#  Data-driven micromorphic mechanics for materials with strain localization

Jacinto Ulloa<EMAIL_ADDRESS>Laurent Stainier  laurent.stainier@ec-
nantes.fr Michael Ortiz<EMAIL_ADDRESS>José E. Andrade
<EMAIL_ADDRESS>Division of Engineering and Applied Science, California
Institute of Technology, Pasadena, CA 91125, USA  Nantes Université, Ecole
Centrale Nantes, CNRS, GeM, UMR 6183, F-44000 Nante, France

###### Abstract

This paper explores the role of generalized continuum mechanics, and the
feasibility of model-free data-driven computing approaches thereof, in solids
undergoing failure by strain localization. Specifically, we set forth a
methodology for capturing material instabilities using data-driven mechanics
without prior information regarding the failure mode. We show numerically
that, in problems involving strain localization, the standard data-driven
framework for Cauchy/Boltzmann continua fails to capture the length scale of
the material, as expected. We address this shortcoming by formulating a
generalized data-driven framework for micromorphic continua that effectively
captures both stiffness and length-scale information, as encoded in the
material data, in a model-free manner. These properties are exhibited
systematically in a one-dimensional softening bar problem and further verified
through selected plane-strain problems.

###### keywords:

Data-driven computing; micromorphic continuum; strain localization; softening
materials

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##  1 Introduction

It is well-acknowledged that computational models based on classical
Cauchy/Boltzmann continua strongly deviate from reality when microstructural
effects modulate the material response. This is evident in the case of solids
exhibiting different forms of strain localization, a ubiquitous process
resulting from material instability, often related to softening responses or
non-associative evolution laws. Examples include shear bands in granular
materials driven by complex particle interactions [1, 2, 3], damage and
fracture in quasi-brittle materials governed by evolving microcracks [4, 5,
6], and ductile failure in metals emerging from the interplay between
dislocations and void growth [7, 8, 9]. The inadequacy of classical theories
to describe such behaviors arises from their inherently local nature,
preserving a continuum representation of the material at arbitrarily small
scales. The de facto approach to this problem in the mechanics community is to
render the constitutive model non-local, introducing, in one way or another, a
material length scale that reflects the heterogeneities of the microstructure.
Aside from strain localization, this approach also allows the description of
size effects [10, 11, 12, 13] and wave dispersion phenomena [14, 15, 16].

From a computational perspective, the central dilemma of standard local models
with strain localization is the loss of ellipticity of the boundary value
problem (BVP), leading to pathological mesh dependence in finite element
simulations with vanishing energy dissipation. This issue emerged in the 1970s
and has since been addressed by constitutive models on various fronts. A
conventional approach involves endowing internal variables with non-local
effects. A substantial body of work exists in this context for gradient
plasticity [17, 18, 19, 20, 21, 22, 23], gradient damage [24, 25, 26], and
integral-type non-local constitutive models [27, 28, 29]. Furthermore, non-
local effects emerge naturally in fracture modeling via phase-field approaches
[30, 26], which can be equipped with additional regularization mechanisms [31,
32, 33, 34, 35, 36, 37], and peridynamics [38, 39, 40].

It is noteworthy that before these developments, the need for generalized
continuum theories for microstructured media had long been recognized,
starting with the seminal framework of _micropolar_ continua developed by the
Cosserats [41]. Therein, a continuum material point is endowed with
independent rotations, associated with the microstructure, whose gradient
enters the formulation and serves as a regularization measure. The more
general theory of _micromorphic_ continua was later elaborated in the works of
Eringen and Suhubi [42], Mindlin [43], and Germain [44], incorporating a full,
generally non–symmetric, micro-deformation tensor as an additional set of
degrees of freedom. This framework recovers several models as special cases;
for instance, the micropolar theory is recovered when the micro-deformation
tensor is purely skew-symmetric, while the second-gradient theory is recovered
when the micro-deformation equals the compatible displacement gradient.
Moreover, the notion of micromorphic degrees of freedom has been extended to
plasticity [45, 46, 47, 48, 49] and damage [50, 32, 51, 52] variables. For a
thorough treatise of micromorphic continuum theories, we refer the reader to
Forest [53] and Neff et al. [54].

All the references above are examples of continuum theories with constitutive
models. In particular, the classical stress-strain relation in Cauchy
continua–a phenomenological equation itself–is complemented by generalized
stress-strain relations. Depending on the modeling assumptions, the number of
additional elasticity parameters may increase significantly. Moreover,
modeling inelastic behaviors, such as plasticity or damage, further requires
an additional set of parameters and, perhaps more importantly, constitutive
functions and evolution equations tailored to match experimental results. The
inevitable caveat of this conventional modeling paradigm is epistemic
uncertainty, particularly for complex material behavior.

The work of Kirchdoerfer and Ortiz [55] on data-driven computing presents a
paradigm shift in continuum mechanics that aims to resolve this shortcoming of
constitutive modeling in a general context. Therein, the fundamental balance
laws of continuum mechanics are enforced while the material behavior is
extracted directly from empirical data rather than from a constitutive model.
This framework has proved effective in capturing both elastic [55, 56] and
inelastic material responses [57, 58] and has been further developed in the
context of finite deformations [59, 60], dynamics [61, 62], viscoelasticity
[63], fracture mechanics [64], and molecular dynamics [65]. Data-driven
computing has also been used in a multi-scale setting to capture the response
of real granular materials [66], including breakage mechanics in crushable
sand [67]. All these works are based on Cauchy continua and are thus unable to
handle material instabilities. An exception is found in Karapiperis et al.
[68], where data-driven computing is extended to micropolar media, a framework
suitable for localization processes driven by significant micro-rotations.

The present study presents a generalized data-driven mechanics framework for
micromorphic continua. The motivation for this framework stems from softening
materials with strain localization, where standard data-driven computing
yields mechanically admissible states that fail to capture the length scale of
the material, i.e., the expected spatial distribution of the state variables.
This limitation and its resolution via micromorphic mechanics are showcased
using synthetic data in a 1D softening bar governed by a gradient damage
model. The framework is further validated in 2D problems considering damage
and non-associative plasticity. Remarkably, no a priori data assignment is
required for the proposed framework to capture the reference response in the
presence of strain localization.

The paper is structured as follows. Section 2 gives an overview of
micromorphic continuum mechanics for the reader’s convenience. Then, section 3
presents the proposed data-driven formulation and numerical implementation.
Section 4 shows the numerical examples and section 5 provides concluding
remarks. Finally, A presents the reference constitutive models used to
generate the synthetic data sets.

##  2 Overview of generalized micromorphic continua

###  2.1 Problem setting

Consider a solid body $\Omega\subset\mathbb{R}^{n}$, $n\in\\{1,2,3\\}$,
evolving in a time interval $\mathrm{I}=[0,T]$. In a classical Cauchy theory
with linearized kinematics, the motion of the continuum solid is fully
characterized by the macro-displacement field
$\bm{u}:\Omega\times\mathrm{I}\to\mathbb{R}^{n}$. Conversely, the theory of
generalized continua [42, 43] further considers that attached to each macro-
coordinate $\bm{x}\in\Omega$ is a _micro-continuum_ body
$\Omega^{\mathrm{m}}$, subject to independent deformations and defined through
micro-coordinate axes $\xi_{i}$ whose origin moves with the macro-displacement
(figure 1). Then, the motion of the microstructured solid is given by a
_micro-displacement_ field, conveniently expressed as the sum of the macro-
displacement and a fluctuation term:

$\bm{\nu}(\bm{x},\bm{\xi})=\bm{u}(\bm{x})+\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}(\bm{x})\cdot\bm{\xi}\,,\quad{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{ij}=\partial\nu_{ij}/\partial\xi_{j}\,.$

Evidently, in this form, the micro-deformation field
$\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}:\Omega\times\mathrm{I}\to\mathbb{R}^{n\times
n}$ is homogeneous in $\Omega^{\mathrm{m}}$ and thus independent of
$\bm{\xi}\in\Omega^{\mathrm{m}}$, resulting in a micromorphic theory of degree
1 [44].

[scale=0.8]bvp

Figure 1: Schematic representation of a BVP in a micromorphic continuum (left)
with a generic microstructure (right). The dashed blue arrows represent
second-order tensors. The first-order theory adopted in this work assumes that
the micro-deformation
${{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}_{ij}$
is homogeneous in $\Omega^{\mathrm{m}}$ but varies in $\Omega$.

