^2 \frac{\overline{C}}{\overline{Q}} u'''(x) + \mathcal{O}(\varepsilon^3). \end{aligned} \qquad (5.12) $$

The generalized strain $e(x)$ agrees with the standard strain $u'(x)$ up to terms $\mathcal{O}(\varepsilon^1)$ in general, and up to terms $\mathcal{O}(\varepsilon^2)$ in the particular case of a symmetric continua, when $\overline{A}=0$—note that both our illustration examples are symmetric.

Multiplying both sides of (5.9) by the vector $(u', \varepsilon u'', \varepsilon^2 u''')$ and discarding the negligible terms indicated by the * symbol, one can rewrite the elastic energy in the bulk as

$$ \int_0^\ell \frac{1}{2} \begin{pmatrix} u' \\ \varepsilon u'' \\ \varepsilon^2 u''' \end{pmatrix} \cdot \overline{\mathbf{K}} \cdot \begin{pmatrix} u' \\ \varepsilon u'' \\ \varepsilon^2 u''' \end{pmatrix} dx = \int_0^\ell \left( \frac{1}{2} \overline{Q} e^2(x) + \varepsilon^2 \frac{G}{2} e'^2(x) \right) dx + \mathcal{O}(\varepsilon^3). \qquad (5.13) $$

The benefit of the method is that the elastic energy in the right-hand side of (5.13) is positive by (3.2)$_1$ and (5.11).

Inserting into (5.8), we obtain the pseudo-energy in the form

$$ \Phi_{\varepsilon}[u] = \int_{0}^{\ell} \left( \frac{1}{2} \overline{Q} e^2(x) + \varepsilon^2 \frac{G}{2} e'^2(x) - \overline{F}_{\varepsilon}(x) u(x) \right) dx \\ + \sum_{b \in \{0, \ell\}} \left[ \varepsilon \tilde{g}_{\varepsilon}^b \begin{pmatrix} u' \\ \varepsilon u'' \end{pmatrix} - \tilde{t}_{\varepsilon}^b \begin{pmatrix} u + \varepsilon \frac{\overline{A}}{\overline{Q}} u' + \varepsilon^2 \frac{\overline{C}}{\overline{Q}} u'' \end{pmatrix} + \varepsilon^2 s^b \frac{\overline{B} - \overline{C}}{\overline{Q}} \overline{F}_{\varepsilon}(b) u' - \varepsilon e_{\varepsilon}^b \right]_{x=b} + \mathcal{O}(\varepsilon^3). \qquad (5.14) $$

**Remark 5.1.** The generalized strain in the spring chain is given by (5.12) and by Table 3.1 as $e(x) = u'(x) + \frac{\varepsilon^2}{24} u'''(x) + \mathcal{O}(\varepsilon^3)$. Up to a scaling factor $\varepsilon$ and with $x$ set at the midpoint of the spring, $x = \varepsilon (i + \frac{1}{2})$, this coincides with the change of spring length $(u(\varepsilon(i+1)) - u(\varepsilon i))$ that was derived in (A.2)$_1$. This coincidence is typical of elastic lattices possessing a single type of elastic elements. In lattices made up of different types of elastic elements, the generalized strain $e(x)$ is a weighted average of the (multiple) microscopic strain measures.

## 5.3. Eliminating the displacement gradient

The final step is to eliminate the first and second gradients of the displacement appearing in the boundary terms of (5.14) in favor of the generalized strain $e(x)$ and its gradient $e'(x)$. To do so, we invert the kinematic relation (5.12) using formal series expansions as

$$ \begin{aligned} u'(x) &= \left( 1 + \varepsilon \frac{\overline{A}}{\overline{Q}} \frac{d}{dx} + \varepsilon^2 \frac{\overline{C}}{\overline{Q}} \frac{d^2}{dx^2} \right)^{-1} e(x) + \mathcal{O}(\varepsilon^3) \\ &= \left( 1 - \varepsilon L \frac{d}{dx} + \varepsilon^2 M \frac{d^2}{dx^2} \right) e(x) + \mathcal{O}(\varepsilon^3), \end{aligned} \qquad (5.15) $$

where the coefficient of the inverse series are given by

$$ \begin{aligned} L &= \frac{\overline{A}}{\overline{Q}}, \\ M &= \left( \frac{\overline{A}}{\overline{Q}} \right)^2 - \frac{\overline{C}}{\overline{Q}}. \end{aligned} \qquad (5.16) $$

In the boundary terms of (5.14), the displacement gradients $u'$ and $u''$ are systematically multiplied by powers of $\varepsilon$ that are at least $\varepsilon^1$ and $\varepsilon^2$, respectively. We can therefore downgrade the order of approximation in (5.15) to

$$ \begin{aligned} u'(b) &= e(b) - \varepsilon L e'(b) + \mathcal{O}(\varepsilon^2) \\ u''(b) &= e'(b) + \mathcal{O}(\varepsilon). \end{aligned} \qquad (5.17) $$

We use this to eliminate $u'(b)$ and $u''(b)$ in the first three boundary terms in (5.14):

* The ghost term $\tilde{g}_{\varepsilon}^b(u'(b), \varepsilon u''(b))$ that appeared in (5.3)–(5.4) is rewritten as

$$ \begin{aligned} \tilde{g}_{\varepsilon}^b \begin{pmatrix} u'(b) \\ \varepsilon u''(b) \end{pmatrix} &= \frac{1}{2} \left( \begin{pmatrix} u'(b) \\ \varepsilon u''(b) \end{pmatrix} - \tilde{\boldsymbol{p}}_{\varepsilon}^b \right) \cdot \tilde{\boldsymbol{k}}^b \cdot \left( \begin{pmatrix} u'(b) \\ \varepsilon u''(b) \end{pmatrix} - \tilde{\boldsymbol{p}}_{\varepsilon}^b \right) + \mathcal{O}(\varepsilon^2) \\ &= \frac{1}{2} \left( \begin{pmatrix} 1 & -L \\ 0 & 1 \end{pmatrix} \cdot \begin{pmatrix} e(b) \\ \varepsilon e'(b) \end{pmatrix} - \tilde{\boldsymbol{p}}_{\varepsilon}^b \right) \cdot \tilde{\boldsymbol{k}}^b \cdot \left( \begin{pmatrix} 1 & -L \\ 0 & 1 \end{pmatrix} \cdot \begin{pmatrix} e(b) \\ \varepsilon e'(b) \end{pmatrix} - \tilde{\boldsymbol{p}}_{\varepsilon}^b \right) + \mathcal{O}(\varepsilon^2) \\ &= \hat{g}_{\varepsilon}^b \begin{pmatrix} e(b) \\ \varepsilon e'(b) \end{pmatrix} + \mathcal{O}(\varepsilon^2) \end{aligned} \qquad (5.18) $$
where $\hat{g}_{\epsilon}^{b}(\mathbf{e})$ is a quadratic form taking $\mathbf{e} = (e(b), \epsilon e'(b))$ as an argument, and $\mathbf{k}^{b}$ and $\hat{\mathbf{p}}_{\epsilon}^{b}$ are variants of the boundary elasticity $\bar{\mathbf{k}}^{b}$ and of the vector $\tilde{\mathbf{p}}_{\epsilon}^{b}$ in the basis $\mathbf{e} = (e(b), \epsilon e'(b))$,

$$ 
\begin{aligned}
\hat{g}_{\epsilon}^{b}(\mathbf{e}) &= \frac{1}{2}(\mathbf{e}-\hat{\mathbf{p}}_{\epsilon}^{b}) \cdot \mathbf{k}^{b} \cdot (\mathbf{e}-\hat{\mathbf{p}}_{\epsilon}^{b}) & \hat{\mathbf{p}}_{\epsilon}^{b} &= \begin{pmatrix} 1 & +L \\ 0 & 1 \end{pmatrix} \cdot \bar{\mathbf{p}}_{\epsilon}^{b} \\
\mathbf{k}^{b} &= \begin{pmatrix} 1 & -L \\ 0 & 1 \end{pmatrix}^{\top} \cdot \bar{\mathbf{k}}^{b} \cdot \begin{pmatrix} 1 & -L \\ 0 & 1 \end{pmatrix} &= \frac{1}{\bar{Q}} \begin{pmatrix} \bar{t}_{\epsilon}^{b} / s^{b} - \epsilon L \bar{F}_{\epsilon}(b) \\ -\epsilon \bar{F}_{\epsilon}(b) \end{pmatrix}.
\end{aligned} 
\quad (5.19) $$

We have used (5.2)$_1$ to obtain an explicit expression of $\hat{\mathbf{p}}_{\epsilon}^{b}$ in the last line.

* The traction term is rewritten with the help of (5.16) and (5.17) as

$$ -\bar{t}_{\epsilon}^{b} \left( u(b) + \epsilon \frac{\bar{A}}{\bar{Q}} u'(b) + \epsilon^2 \frac{\bar{C}}{\bar{Q}} u''(b) \right) = -\bar{t}_{\epsilon}^{b} \left( u(b) + \epsilon L (e(b) - \epsilon \mathcal{L} e'(b)) + \epsilon^2 (\mathcal{L}^2 - M) e'(b) \right) + \mathcal{O}(\epsilon^3). \quad (5.20) $$

* Using (5.16)$_2$ and (5.10), we have $\frac{\bar{B}-\bar{C}}{\bar{Q}} = \frac{(\bar{A}^2-\bar{C}\bar{Q})+(\bar{Q}\bar{B}-\bar{A}^2)}{\bar{Q}^2} = M + \frac{G}{\bar{Q}}$ which, together with (5.17)$_1$, makes it possible to rewrite the third boundary term as

$$ \epsilon^2 s^b \frac{\bar{B}-\bar{C}}{\bar{Q}} \bar{F}_{\epsilon}(b) u'(b) = \epsilon^2 s^b \left( M + \frac{G}{\bar{Q}} \right) \bar{F}_{\epsilon}(b) e(b) + \mathcal{O}(\epsilon^3). \quad (5.21) $$

Inserting this into (5.14), we get the final expression of the pseudo-energy in the form

$$ \Phi_{\epsilon}[u,e] = \int_{0}^{\ell} \left( \frac{1}{2} \bar{Q} e^2(x) + \epsilon^2 \frac{G}{2} e'^2(x) - \bar{F}_{\epsilon}(x) u(x) \right) dx \\ + \sum_{b \in \{0, \ell\}} \left[ \epsilon \hat{g}_{\epsilon}^{b} \begin{pmatrix} e \\ \epsilon e' \end{pmatrix} - \bar{t}_{\epsilon}^{b} (u + \epsilon L e - \epsilon^2 M e') + \epsilon^2 s^b \left( M + \frac{G}{\bar{Q}} \right) \bar{F}_{\epsilon}(b) e - \epsilon \epsilon c_{\epsilon}^{b} \right]_{x=b} + \mathcal{O}(\epsilon^3). \quad (5.22) $$

It is more compact than the expression in (3.1) which we started from. Both are equivalent up to an error $\mathcal{O}(\epsilon^3)$, and Equation (5.22) therefore generates the same perturbative boundary-value problem as earlier in (4.4).

The pseudo-energy in (5.22) uses the generalized strain $e$ as a dependent variable. It is slaved to the main unknown $u$ by the kinematical relation (5.15).

**Remark 5.2.** In hindsight, the coefficient $\bar{B}-2\bar{C}$ appearing in the constitutive law (4.4)$_3$ can be expressed in terms of the coefficients produced by the Cholesky decomposition in (5.10) and (5.16) as

$$ 
\begin{aligned}
\bar{B}-2\bar{C} &= \bar{Q} \left( \frac{\bar{Q}\bar{B}-\bar{A}^2}{\bar{Q}^2} + 2 \frac{\bar{A}^2-\bar{Q}\bar{C}}{\bar{Q}^2} - \frac{\bar{A}^2}{\bar{Q}^2} \right) \\
&= \bar{Q} \left( \frac{G}{\bar{Q}} + 2M-L^2 \right).
\end{aligned} 
\quad (5.23) $$

## 5.4. Illustrations

In Table 5.1, we provide the values of the parameters defining the reduced form (5.22) of the pseudo-energy for each of the illustration examples, namely the pure-loading energy $c_{\epsilon}^{b}$, the second-gradient modulus $G \ge 0$, and the coefficients $L$ and $M$ produced by the Cholesky factorization. The expressions of the quantities $\mathbf{k}^{b}$ and $\hat{\mathbf{p}}_{\epsilon}^{b}$ entering in the ghost term $\hat{g}_{\epsilon}^{b}$ are omitted, as they will not appear in the variational principle stated in Section 6. They are available from the companion notebook along with the detailed derivation of all coefficients.

<table><caption>**Table 5.1.** Values of some of the key quantities appearing in the reduced form (5.22) of the pseudo-energy, for each of the illustration examples. Note that both examples are symmetric ($L=0$) and that the spring chain features no gradient effect ($G=0$).</caption><thead><tr><th></th><th>$c_{\epsilon}^{b} = \dots + \mathcal{O}(\epsilon^2)$</th><th>$G$</th><th>$L$</th><th>$M$</th></tr></thead><tbody><tr><td>introductory example (2.1)</td><td>$-\frac{\epsilon}{\bar{Q}} \rho^{b} \mathcal{F}(b)$</td><td>$\mathcal{Q}$</td><td>0</td><td>$-5/8$</td></tr><tr><td>spring chain (3.6)</td><td>$\frac{\epsilon}{24\bar{Q}} \rho^{b} \mathcal{F}(b)$</td><td>0</td><td>0</td><td>$-1/24$</td></tr></tbody></table>

# 6. VARIATIONAL PRINCIPLE

So far, we have dealt with a pseudo-energy $\Phi_{\epsilon}[u]$ that includes an unspecified remainder $\mathcal{O}(\epsilon^3)$, we have postulated a formal asymptotic expansion for the equilibrium solution, and we have made the pseudo-energy stationary order by order. In the present section, we establish a proper variational principle, by defining an energy functional $\Psi_{\epsilon}$ which is positive for any value of $\epsilon > 0$. Its minimizers agree to second order with the perturbative solution in the limit $\epsilon \to 0$, in a sense which will be made precise in Section 6.4.
## 6.1. Energy formulation

The variational principle is based on the following energy formulation in terms of the macroscopic displacement $u$ and generalized strain $e$,

$$ egin{cases} 
\forall x, \quad u'(x) = e(x) - \varepsilon L e'(x) + \varepsilon^2 M e''(x), \\
\Psi_{\varepsilon}[u, e] = \int_{0}^{\ell} \left( \frac{1}{2} \bar{Q} e^2(x) + \varepsilon^2 \frac{G}{2} e'^2(x) - \bar{F}_{\varepsilon}(x) u(x) \right) dx \\
+ \sum_{b \in \{0, \ell\}} \left[ -\bar{t}_{\varepsilon}^b (u + \varepsilon L e - \varepsilon^2 M e') + \varepsilon^2 s^b \left( M + \frac{G}{\bar{Q}} \right) \bar{F}_{\varepsilon}(b) e - \varepsilon c_{\varepsilon}^b \right]_{x=b}. 
\end{cases} 
\quad (6.1) $$

This is a constrained energy formulation: we use the inverse form of the kinematical constraint (5.15) to relate the displacement gradient $u'$ and the generalized strain $e$. The benefit of doing so is discussed in the conclusion.

There is no remainder in the energy formulation (6.1) and we stop truncating systematically beyond order $\varepsilon^2$ from now on: in the remainder of this section, we derive the minimizer of $\Psi_{\varepsilon}$ without any approximation.

The energy $\Psi_{\varepsilon}$ proposed in (6.1)$_2$ is based on the reduced form (5.22) of the pseudo-energy $\Phi_{\varepsilon}$: we have discarded the ghost terms $\varepsilon \hat{g}_{\varepsilon}^b$ and the remainder $\mathcal{O}(\varepsilon^3)$. The reduction carried out in Section 5 modifies the remainder, and the truncation in (6.1) is different from the ‘uninformed’ (and unsuccessful) truncation which we tried in Section 2.4. In Section 6.5, we justify why the ghost terms can be discarded, by extending the argument proposed in Section 2.6.

The only quadratic terms in (6.1) are the classical elasticity $\bar{Q}e^2/2$ and the strain-gradient elasticity $\varepsilon^2 Ge'^2/2$, both being found under the integral sign: the boundary elasticity $\bar{K}^b$ has been first absorbed in the ghost terms and next discarded. As the elastic moduli $\bar{Q}>0$ and $G \ge 0$ are positive by (3.2)$_1$ and (5.11), the energy $\Psi_{\varepsilon}[u]$ is positive.

In the remainder of this section, we establish the second key property of the energy formulation above: its minimizer agrees with the perturbative solution, their difference being characterized quantitatively in Section 6.4.

## 6.2. Equilibrium boundary-value problem

In Appendix B, we work out the stationarity condition produced by the constrained energy formulation (6.1) using the method of Lagrange multipliers. The first variation of the augmented functional obtained at the end of the appendix, see Equation (B.3), yields the equilibrium boundary-value problem in the form

$$ egin{array}{l@{\quad}l}
\begin{array}{c} \text{equilibrium} \\ \text{constitutive law} \\ \text{generalized strain} \end{array} &
\begin{array}{l} \forall x \in [0, \ell], \quad n'(x) + \bar{F}_{\varepsilon}(x) = 0 \\ \\ \forall b \in \{0, \ell\}, \quad n(b) = \frac{\bar{t}_{\varepsilon}^b}{s^b} \\ \\ \forall x \in [0, \ell], \quad \left(\bar{Q} - \varepsilon^2 G \frac{d^2}{dx^2}\right) e(x) = \left(1 + \varepsilon L \frac{d}{dx} + \varepsilon^2 M \frac{d^2}{dx^2}\right) n(x) \\ \\ \forall b \in \{0, \ell\}, \quad G(\bar{Q}e'(b) + \bar{F}_{\varepsilon}(b)) = 0 \\ \\ \forall x \in [0, \ell], \quad u'(x) = \left(1 - \varepsilon L \frac{d}{dx} + \varepsilon^2 M \frac{d^2}{dx^2}\right) e(x), \end{array} \\
& \qquad (6.2)
\end{array} $$

where the kinematic relation (6.2)$_5$ connecting the generalized strain $e$ and the displacement gradient $u'$ first appeared in (6.1)$_1$.

The equilibrium equations (6.1)$_{2,1}$ in both the interior and at the endpoints are identical to those delivered by the perturbative approach (4.4).

The main novelty is that the constitutive law (6.2)$_3$ is now a differential equation of order 2 for the generalized strain $e(x)$. It comes with one boundary condition at each endpoint, see (6.2)$_4$, hence 2 boundary conditions overall, which warrants that the problem is well posed. The boundary conditions (6.2)$_4$ are similar to those which we derived in earlier work [TAL24] by applying the technique of matched asymptotic expansions to a periodic one-dimensional lattice. They are redundant with the leading order of the equilibrium in the bulk, in the sense that the constitutive law and the equilibrium yield independently $G(\bar{Q}e'(x) + \bar{F}_{\varepsilon}(x)) = G(n'(x) + \bar{F}_{\varepsilon}(x)) + \mathcal{O}(\varepsilon) = \mathcal{O}(\varepsilon)$. In this earlier work [TAL24], we have shown that the redundancy ensures that the boundary-value problem (6.2) admits a solution in the form of a formal series expansion in $\varepsilon$ (perturbative solution).

Besides, we observe that the constitutive law can be rewritten, upon elimination of the generalized strain $e(x)$, as

$$ u'(x) = \left(1 - \varepsilon L \frac{d}{dx} + \varepsilon^2 M \frac{d^2}{dx^2}\right) \left(\bar{Q} - \varepsilon^2 G \frac{d^2}{dx^2}\right)^{-1} \left(1 + \varepsilon L \frac{d}{dx} + \varepsilon^2 M \frac{d^2}{dx^2}\right) n(x). \quad (6.3) $$

The inverse of the differential operator is a formal notation referring to the solution of the differential equation, carried out in detail in Section 6.3 below. Expanding formally the right-hand side of (6.3) in series, we obtain $u'(x) = \bar{Q} \left(1 + \varepsilon^2 \left(\frac{G}{\bar{Q}} + 2M - L^2\right) \frac{d^2}{dx^2}\right) n(x) + \mathcal{O}(\varepsilon^3)$, which agrees with the perturbative constitutive law in (4.4)$_3$ when the identity (5.23) is used.

These arguments show that the variational equilibrium problem (6.2) is formally consistent with existing results. We proceed to show that its solution is also quantitatively correct.

## 6.3. Explicit solution

The solution $(n_{\varepsilon}, e_{\varepsilon}, u_{\varepsilon})$ to the variational boundary-value problem (6.2) can be derived as follows.
**Internal force** The equilibrium equations (6.2)$_{1,2}$ being identical to those found in the perturbation approach, see (4.4)$_{1,2}$, the solution for $n_\epsilon$ is the same as that obtained earlier in (4.7) but without the remainder $\mathcal{O}(\epsilon^3)$,

$$n_{\epsilon}(x) = -\bar{t}_{\epsilon}^{0} - \int_{0}^{x} \bar{F}_{\epsilon} dx. \qquad (6.4)$$

The applicable global balance of forces is also similar to (4.8) but again without any error term $\mathcal{O}(\epsilon^3)$,

$$\bar{t}_{\epsilon}^{0} + \int_{0}^{\ell} \bar{F}_{\epsilon} dx + \bar{t}_{\epsilon}^{\ell} = 0. \qquad (6.5)$$

**Special case G=0** The constitutive relation is in general a second-order differential equation, see (6.2)$_3$, but in the special case $G=0$ it degenerates to an algebraic (non-differential) relation. The problem remains well posed for $G=0$ as the boundary conditions (6.2)$_4$ are then automatically satisfied. Using Equations (6.2)$_3$ and (6.3), we obtain the solution in the form

$$\begin{align*}
e_{\epsilon}(x) &= \frac{1}{\bar{Q}} \left( 1 + \epsilon L \frac{d}{dx} + \epsilon^2 M \frac{d^2}{dx^2} \right) n_{\epsilon}(x) \\
u_{\epsilon}(x) &= u_{\epsilon}(0) + \frac{1}{\bar{Q}} \int_{0}^{x} \left( 1 - \epsilon L \frac{d}{dx} + \epsilon^2 M \frac{d^2}{dx^2} \right) \left( 1 + \epsilon L \frac{d}{dx} + \epsilon^2 M \frac{d^2}{dx^2} \right) n_{\epsilon} dx \quad (\text{case } G=0). \tag{6.6} \\
&= u_{\epsilon}(0) + \frac{1}{\bar{Q}} \int_{0}^{x} \left( 1 + \epsilon^2 (2M - L^2) \frac{d^2}{dx^2} + \epsilon^4 M^2 \frac{d^4}{dx^4} \right) n_{\epsilon} dx
\end{align*}$$

where the constant of integration $u_\epsilon(0)$ represents a rigid translation. The solution $u_\epsilon$ of the variational problem in (6.6)$_3$ agrees with the perturbative solution (4.9) up to terms of order $\mathcal{O}(\epsilon^3)$, as can be checked by using the identity (5.23) with $G=0$. The term of order $\epsilon^4$ in the displacement solution (6.6)$_3$ is not asymptotic, i.e., beyond what the second-order method can resolve.

In the remainder of this section, we analyze the general case $G \neq 0$, when the gradient effect is present.

**Generalized strain** In Appendix C, we derive the explicit solution of the problem (6.2)$_{3,4}$ for $e(x)$. We use scaled variables that are obtained by identifying (6.2)$_{3,4}$ and (C.1) as

$$\eta \equiv \epsilon \sqrt{\frac{G}{\bar{Q}}}, \quad a_{\eta}(x) \equiv \frac{1}{\bar{Q}} \left( 1 + \epsilon L \frac{d}{dx} + \epsilon^2 M \frac{d^2}{dx^2} \right) n_{\epsilon}(x), \quad j_{\eta}(b) \equiv \frac{1}{\bar{Q}} \bar{F}_{\epsilon}(b). \qquad (6.7)$$

The solution is obtained in (C.19) and is the sum of a particular solution $\tilde{e}_\epsilon(x)$ of the differential equation (6.2)$_3$ which is smooth, plus boundary layers that are produced by the boundary conditions (6.2)$_4$,

$$e_{\epsilon}(x) = \tilde{e}_{\epsilon}(x) - \sum_{b \in \{(0, \ell)\}} s^{b} C^{b}(\epsilon) \exp \left( - \frac{|b - x|}{\epsilon \sqrt{G/\bar{Q}}} \right). \qquad (6.8)$$

For the introductory example, the smooth term $\tilde{e}_\epsilon(x)$ is the one that appeared in the first column of the explicit solution (2.20), and the boundary layer is the one in the second column. We emphasized then that the boundary layers are mathematical artifacts introduced by the truncation, and that they are unrelated to the boundary layers present in the lattice. This has also been discussed earlier in Remark 2.1.

The two terms in the right-hand side of (6.8) are obtained in Appendix C as follows.

* The smooth part $\tilde{e}_\epsilon(x)$ is given by a convolution of the right-hand side of (6.2)$_3$ (that is $a_\eta(x)$) with the Green's function of the differential operator in the left-hand side, see Equations (C.2) and (C.4),

$$\tilde{e}_{\epsilon}(x) = \frac{1}{\bar{Q}} \int_{-\infty}^{+\infty} \left[ \left( 1 + \epsilon L \frac{d}{dx'} + \epsilon^2 M \frac{d^2}{dx'^2} \right) n_{\epsilon}(x') \right] \frac{1}{2} \exp \left( - \frac{|x-x'|}{\epsilon \sqrt{G/\bar{Q}}} \right) \frac{dx'}{\epsilon \sqrt{G/\bar{Q}}}. \qquad (6.9)$$

It is nothing but the perturbative solution to the differential equation (6.2)$_3$, see Section C.3.