Consequently, we may formulate a BVP at the macro-scale in terms of the
primary fields $\bm{u}$ and $\textstyle\chi$, with the latter representing an
additional set of degrees of freedom (DOFs). The solid boundary $\Gamma$ is
then decomposed into a Dirichlet part $\Gamma^{u}_{\mathrm{D}}$ with imposed
displacements
$\bar{\bm{u}}:\Gamma^{u}_{\mathrm{D}}\times\mathrm{I}\to\mathbb{R}^{n}$ and a
Neumann part $\Gamma^{u}_{\mathrm{N}}$ with applied forces
$\bar{\bm{t}}:\Gamma^{u}_{\mathrm{N}}\times\mathrm{I}\to\mathbb{R}^{n}$, with
$\Gamma^{u}_{\mathrm{D}}\cup\Gamma^{u}_{\mathrm{N}}=\Gamma$ and
$\Gamma^{u}_{\mathrm{D}}\cap\Gamma^{u}_{\mathrm{N}}=\emptyset$. Similarly, the
boundary conditions for $\textstyle\chi$ consist of applied micro-deformations
$\bar{\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}}:\Gamma^{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{\mathrm{D}}\times\mathrm{I}\to\mathbb{R}^{n\times
n}$ and applied double forces
$\bar{\bm{T}}:\Gamma^{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{\mathrm{N}}\times\mathrm{I}\to\mathbb{R}^{n\times
n}$, with
$\Gamma^{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{\mathrm{D}}\cup\Gamma^{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{\mathrm{N}}=\Gamma$
and
$\Gamma^{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{\mathrm{D}}\cap\Gamma^{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{\mathrm{N}}=\emptyset$.111We
note that the physical significance of Dirichlet boundary conditions on
$\textstyle\chi$ is debatable. For the sake of conciseness, volume forces and
volume double forces are omitted, without losing generality.

###  2.2 Kinematics and balance laws

The micromorphic continuum of degree 1 involves the following generalized
strain measures:

Compatible strain
$\displaystyle\bm{\varepsilon}=\operatorname*{sym}\,\nabla\bm{u}\,,$
$\displaystyle\varepsilon_{ij}=\frac{1}{2}(u_{i,j}+u_{j,i})\,;$ (1) Relative
strain
$\displaystyle\bm{\gamma}=\nabla\bm{u}-\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}\,,$
$\displaystyle\gamma_{ij}=u_{i,j}-{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{ij}\,;$
(2) Micro-deformation gradient
$\displaystyle\bm{\zeta}=\nabla\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}\,,$
$\displaystyle\zeta_{ijk}={\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{ij,k}\,.$
(3)

The power density of internal mechanical forces reads

$p^{\mathrm{int}}=\bm{\sigma}:\dot{\bm{\varepsilon}}+\bm{\tau}:\dot{\bm{\gamma}}+\bm{\mu}\,\Shortstack{...}\,\dot{\bm{\zeta}}\,,$
(4)

where $\bm{\sigma}$ is the second-order, symmetric Cauchy stress tensor;
$\bm{\tau}$ is the second-order, generally non-symmetric relative stress
tensor; and $\bm{\mu}$ is the third-order double stress tensor. The principle
of virtual power

$\int_{\Omega}\big{(}\bm{\sigma}:\dot{\bm{\varepsilon}}+\bm{\tau}:\dot{\bm{\gamma}}+\bm{\mu}\,\Shortstack{...}\,\dot{\bm{\zeta}}\big{)}\,\mathrm{d}\bm{x}-\int_{\Gamma}\big{(}\bar{\bm{t}}\cdot\dot{\bm{u}}+\bar{\bm{T}}:\dot{\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}}\big{)}\,\mathrm{d}S=0$
(5)

then yields the following balance equations $\forall\,t\in\mathrm{I}$:

Stress equilibrium
$\displaystyle\operatorname*{div}\,(\bm{\sigma}+\bm{\tau})=\bm{0}\ \ \
\text{in}\ \ \ \Omega\,,$
$\displaystyle\bm{n}\cdot(\bm{\sigma}+\bm{\tau})=\bar{\bm{t}}\ \ \ \text{on}\
\ \ \Gamma_{\mathrm{N}}^{u}\,;$ (6) Double stress equilibirum
$\displaystyle\operatorname*{div}\,\bm{\mu}+\bm{\tau}=\bm{0}\ \ \ \text{in}\ \
\ \Omega\,,$ $\displaystyle\bm{n}\cdot\bm{\mu}=\bar{\bm{T}}\ \ \ \text{on}\ \
\
\Gamma_{\mathrm{N}}^{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}\,.$
(7)

This system must be solved for the primary fields
$\bm{u}\in\mathrm{H}^{1}(\Omega;\mathbb{R}^{n})$ and
$\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}\in\mathrm{H}^{1}(\Omega;\mathbb{R}^{n\times
n})$ subject to the boundary conditions $\bm{u}=\bar{\bm{u}}$ on
$\Gamma^{u}_{\mathrm{D}}$ and
$\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}=\bar{\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}}$
on
$\Gamma^{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{\mathrm{D}}$.222When
$\bm{\gamma}\equiv\bm{0}$, we recover the second-gradient theory with a single
balance equation in terms of $\bm{u}\in\mathrm{H}^{2}(\Omega;\mathbb{R}^{n})$.
Clearly, at this point, we require relations between the generalized stresses
$\\{\bm{\sigma},\bm{\tau},\bm{\mu}\\}$ and the conjugate generalized strains
$\\{\bm{\varepsilon},\bm{\gamma},\bm{\zeta}\\}$.

###  2.3 Material behavior

####  2.3.1 Elasticity

As in classical Cauchy continua, conventional theories for generalized
continua invoke constitutive models to establish the relation between
$\\{\bm{\sigma},\bm{\tau},\bm{\mu}\\}$ and
$\\{\bm{\varepsilon},\bm{\gamma},\bm{\zeta}\\}$. In the case of linear
elasticity, a linear mapping
$(\bm{\varepsilon},\bm{\gamma},\bm{\zeta})\mapsto\\{\bm{\sigma},\bm{\tau},\bm{\mu}\\}(\bm{\varepsilon},\bm{\gamma},\bm{\zeta})$
is employed, usually in terms of a Helmholtz energy density
$\psi\coloneqq{\psi}(\bm{\varepsilon},\bm{\gamma},\bm{\zeta})$, such that

$\bm{\sigma}=\frac{\partial\psi}{\partial\bm{\varepsilon}}\,,\qquad\bm{\tau}=\frac{\partial\psi}{\partial\bm{\gamma}}\,,\qquad\bm{\mu}=\frac{\partial\psi}{\partial\bm{\zeta}}\,.$
(8)

A fairly general form can be written as

${\psi}(\bm{\varepsilon},\bm{\gamma},\bm{\zeta})\coloneqq\frac{1}{2}\big{(}\bm{\mathsf{C}}\bm{\varepsilon}:\bm{\varepsilon}+\bm{\mathsf{D}}\bm{\gamma}:\bm{\gamma}+\bm{\mathsf{E}}\bm{\varepsilon}:\bm{\gamma}+\bm{\mathsf{A}}\bm{\zeta}\,\Shortstack{...}\,\bm{\zeta}\big{)}\,,$
(9)

where $\bm{\mathsf{C}}$, $\bm{\mathsf{D}}$, $\bm{\mathsf{E}}$ (4th-order), and
$\bm{\mathsf{A}}$ (6th-order) are generalized elasticity tensors. For
isotropic materials, these tensors involve up to 18 parameters [43], most of
which are difficult to obtain experimentally. Of course, this general model
includes several special cases with fewer parameters (table 1). Moreover, the
coupling tensor $\bm{\mathsf{E}}$ is generally assumed to vanish due to its
unclear physical meaning [54], leaving 16 parameters in the general theory.
Then, the micropolar model, with $\textstyle\chi$ purely skew-symmetric,
preserves 6 parameters, while the microstrain model, with $\textstyle\chi$
purely symmetric, preserves 9 parameters. By using similar considerations, the
relaxed micromorphic model of Neff et al. [54] is reduced to 7 parameters.

######  Remark 2.1.

In previous works [69, 70, 71], the following forms are adopted for the
microstrain simplification of the general theory:
$\mathsf{C}_{ijkl}=\lambda\delta_{ij}\delta_{kl}+G(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})$,
where $\lambda$ and $G$ are the Lamé coefficients;
$\mathsf{D}_{ijkl}=c_{1}\mathsf{C}_{ijkl}$, with $c_{1}\geq 1$; and
$\mathsf{A}_{ijklmn}=c_{1}\ell_{\chi}^{2}\mathsf{C}_{ijkl}\delta_{mn}$, where
$\ell_{\chi}\geq 0$ is a length scale parameter. A micromorphic continuum of
this type, with only 4 elasticity parameters, is considered for the sake of
simplicity in the models presented in A and employed in the numerical
simulations of section 4.