* The amplitudes of the boundary-layer artifacts are given in exact form in (C.17) as

$$C^b(\epsilon) = \frac{\epsilon \sqrt{G/\bar{Q}}}{1 - \exp\left(-\frac{2\ell}{\epsilon \sqrt{G/\bar{Q}}}\right)} \left( \tilde{e}'_\epsilon(b) + \frac{1}{\bar{Q}} \bar{F}_\epsilon(b) - \exp\left(-\frac{\ell}{\epsilon \sqrt{G/\bar{Q}}}\right) \left( \tilde{e}'_\epsilon(\ell-b) + \frac{1}{\bar{Q}} \bar{F}_\epsilon(\ell-b) \right) \right), \qquad (6.10)$$

and can be estimated using (C.18) as

$$\begin{align*}
C^b(\epsilon) &= \epsilon \frac{G^{1/2}}{\bar{Q}^{3/2}} \left( (\ell + \epsilon L \frac{d}{dx}) n'_\epsilon(b) + \bar{F}_\epsilon(b) + \mathcal{O}(\epsilon^2) \right) \tag{6.11} \\
&= -\epsilon^2 \frac{G^{1/2}}{\bar{Q}^{3/2}} L \bar{F}'_\epsilon(b) + \mathcal{O}(\epsilon^3) \\
&= \mathcal{O}(\epsilon^2)
\end{align*}$$

The cancellation of the struck terms in (6.11)$_1$ follows from the equilibrium equation (6.2)$_1$. In Section 6.4, we show that boundary-layer artifacts having amplitude $C^b(\epsilon) = \mathcal{O}(\epsilon^2)$ are negligible in the context of second-order homogenization.
**Displacement** Integrating (6.2)$_5$, we obtain the displacement in the form

$$u_{\varepsilon}(x) = u_{\varepsilon}(0) + \int_{0}^{x} \left( 1 - \varepsilon L \frac{d}{dx} + \varepsilon^2 M \frac{d^2}{dx^2} \right) e_{\varepsilon} dx, \qquad (6.12)$$

where the constant of integration represents a rigid-body translation, as earlier.

Overall, there is only one significant difference between the above solution for the strain-gradient continuum and that of a classical elastic bar: inverting the constitutive law requires solving a boundary-value problem of order 2. The additional effort to capture the gradient effect is thus minimal, as noted earlier in Section 2.5.

**Remark 6.1.** The introductory example is symmetric, as $L=0$ in Table 5.1. Equation (6.11)$_2$ shows the boundary-layer artifacts are even smaller in this case, $C^b(\varepsilon) = \mathcal{O}(\varepsilon^3)$, in agreement with the results obtained earlier in (2.20).

## 6.4. Comparison to the perturbative solution

In this section, we quantify how the solution to the variational principle differs from the perturbative solution, which we use as a reference. Our goal is to show that the difference is negligible. Equations (6.8) and (6.11) are instrumental to this comparison: the macroscopic strains $e_{\varepsilon}$ and $\tilde{e}_{\varepsilon}(x)$ predicted, respectively, by the variational and perturbative approaches differ by the boundary-layer artifacts. These artifacts are exponential terms of magnitude $C^b = \mathcal{O}(\varepsilon^2)$ decaying away from the endpoints over a microscopic typical length $\sim \varepsilon \lambda$ where $\lambda = \sqrt{G/\bar{Q}} = \mathcal{O}(1)$ is fixed. We ignore the trivial case $G=0$, such that these exponential terms disappear entirely.

In Table 6.1, we analyze how the boundary-layer artifacts affect the successive gradients of the displacement. We denote as $\Delta_{\text{var}}u'_{\varepsilon}$, $\Delta_{\text{var}}u''_{\varepsilon}$ and $\Delta_{\text{var}}u'''_{\varepsilon}$ the corresponding errors.

<table><caption>**Table 6.1.** Errors introduced by the boundary-layer artifacts in the variational model. The boundary-layer artifacts involve exponentials decaying over a scaled length $\lambda = \sqrt{G/\bar{Q}}$.</caption><thead><tr><th>quantity</th><th>error</th><th>origin</th></tr></thead><tbody><tr><td>$\Delta_{\text{var}}e_{\varepsilon}$</td><td>$\mathcal{O}(\varepsilon^2 \exp(-\frac{\ell-x}{\varepsilon\lambda}), \varepsilon^2 \exp(-\frac{x}{\varepsilon\lambda}))$</td><td>Eqs. (6.8) and (6.11)</td></tr><tr><td>$\Delta_{\text{var}}e'_{\varepsilon}$</td><td>$\mathcal{O}(\varepsilon^1 \exp(-\frac{\ell-x}{\varepsilon\lambda}), \varepsilon^1 \exp(-\frac{x}{\varepsilon\lambda}))$</td><td>differentiation of fast-varying exponential in $e_{\varepsilon}$</td></tr><tr><td>$\Delta_{\text{var}}e''_{\varepsilon}$</td><td>$\mathcal{O}(\varepsilon^0 \exp(-\frac{\ell-x}{\varepsilon\lambda}), \varepsilon^0 \exp(-\frac{x}{\varepsilon\lambda}))$</td><td>idem</td></tr><tr><td>$\Delta_{\text{var}}u'_{\varepsilon}$</td><td>$\mathcal{O}(\varepsilon^2 \exp(-\frac{\ell-x}{\varepsilon\lambda}), \varepsilon^2 \exp(-\frac{x}{\varepsilon\lambda}))$</td><td>Eq. (6.1)$_1$</td></tr><tr><td>$\Delta_{\text{var}}u''_{\varepsilon}$</td><td>$\mathcal{O}(\varepsilon^1 \exp(-\frac{\ell-x}{\varepsilon\lambda}), \varepsilon^1 \exp(-\frac{x}{\varepsilon\lambda}))$</td><td>differentiation of fast-varying exponential in $u'_{\varepsilon}$</td></tr><tr><td>$\Delta_{\text{var}}u'''_{\varepsilon}$</td><td>$\mathcal{O}(\varepsilon^0 \exp(-\frac{\ell-x}{\varepsilon\lambda}), \varepsilon^0 \exp(-\frac{x}{\varepsilon\lambda}))$</td><td>idem</td></tr></tbody></table>

To decide whether this error is acceptable, we rely Appendix [H] of our previous paper on elastic lattices [TAL25a]. In the context of second-order methods, a sufficient condition for a perturbation to be negligible is derived in Equation [H.1] in the form

$$ \left\{ \begin{array}{l} \| (\Delta_{\text{var}}u'_{\varepsilon}, \varepsilon \Delta_{\text{var}}u''_{\varepsilon}, \varepsilon^2 \Delta_{\text{var}}u'''_{\varepsilon}) \|_{2} = \mathcal{O}(\varepsilon^2), \\ \forall b \in \{0, \ell\}, \quad \| (\Delta_{\text{var}}u'_{\varepsilon}, \varepsilon \Delta_{\text{var}}u''_{\varepsilon}) (b) \| = \mathcal{O}(\varepsilon^{3/2}). \end{array} \right. \qquad (6.13) $$

Note that slightly more permissive bounds are permitted at the endpoints (second line) than in the interior (first line). Specifically, Equation (6.13) warrants that the energy of the lattice equilibrium predicted by the effective model (including the perturbation $\Delta_{\text{var}}u_{\varepsilon}$ produced by the boundary-layer artifacts) remains at most $\mathcal{O}(\varepsilon^4)$ higher than the (minimum) energy of the exact discrete lattice equilibrium. We argued in Section [10.2] of [TAL25a] that this is a good test for second-order accuracy, and demonstrated that the perturbative solution ($\Delta_{\text{var}}u_{\varepsilon}=0$), in particular, passes this test. An equivalent criterion is that the discrete $L_2$-norm of the error on the relocalized strain remains $\mathcal{O}(\varepsilon^2)$.

Using the estimates in Table 6.1, we find that $\| (\Delta_{\text{var}}u'_{\varepsilon}, \varepsilon \Delta_{\text{var}}u''_{\varepsilon}, \varepsilon^2 \Delta_{\text{var}}u'''_{\varepsilon}) \|_2 = \mathcal{O}(\varepsilon^{5/2})$, and that $\| (\Delta_{\text{var}}u'_{\varepsilon}, \varepsilon \Delta_{\text{var}}u''_{\varepsilon}) (b) \| = \mathcal{O}(\varepsilon^2)$ at both endpoints $b \in \{0, \ell\}$. The assumptions (6.13) are thus satisfied: the boundary-layer artifacts brought about by the variational principle are negligible, and the solution to the variational principle is accurate to second order in $\varepsilon$.

In Section 7, we carry out a detailed convergence test of the solution predicted by the variational model in a particular lattice geometry, and confirm the second-order accuracy of the variational approach.

## 6.5. Pseudo-energy and energy differ by ghost terms

In this section, we justify why the ghost terms $\hat{g}_{\varepsilon}^b$ present in the pseudo-energy (5.22) could be ignored in the variational principle (6.1), even though they are larger than $\mathcal{O}(\varepsilon^3)$. The argument extends that given for the introductory example in Section 2.6.

First, we observe that Equation (4.5) predicting the boundary values of $e(b)$ and $e'(b)$ in the perturbative setting can be extended to the variational setting in the form

$$ \left( \begin{pmatrix} 1 - \varepsilon^2 \frac{G}{\bar{Q}} \frac{d^2}{dx^2} \\ \varepsilon e'(b) \end{pmatrix} e(b) \right) = \boldsymbol{p}_{\varepsilon}^b \quad \text{where} \quad \boldsymbol{p}_{\varepsilon}^b = \frac{1}{\bar{Q}} \left( -\varepsilon \begin{pmatrix} \frac{\bar{p}_{\varepsilon}^b}{s^b} - (\varepsilon L + \varepsilon^2 M \frac{d}{dx}) \bar{F}_{\varepsilon}(b) \\ \begin{pmatrix} 0 \\ (\varepsilon L \frac{d}{dx} + \varepsilon^2 M \frac{d^2}{dx^2}) \bar{F}_{\varepsilon}(b) \end{pmatrix} & \text{if } G \neq 0 \\ \end{pmatrix} \right). \qquad (6.14) $$
This equality can be established by a variant of the argument given after Equation (2.23):

* the equality concerning the first components is obtained by inserting the bulk equilibrium (6.2)$_1$ and the boundary equilibrium (6.2)$_2$ into the constitutive law (6.2)$_3$;

* that concerning the second components is obtained

* directly from the boundary condition (6.2)$_4$ if $G \neq 0$,

* by inserting the bulk equilibrium (6.2)$_1$ into the differentiated form of the constitutive law (6.2)$_3$ if $G=0$.

Next, we introduce a new variant of the ghost term $\hat{g}_{\varepsilon}^b$ in (5.19)$_1$, by replacing the quantity $\hat{p}_{\varepsilon}^b$ related to the perturbative approach with the new one, $p_{\varepsilon}^b$, defined above,

$$g_{\varepsilon}^{b}(\tilde{e}) = \frac{1}{2}(\tilde{e} - p_{\varepsilon}^{b}) \cdot k^{b} \cdot (\tilde{e} - p_{\varepsilon}^{b}). \quad (6.15)$$

As earlier, the argument $\tilde{e}$ is a vector of dimension 2. The expression of the old $\hat{p}_{\varepsilon}^b$ in (5.19)$_3$ and that of the new $p_{\varepsilon}^b$ in (6.14) are consistent, $p_{\varepsilon}^b = \hat{p}_{\varepsilon}^b + \mathcal{O}(\varepsilon^2)$, and so are therefore the old and the new ghost terms, $g_{\varepsilon}^b(\tilde{e}) = \hat{g}_{\varepsilon}^b(\tilde{e}) + \mathcal{O}(\varepsilon^2)$, for any choice of $\tilde{e} = \mathcal{O}(\varepsilon^0)$.

In view of this, the difference $\varepsilon \hat{g}_{\varepsilon}^b(e(b) \ \varepsilon e'(b))$ between the pseudo-energy $\Phi_{\varepsilon}[u]$ in (5.22) and the energy $\Psi_{\varepsilon}[u]$ in (6.1) can be rewritten as $\varepsilon \hat{g}_{\varepsilon}^b(e(b) \ \varepsilon e'(b)) = \varepsilon g_{\varepsilon}^b(e(b) \ \varepsilon e'(b)) + \mathcal{O}(\varepsilon^3) = \varepsilon g_{\varepsilon}^b(1 - \varepsilon^2 G / \bar{q}^2 \frac{d^2}{dx^2})e(b) \ \varepsilon e'(b) + \mathcal{O}(\varepsilon^3)$: in the last equality, we have brought up the left-hand side of (6.14)$_1$. The pseudo-energy and the energy are therefore related by

$$\Phi_{\varepsilon}[u] = \Psi_{\varepsilon}[u, e] + \Gamma_{\varepsilon}[e] + \mathcal{O}(\varepsilon^3), \quad \text{where } \Gamma_{\varepsilon}[e] = \sum_{b \in \{0, \ell\}} \varepsilon g_{\varepsilon}^{b} \left( \left( 1 - \varepsilon^2 \frac{G}{\bar{q}} \frac{d^2}{dx^2} \right) e(b) \ \varepsilon e'(b) \right). \quad (6.16)$$

Using Equations (6.14)–(6.16), one can now easily adapt the argument proposed in Section 2.6. In brief, Equation (6.15) shows that the first variation of $g_{\varepsilon}^b(1 - \varepsilon^2 G / \bar{q}^2 \frac{d^2}{dx^2})e(b) \ \varepsilon e'(b)$ is proportional to $(1 - \varepsilon^2 G / \bar{q}^2 \frac{d^2}{dx^2})e(b) \ \varepsilon e'(b) - p_{\varepsilon}^b$, and by (6.14) the first variation vanishes when evaluated with the minimizer of $\Psi_{\varepsilon}$ such that $\mathrm{D}\Gamma_{\varepsilon}[e, \delta e] = 0$, for any $\delta e$. This shows that the minimizer of $\Psi_{\varepsilon}$ makes $\Phi_{\varepsilon}$ stationary in the sense of Equation (2.26). This in turn explains why $\Phi_{\varepsilon}$ and $\Psi_{\varepsilon}$ predict quantitatively close solutions, see Section 6.4, even though they differ by non-negligible ghost terms, of orders $\varepsilon$.

## 6.6. Illustration: spring chain

The variational principle has been applied by anticipation in Section 2.5 to the introductory example. This example features a gradient effect ($G \neq 0$).

Here, we discuss the variational principle for the spring chain, which is peculiar as it has no gradient effect ($G=0$). The energy formulation is obtained by inserting the constants identified in Tables 3.1 and 5.1 into Equation (6.1) as

$$\left\{
\begin{array}{ll}
\forall x \quad u'(x) = \left(1 - \frac{\varepsilon^2}{24} \frac{d^2}{dx^2}\right) e(x), & \\
\Psi_{\varepsilon}[u, e] = \int_0^{\ell} \left(\frac{Q}{2} e^2(x) - \mathcal{F}(x) u(x)\right) dx & (6.17) \\
\qquad + \sum_{b \in \{0, \ell\}} \left(-\bar{t}_{\varepsilon}^b \left(u(b) + \frac{\varepsilon^2}{24} e'(b)\right) - \frac{\varepsilon^2 s^b \mathcal{F}(b)}{24} e(b) - \frac{\varepsilon^2 \rho^b \mathcal{F}(b)}{24 Q}\right). &
\end{array}
\right.$$

Note the absence of a strain-gradient contribution proportional to $e'^2$ in the integral ($G=0$). The effective traction has been identified in Table 3.1 as $\bar{t}_{\varepsilon}^b = \rho^b + \frac{\varepsilon}{2} \mathcal{F}(b) + \frac{\varepsilon^2 s^b}{12} \mathcal{F}'(b)$.

This energy formulation fixes the non-positivity coming from both the bulk and the boundary elasticity in the pseudo-energy (3.6), and predicts the same macroscopic solution to order $\varepsilon^2$ included (see below). To the best of our knowledge, no positive energy functional of this kind has been proposed in prior work, even for a system as simple as the chain of springs.

The variational boundary-value problem is obtained by inserting the relevant parameter values from Tables 3.1 and 5.1 into (6.2),

$$\left[
\begin{array}{ll}
\text{equilibrium} & \forall x \in [0, \ell], \quad n'(x) + \mathcal{F}(x) = 0 \\
& \forall b \in \{0, \ell\}, \quad n(b) = \frac{\bar{t}_{\varepsilon}^b}{s^b} \\
\text{constitutive law} & \forall x \in [0, \ell], \quad Q e(x) = \left(1 - \frac{\varepsilon^2}{24} \frac{d^2}{dx^2}\right) n(x) \\
\text{generalized strain} & \forall x \in [0, \ell], \quad u'(x) = \left(1 - \frac{\varepsilon^2}{24} \frac{d^2}{dx^2}\right) e(x).
\end{array}
\right.
\quad (6.18)$$

Given that $G=0$, the constitutive law is algebraic (and not differential) and the associated boundary conditions are automatically satisfied: we have omitted the latter.
The equilibrium solution is furnished by (6.6) as

$$ 
\begin{align} 
n_{\epsilon}(x) &= -\bar{t}_{\epsilon}^{0} - \int_{0}^{x} \mathcal{F} \, dx' \nonumber \\ 
e_{\epsilon}(x) &= \frac{1}{\bar{Q}} \left( 1 - \frac{\epsilon^2}{24} \frac{d^2}{dx^2} \right) n_{\epsilon}(x) \tag{6.19} \\ 
u_{\epsilon}(x) &= u_{\epsilon}(0) + \frac{1}{\bar{Q}} \int_{0}^{x} \left( 1 - \frac{\epsilon^2}{24} \frac{d^2}{dx^2} \right)^2 n_{\epsilon}(x) \, dx. \nonumber 
\end{align} 
$$

There are no boundary-layer artifacts, as could be expected for $G=0$.

By expanding the square in the integral in the right-hand side of (6.19)₃, one recovers the perturbative solution $u_{\epsilon} = u_{\epsilon}(0) + \int_{0}^{x} \frac{1}{\bar{Q}} (1 - \epsilon^2/12 \frac{d^2}{dx^2}) n_{\epsilon}(x) d\epsilon + \mathcal{O}(\epsilon^3)$ obtained in Table 4.1, up to a negligible term of order $\epsilon^4$.

In the companion notebook, we use (6.14)–(6.16) to obtain the explicit expression of the ghost terms in the form

$$ \Gamma_{\epsilon}[e] = \sum_{b \in (0, \ell)} \frac{\epsilon^2 s^b}{24 \bar{Q}} \left\{ \mathcal{Q} e(b) - \left( \frac{1}{s^b} (\ell^b + \epsilon/2 \mathcal{F}(b)) + \epsilon^2/8 \mathcal{F}'(b) \right) \right\} \left\{ \mathcal{Q} e'(b) + \left( 1 - \frac{\epsilon^2}{24} \frac{d^2}{dx^2} \right) \mathcal{F}(b) \right\}. \quad (6.20) $$

By design, each one of the factors inside the curly braces vanishes when the equilibrium solution (6.19) is used and the global force balance condition (4.8) is taken into account, see Section 6.5. This ensures that the ghost terms in $\Gamma_{\epsilon}[e]$ do not affect the equilibrium. Their quadratic content being proportional to $e(b)e'(b)$, the ghost terms are non-positive. Indeed, they were specifically designed to absorb the non-positive boundary elasticity present in the original pseudo-energy (3.6).

## 7. VALIDATION ON A LATTICE WITH NEXT-NEAREST-NEIGHBOR INTERACTIONS

In this section, we validate our results using the one-dimensional lattice shown in Figure 7.1. The lattice features both nearest-neighbor and next-nearest-neighbor interactions. It is made up of $N$ identical unit cells (dashed rec-tangles in the figure) that are complemented by a different arrangement of springs at both ends. The cell length is set as $\epsilon = N^{-1}$, so that the total length of the lattice is $1 + \epsilon/2$. The spring constants are identical for all springs of a given type (colored boxes in the figure): their values are indicated in the legend of the figure. The spring constants are chosen such that the equivalent traction modulus is equal to $\bar{Q} = 1$ at leading order. This requires that the spring constants scale as $\epsilon^{-1}$.

<figure><img src="image_3.png" /><figcaption>**Figure 7.1.** One-dimensional lattice used for validation. The lattice is made up of $N$ identical unit cells with length $\epsilon = N^{-1}$ (dashed rounded rectangles), that are complemented by a special geometry at both ends (for $x \le 0$ and $x \ge 1$). The lattice comprises 4 different types of linearly elastic springs (colored boxes) and there are two types of nodes (black circles and disks). The initial spring positions extends from $x = -\epsilon/4$ to $x = 1 + \epsilon/4$. The first node is blocked and the other nodes are subjected to a distributed force $\bar{F}_{\epsilon}(x)$ per unit length. The lattice is represented by an effective, second-order continua covering the domain $0 \le x \le 1$.</figcaption></figure>

The first node, with initial position $x = -\epsilon/4$, is blocked. The other nodes are subjected to a force $\epsilon/2 \bar{F}_{\epsilon}(x_i)$, where $x_i$ is the node initial position and $\bar{F}_{\epsilon}(x)$ is the applied force per unit length. We use a quadratic profile of distributed force for validation purposes,

$$ \bar{F}_{\epsilon}(x) = \frac{x^2}{2}. \quad (7.1) $$

The force profile $\bar{F}_{\epsilon}(x) = x$ used in the earlier paper [TAL25a] has been abandoned as it produces no boundary-layer artifacts (see below) and is thus peculiar.

In our previous work [TAL25a], we analyzed lattices of this kind and obtained:

* An effective pseudo-energy $\Phi_{\epsilon}$ of the form (3.1) characterizing the equivalent second-order continuum covering the domain $0 \le x \le 1$, shown as the brown thick line in Figure 7.1. All effective constants were calculated using the symbolic language Wolfram Mathematica [Wol21], and we distribute the source code publicly [TA25]. We checked that the homogenized elastic constant in the bulk satisfy the positivity property (3.2), and that the higher-order boundary tractions are of the form (3.3)–(3.4).

* The perturbative solution $u_{\epsilon}(x)$ that makes the pseudo-energy stationary to order $\epsilon^2$ included.

* Relocalization formulas for the microscopic displacement and strain, producing an approximation of the (discrete) solution to the original lattice problem based on the perturbative solution $u_{\epsilon}(x)$.
* A convergence test, based on an error $||\mathbf{E}_j^e - \mathbf{E}_j^d||_2$ quantifying the accuracy of the effective model: the error $||\mathbf{E}_j^e - \mathbf{E}_j^d||_2$ is the discrete $L_2$-norm of the difference in the microscopic strain predicted by the effective model via relocalization $(\mathbf{E}_j^e)$, and the exact one produced by the discrete lattice problem $(\mathbf{E}_j^d)$. The second-order accuracy of the effective model is established when the error decreases as $||\mathbf{E}_j^e - \mathbf{E}_j^d||_2 = \mathcal{O}(N^{-2})$ for large $N$.

In the present section, we upgrade the pseudo-energy $\Phi_{\epsilon}$ to a strain-gradient theory $\Psi_{\epsilon}$, we derive the minimizer $u_{\epsilon}$ of $\Psi_{\epsilon}$, and we apply the convergence test to verify the second-order accuracy of the relocalized solution produced by this minimizer $u_{\epsilon}$. The fixed-endpoint condition $w_{\epsilon}^0 = 0$ of the first node is treated using the method of Legendre transform presented in Appendix D. The implementation is distributed in the companion notebook [TA25].

The energy formulation of the lattice is of the form derived in the Appendix, see (D.9),

$$ 
\left\{ 
\begin{array}{l} 
\forall x, \quad u'(x) = e(x) - \epsilon L e'(x) + \epsilon^2 M e''(x), \\ 
\Psi_{\epsilon}^*[u, e] = \int_0^{\ell} \left( \frac{1}{2} \bar{Q} e^2(x) + \epsilon^2 \frac{G}{2} e'^2(x) - \bar{F}_{\epsilon}(x) u(x) \right) dx \\ 
\qquad + \sum_{b \in \{0, \ell\}} \epsilon^2 s^b \left[ \left( M + \frac{G}{\bar{Q}} \left( 1 + \epsilon L \frac{d}{dx} \right) \bar{F}_{\epsilon} \right) (b) e(b) \right. \\ 
\qquad \left. + \left[ \frac{m_{\epsilon}^0}{2\epsilon} (u(0) + \epsilon L e(0) - \epsilon^2 M e'(0) - \epsilon d_{\epsilon}^0)^2 - \hat{c}_{\epsilon}^0 \right] + \left[ -\bar{t}_{\epsilon}^1 (u + \epsilon L e - \epsilon^2 M e') - \epsilon c_{\epsilon}^1 \right] \right]_{x=1}, 
\end{array} 
\right. 
\quad (7.2) 
$$

where the effective parameters are listed in Table 7.1 and we recall that $s^b = \pm 1$ is the unit outer normal at the endpoint $b \in \{0, \ell\}$, see (2.8). The boundary terms in the two lines in (7.2) represent, respectively, an effective fixed-endpoint condition at $x=0$ with stiffness $m_{\epsilon}^0 / \epsilon = \mathcal{O}(\epsilon^{-1})$ and offset $\epsilon d_{\epsilon}^0$, and an effective applied traction $\bar{t}_{\epsilon}^1 = \mathcal{O}(\epsilon)$ at the endpoint $x=1$.