Table 1: Some special cases of the first-order micromorphic theory for isotropic elastic materials with a free energy of the form (9). See Forest and Sievert [72] and Neff et al. [54] for thorough discussions. Model | Special case | No. DOFs | No. Params.
---|---|---|---
Cauchy | $\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}\rightsquigarrow\bm{0}$ |  3 $(\bm{u})$ | 2
Second-gradient [73] | $\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}\rightsquigarrow\bm{\bm{\varepsilon}}$ |  3 $(\bm{u})$ | 3
Micropolar [41] | $\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}\rightsquigarrow\mathrm{skew}\,\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}$ |  6 $(\bm{u},\mathrm{skew}\,\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}})$ | 6
Microstrain [72] | $\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}\rightsquigarrow\mathrm{sym}\,\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}$ |  9 $(\bm{u},\operatorname*{sym}\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}})$ | 11
Relaxed microm. [54] | $\nabla\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}\rightsquigarrow\mathrm{curl}\,\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}$ | 12 $(\bm{u},\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}})$ | 9
$\bm{\gamma}\rightsquigarrow\operatorname*{sym}\,\bm{\bm{\gamma}}$

####  2.3.2 Inelasticity

Extensions of generalized continuum theories to plasticity, damage, and other
inelastic phenomena [53] typically resort to the framework of thermomechanics
with internal variables. In this setting, we introduce a generic set of
internal variables $\mathbf{q}$ and consider free energies depending on both
$\mathbf{q}$ and (possibly) its gradient, i.e.,
$\psi\coloneqq{\psi}(\bm{\varepsilon},\bm{\gamma},\bm{\zeta},\mathbf{q},\nabla\mathbf{q})$.
The internal variables obey evolution laws, generally stated as differential
inclusions of the form

$\frac{\delta\psi}{\delta\mathbf{q}}(\bm{\varepsilon},\bm{\gamma},\bm{\zeta},\mathbf{q},\nabla\mathbf{q})\;+\;\partial{\phi}(\dot{\mathbf{q}})\ni\bm{0}\,.$
(10)

Here, $\phi\coloneqq{\phi}(\dot{\mathbf{q}})\geq 0$ is a convex dissipation
potential complying with the second law of thermodynamics. Evidently, the
constitutive relations (8) are now functions of $\mathbf{q}$ and thus become
history-dependent. Models of this type can be derived rigorously from the
principle of virtual power [74] or, more generally, from the energetic
formulation [75, 76], as shown for a gradient damage model in A.1. It is worth
noting that the dissipation potential may also depend on the current state,
which allows us to consider non-associative models [77, 78]. A.2 presents a
plasticity model of this type.

###  2.4 Data-driven approach

In the standard formulation outlined above, even classical Cauchy continuum
models are susceptible to epistemic uncertainty, particularly when dealing
with complex inelastic behavior. This challenge is further amplified in the
framework of generalized continua, requiring additional constitutive functions
and parameters. Thus, in this study, we shift our focus towards the realm of
data-driven mechanics [55] for inelastic materials [57] and extend this
approach to encompass generalized micromorphic continua. In this setting,
instead of relying on constitutive relations and evolution equations such as
(8) and (10), we assume that the material response in terms of generalized
strains $\\{\bm{\varepsilon},\bm{\gamma},\bm{\zeta}\\}$ and stresses
$\\{\bm{\sigma},\bm{\tau},\bm{\mu}\\}$ is encapsulated in a finite data set,
presumably obtained from experimental measurements or high-fidelity numerical
simulations. We shall then make direct use of this data to solve the
fundamental equations (1)–(3) and (6)–(7), without resorting to an empirical
model.

Of course, embarking on this path presupposes the availability of data sets
comprising generalized coordinates
$\\{(\bm{\varepsilon},\bm{\gamma},\bm{\zeta}),(\bm{\sigma},\bm{\tau},\bm{\mu})\\}$.
Measurements of compatible strains and the corresponding Cauchy stresses,
$(\bm{\varepsilon},\bm{\sigma})$, may be obtained, for instance, from high-
fidelity virtual simulations and standard homogenization theory, as done in
recent works on granular media [66, 79, 67]. Acquiring data for the
generalized quantities $(\bm{\gamma},\bm{\tau})$ and $(\bm{\zeta},\bm{\mu})$
poses an additional challenge. In this context, a data-driven framework for
micropolar continua using micro-rotation and couple-stress data directly
extracted from LS-DEM simulations of sand has been recently presented [68].
Extending these virtual measurements to encompass generalized micromorphic
continua appears feasible in light of previous works on this topic, e.g.,
considering granular media [80, 81] or chiral metamaterials [82].

In this study, we focus not on data acquisition but rather on developing a
data-driven framework that incorporates generalized stress-strain data to
simulate materials with strain localization. Consequently, we limit our
studies in the sequel to synthetic data obtained from reference micromorphic
constitutive models presented in A.1 and A.2, leaving the task of identifying
material data from other sources for future research. The following section
presents the proposed data-driven formulation.

##  3 Set-oriented data-driven formulation of micromorphic continua

This section presents the extension of data-driven computing to micromorphic
materials. In agreement with most references on data-driven mechanics, we
express the formulation in a discretized form, i.e., considering a finite-
dimensional system. Consequently, the fundamental kinematic relations (1)–(3)
and balance laws (6)–(7) are restated in a finite element setting. The
resulting equations are employed in the statement of the data-driven problem
that follows.

###  3.1 Discrete kinematics and balance equations

Let us consider hereafter an isotropic solid discretized into $N$ nodes and
$M$ material points, leading to the assembly of nodal displacement and
external force vectors denoted as $\mkern
1.2mu\underline{\mkern-1.2mu\mathbf{u}\mkern-1.2mu}\mkern
1.2mu=\\{{\bm{u}}_{i}\in\mathbb{R}^{n}\\}_{i=1}^{N}$ and $\mkern
1.2mu\underline{\mkern-1.2mu\mathbf{f}\mkern-1.2mu}\mkern
1.2mu^{\mathrm{ext}}=\\{{\bm{f}}^{\mathrm{ext}}_{i}\in\mathbb{R}^{n}\\}_{i=1}^{N}$,
respectively. Similarly, the nodal micro-deformations and double forces are
collected in vectors denoted as $\mkern
1.2mu\underline{\mkern-1.2mu\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}\mkern-1.2mu}\mkern
1.2mu=\\{{\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}}_{i}\in\mathbb{R}^{n_{\chi}}\\}_{i=1}^{N}$
and $\mkern 1.2mu\underline{\mkern-1.2mu\mathbf{m}\mkern-1.2mu}\mkern
1.2mu^{\mathrm{ext}}=\\{{\bm{m}}^{\mathrm{ext}}_{i}\in\mathbb{R}^{n_{\chi}}\\}_{i=1}^{N}$,
respectively. Here, the size of the nodal micro-deformation vector,
$n_{\chi}$, depends on the symmetry conditions. For instance, $n_{\chi}=n^{2}$
for the full first-order micromorphic model and $n_{\chi}=n(n+1)/2$ for the
microstrain simplification (see table 1).

On the other hand, the time interval $\mathrm{I}$ is discretized into a finite
number of time steps $0=t^{0}<\dots<t^{k}<t^{k+1}<\dots<t^{K}=T$. Hereinafter,
unless required for clarity, we denote a quantity at a given time $t^{k}$ as
$\Box^{k}\coloneqq\Box(t^{k})$ and omit the superscript at the next time step,
so that $\Box\coloneqq\Box^{k+1}$.

In this section, second- and higher-order tensors evaluated at a point in
space are expressed in their Voigt representation. Then, the discrete form of
the kinematic relations (1)–(3) reads

$\displaystyle\bm{\varepsilon}_{e}=\mathbf{B}^{\varepsilon}_{e}\,\mkern
1.2mu\underline{\mkern-1.2mu\mathbf{u}\mkern-1.2mu}\mkern
1.2mu\;+\;\bm{g}^{u}_{e}\,,$ (11)
$\displaystyle\bm{\gamma}_{e}=\mathbf{B}^{\gamma}_{e}\,\mkern
1.2mu\underline{\mkern-1.2mu\mathbf{u}\mkern-1.2mu}\mkern
1.2mu\;-\;\mathbf{N}^{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}_{e}\,\mkern
1.2mu\underline{\mkern-1.2mu\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}\mkern-1.2mu}\mkern
1.2mu\,,$ (12) $\displaystyle\bm{\zeta}_{e}=\mathbf{B}^{\zeta}_{e}\,\mkern
1.2mu\underline{\mkern-1.2mu\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}\mkern-1.2mu}\mkern
1.2mu\;+\;\bm{g}^{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{e}\,,$
(13)

where $\bm{g}^{u}_{e}\in\mathbb{R}^{n_{\varepsilon}}$ and
$\bm{g}^{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{e}\in\mathbb{R}^{n_{\zeta}}$
follow from Dirichlet boundary conditions, with $n_{\varepsilon}=n(n+1)/{2}$
and $n_{\zeta}=n_{\chi}(n_{\chi}+1)/2$. Similarly, the balance equations
(6)–(7) take the discrete forms