The effective properties listed in Table 7.1 have been obtained in the companion notebook [TA25]:

* by identifying all the constants entering in the expression (3.1) of the pseudo-energy relevant to this particular lattice, including $\bar{t}_{\epsilon}^1$, using a notebook distributed along with our previous work [TAL25a],

* by calculating $G$, $L$ and $M$ using (5.10) and (5.16),

* by identifying $c_{\epsilon}^b$ using (5.2)$_1$ and (5.6) and then $m_{\epsilon}^0$ using (D.7)$_2$,

* and finally by using (D.10) to get $d_{\epsilon}^0$.

$$ 
\begin{array}{l@{\quad}l@{\quad}l}
\text{bulk} & \text{fixed endpoint} & \text{free endpoint} \\
\bar{Q} = 1 & m_{\epsilon}^{0} = 0.681059 & \bar{t}_{\epsilon}^{1} = \frac{\epsilon}{2} + \frac{11\epsilon^{2}}{96} \\
G = \frac{47}{300} & d_{\epsilon}^{0} = -0.168243 \epsilon^{2} & \\
L = \frac{1}{10} & & \\
M = -\frac{133}{1200} & & \\
\end{array}
 $$

<caption>**Table 7.1.** Effective parameters of the strain-gradient theory (7.2) for the lattice shown in Figure 7.1. The values are all furnished by the two-scale analysis of [TAL25a] in terms of the lattice properties and the distributed force profile $\bar{F}_{\epsilon}(x)$ (7.1), see the companion notebook [TA25]. The constant $\hat{c}_{\epsilon}^0$ and $c_{\epsilon}^1$ that do not enter into the effective equilibrium problem are omitted here.</caption>

Next, the minimizer $u_{\epsilon}$ of (7.2) is obtained by solving the boundary-value problem in (D.11), following the method in Section 6.3. The very final step is adapted to reflect the fixed-endpoint condition: the constant of integration for $u_{\epsilon}(x)$ is no longer unspecified. It is instead calculated using the fixed-endpoint condition in (D.11)$_3$. The solution $u_{\epsilon}(x)$ is obtained in closed analytical form in the companion notebook [TA25].

The solution features the typical boundary-layer artifacts produced by the variational formulation: they are of magnitude $\epsilon^2$ in the generalized strain $e_{\epsilon}(x)$, see (6.8), and of order $\epsilon^3$ in the displacement $u_{\epsilon}(x)$, which involves an additional integration, see (6.12). The analysis in Section 6.4 suggests that these artifacts are small enough to be insignificant. We check this by applying the convergence test to the strain-gradient theory (7.2), see Figure 7.2: the error decreases as $||\mathbf{E}_j^e - \mathbf{E}_j^d||_2 = \mathcal{O}(N^{-2})$ for large $N$, which confirms the second-order accuracy of the strain-gradient theory.

<figure><img src="image_4.png" /><figcaption>**Figure 7.2.** The error $||\mathbf{E}_j^e - \mathbf{E}_j^d||_2$ associated with the predictions of the strain-gradient theory (7.2) decreases for large $N$ as $N^{-2}$. This confirms that it is accurate to second order in $\epsilon = N^{-1}$.</figcaption></figure>
## 8. INAPPLICABILITY OF THE TRUNCATION-LIMITED-TO-STRAIN APPROACH

In this Section, we examine the method of [Pan09] who obtained a strain-gradient model for a laminate, including boundary terms that are asymptotic to order $\epsilon^2$ included. The author considered a laminate whose layers are oriented perpendicular to the boundary (i.e., which is invariant in the direction perpendicular to the boundary). We show that the method is only applicable for this particular microstructure orientation. The 1D lattices we consider, for instance, are not invariant (but periodic) in the direction x perpendicular to the boundary.

Below, we apply the first steps of Panasenko's method to a simple chain of springs, and highlight the source of the difficulties.

From Appendix A, see Equations (3.5) and (A.2), the discrete energy of the simple chain of springs can be written as

$$ \Phi_{\epsilon}^{\text{spr}} = \frac{\epsilon}{2} \mathcal{Q} \sum_{i=0}^{N-1} \left( u' + \frac{\epsilon^2}{24} u''' \right)^2_{x=\epsilon(i+\frac{1}{2})} - \mathcal{L}[u] + \mathcal{O}(\epsilon^3) \quad (8.1) $$

where $\mathcal{L}$ are linear terms in the nodal displacements ($u_0, \dots, u_N$). In the sum, the truncation has been limited to the strain, which is the quantity appearing inside the square. The key idea of [SC00, Pan09] is to refrain from doing any additional truncation, so as to preserve positivity.

To turn this discrete sum into an integral, one has to use the Euler-Maclaurin formula, see (A.4)$_2$. It produces a boundary term,

$$ \Phi_{\epsilon}^{\text{spr}} = \frac{1}{2} \int_{0}^{N\epsilon} \mathcal{Q} \left( u' + \frac{\epsilon^2}{24} u''' \right)^2 dx - \frac{\epsilon^2}{24} \left[ \mathcal{Q} u' u''' \right]_{0}^{N\epsilon} - \mathcal{L}[u] + \mathcal{O}(\epsilon^3). \quad (8.2) $$

The new boundary term $-\frac{\epsilon^2}{24} \left[ \mathcal{Q} u' u''' \right]_0^{N\epsilon}$ is quadratic but not positive, potentially ruining the positivity of the functional obtained by removing the $\mathcal{O}(\epsilon^3)$ remainder.

Panasenko analyzes a special circumstance: when the continuum is invariant in the direction x perpendicular to the boundary, its strain energy is readily available as the integral over x and the Euler-Maclaurin step is absent.

A laminate whose layers are oriented along the boundary raises the same difficulty as the spring lattice. Indeed, its strain energy is first available as an integral over x mixing the slow (x) and fast (x/η) variables. Homogenization involves a mathematical step that turns this mixed integration into a sequential integration over y=x/η and x. This step produces the same kind of undesirable boundary terms as the Euler-Maclaurin formula.

## 9. DISCUSSION AND CONCLUSION

We have revisited second-order homogenization as a variational problem, considering one-dimensional elastic lattices as an archetypel of periodic elastic continua. A strain-gradient theory governing the macroscopic displacement u has been obtained. Its second-order accuracy has been established by comparison with the discrete solution to the lattice problem, when the number of cells is large.

The derivation of the strain-gradient theory is spread over two papers. In our previous work [TAL25a], we have derived the pseudo-energy of the lattice, along with relocalization formulas in terms of u. The pseudo-energy is a functional of u, taking the form of a formal series expansion in powers of the scale separation parameter $\epsilon$. When truncated beyond order $\epsilon^2$ in an uninformed way, the pseudo-energy is non-positive and thus fails to deliver a variational principle. In the present work, we have presented a recipe that upgrades the pseudo-energy into a proper strain-gradient theory. The recipe leverages two important properties of the pseudo-energy which we have identified in [TAL25a]: minimalistic positivity properties for the effective elastic parameters in the bulk, see (3.2), and an expression of the effective higher-order tractions in terms of the other effective properties, see (3.3)–(3.4). We argued that these two crucial properties hold in periodic continua in general, and not just in one-dimensional lattices.

The derivation of the strain-gradient theory involves three key ingredients: (i) building up on our previous work [TAL25b], we have restored the positivity of the strain-gradient elasticity in the bulk using a Cholesky LDLT decomposition; (ii) the residual non-positive boundary elasticity has been absorbed into ghost terms and eliminated; (iii) we have rewritten in inverse form the definition of the generalized strain $e(x)$ produced by the Cholesky decomposition, see (5.15).

The variational boundary-value problem associated with the strain-gradient theory takes a compact and legible form, see (6.2). It involves a limited number of effective parameters whose interpretation is clear. Remarkably, the solution to this high-order equilibrium problem can be split into three lower-order problems, each corresponding to a block delimited by horizontal lines in (6.2): force balance equations (first order), constitutive law (second order), and integration of generalized strain into displacement (first order). This eases the derivation of the equilibrium solution considerably. The strain-gradient theory proposed in the seminal paper of Panasenko [Pan09] does not enjoy these features.

The kinematic relation defining the generalized strain $e(x)$ in terms of the displacement gradient $u'(x)$ has been first derived in (5.12) as $e(x) = \mathcal{L}_{\epsilon}[u'](x) + \mathcal{O}(\epsilon^3)$ where $\mathcal{L}$ is the differential operator $\mathcal{L}_{\epsilon} = (1 + \epsilon \frac{\bar{A}}{Q} \frac{d}{dx} + \epsilon^2 \frac{\bar{C}}{Q} \frac{d^2}{dx^2})$. In the strain-gradient theory, we have inverted this kinematic relation before truncating it, see (5.15). If we had omitted the inversion, any solution $u$ to the differential equation $\mathcal{L}_{\epsilon}[u'] = 0$ would have become a soft mode having zero strain energy ($\epsilon \equiv 0$). The existence of a space of dimension 2 of such soft modes (in addition to the legitimate rigid-body translations, $u' \equiv 0$) would be problematic. To make things worse, the soft modes vary on the microscopic scale $\epsilon$, thus defeating the basic assumption underlying the two-scale analysis. The inverted form (5.15) of the kinematic relation thus ensures that the minimum of the strain-gradient theory is unique and smooth.
It is remarkable that the displacement $v(x) = u(x) + \varepsilon L e(x) - \varepsilon^2 M e'(x)$ dual to the traction $\tilde{t}_e^b$ in the strain-gradient energy $(6.1)_2$ yields upon differentiation $v'(x) = u'(x) + \varepsilon L e'(x) - \varepsilon^2 M e''(x)$, which is nothing but the generalized strain $v'(x) = e(x)$ by $(6.1)_1$. This suggests a compact reformulation of the strain-gradient theory,

$$ \Psi_{\varepsilon}[v] = \int_{0}^{\ell} \left( \frac{1}{2} \bar{Q} v'^{2} + \varepsilon^{2} \frac{G}{2} v''^{2} - \tilde{F}_{\varepsilon} v \right) dx + \sum_{b \in (0, \ell)} \left[ -\tilde{t}_{\varepsilon}^{b} v + \varepsilon^{2} s^{b} \frac{G}{\bar{Q}} \tilde{F}_{\varepsilon} v' - \varepsilon c_{\varepsilon}^{b} \right]_{x=b} \quad (9.1) $$

where $\tilde{t}_e^b = \tilde{t}_e^b - \varepsilon s^b (L\bar{F}_e(b) + \varepsilon M\bar{F}_e'(b))$ and $\tilde{F}_e(x) = \bar{F}_e(x) + \varepsilon L\bar{F}_e'(x) + \varepsilon^2 M\bar{F}_e''(x)$ are an alternative traction and distributed force, respectively. This expression has been obtained by eliminating the original macroscopic displacement $u$ in favor of the generalized one $v$ in (6.1), and by integrating by parts the $\bar{F}_e v'$ and $\bar{F}_e v''$ terms appearing in the integral. The functional $\Psi_{\varepsilon}[v]$ is remarkably compact. An additional advantage over the formulation (6.1) is that it is free of any kinematical constraint, at least for force-controlled endpoints: once $v$ has been obtained by minimizing $\Psi_{\varepsilon}[v]$, the original macroscopic displacement is obtained as $u = v - \varepsilon L e + \varepsilon^2 M e' = v - \varepsilon L v' + \varepsilon^2 M v''$. We did not promote this alternative formulation as it is specific to dimension 1: in higher dimension, the generalized strain is not geometrically compatible [TAL25b], i.e., there is no equivalent for the equality $e = v'$.

In future work, we will investigate in more details the concrete benefits of our method. An obvious one is the possibility to set up simulations of the effective model using numerical methods that rely on the energy formulation, such as the finite-element method. Generating a prediction for a finite value $\varepsilon > 0$ of the scale-separation parameter directly, and not just via a formal expansion for $\varepsilon \to 0$, is highly appealing for applications. As with other higher-order problems, simulating the strain-gradient theory will require special care, particularly in the presence of the kinematic constraint (6.1)$_1$. One can hope that the splitting of the equilibrium problem into lower-order problems observed in (6.2) will mitigate these difficulties. As discussed in Section [8] of our previous work [TAL25b], one might anticipate that the regularizing nature of the strain-gradient term ($G > 0$) makes it possible for the strain-gradient theory to handle distributed force profiles $\bar{F}_e(x)$ having low regularity. This will need to be properly analyzed.

Another important question to be investigated in future work is the extension of our method to continua of dimension $d=2$ or 3. Our presentation has been designed to make this extension as natural as possible. Specifically, (i) the derivation of the pseudo-energy is similar in higher dimension, except for the fact that the boundary may display other types of layers such as those existing at corners or wedges, (ii) the characterization of higher-order tractions consistent with the perturbative approach is not expected to raise any difficulty, (iii) the identification of ghost terms is likely to follow from a straightforward extension of the 1D case, but it is difficult to be definite on this point, (iv) the Cholesky truncation ensuring positivity of the bulk term has already been formulated in a dimension-independent setting [TAL25b], (v) the final tuning of the boundary terms proposed in Section 5.3 may involve further adjustments given that the generalized strain is not necessarily geometrically compatible in higher dimension [TAL25b]. The extension to dynamical homogenization will also be explored in future work.

**Acknowledgements** This manuscript has been typeset using TeX\_MACS, a powerful, multi-platform and freely distributed scientific editor [vdHGG+13].

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## APPENDIX A. PSEUDO-ENERGY OF THE CHAIN OF SPRINGS

In this Appendix, we derive a continuous version of the energy (3.5) of the spring chain shown in Figure 3.1. For this simple lattice, there are neither boundary layers nor a microscopic shift in interior, and the continuous model is derived by postulating that the nodal displacement $u_i$ derives from a smooth macroscopic displacement $u(x)$,

$$u_i = u(\epsilon i). \qquad (A.1)$$

This relocalization formula has a particularly simple form. The more general form applicable to one-dimensional lattices with, e.g., next-neighbor interactions, can be found in Equation [4.2] in [TAL25a].