$\displaystyle\sum_{e=1}^{M}w_{e}\Big{[}\mathbf{B}^{\varepsilon\,\mathrm{T}}_{e}\,\bm{\sigma}_{e}\;+\;\mathbf{B}^{\gamma\,\mathrm{T}}_{e}\,\bm{\tau}_{e}\Big{]}-\mkern
1.2mu\underline{\mkern-1.2mu\mathbf{f}\mkern-1.2mu}\mkern
1.2mu^{\mathrm{ext}}=\bm{0}\,,$ (14)
$\displaystyle\sum_{e=1}^{M}w_{e}\Big{[}\mathbf{B}^{\zeta\,\mathrm{T}}_{e}\,\bm{\mu}_{e}\;-\;\mathbf{N}^{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}\,\mathrm{T}}_{e}\,\bm{\tau}_{e}\Big{]}-\mkern
1.2mu\underline{\mkern-1.2mu\mathbf{m}\mkern-1.2mu}\mkern
1.2mu^{\mathrm{ext}}=\bm{0}\,,$ (15)

where $w_{e}$ is an integration weight. In these expressions,
$\mathbf{N}^{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{e}\in\mathbb{R}^{n_{\chi}\times
Nn_{\chi}}$ is the shape function matrix for the nodal micro-deformation
vector $\bm{\mkern
1.2mu\underline{\mkern-1.2mu{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}\mkern-1.2mu}\mkern
1.2mu}$ while
$\mathbf{B}^{\varepsilon}_{e}\in\mathbb{R}^{n_{\varepsilon}\times Nn}$,
$\mathbf{B}^{\gamma}_{e}\in\mathbb{R}^{n_{\chi}\times Nn_{\chi}}$, and
$\mathbf{B}^{\zeta}_{e}\in\mathbb{R}^{n_{\zeta}\times Nn_{\chi}}$ are standard
discrete differential operators.

###  3.2 The data-driven problem

The data-driven problem requires a set-oriented reformulation of the discrete
system described above. In this setting, the state of the solid at a material
point $e$ is defined in terms of generalized coordinates $\mathbf{z}_{e}$ in a
local phase space $\mathrm{Z}_{e}$. For the present micromorphic formulation,
we have, in general,

$\mathbf{z}_{e}\coloneqq(\bm{\varepsilon}_{e},\bm{\gamma}_{e},\bm{\zeta}_{e},\bm{\sigma}_{e},\bm{\tau}_{e},\bm{\mu}_{e})\in\mathrm{Z}_{e},\quad\mathrm{Z}_{e}\coloneqq\mathbb{R}^{n_{\varepsilon}}\times\mathbb{R}^{n_{\chi}}\times\mathbb{R}^{n_{\zeta}}\times\mathbb{R}^{n_{\varepsilon}}\times\mathbb{R}^{n_{\chi}}\times\mathbb{R}^{n_{\zeta}}\,.$

The local phase space is endowed with the following metric (cf. (9) with
$\bm{\mathsf{E}}=\bm{0}$):

$\|\mathbf{z}_{e}\|^{2}\coloneqq\frac{1}{2}\Big{(}\bm{\mathsf{C}}_{e}\bm{\varepsilon}_{e}\cdot\bm{\varepsilon}_{e}+\bm{\mathsf{C}}_{e}^{-1}\bm{\sigma}_{e}\cdot\bm{\sigma}_{e}+\bm{\mathsf{D}}_{e}\bm{\gamma}_{e}\cdot\bm{\gamma}_{e}+\bm{\mathsf{D}}_{e}^{-1}\bm{\tau}_{e}\cdot\bm{\tau}_{e}+\bm{\mathsf{A}}_{e}\bm{\zeta}_{e}\cdot\bm{\zeta}_{e}+\bm{\mathsf{A}}_{e}^{-1}\bm{\mu}_{e}\cdot\bm{\mu}_{e}\Big{)}\,.$
(16)

Here, $\bm{\mathsf{C}}_{e}\in\mathbb{R}^{n_{\varepsilon}\times
n_{\varepsilon}}_{\mathrm{sym},+}$,
$\bm{\mathsf{D}}_{e}\in\mathbb{R}^{n_{\chi}\times n_{\chi}}_{\mathrm{sym},+}$,
and $\bm{\mathsf{A}}_{e}\in\mathbb{R}^{n_{\zeta}\times
n_{\zeta}}_{\mathrm{sym},+}$ are positive-definite matrices of a purely
numerical nature, in principle, unrelated to the material behavior. Globally,
the local coordinates
$\mathbf{z}\coloneqq\\{\mathbf{z}_{e}\in\mathrm{Z}_{e}\\}_{e=1}^{M}\in\mathrm{Z}$
represent points in the global phase space
$\mathrm{Z}=\mathrm{Z}_{1}\times\dots\times\mathrm{Z}_{M}$. In view of the
discrete formulation of section 3.1, the global distance metric reads

$\|\mathbf{z}\|^{2}\coloneqq\sum_{e=1}^{M}w_{e}\|\mathbf{z}_{e}\|^{2}\,.$ (17)

We proceed to define a subset of $\mathrm{Z}$ containing all mechanically
admissible states:

$\mathrm{E}^{k+1}\coloneqq\Big{\\{}\mathbf{z}\in\mathrm{Z}\ \colon\
\text{\eqref{eq:epsilon_disc}--\eqref{eq:eqmu_disc} }\ \text{at }\
t=t^{k+1}\Big{\\}}\,,$ (18)

which may be viewed as a _generalized equilibrium set_ evaluated at time
$t^{k+1}$. On the other hand, rather than finding closure to the discrete
system (11)–(15) by invoking constitutive relations (section 2.3), we assume
that the material response is encoded in a _material data set_
$\mathrm{D}=\mathrm{D}_{1}\times\dots\times\mathrm{D}_{M}$, where
$\mathrm{D}_{e}\subset\mathrm{Z}_{e}$ is the local data set containing
generalized stress-strain coordinates accessible to the material point $e$. We
account for history-dependent, inelastic responses by defining a suitable
subset $\mathrm{D}^{k+1}\subset\mathrm{D}$ that contains, at time $t^{k+1}$,
data points compatible with certain constraints imposed on the stress-strain
history [57]. In this context, let us assume that a generic internal variable
$\mathbf{q}$ is appended to the data set, such that a realization
$\mathbf{q}_{\,e}$ is attached to every data point
$\mathbf{z}_{e}\in\mathrm{D}_{e}$. Then, we consider a time-evolving data set
evaluated at time $t^{k+1}$:

$\mathrm{D}_{e}^{k+1}\coloneqq\Big{\\{}\mathbf{z}_{e}\in\mathrm{D}_{e}\
\colon\
\Phi\big{(}\mathbf{q}_{\,e}^{k+1},\\{(\bm{\varepsilon}_{e},\bm{\gamma}_{e},\bm{\zeta}_{e},\bm{\sigma}_{e},\bm{\tau}_{e},\bm{\mu}_{e})^{s}\\}_{s\leq
k+1}\big{)}\leq\bm{0}\Big{\\}}\,,$ (19)

where $\Phi$ is a generic function establishing a constraint to be fulfilled
by the generalized stress-strain history and the internal variable state.
Then, an evolving global data set can be defined at time $t^{k+1}$ as
$\mathrm{D}^{k+1}=\mathrm{D}_{1}^{k+1}\times\dots\times\mathrm{D}_{M}^{k+1}$.
It bears emphasis that the availability of an internal state variable
$\mathbf{q}_{\,e}$ does not entail the existence of a constitutive model; this
variable must be measurable and included in the database alongside
$\mathbf{z}_{e}\in\mathrm{D}_{e}$. For instance, in recent works on data-
driven computing for granular materials, LS-DEM simulations [83] have been
used by Karapiperis et al. [66] to obtain energy quantities and enforce
thermodynamic constraints, and by Ulloa et al. [67] to measure a breakage
state variable and enforce irreversibility conditions in crushable media.