Using (A.1), one obtains the Taylor expansion of the squared microscopic strain as

$$ \begin{aligned} (u(\epsilon(i+1)) - u(\epsilon i))^2 &= \epsilon^2 \left( u' + \frac{\epsilon^2}{24} u''' + \mathcal{O}(\epsilon^4) \right)^2_{x=\epsilon(i+\frac{1}{2})} \\ &= \epsilon^2 \left( u'^2 + \frac{\epsilon^2}{12} u'u''' + \mathcal{O}(\epsilon^4) \right)_{x=\epsilon(i+\frac{1}{2})}. \end{aligned} \qquad (A.2) $$

Inserting into the energy (3.5), we get

$$ \Phi_{\epsilon}^{\text{spr}} = \frac{\epsilon}{2} \sum_{i=0}^{N-1} \left( u'^2 + \frac{\epsilon^2}{12} u'u''' \right)_{x=\epsilon(i+\frac{1}{2})} - \rho^0 u(0) - \rho^l u(l) - \epsilon \sum_{i=0}^{N} (\mathcal{F} u)_{x=\epsilon i} + \mathcal{O}(\epsilon^3) \qquad (A.3) $$

To replace the discrete sums with integrals, we use the following variants of the Euler–Maclaurin summation formula [Apo99], which are established in Appendix [A] of our earlier paper [TAL25a], see Equations [A.5] and [A.6],

$$ \begin{aligned} \epsilon \sum_{i=0}^{N-1} j\left(\epsilon\left(i+\frac{1}{2}\right)\right) &= \int_{0}^{N\epsilon} j(x) \, dx - \frac{\epsilon^2}{24} [j']_{0}^{N\epsilon} + \mathcal{O}(\epsilon^3) \\ \epsilon \sum_{i=0}^{N} l(\epsilon i) &= \int_{0}^{N\epsilon} l(x) \, dx + \frac{\epsilon}{2} (l(0) + l(N\epsilon)) + \frac{\epsilon^2}{12} [l']_{0}^{N\epsilon} + \mathcal{O}(\epsilon^3), \end{aligned} \qquad (A.4) $$

where $[h]_0^{N\epsilon} = h(N\epsilon) - h(0)$ denotes the variation of a function $h$.

Choosing $j(x) = \frac{Q}{2}(u'^2 + \frac{\epsilon^2}{12}u'u'')$ and $l(x) = -\mathcal{F}(x)u(x)$ in (A.4), we rewrite the pseudo-energy (A.3) as

$$ \Phi_{\epsilon}^{\text{spr}}[u] = \int_{0}^{N\epsilon} \left( \frac{Q}{2} \left( u'^2 + \frac{\epsilon^2}{12} u' u'' \right) - \mathcal{F} u \right) dx \\ - \rho^0 u(0) - \rho^l u(l) - \frac{\epsilon^2}{24} \left[ \mathcal{Q} u' u'' \right]_{0}^{N\epsilon} - \frac{\epsilon}{2} (\mathcal{F}(0)u(0) + \mathcal{F}(l)u(l)) + \frac{\epsilon^2}{12} \left[ -\mathcal{F}'u - \mathcal{F}u' \right]_{0}^{N\epsilon} + \mathcal{O}(\epsilon^3). $$
Rewriting the variations $[h]_0^{N\epsilon} = h(N\epsilon) - h(0)$ produced by the integration by parts as

$$[h]_0^{N\epsilon} = \sum_{b \in \{0, \ell\}} s^b h(b), \qquad (A.5)$$

where $s^b = \pm 1$ is the outward unit normal defined in (2.8), and recalling that the lattice length is denoted as $\ell = N\epsilon$, we obtain the continuous pseudo-energy in the form announced in (3.6).

# APPENDIX B. STATIONARITY CONDITION FOR $\Psi_{\epsilon}$

In this Appendix, we derive the stationarity condition associated with the constrained energy formulation (6.1). We treat the kinematical constraint (6.1)$_1$ using a Lagrange multiplier $n(x)$, which can be interpreted as the internal force, i.e., we render stationary the augmented Lagrangian

$$\Psi_{\epsilon}^{\text{aug}}[u, e, n] = \Psi_{\epsilon}[u, e] + \int_{0}^{\ell} n(x) (u'(x) - e(x) + \epsilon L e'(x) - \epsilon^2 M e''(x)) dx. \qquad (B.1)$$

The first variation with respect to $u$ and $e$ is given by

$$D\Psi_{\epsilon}^{\text{aug}} = \int_{0}^{\ell} (\bar{Q}e \delta e + \epsilon^2 Ge' \delta e' - \bar{F}_{\epsilon} \delta u + n (\delta u' - \delta e + \epsilon L \delta e' - \epsilon^2 M \delta e'')) dx \\ + \sum_{b \in \{0, \ell\}} \left[ -\bar{t}_{\epsilon}^{b} (\delta u + \epsilon L \delta e - \epsilon^2 M \delta e') + \epsilon^2 s^b \left[ \left(M + \frac{G}{Q}\right) \bar{F}_{\epsilon} \right] (b) \delta e \right]_{x=b}. \qquad (B.2)$$

Integrating by parts the terms proportional to $\delta u'$, $\delta e'$ and $\delta e''$ in the integral, we get

$$D\Psi_{\epsilon}^{\text{aug}} = \int_{0}^{\ell} \left( \left( \left( \bar{Q} - \epsilon^2 G \frac{d^2}{dx^2} \right) e(x) - \left( 1 + \epsilon L \frac{d}{dx} + \epsilon^2 M \frac{d^2}{dx^2} \right) n(x) \right) \delta e(x) - (n'(x) + \bar{F}_{\epsilon}(x)) \delta u(x) \right) dx \\ + \sum_{b \in \{0, \ell\}} \left[ - (\bar{t}_{\epsilon}^{b} - s^b n)(\delta u + \epsilon L \delta e - \epsilon^2 M \delta e') + \epsilon^2 s^b \left( M(n' + \bar{F}_{\epsilon}(b)) + \frac{G}{Q}(\bar{Q}e' + \bar{F}_{\epsilon}(b)) \right) \delta e \right]_{x=b}, \qquad (B.3)$$

where all the boundary terms produced by the integration by parts have been rewritten as earlier in (A.5).

By the fundamental lemma in the calculus of variations, we obtain the equilibrium boundary-value problem by setting to zero the coefficients of $\delta e$ and $\delta u$ in the integral, and of $\delta u(b)$, $\delta e(b)$ and $\delta e'(b)$ at the boundaries. This yields the boundary-value problem in (6.2).

It is remarkable that the boundary terms in (B.3) produce redundant boundary conditions:

* in factor of the $M \delta e$ term, we obtain the same equilibrium condition $n'(b) + \bar{F}_{\epsilon}(b) = 0$ as found in factor of $\delta u$ in the bulk,

* the same equilibrium condition $\bar{t}_{\epsilon}^{b} = s^b n(b)$ warrants stationarity with respect to $\delta u(b)$ and $\delta e'(b)$, thanks to the factorization by $(\delta u + \epsilon L \delta e - \epsilon^2 M \delta e')$.

These remarkable properties are a consequence of the special form (3.3) of the higher-order boundary tractions which we used as a starting point.

# APPENDIX C. EXPLICIT SOLUTION TO THE VARIATIONAL PROBLEM

## C.1. Scaled boundary-value problem for the generalized strain

In this appendix, we derive the solution to the following boundary-value problem,

$$\begin{aligned} \forall x \in [0, \ell], \quad & \left(1 - \eta^2 \frac{d^2}{dx^2}\right) e(x) = a_{\eta}(x) \\ \forall b \in \{0, \ell\}, \quad & e'(b) + j_{\eta}(b) = 0. \end{aligned} \qquad (C.1)$$

This problem is a scaled version of that produced by the variational principle in (6.2)$_{3,4}$, with the scaled functions and parameters defined in (6.7). The functions $a_{\eta}(x)$ and $j_{\eta}(x)$ are two smooth functions parameterized by the scalar $\eta > 0$, which is a scaled version of the separation parameter $\epsilon \ll 1$ used in the main text: we consider the limit $\eta \to 0$.

We assume here that $a_{\eta}$, which was initially defined on the interval $x \in [0, \ell]$, can be extended as a smooth function defined over the entire set of real numbers. This assumption is an artifact that simplifies the argument considerably, by saving us from introducing the Green's functions of the boundary-value problem on a finite interval. We will only introduce the Green's function of the differential equation (C.1)$_1$ in an infinite domain, which is simpler.

We define the auxiliary function

$$h(x) = \frac{1}{2} \exp(-|x|), \qquad (C.2)$$

which satisfies

$$\begin{array}{l} \forall x \neq 0 \quad h(x) - h''(x) = 0 \\ h'(0^{+}) - h'(0^{-}) = -1 \quad \Leftrightarrow \quad h - h'' = \delta_0, \end{array} \qquad (C.3)$$
where $\delta_0$ is the Dirac distribution centered around $x=0$. The equivalence is based on the fact that, when integrated over a small segment surrounding $x=0$, the equality $h-h'' = \delta_0$ yields the jump of $h'$ as $h'(0^+) - h'(0^-) = -1$ (given that $h$ is continuous at $x=0$). The equality between distributions $h-h'' = \delta_0$ will be used in a formal way, but we offer an alternative proof that does not use distributions.

## C.2. Particular solution of the differential equation

For $x \in [0, \ell]$, we consider the function $\tilde{e}(x)$ obtained by convolving the right-hand side of (C.1)$_1$ with the function $x \mapsto \frac{1}{\eta}h(\frac{x}{\eta})$, that appears to be the Green's function of the differential operator $[1 - \eta^2 d^2/dx^2]$ in an infinite domain:

$$ \tilde{e}_{\eta}(x) := \int_{-\infty}^{+\infty} a_{\eta}(x') \frac{1}{\eta} h\left(\frac{x-x'}{\eta}\right) dx'. \quad (C.4) $$

This is a particular solution of the differential equation (C.1)$_1$, as can be expected by applying the principle of linear superposition to the point-like sources $a_\eta(x')$ at $x'$ present in the right-hand side of (C.1)$_1$. We offer two arguments in support of this:

* By a formal argument, we can differentiate $\tilde{e}_\eta(x)$ in (C.4) twice and use $h'' = h - \delta_0$, see (C.3) to get

$$ \begin{aligned} \tilde{e}_{\eta}''(x) &= \int_{-\infty}^{+\infty} a_{\eta}(x') \frac{1}{\eta^3} h''\left(\frac{x-x'}{\eta}\right) dx' \\ &= \frac{1}{\eta^2} \int_{-\infty}^{+\infty} a_{\eta}(x') \frac{1}{\eta} \left( h\left(\frac{x-x'}{\eta}\right) - \delta_0\left(\frac{x-x'}{\eta}\right) \right) dx' \end{aligned} \quad (C.5) \\ &= \frac{1}{\eta^2} \left( \tilde{e}_{\eta}(x) - \int_{-\infty}^{+\infty} a_{\eta}(x-\eta y) \delta_0(y) dy \right) \\ &= \frac{1}{\eta^2} (\tilde{e}_{\eta}(x) - a_{\eta}(x)). \end{aligned} $$

* More rigorously, we note that the integrand in (C.4) is not smooth at $x'=x$. The derivative with respect to $x$ of an integral parameterized by $x$ and having a *moving singularity* at $x'=x_{ns}(x)=x$ includes a term proportional to the jump of the integrand at the discontinuity, and is given by

$$ \frac{d\tilde{e}_{\eta}}{dx}(x) = \int_{\mathbb{R} \setminus \{x_{ns}(x)\}} \frac{d}{dx} \left( a_{\eta}(x') \frac{1}{\eta} h\left(\frac{x-x'}{\eta}\right) \right) dx' - \frac{dx_{ns}}{dx}(x) \left[ \int_{x'=x_{ns}(x)}^{x'} a_{\eta}(x') \frac{1}{\eta} h\left(\frac{x-x'}{\eta}\right) dx' \right]. \quad (C.6) $$

There, $\|q(x, x')\|_{x'=x_d} = q(x, (x_d')^+) - q(x, (x_d')^-)$ denotes the jump of a function $q$ across $x' = x_d$. The jump is actually zero for this particular function, as both $a_\eta$ and $h$ are continuous. This yields

$$ \frac{d\tilde{e}_{\eta}}{dx}(x) = \frac{1}{\eta} \int_{\mathbb{R} \setminus \{x_{ns}(x)\}} a_{\eta}(x') \frac{1}{\eta} h'\left(\frac{x-x'}{\eta}\right) dx'. \quad (C.7) $$

Differentiating $\frac{d\tilde{e}_s}{dx}$ by the same argument, and noting that it has a true moving discontinuity in the integrand, we get

$$ \frac{d^2 \tilde{e}_{\eta}}{dx^2}(x) = \frac{1}{\eta} \left( \int_{\mathbb{R} \setminus \{x_{ns}(x)\}} \frac{d}{dx} \left( a_{\eta}(x') \frac{1}{\eta} h'\left(\frac{x-x'}{\eta}\right) \right) dx' - \frac{1}{\eta} \frac{dx_{ns}}{dx}(x) \left[ \int_{x'=x} a_{\eta}(x') h'\left(\frac{x-x'}{\eta}\right) dx' \right] \right). \quad (C.8) $$

With $\frac{dx_{ns}}{dx}(x) = \frac{dx}{dx} = 1$ and $\|a_\eta(x') h'((x-x')/\eta)\|_{x'=x} = a_\eta(x) (h'(0^-) - h'(0^+)) = -a_\eta(x) (h'(0^+) - h'(0^-)) = a_\eta(x)$, see (C.3), we have

$$ \frac{d^2 \tilde{e}_{\eta}}{dx^2}(x) = \frac{1}{\eta^2} \left( \int_{\mathbb{R} \setminus \{x\}} a_{\eta}(x') \frac{1}{\eta} h''\left(\frac{x-x'}{\eta}\right) dx' - a_{\eta}(x) \right). \quad (C.9) $$

On the domain $x' \in \mathbb{R} \setminus \{x_{ns}(x)\} = \mathbb{R} \setminus \{x\}$, we can use $h'' = h$, see (C.3),

$$ \begin{aligned} \frac{d^2 \tilde{e}_{\eta}}{dx^2}(x) &= \frac{1}{\eta^2} \left( \int_{\mathbb{R} \setminus \{x\}} a_{\eta}(x') \frac{1}{\eta} h\left(\frac{x-x'}{\eta}\right) dx' - a_{\eta}(x) \right) \\ &= \frac{1}{\eta^2} (\tilde{e}_{\eta}(x) - a_{\eta}(x)). \end{aligned} \quad (C.10) $$

The identical expressions of $\tilde{e}_{\eta}''(x)$ in (C.5) and (C.10) confirm that $\tilde{e}_{\eta}(x)$ is a particular solution of the differential equation (C.1)$_1$,

$$ \forall x \in [0, \ell] \qquad \tilde{e}_{\eta}(x) - \eta^2 \tilde{e}_{\eta}''(x) = a_{\eta}(x). \quad (C.11) $$

**Remark C.1.** Like the differential equation (C.1)$_1$, the convolution has regularizing properties and the particular solution $\tilde{e}(x)$ in (C.4) is smoother than $a_\eta(x)$.