With the above definitions in hand, the time-discrete data-driven problem may
be written as

$\inf_{\displaystyle\mathbf{y}\in\mathrm{D}^{k+1}}\,\inf_{\displaystyle\mathbf{z}\in\mathrm{E}^{k+1}}\|\mathbf{y}-\mathbf{z}\|^{2}=\inf_{\displaystyle\mathbf{z}\in\mathrm{E}^{k+1}}\,\inf_{\displaystyle\mathbf{y}\in\mathrm{D}^{k+1}}\|\mathbf{y}-\mathbf{z}\|^{2}\,.$
(20)

This problem is typically solved using a fixed-point iteration scheme. Fixing
an estimate for the mechanically admissible state
$\mathbf{z}^{*}\in\mathrm{E}^{k+1}$, the problem

$\mathbf{y}^{*}=\mathrm{arg}\inf_{\displaystyle\mathbf{y}\in\mathrm{D}^{k+1}}\,\|\mathbf{y}-\mathbf{z}^{*}\|^{2}$
(21)

is solved using a local nearest neighbor search for the data points
$\mathbf{y}_{e}\in\mathrm{D}_{e}^{k+1}$. Then, fixing the estimate
$\mathbf{y}^{*}\in\mathrm{D}^{k+1}$, the problem

$\mathbf{z}^{*}=\mathrm{arg}\inf_{\displaystyle\mathbf{z}\in\mathrm{E}^{k+1}}\,\|\mathbf{y}^{*}-\mathbf{z}\|^{2}$
(22)

is solved enforcing the mechanical constrains embedded in (18). The kinematics
are enforced directly by replacing the generalized strains (11)–(13) in the
local metric $\|\mathbf{z}_{e}-\mathbf{y}^{*}_{e}\|$, using equations
(16)–(15). On the other hand, the balance equations (14)–(15) are enforced by
means of Lagrange multipliers collected in vectors $\mkern
1.2mu\underline{\mkern-1.2mu\bm{\lambda}\mkern-1.2mu}\mkern 1.2mu^{u}$ and
$\mkern 1.2mu\underline{\mkern-1.2mu\bm{\lambda}\mkern-1.2mu}\mkern
1.2mu^{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}$.
In light of equation (17), we obtain the Euler-Lagrange equations

$\displaystyle\sum_{e=1}^{M}w_{e}\Big{[}\mathbf{B}^{\varepsilon\,\mathrm{T}}_{e}\,\bm{\mathsf{C}}_{e}\,\big{(}\mathbf{B}^{\varepsilon}_{e}\,\mkern
1.2mu\underline{\mkern-1.2mu\mathbf{u}\mkern-1.2mu}\mkern
1.2mu-\bm{\varepsilon}_{e}^{*}\big{)}+\mathbf{B}^{\gamma\,\mathrm{T}}_{e}\,\bm{\mathsf{D}}_{e}\,\big{(}\mathbf{B}^{\gamma}_{e}\,\mkern
1.2mu\underline{\mkern-1.2mu\mathbf{u}\mkern-1.2mu}\mkern
1.2mu-\mathbf{B}^{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}_{e}\,\mkern
1.2mu\underline{\mkern-1.2mu\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}\mkern-1.2mu}\mkern
1.2mu-\bm{\gamma}^{*}_{e}\big{)}\Big{]}=\bm{0}\,,$ (23)
$\displaystyle\sum_{e=1}^{M}w_{e}\Big{[}\mathbf{B}^{\zeta\,\mathrm{T}}_{e}\,\bm{\mathsf{A}}_{e}\,\big{(}\mathbf{B}^{\zeta}_{e}\,\mkern
1.2mu\underline{\mkern-1.2mu\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}\mkern-1.2mu}\mkern
1.2mu-\bm{\zeta}^{*}_{e}\big{)}-\mathbf{N}^{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}\,\mathrm{T}}_{e}\,\bm{\mathsf{D}}_{e}\,\big{(}\mathbf{B}^{\gamma}_{e}\,\mkern
1.2mu\underline{\mkern-1.2mu\mathbf{u}\mkern-1.2mu}\mkern
1.2mu-\mathbf{N}^{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}_{e}\,\mkern
1.2mu\underline{\mkern-1.2mu\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}\mkern-1.2mu}\mkern
1.2mu-\bm{\gamma}_{e}^{*}\big{)}\Big{]}=\bm{0}\,,$ (24)
$\displaystyle\sum_{e=1}^{M}w_{e}\Big{[}\mathbf{B}^{\varepsilon\,\mathrm{T}}_{e}\,\bm{\sigma}_{e}+\mathbf{B}^{\gamma\,\mathrm{T}}_{e}\,\bm{\tau}_{e}\Big{]}-\mkern
1.2mu\underline{\mkern-1.2mu\mathbf{f}\mkern-1.2mu}\mkern
1.2mu^{\mathrm{ext}}=\bm{0}\,,$ (25)
$\displaystyle\sum_{e=1}^{M}w_{e}\Big{[}\mathbf{B}^{\zeta\,\mathrm{T}}_{e}\,\bm{\mu}_{e}-\mathbf{N}^{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}\,\mathrm{T}}_{e}\,\bm{\tau}_{e}\Big{]}-\mkern
1.2mu\underline{\mkern-1.2mu\mathbf{m}\mkern-1.2mu}\mkern
1.2mu^{\mathrm{ext}}=\bm{0}\,,$ (26) $\displaystyle
w_{e}\bm{\mathsf{C}}_{e}^{-1}\big{(}\bm{\sigma}_{e}-\bm{\sigma}^{*}_{e}\big{)}-w_{e}\mathbf{B}^{\varepsilon}_{e}\,\mkern
1.2mu\underline{\mkern-1.2mu\bm{\lambda}\mkern-1.2mu}\mkern
1.2mu^{u}=\bm{0}\,,$ (27) $\displaystyle
w_{e}\bm{\mathsf{D}}_{e}^{-1}\big{(}\bm{\sigma}_{e}-\bm{\sigma}^{*}_{e}\big{)}-w_{e}\mathbf{B}^{\gamma}_{e}\,\mkern
1.2mu\underline{\mkern-1.2mu\bm{\lambda}\mkern-1.2mu}\mkern
1.2mu^{u}+w_{e}\mathbf{N}^{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}_{e}\,\mkern
1.2mu\underline{\mkern-1.2mu\bm{\lambda}\mkern-1.2mu}\mkern
1.2mu^{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}=\bm{0}\,,$
(28) $\displaystyle
w_{e}\bm{\mathsf{A}}_{e}^{-1}\big{(}\bm{\mu}_{e}-\bm{\mu}^{*}_{e}\big{)}-w_{e}\mathbf{B}^{\zeta}_{e}\,\mkern
1.2mu\underline{\mkern-1.2mu\bm{\lambda}\mkern-1.2mu}\mkern
1.2mu^{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}=\bm{0}\,.$
(29)

Upon substitution and rearrangement, we obtain the following linear system to
be solved for $(\mkern
1.2mu\underline{\mkern-1.2mu\mathbf{u}\mkern-1.2mu}\mkern 1.2mu,\mkern
1.2mu\underline{\mkern-1.2mu\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}\mkern-1.2mu}\mkern
1.2mu)$:

$\displaystyle\sum_{e=1}^{M}w_{e}\Big{[}\mathbf{B}^{\varepsilon\,\mathrm{T}}_{e}\bm{\mathsf{C}}_{e}\mathbf{B}^{\varepsilon}_{e}+\mathbf{B}^{\gamma\,\mathrm{T}}_{e}\bm{\mathsf{D}}_{e}\mathbf{B}^{\gamma}_{e}\Big{]}\mkern
1.2mu\underline{\mkern-1.2mu\mathbf{u}\mkern-1.2mu}\mkern
1.2mu-\sum_{e=1}^{M}w_{e}\Big{[}\mathbf{B}^{\gamma\,\mathrm{T}}_{e}\bm{\mathsf{D}}_{e}\mathbf{N}^{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}_{e}\Big{]}\mkern
1.2mu\underline{\mkern-1.2mu\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}\mkern-1.2mu}\mkern
1.2mu=\sum_{e=1}^{M}w_{e}\Big{[}\mathbf{B}^{\varepsilon\,\mathrm{T}}_{e}\bm{\mathsf{C}}_{e}\,\bm{\varepsilon}^{*}_{e}+\mathbf{B}^{\gamma\,\mathrm{T}}_{e}\bm{\mathsf{D}}_{e}\bm{\gamma}^{*}_{e}\Big{]},$
(30)
$\displaystyle\sum_{e=1}^{M}w_{e}\Big{[}\mathbf{N}^{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}\,\mathrm{T}}_{e}\bm{\mathsf{D}}_{e}\mathbf{B}^{\gamma}_{e}\Big{]}\mkern
1.2mu\underline{\mkern-1.2mu\mathbf{u}\mkern-1.2mu}\mkern
1.2mu-\sum_{e=1}^{M}w_{e}\Big{[}\mathbf{B}^{\zeta\,\mathrm{T}}_{e}\bm{\mathsf{A}}_{e}\mathbf{B}^{\zeta}_{e}+\mathbf{N}^{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}\,\mathrm{T}}_{e}\bm{\mathsf{D}}_{e}\mathbf{N}^{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}_{e}\Big{]}\mkern
1.2mu\underline{\mkern-1.2mu\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}\mkern-1.2mu}\mkern
1.2mu=\sum_{e=1}^{M}w_{e}\Big{[}\mathbf{N}^{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}\,\mathrm{T}}_{e}\bm{\mathsf{D}}_{e}\bm{\gamma}^{*}_{e}-\mathbf{B}^{\zeta\,\mathrm{T}}_{e}\bm{\mathsf{A}}_{e}\,\bm{\zeta}^{*}_{e}\Big{]},$
(31)

and the following linear system to be solved for $(\mkern
1.2mu\underline{\mkern-1.2mu\bm{\lambda}\mkern-1.2mu}\mkern 1.2mu^{u},\mkern
1.2mu\underline{\mkern-1.2mu\bm{\lambda}\mkern-1.2mu}\mkern
1.2mu^{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}})$:

$\displaystyle\sum_{e=1}^{M}w_{e}\Big{[}\mathbf{B}^{\varepsilon\,\mathrm{T}}_{e}\bm{\mathsf{C}}_{e}\mathbf{B}^{\varepsilon}_{e}+\mathbf{B}^{\gamma\,\mathrm{T}}_{e}\bm{\mathsf{D}}_{e}\mathbf{B}^{\gamma}_{e}\Big{]}\mkern
1.2mu\underline{\mkern-1.2mu\bm{\lambda}\mkern-1.2mu}\mkern
1.2mu^{u}-\sum_{e=1}^{M}w_{e}\Big{[}\mathbf{B}^{\gamma\,\mathrm{T}}_{e}\bm{\mathsf{D}}_{e}\mathbf{N}^{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}_{e}\Big{]}\mkern
1.2mu\underline{\mkern-1.2mu\bm{\lambda}\mkern-1.2mu}\mkern
1.2mu^{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}=\mkern
1.2mu\underline{\mkern-1.2mu\mathbf{f}\mkern-1.2mu}\mkern
1.2mu^{\mathrm{ext}}-\sum_{e=1}^{M}w_{e}\Big{[}\mathbf{B}^{\varepsilon\,\mathrm{T}}_{e}\bm{\sigma}^{*}_{e}+\mathbf{B}^{\gamma\,\mathrm{T}}_{e}\bm{\tau}^{*}_{e}\Big{]},$
(32)
$\displaystyle\sum_{e=1}^{M}w_{e}\Big{[}\mathbf{N}^{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}\,\mathrm{T}}_{e}\bm{\mathsf{D}}_{e}\mathbf{B}^{\gamma}_{e}\Big{]}\mkern
1.2mu\underline{\mkern-1.2mu\bm{\lambda}\mkern-1.2mu}\mkern
1.2mu^{u}-\sum_{e=1}^{M}w_{e}\Big{[}\mathbf{B}^{\zeta\,\mathrm{T}}_{e}\bm{\mathsf{A}}_{e}\mathbf{B}^{\zeta}_{e}+\mathbf{N}^{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}\,\mathrm{T}}_{e}\bm{\mathsf{D}}_{e}\mathbf{N}^{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}_{e}\Big{]}\mkern
1.2mu\underline{\mkern-1.2mu\bm{\lambda}\mkern-1.2mu}\mkern
1.2mu^{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}=\sum_{e=1}^{M}w_{e}\Big{[}\mathbf{B}^{\zeta\,\mathrm{T}}_{e}\bm{\mu}^{*}_{e}-\mathbf{N}^{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}\,\mathrm{T}}_{e}\bm{\tau}^{*}_{e}\Big{]}-\mkern
1.2mu\underline{\mkern-1.2mu\mathbf{m}\mkern-1.2mu}\mkern
1.2mu^{\mathrm{ext}}.$ (33)

The coupled systems (30)–(31) and (32)–(33) are standard forms for linear
micromorphic elasticity and can be easily implemented in conventional finite
element programs. The mechanical state is then updated as

$\displaystyle\begin{dcases}\bm{\varepsilon}_{e}=\mathbf{B}^{\varepsilon}_{e}\,\mkern
1.2mu\underline{\mkern-1.2mu\mathbf{u}\mkern-1.2mu}\mkern 1.2mu\,,\\\
\bm{\sigma}_{e}=\bm{\mathsf{C}}_{e}\mathbf{B}^{\varepsilon}_{e}\,\mkern
1.2mu\underline{\mkern-1.2mu\bm{\lambda}\mkern-1.2mu}\mkern
1.2mu^{u}+\bm{\sigma}_{e}^{*}\,;\end{dcases}$ (34)
$\displaystyle\begin{dcases}\bm{\gamma}_{e}=\mathbf{B}^{\gamma}_{e}\,\mkern
1.2mu\underline{\mkern-1.2mu\mathbf{u}\mkern-1.2mu}\mkern
1.2mu-\mathbf{N}^{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}_{e}\,\mkern
1.2mu\underline{\mkern-1.2mu\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}\mkern-1.2mu}\mkern
1.2mu\,,\\\ \bm{\tau}_{e}=\bm{\mathsf{D}}_{e}\mathbf{B}^{\gamma}_{e}\,\mkern
1.2mu\underline{\mkern-1.2mu\bm{\lambda}\mkern-1.2mu}\mkern
1.2mu^{u}-\bm{\mathsf{D}}_{e}\mathbf{N}^{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}_{e}\,\mkern
1.2mu\underline{\mkern-1.2mu\bm{\lambda}\mkern-1.2mu}\mkern
1.2mu^{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}+\bm{\tau}_{e}^{*}\,;\end{dcases}$
(35) $\displaystyle\begin{dcases}\bm{\zeta}_{e}=\mathbf{B}^{\zeta}_{e}\,\mkern
1.2mu\underline{\mkern-1.2mu\mathbf{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}\mkern-1.2mu}\mkern
1.2mu\,,\\\ \bm{\mu}_{e}=\bm{\mathsf{A}}_{e}\mathbf{B}^{\zeta}_{e}\,\mkern
1.2mu\underline{\mkern-1.2mu\bm{\lambda}\mkern-1.2mu}\mkern
1.2mu^{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}+\bm{\mu}_{e}^{*}\,.\end{dcases}$
(36)

This general formulation involves 12 (6) DOFs per node in 3D (2D), with $n=3$
($n=2$) and $n_{\chi}=9$ ($n_{\chi}=4$), while the numerical operators
$\bm{\mathsf{C}}_{e}$, $\bm{\mathsf{D}}_{e}$, and $\bm{\mathsf{A}}_{e}$
involve 2, 3, and 11 parameters, respectively. However, as discussed in
section 2.3, simplifications of the general framework are possible. Hereafter,
we consider, for the sake of simplicity, a microstrain continuum with
$\textstyle\chi$ purely symmetric (table 1). Then, the number of DOFs in 3D
(2D) is reduced to 9 (5), with $n_{\chi}=6$ ($n_{\chi}=3$). Moreover, in
agreement with remark 2.1, the total number of parameters in the operators
$\bm{\mathsf{C}}_{e}$, $\bm{\mathsf{D}}_{e}$, and $\bm{\mathsf{A}}_{e}$ is
reduced to 4.

##  4 Numerical simulations

This section presents representative numerical simulations that showcase the
ability of data-driven micromorphic computing to capture responses with strain
localization. For the sake of conciseness, we limit the study to synthetic
data sampled from the solution of reference BVPs. These data are then employed
in data-driven computations that aim to replicate the reference model
response. Hence, the main challenge for the data-driven algorithm will be to
assign suitable data to each material point.

In the examples that follow, a single data set is generated for all material
points, such that $\mathrm{D}_{e}^{k+1}\equiv\mathrm{D}^{k+1}$, and the
nearest neighbor search (21) is performed in a random order. Moreover, the
history dependence of the material response is considered by imposing
irreversibility conditions in the data set (19):

$\mathrm{D}_{e}^{k+1}\coloneqq\Big{\\{}\mathbf{z}_{e}\in\mathrm{D}_{e}\
\colon\ 0\leq q^{k+1}-q^{k}\leq\mathtt{TOL}_{q}\Big{\\}}\,.$ (37)

Here, $q$ is an irreversible internal state variable, taken as the damage
variable in sections 4.1 and 4.2 and as the equivalent plastic strain in
section 4.3. Moreover, $\mathtt{TOL}_{q}$ is a tolerance parameter introduced
to avoid large jumps in data sets that include loading-unloading and softening
branches.

######  Remark 4.1.

The data set parametrization (37) assumes access to the internal state
variable $q$, computed from the reference model in the current numerical
study. However, for the application of this framework to real material data,
this irreversible quantity must be accessible through empirical measurements.
In this context, a measure of energy dissipation, computed from grain-scale
simulations of granular materials, has been effectively employed [66].
Additionally, a breakage state variable, measurable in experiments or grain-
scale simulations, was used in a recent study [67] focusing on crushable
granular media.