**Remark C.2.** Using the convolution operator appearing in (C.4), we can assign an accurate meaning to the formal inversion of the differential operator $(\bar{Q} - \varepsilon^2 G \frac{d^2}{dx^2})^{-1}$ used earlier in (6.3): ignoring the boundary-layer artifacts, one can identify in (6.3)

$$ \tilde{e}_{\eta}(x) \equiv \left( \bar{Q} - \varepsilon^2 G \frac{d^2}{dx^2} \right)^{-1} \left( 1 + \varepsilon L \frac{d}{dx} + \varepsilon^2 M \frac{d^2}{dx^2} \right) n_{\varepsilon}. \quad (C.12) $$
### C.3. Small-$\eta$ expansion of the particular solution

In the limit $\eta \to 0$ and for fixed $x \in [0, \ell]$, we can obtain a series expansion of $\tilde{e}_\eta(x)$ in powers of $\eta$ by using the Taylor expansion of $a_\eta(x')$ around $x$ in (C.4), taking advantage of the fact that the exponential decays as soon as $|x'-x| \gg \eta$ and that $h$ is an even function:

$$
\begin{align*}
\tilde{e}_{\eta}(x) &= \int_{-\infty}^{+\infty} \left( \sum_{p=0}^{2} \frac{1}{p!} \frac{d^p a_{\eta}}{dx^p}(x) (x'-x)^p + \mathcal{O}((x-x')^3) \right) \frac{1}{\eta} h\left(\frac{x-x'}{\eta}\right) dx' \\
&= \sum_{p=0}^{2} \frac{1}{p!} \frac{d^p a_{\eta}}{dx^p}(x) \int_{-\infty}^{+\infty} (x'-x)^p h\left(\frac{x'-x}{\eta}\right) \frac{dx'}{\eta} + \mathcal{O}(\eta^3) \\
&= \sum_{p=0}^{2} \frac{\eta^p}{p!} \frac{d^p a_{\eta}}{dx^p}(x) \int_{-\infty}^{+\infty} y^p h(y) dy + \mathcal{O}(\eta^3) \tag{C.13} \\
&= \sum_{p=0}^{2} \frac{\eta^p}{p!} \frac{d^p a_{\eta}}{dx^p}(x) \begin{cases} p! & \text{if } p \text{ is even} \\ 0 & \text{if } p \text{ is odd} \end{cases} + \mathcal{O}(\eta^3) \\
&= \sum_{p'=0}^{1} \eta^{2p'} \frac{d^{2p'} a_{\eta}}{dx^{2p'}}(x) + \mathcal{O}(\eta^3) \\
&= a_{\eta}(x) + \eta^2 a_{\eta}''(x) + \mathcal{O}(\eta^3).
\end{align*}
$$

This is nothing but the perturbative solution, at order $\eta^2$, to the differential equation (C.1).

The expansion of $\tilde{e}'_\eta(x)$ in powers of $\eta$ is given by a similar argument, with an additional integration by parts,

$$
\begin{align*}
\tilde{e}'_{\eta}(x) &= \frac{1}{\eta} \int_{-\infty}^{+\infty} a_{\eta}(x') \frac{1}{\eta} h'\left(\frac{x-x'}{\eta}\right) dx' \\
&= \int_{-\infty}^{+\infty} a'_{\eta}(x') \frac{1}{\eta} h\left(\frac{x-x'}{\eta}\right) dx' \tag{C.14} \\
&= a'_{\eta}(x) + \eta^2 a'''_{\eta}(x) + \mathcal{O}(\eta^3).
\end{align*}
$$

Note that (C.14) can be recovered by formally differentiating both sides of (C.13), assuming that the remainder $\mathcal{O}(\eta^3)$ does not blow up upon differentiation.

Equation (C.14) will provide an estimate of the boundary-layer artifacts, see Equation (C.18) below.

### C.4. Enforcing boundary conditions

We proceed to derive the solution of the complete problem (C.1) including the boundary conditions. It is found by combining the smooth, particular solution $\tilde{e}(x)$ of the differential equation with two independent homogeneous solutions

$$e(x) = \tilde{e}_{\eta}(x) + C^0(\eta) \exp\left(-\frac{x}{\eta}\right) - C^{\ell}(\eta) \exp\left(-\frac{\ell-x}{\eta}\right). \quad (C.15)$$

The coefficients $C^0(\eta)$ and $C^\ell(\eta)$ are set by the boundary conditions (C.1)$_2$ as

$$
\begin{align}
\tilde{e}'_{\eta}(0) + j_{\eta}(0) - \frac{C^0(\eta)}{\eta} - \frac{C^{\ell}(\eta)}{\eta} \exp\left(-\frac{\ell}{\eta}\right) &= 0 \tag{C.16} \\
\tilde{e}'_{\eta}(\ell) + j_{\eta}(\ell) - \frac{C^0(\eta)}{\eta} \exp\left(-\frac{\ell}{\eta}\right) - \frac{C^{\ell}(\eta)}{\eta} &= 0. \notag
\end{align}
$$

The solution of this linear set of equations is

$$
\begin{align}
C^0(\eta) &= \frac{\eta}{1 - \exp\left(-\frac{2\ell}{\eta}\right)} \left( (\tilde{e}'_\eta(0) + j_\eta(0)) - \exp\left(-\frac{\ell}{\eta}\right) (\tilde{e}'(\ell) + j_\eta(\ell)) \right), \tag{C.17} \\
C^\ell(\eta) &= \frac{\eta}{1 - \exp\left(-\frac{2\ell}{\eta}\right)} \left( (\tilde{e}'_\eta(\ell) + j_\eta(\ell)) - \exp\left(-\frac{\ell}{\eta}\right) (\tilde{e}'(0) + j_\eta(0)) \right). \notag
\end{align}
$$

Equation (C.14) yields the following approximation of the coefficients $C^b(\eta)$ in the limit $\eta \to 0$, for $b \in \{0, \ell\}$,

$$
\begin{align}
C^b(\eta) &= \eta (\tilde{e}'_\eta(b) + j_\eta(b)) + \mathcal{O}\left(\eta \exp\left(-\frac{\ell}{\eta}\right)\right) \nonumber \\
&= \eta (a'_\eta(b) + j_\eta(b)) + \eta^3 a'''_\eta(b) + \mathcal{O}\left(\eta^4, \eta \exp\left(-\frac{\ell}{\eta}\right)\right). \tag{C.18}
\end{align}
$$

In the problem at hand, the leading term of order $\eta$ will turn out to vanish, making the boundary-layer amplitudes $\mathcal{O}(\eta^2) = \mathcal{O}(\epsilon^2)$ and not $\mathcal{O}(\eta) = \mathcal{O}(\epsilon)$, see (6.11).
### C.5. Summary

The solution (C.15) of the boundary-value problem (C.1) has been obtained in the interval $x \in [0, \ell]$ in the form

$$e_{\eta}(x) = \underbrace{\int_{-\infty}^{+\infty} a_{\eta}(x') \frac{1}{2\eta} \exp(-|x-x'|/\eta)}_{\tilde{e}_{\eta}(x)} dx' - \sum_{b \in \{0, \ell\}} s^b \underbrace{\left( \eta(a'_{\eta}(b) + j_{\eta}(b)) + \eta^3 a'''_{\eta}(b) + \mathcal{O}(\eta^4) \right)}_{c^b(\eta)} \exp\left(-\frac{|b-x|}{\eta}\right) \quad (C.19)$$

As noted earlier,

* the particular solution $\tilde{e}_{\eta}(x)$ is defined by a convolution regularizing the right-hand side $a_{\eta}$, see (C.4),

* to order $\eta^2$, $\tilde{e}_{\eta}(x)$ matches the perturbative solution to the differential equation (C.1)$_1$,

* the other terms in the right-hand side of (C.19) are boundary-layer solutions localized near each of the end-points $b \in \{0, \ell\}$,

* the amplitude of each boundary layer is given, to leading order, by $\pm\eta(a'_{\eta}(b) + j_{\eta}(b))$,

* the boundary layers arise as a result of the incompatibility between the differential equation and the boundary condition in (C.1).

**Remark C.3.** For the sake of convenience, we have assumed that $a_{\eta}$ is defined over the entire real axis, $x \in \mathbb{R}$ but the values of $a_{\eta}$ that matter for the boundary-value problem (C.1) are those in the interval $[0, \ell]$, as we show now. The particular solution $\tilde{e}_{\eta}(x)$ in (C.4) depends on the values of $a_{\eta}$ outside the interval $[0, \ell]$, but this dependence is exponentially small for $\eta \to 0$. If we were to change the definition of $a_{\eta}$ outside the interval $[0, \ell]$, this would change the value of both $\tilde{e}_{\eta}(x)$ (by exponentially small terms) and of the remainder $\mathcal{O}(\eta^4)$ in factor of the exponential boundary layers in (C.19). This two changes would compensate each other, so that $e_{\eta}(x)$ in the left-hand side of (C.19) would remain unchanged as required.

## APPENDIX D. EXTENSION TO A FIXED ENDPOINT

In the main text, we have considered endpoints subjected to force control. In this Appendix, we show that the strain-gradient theory can be extended to the case of displacement control at an endpoint. The case where both endpoints are fixed can be treated similarly.

### D.1. Strategy

We denote as $b \in \{0, \ell\}$ the endpoint where the displacement is fixed, and by $w^b_\epsilon$ its imposed value. We denote as $P^b$ the reaction force enforcing the fixed-endpoint boundary condition. The quantity $P^b$ plays a similar role as the point-like force $\mu^b$ applied at the endpoint of the spring chain, see Figure 3.1, but it is now unknown.

We denote as $\Phi_\epsilon$ the pseudo-energy for force control at the endpoints, as studied in the main text. The terminal force $P^b$ enters in $\Phi_\epsilon$ in factor of minus the endpoint displacement, via the potential energy: this is clear from the spring chain example in (3.5) and holds true in general, see Equations [3.10] and [3.12]$_2$ in [TAL25a]. As a result, the kinematic condition prescribing the position $w^b_\epsilon$ of the endpoint $b$ can be written as

$$-\frac{\partial \Phi_{\epsilon}[u, e, P^{b}]}{\partial P^{b}} = w_{\epsilon}^{b}, \qquad (D.1)$$

where we have made explicit the previously implicit dependence of $\Phi_\epsilon$ on the point-like force $P^b$.

We denote as $\Psi_\epsilon$ the strain-gradient theory obtained in Section 6 for force control at the endpoints. With the aim to generate the variational formulation for the fixed-endpoint case, we introduce the Legendre transform of $\Psi_\epsilon[u, e, P^b]$ as the functional

$$\Psi_{\epsilon}^{*}[u, e, w] = \Psi_{\epsilon}[u, e, P^{b}(w)] + w P^{b}(w), \qquad (D.2)$$

where $P^b(w)$ is the value $P^b = P^b(w)$ of the reaction force making the right-hand side stationary,

$$\frac{\partial \Psi_{\epsilon}}{\partial P^{b}}[u, e, P^{b}] + w = 0. \qquad (D.3)$$

We claim that a stationary point $(u, e)$ of $\Psi^*_{\epsilon}[u, e, w^b_\epsilon]$, subjected to the kinematic constraint $u'(x) = e(x) - \epsilon L e'(x) + \epsilon^2 M e''(x)$ in (6.1)$_1$ as always, satisfies the boundary-value problem (6.2) but with the equilibrium $s^b n(b) = \tilde{e}^b_\epsilon$ of the displacement-controlled boundary replaced by the kinematic condition (D.1), up to terms of order $\epsilon^3$. The Legendre transform $\Psi^*_{\epsilon}[u, e, w^b_\epsilon]$ therefore generates variationally the equilibrium problem with prescribed endpoint displacement. The proof is given in Section D.2.