###  4.1 One-dimensional softening bar with damage

Let us first examine a simple 1D problem that allows us to identify the key
issues related to softening responses and strain localization. In particular,
we assess the standard data-driven framework for Cauchy continua in a
softening bar, isolating the underlying challenges. These challenges are then
addressed through the proposed extension to micromorphic continua.

Consider a 1D bar with constant unitary cross-section and length $L$ under
tension, with applied displacements $u=\bar{u}(t)$ on one end and fixed
displacements $u=0$ on the other (figure 2). The displacements are imposed in
153 increments from $\bar{u}=0$ to $\bar{u}=2.5$. The material is assumed to
obey the gradient damage model with micromorphic elasticity presented in A.1,
reduced to the 1D case ($n=1$ and $n_{\chi}=1$) through obvious model
simplifications. The balance equations (46) and (47) are solved together with
the damage evolution equations (48) using a standard staggered scheme. The
constitutive parameters are chosen as $\mathsf{C}_{1111}/L=1$, $w_{1}L=1$,
$\ell/L=0.8$, $c_{1}=1$, and
$\ell_{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}/L=0.1$.
The bar is discretized into 100 finite elements using standard linear shape
functions. Homogeneous Dirichlet boundary conditions are imposed on the damage
field for the sake of obtaining a symmetric spatial response.

######  Remark 4.2.

In this normalized setting, arbitrary values of length $L$ will yield the same
response of the model in terms of the nodal quantities
$(u,L{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}},\alpha)$
and state variables $(L\varepsilon,\sigma,L\gamma,\tau,L^{2}\zeta,\mu/L)$. For
ease of notation, in this subsection, we let
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}\leftarrow
L{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}$,
$\varepsilon\leftarrow L\varepsilon$, $\gamma\leftarrow L\gamma$,
$\zeta\leftarrow L^{2}\zeta$, and $\mu\leftarrow\mu/L$.

[scale=0.95]bar1_bvp

Figure 2: BVP for the 1D bar with constant cross-section.

[scale=0.85]bardata_loc

Figure 3: Data gathered from the solution of the 1D bar with constant cross-
section, using the damage model in A.1 simplified to the 1D case, to be used
in standard data-driven computations. The data is taken from all the material
points in the bar, yielding a total of 15300 points (100 elements and 153 load
steps). Note that $\sigma+\tau$ plays the role of Cauchy’s stress.

The standard data-driven framework for Cauchy continua is recovered from the
general framework presented in section 3 by enforcing only the compatibility
equation (11) and the stress balance equation (6) in the equilibrium set (17).
The phase-space coordinates are then reduced to
$\mathbf{z}_{e}=(\varepsilon_{e},\sigma_{e}+\tau_{e})$. As such, the stress
quantity $\sigma_{e}+\tau_{e}$ is taken as an effective Cauchy stress,
computed directly from the data to determine the material states and enforced
to satisfy the standard equilibrium equations to determine the mechanical
states. Accordingly, the distance metric (16) takes the standard form

$2\,\|\mathbf{z}_{e}\|^{2}=E\varepsilon_{e}^{2}+\frac{1}{E}(\sigma_{e}+\tau_{e})^{2}\,,\
$ (38)

where $E\coloneqq\mathsf{C}_{1111}$. Figure 3 presents the data points
gathered from the solution of the BVP, showing softening in the central
element with a transition to elastic unloading at $x=0$ and $x=L$. Note that
the irreversible damage variable $\alpha$ is only employed to parametrize the
history-dependent data set (37).

[scale=0.85]bar1_loc

Figure 4: Data-driven results using the standard framework and comparison with
the reference simulation (solid lines) for the 1D bar with constant cross-
section, showing (a) the force-displacement response and the spatial
distribution of (b) damage, (c) displacements, and (d) strains. The strain
profile shows both the mechanical and material states, which visually overlap.

Figure 4 shows the result of applying these data points in the standard data-
driven framework for Cauchy continua. The initial undamaged parameter
$E=C_{1111}=1$ is employed in the distance metric (38). The global force-
displacement curve shows a virtually perfect match between the reference
response and the data-driven predictions, considering both the material state
(i.e., the selected data points $\mathbf{y}^{*}\in\mathrm{D}^{k+1}$) and the
mechanical solution (i.e., $\mathbf{z}^{*}\in\mathrm{E}^{k+1}$). On the other
hand, the displacement, strain, and damage profiles (figures 4b,c,d) are
chaotic, in sharp contrast with the reference response.

Note, nevertheless, that this solution is mechanically admissible, and
corresponds to a local minimum of the strongly non-convex data-driven problem
(20). Notably, figure 5a shows that the balance equation (14) is approximately
satisfied at all time steps by the material state, up to the allowable error
in figure 6a. Indeed, any strain profile with constant stress values
$\sigma_{e}+\tau_{e}$ along the length of the bar will comply with this
condition. Therefore, different admissible profiles may emerge depending on
the order in which the material points are read in the nearest neighbor search
(21). Then, unless we force the material points to only read the data
corresponding to similar locations in the reference simulation, the standard
algorithm has no reason to converge to a strain profile that captures the
length scale of the reference response.

[scale=0.85]bar1_eq

Figure 5: Verification of equilibrium conditions in the data-driven
simulations of the 1D bar with constant cross-section, showing the results of
the standard framework (dots) and the micromorphic framework (x marks). The
color evolution corresponds to the load step $\bar{u}$. The spatial
distribution of (a) $\sigma+\tau$ satisfies the equilibrium equation (14) with
uniform profiles. The spatial distribution of (b)
$\mathrm{d}\mu/\mathrm{d}\mathrm{x}+\tau$ satisfies the double-stress
equilibrium equation (15) with vanishing profiles.

[scale=0.87]bar1_dists

Figure 6: Evolution of the distance metric $\|\mathbf{z}^{*}-\mathbf{y}^{*}\|$
between material and mechanical states and individual contributions of the
state variables for the data-driven simulations of the 1D bar with constant
cross-section, showing the results of (a) the standard local framework and (b)
the micromorphic framework.

[scale=0.85]bardata

Figure 7: Data gathered from the solution of the 1D bar with constant cross-
section, using the damage model in A.1 simplified to the 1D case, to be used
in data-driven micromorphic computations. The data is taken from all the
material points in the bar, yielding a total of 15300 points (100 elements and
153 load steps).

Let us now apply the proposed micromorphic formulation to this 1D problem. For
this purpose, we proceed with the generalized phase-space coordinates
$\mathbf{z}_{e}=(\varepsilon_{e},\gamma_{e},\zeta_{e},\sigma_{e},\tau_{e},\mu_{e})$.
The full distance metric (16) then takes the form

$2\,\|\mathbf{z}_{e}\|^{2}=E\varepsilon_{e}^{2}+\frac{1}{E}\sigma_{e}^{2}+c_{1}E\gamma_{e}^{2}+\frac{1}{c_{1}E}\tau_{e}^{2}+c_{1}\ell_{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}^{2}E\zeta_{e}^{2}+\frac{1}{c_{1}\ell_{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}^{2}E}\mu_{e}^{2}\,.$
(39)

Figure 7 presents the generalized data points collected from the reference
solution. Again, the irreversible damage variable $\alpha$ is only employed to
parametrize the history-dependent data set (37). As in figure 3, the central
element experiences softening, with a transition to elastic unloading at $x=0$
and $x=L$ in $(\varepsilon,\sigma)$ space. On the other hand, the central
element exhibits a monotonically increasing behavior in $(\gamma,\tau)$ space
and remains small in $(\zeta,\mu)$ space. For the latter, peak absolute values
in $\zeta$ and $\mu$ occur at a distance from the center that evolves during
the simulation, ending at $x/L\approx 0.5\pm 0.085$ and $x/L\approx 0.5\pm
0.25$, respectively. These profiles are better illustrated in the reference
model response (figure 8).

[scale=0.85]bar1_mmorph

Figure 8: Data-driven results using the proposed micromorphic framework and
comparison with the reference simulation (solid lines) for the 1D bar with
constant cross-section, showing (a) the force-displacement response and the
spatial distribution of (b) damage, (c) displacements, (d) strains, (e) micro-
deformations, and (e) micro-deformation gradients. The strain and micro-
deformation gradient profiles show both the mechanical and material states,
which visually overlap.

Figure 8 shows the result of applying these data points in the generalized
data-driven framework for micromorphic continua. Again, the undamaged initial
constitutive parameters are used in the distance metric (39). As in figure 4,
the global force-displacement curve shows a very close agreement between the
reference solution and both the material ($\mathbf{y}^{*}\in\mathrm{D}^{k+1}$)
and mechanical ($\mathbf{z}^{*}\in\mathrm{E}^{k+1}$) data-driven predictions.
However, in this case, a realistic spatial distribution profile is predicted
for all variables. In particular, figures 8b–f show that the damage,
displacement, strain, microstrain, and microstrain gradient profiles predicted
by the data-driven algorithm are in close agreement with the reference
response.