We will confirm in (D.9) the well-known fact that the Legendre transform $\Psi^*_{\epsilon}[u, e, w^b_\epsilon]$ inherits the positivity of $\Psi_\epsilon$. The functional $\Psi^*_{\epsilon}$ therefore extends the variational principle to the case of displacement control at one of the endpoints.

**Remark D.1.** In Equation (3.1), the pure-loading contribution $\tilde{c}^b_\epsilon$ depends in general on $P^b$, and the same holds for $c^b_\epsilon$ by (5.6). As a result the endpoint position in the left-hand side of (D.1) includes a term $\epsilon \partial c^b_\epsilon / \partial P^b$. Contrary to what happened with prescribed endpoint forces, the pure-loading energy $c^b_\epsilon$ thus enters into the equilibrium problem with fixed endpoint. Indeed, it encodes important information about the displacement of the endpoint (relocalization).
## D.2. Justification

Making the Legendre transform $\Psi_{\epsilon}^{*}$ in (D.2) stationary with respect to $u$ and $e$ generates the same equilibrium problem as that produced by the original function $\Psi_{\epsilon}$. The only difference is that the boundary load $P^{b} = P^{b}(w_{\epsilon}^{b})$ is now an unknown set by Equation (D.3). This is precisely the point of the Legendre transform.

We proceed to analyze the boundary condition (D.3) produced by the Legendre transform, and to check that it agrees with the fixed-endpoint condition (D.1). For any stationary point $(u, e)$ of $\Psi_{\epsilon}^{*}[u, e, w_{\epsilon}^{b}]$, the stationarity condition for $P^{b}$ in (D.3) can be combined with the equivalence between $\Phi_{\epsilon}$ and $\Psi_{\epsilon}$ in (6.16) to yield

$$ 
\begin{aligned}
0 &= w_{\epsilon}^{b} + \frac{\partial \Psi_{\epsilon}}{\partial P^{b}}[u, e, P^{b}] \\
&= w_{\epsilon}^{b} + \frac{\partial \Phi_{\epsilon}}{\partial P^{b}}[u, e, P^{b}] - \epsilon \frac{\partial g_{\epsilon}^{b}}{\partial P^{b}} \left( \frac{1 - \epsilon^2 \frac{G}{\bar{Q}} \frac{d^2}{dx^2}}{\epsilon e'(b)} e(b) \right) + \mathcal{O}(\epsilon^3).
\end{aligned} 
\quad (D.4) $$

Note that the gradient of the ghost term is limited to the endpoint $b$ with displacement control, since the ghost term at the other boundary does not depend on $P^{b}$. The derivative of the ghost term can be calculated using the expressions (6.15) of $g_{\epsilon}^{b}$ as

$$ \epsilon \frac{\partial g_{\epsilon}^{b}}{\partial P^{b}} \left( \frac{1 - \epsilon^2 \frac{G}{\bar{Q}} \frac{d^2}{dx^2}}{\epsilon e'(b)} e(b) \right) = - \left[ \left( \frac{1 - \epsilon^2 \frac{G}{\bar{Q}} \frac{d^2}{dx^2}}{\epsilon e'(b)} e(b) \right) - \mathbf{p}_{\epsilon}^{b} \right] \cdot \mathbf{k}^{b} \cdot \frac{\partial \mathbf{p}_{\epsilon}^{b}}{\partial P^{b}}. \quad (D.5) $$

As we have noted multiple times, the vector $\mathbf{p}_{\epsilon}^{b}$ has been crafted in such a way that the vector in the square bracket in the right-hand side above is zero whenever $(u, e)$ satisfy Equations (6.2)$_{1-3}$ of the boundary-value problem. We conclude that for any stationary point $(u, e)$ of $\Psi_{\epsilon}^{*}$, the gradient of the ghost term vanishes,

$$ \epsilon \frac{\partial g_{\epsilon}^{b}}{\partial P^{b}} \left( \frac{1 - \epsilon^2 \frac{G}{\bar{Q}} \frac{d^2}{dx^2}}{\epsilon e'(b)} e(b) \right) = 0. \quad (D.6) $$

As a result, Equation (D.4) yields the kinematic boundary condition (D.1), up to a small remainder of order $\mathcal{O}(\epsilon^3)$.

**Remark D.2.** The energy $\Psi_{\epsilon}[u, e, P^{b}] + \Gamma_{\epsilon}(e)$ including the ghost terms is by (6.16) asymptotically equivalent to the original pseudo-energy $\Phi_{\epsilon}$. One could therefore Legendre-transform this energy and not just $\Psi_{\epsilon}[u, e, P^{b}]$. The above argument can be adapted to prove that the Legendre transform of $\Psi_{\epsilon}[u, e, P^{b}] + \Gamma_{\epsilon}(e)$ is $\Psi_{\epsilon}^{*}[u, e, w_{\epsilon}^{b}]$ plus (different) ghost terms, that do not affect the stationarity conditions of $\Psi_{\epsilon}^{*}[u, e, w_{\epsilon}^{b}]$. The Legendre transform therefore maps ghost terms to ghost terms. This is what make possible to ignore the ghost terms in the Legendre transform in (D.2).

## D.3. Explicit Legendre transform

To calculate the Legendre transform $\Psi_{\epsilon}^{*}$, we start by making explicit the dependence of $\Psi_{\epsilon}[u, e, P^{b}]$ on the unknown reaction $P^{b}$, which takes place via the effective traction $\bar{t}_{\epsilon}^{b}$ and of the pure-loading potential $c_{\epsilon}^{b}$. In the context of linear elasticity, the dependence on the force $P^{b}$ of the effective traction $\bar{t}_{\epsilon}^{b}$ is affine, and that of the energy $c_{\epsilon}^{b}$ is quadratic,

$$ 
\begin{aligned}
\bar{t}_{\epsilon}^{b}(P^{b}) &= t_{\epsilon}^{b} + P^{b} \\
c_{\epsilon}^{b}(P^{b}) &= \frac{1}{2} \left( c_{\epsilon}^{b} + 2 \ddot{c}_{\epsilon}^{b} P^{b} + \frac{(P^{b})^{2}}{m_{\epsilon}^{b}} \right).
\end{aligned} 
\quad (D.7) $$

The coefficients $t_{\epsilon}^{b}$, $\dot{c}_{\epsilon}^{b}$, $\ddot{c}_{\epsilon}^{b}$ and $1/m_{\epsilon}^{b}$ are independent of the reaction force $P^{b}$, but may depend on the other known loading parameters and on the lattice properties (and hence on $\epsilon$). For the spring chain, for instance, identifying the terminal point-like force as $P^{b} = \rho^{b}$, we have from Table 3.1, $t_{\epsilon}^{b} = \frac{\epsilon}{2} \mathcal{F}(b) + \frac{\epsilon^2 s^{b}}{12} \mathcal{F}'(b)$ and from Table 5.1, $\dot{c}_{\epsilon}^{b} = 0$, $\ddot{c}_{\epsilon}^{b} = \frac{\epsilon}{24\bar{Q}} \mathcal{F}(b)$ and $m_{\epsilon}^{b} = \infty$. For a general lattice, the expressions of $t_{\epsilon}^{b}$, $\dot{c}_{\epsilon}^{b}$, $\ddot{c}_{\epsilon}^{b}$ and $1/m_{\epsilon}^{b}$ are outcomes of the two-scale analysis proposed in [TAL25a], see for instance the expression of $m_{\epsilon}^{b}$ obtained in Equation [9.5].

Inserting (D.7) into (D.3) and using the expression of $\Psi_{\epsilon}$ in (6.1), we get the equilibrium value of the Lagrange multiplier as

$$ P^{b} = -\frac{m_{\epsilon}^{b}}{\epsilon} (-w_{\epsilon}^{b} + \epsilon \ddot{c}_{\epsilon}^{b} + u(b) + \epsilon Le(b) - \epsilon^2 Me'(b)). \quad (D.8) $$

Inserting into the definition (D.2) of the Legendre transform and using (6.1) again, we obtain the variational formulation

$$ 
\left\{ 
\begin{array}{ll}
\forall x, & u'(x) = e(x) - \epsilon L e'(x) + \epsilon^2 M e''(x), \\
\Psi_{\epsilon}^{*}[u, e, w_{\epsilon}^{b}] = & \displaystyle\int_{0}^{\ell} \left( \frac{1}{2} \bar{Q} e^2(x) + \epsilon^2 \frac{G}{2} e'^2(x) - \bar{F}_{\epsilon}(x) u(x) \right) dx \\
& + \displaystyle\sum_{b'' \in \{0, \ell\}} \epsilon^2 s^{b''} \left[ \left( M + \frac{G}{\bar{Q}} \right) \bar{F}_{\epsilon} \right] (b'') e(b'') \\
& + \left[ -\bar{t}_{\epsilon}^{b'} (u + \epsilon L e - \epsilon^2 M e') - \epsilon c_{\epsilon}^{b'} \right]_{x=\ell-b} \\
& + \left[ \frac{m_{\epsilon}^{b}}{2\epsilon} (u(b) + \epsilon L e(b) - \epsilon^2 M e'(b) - (w_{\epsilon}^{b} + \epsilon d_{\epsilon}^{b}))^2 - \ddot{c}_{\epsilon}^{b} \right].
\end{array}
\right. 
\quad (D.9) $$
We have included in the first line the kinematical constraint (6.1)$_1$ which we use throughout. In the energy $\Psi_{\epsilon}^*$, the index $b''$ runs over all boundaries, the index $b' = \ell - b$ denotes the boundary with force control, and $b$ the other boundary, with displacement control.

In the boundary term on the fourth line, an apparent shift of the pinned boundary $\epsilon d_{\epsilon}^b$ has appeared, given in scaled form by

$$d_{\epsilon}^{b} = \frac{\dot{i}_{\epsilon}^{b}}{m_{\epsilon}^{b}} - \ddot{c}_{\epsilon}. \qquad (D.10)$$

The quantity $\hat{c}_{\epsilon}^b$ appearing in (D.9) is given by $\hat{c}_{\epsilon}^b = \dot{i}_{\epsilon}^b(w_{\epsilon}^b - \epsilon \dot{c}_{\epsilon}^b) + \frac{\epsilon}{2} \left( \dot{c}_{\epsilon}^b + \frac{(\dot{i}_{\epsilon}^b)^2}{m_{\epsilon}^b} \right)$ and does not depend on any of the unknowns $u$ and $e$. As a result, it has no influence on the boundary-value problem and can be ignored. It would only be useful if we were to Legendre-transform $\Psi_{\epsilon}^*$ back into the original $\Psi_{\epsilon}$, which we will not attempt.

## D.4. Boundary-value problem

By a calculation similar to that done in Appendix B, one can derive the equilibrium problem generated variationally by the energy formulation (D.9). The result is a variant of (6.2) in which the boundary condition $n(b) = \bar{t}_{\epsilon}^b/s^b$ is modified to account for the fixed endpoint,

$$ \begin{array}{l|l} \text{equilibrium} & \begin{array}{l} \forall x \in [0, \ell], \quad n'(x) + \bar{F}_{\epsilon}(x) = 0 \\ b' = \ell - b, \quad n(b') = \bar{t}_{\epsilon}^b/s^{b'} \end{array} \\ \text{fixed endpoint} & b, \quad n(b) + \frac{m_{\epsilon}^b}{\epsilon s^b} (u(b) + \epsilon L e(b) - \epsilon^2 M e'(b) - (w_{\epsilon}^b + \epsilon d_{\epsilon}^b)) = 0 \\ \text{constitutive law} & \begin{array}{l} \forall x \in [0, \ell], \quad (\bar{Q} - \epsilon^2 G \frac{d^2}{dx^2}) e(x) = (1 + \epsilon L \frac{d}{dx} + \epsilon^2 M \frac{d^2}{dx^2}) n(x) \\ \forall b'' \in \{0, \ell\}, \quad G(\bar{Q} e'(b'') + \bar{F}_{\epsilon}(b'')) = 0 \end{array} \\ \text{kinematics} & \forall x \in [0, \ell], \quad u'(x) = (1 - \epsilon L \frac{d}{dx} + \epsilon^2 M \frac{d^2}{dx^2}) e(x), \end{array} \qquad (D.11) $$

A perturbative version of this boundary-value problem has been obtained in Section [9.2] of our previous paper [TAL25a], and validated against discrete solutions. By eliminating the stress variables, we obtained in Equation [9.8] of this previous paper a variant of the fixed-endpoint condition (D.11)$_3$ that is easier to interpret.

The equilibrium problem (D.11) derives from the energy $\Psi_{\epsilon}^*[u, e, w]$ without any approximation, and is thus variational. It extends to the case of a fixed endpoint the variational structure identified in the main text.