Note, nevertheless, that a slight shift to the left is observed. As in the
standard case, this result is a local minimum of the strongly non-convex data-
driven problem (20). However, the possible profiles are now restricted by the
fulfillment of the double balance equation (15). For illustration, Figure 5b
shows that the material state predicted by the micromorphic framework complies
with equation (15), up to the allowable error in figure 6b, while the state
predicted by the standard framework does not. Hence, while profiles with peaks
at different locations are possible and observed in data-driven micromorphic
simulations, the response must satisfy all the constraints embedded in the
generalized mechanical manifold (18), rendering profiles like those in figure
4 no longer admissible. We further note that the length scale of the reference
response is now captured: while the stiffness encoded in the
$(\varepsilon,\sigma)$ data is extracted by both frameworks, the length scale
encoded in the $(\zeta,\mu)$ data is only extracted in the micromorphic
simulations.

[scale=0.87]bar1_conv

Figure 9: Relative distance between the material and mechanical states with
increasing number of data points $|\mathrm{D}|$ in the data-driven
micromorphic simulations of the 1D bar with constant cross-section, showing
(a) the evolution of the relative distance metric and (b) the convergence rate
in logarithmic space at different load steps $\bar{u}$. The cardinality
$|\mathrm{D}|$ is divided by the number of load steps (153) to yield the
number of sampling points.

Figure 9 shows the convergence of the data-driven micromorphic simulations
with respect to the number of data points. As expected, the relative error,
measured as the relative distance between the material and mechanical states,
i.e., $\|\mathbf{z}^{*}-\mathbf{y}^{*}\|/\|\mathbf{z}^{*}\|$, is smallest when
the data is sampled from all the material points in the reference solution
($|\mathrm{D}|/153=100$). The data-driven predictions become less accurate as
the applied displacement $\bar{u}$ increases due to the increasingly non-
homogeneous response. Moreover, figure 9b shows that the importance of the
data size increases with $\bar{u}$. At the peak load $\bar{u}=1.5$, the
relative error is approximately $2\%$ with only 13 sampled material points and
approximately $0.9\%$ with all the data points. The corresponding error values
increase to $8\%$ and $2\%$ at the final load step.

Finally, we repeat the calculations above for a bar with a central geometrical
notch (figure 10), imposing displacements in 145 increments from $\bar{u}=0$
to $\bar{u}=2.35$. Again, the standard framework fails to capture the spatial
distribution of the reference response (these results are not shown for the
sake of brevity). On the other hand, the data-driven prediction obtained with
the micromorphic framework exhibits, in this case, peak strains at the center
of the bar, coinciding with the reference solution (figure 11). Note that the
data is now taken from the reference solution of the notched bar (plots
analogous to figures 3 and 7, also not shown for the sake of brevity).

[scale=0.95]bar2_bvp

Figure 10: BVP for the 1D bar with a central notch. The cross-section is
reduced to 0.5 in the middle along the coordinates $x/L\in[0.45,0.55]$.

[scale=0.85]bar2_mmorph

Figure 11: Data-driven results using the proposed micromorphic framework and
comparison with the reference simulation (solid lines) for the 1D notched bar,
showing (a) the force-displacement response and the spatial distribution of
(b) damage, (c) displacements, (d) strains, (e) micro-deformations, and (f)
micro-deformation gradients. The strain and micro-deformation gradient
profiles show both the mechanical and material states, which visually overlap.

[scale=0.87]bar2_conv

Figure 12: Relative distance with increasing number of data points
$|\mathrm{D}|$ in the data-driven micromorphic simulations of the 1D notched
bar, showing (a) the evolution of the relative distance metric between
material and mechanical states, and (b) the evolution of the relative distance
metric between the mechanical states and the reference solution
$\mathbf{z}^{\mathrm{ref}}$.

Figure 12a shows the relative distance metric
$\|\mathbf{z}^{*}-\mathbf{y}^{*}\|/\|\mathbf{z}^{*}\|$, quantifying the
relative error between the material and mechanical states in the notched bar
for different numbers of data points. At the end of the simulation, this value
is approximately $5\%$ when only 13 material points are sampled and
approximately $0.5\%$ when all 100 material points are sampled. Note that the
error no longer increases monotonically as the load $\bar{u}$ increases. On
the other hand, figure 12b shows the relative errors with respect to the
reference solution $\mathbf{z}^{\mathrm{ref}}$, i.e., the state extracted
directly from the model response. At $\bar{u}=1.5$, the error ranges from
$11.5\%$ with 13 sampled material points to $0.6\%$ with all the data points.
At the final time step, these values increase, respectively, to approximately
$35\%$ and $3\%$. Of course, because the relative distance between the
mechanical and material states is much smaller, these responses are always
mechanically admissible up to a certain tolerance.

At this point, we have shown that the proposed extension of data-driven
mechanics to micromorphic continua can capture the non-homogeneous response of
softening bars with no prior data assignment, whereas the (mechanically
admissible) state predicted by the standard Cauchy continuum framework does
not yield the expected profiles. The next sections evaluate the proposed
framework in 2D scenarios.

######  Remark 4.3.

So far, the reference strain profiles have shown moderate localization, with
none of the bars studied above reaching a damage level of 1 (i.e., the bars
did not crack). While the proposed framework can handle scenarios with
complete damage, as shown in the following example, this problem merits a word
of caution. Specifically, the data set becomes sparse in these stages, with
very low stresses and high strains. A dedicated approach may be considered in
future works to properly navigate through such sparse data sets, e.g.,
performing adaptive sampling [79].

###  4.2 V-notched specimen with damage

We now apply the proposed data-driven micromorphic framework to a 2D BVP where
the material experiences complete damage. In particular, consider a V-notched
sample under plane strain conditions and tensile loading (figure 13a).
Displacements are applied incrementally from $\bar{u}=0$ to $\bar{u}=0.106$ mm
in 528 load steps. The material is assumed to obey the gradient damage model
with micromorphic elasticity presented in A.1. We employ the following
material parameters: Young’s modulus $E=25.85$ GPa, Poisson’s ratio
$\nu=0.18$, damage threshold $w_{1}=4.8$ kPa, damage length scale $\ell=12.5$
mm, micromorphic constant $c_{1}=1$, and micromorphic length scale
$\ell_{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}=12.5$
mm. The 2D specimen is discretized into 1664 quadrilateral elements, refining
the region between the notches where fracture is expected, and using standard
bilinear shape functions.

[scale=0.74]V_bvp_fd_

Figure 13: BVP for the V-notched specimen with dimensions in mm (left) and
force-displacement response comparing the data-driven solution to the
reference simulation (right).

The reference solution of the BVP is obtained by solving the governing
equations (46)–(48) of the micromorphic gradient damage model using a standard
staggered scheme. Figure 13b shows the force-displacement curve of the
reference simulation (red line). The curve shows an initial, seemingly elastic
stage. However, since the AT-2 formulation is used, with
$w(\alpha)=w_{1}\alpha^{2}$, a slight level of damage evolves from the onset
of loading and the response is never fully linear. The curve reaches a peak
load at around $\bar{u}=0.1$ mm, followed by a softening response that leads
to abrupt failure, characterizing brittle fracture. To reduce the brutal
damage evolution at this stage, we include a slight numerical viscosity in the
simulation, as commonly done in the phase-field literature [32].
Alternatively, dissipation-based arc-length techniques can be employed [84].
Figure 14 (top) shows snapshots of the vertical displacement component
$u_{y}$, the microstrain component
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{yy}$,
and the damage variable $\alpha$. Cracks nucleate at the notches and propagate
towards the middle of the sample. During this stage, the smoothness of the
displacement field progressively decreases while the micro-deformation field
concentrates at the crack tips. Figure 15 (top) presents the corresponding
stress fields, showing the $\sigma_{yy}+\tau_{yy}$, $\mu_{yyy}$, and
$\mu_{yyx}$ components.

[width=0.97]V_kin_v2_

Figure 14: Reference simulation results for the V-notched specimen (top) and
data-driven predictions (bottom), showing the vertical displacement component
$u_{y}$ (left), the micro-deformation component
$\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}_{yy}$
(middle), and the damage field $\alpha$ (right). The snapshots correspond to
initial crack propagation at the peak load and crack merging at failure.

We proceed to reproduce the reference simulation using data-driven
micromorphic computing. The generalized stress-strain states are extracted
from the reference simulation at intervals of every 4 material points,
equivalent to utilizing 25% of the available information. In total, we collect
time-history data (load steps) from 1664 sampling points, each yielding 528
generalized stress-strain states. It is worth noting that the data becomes
considerably sparse during the abrupt crack propagation stage, particularly at
the final fracture steps when the specimen undergoes total failure. Addressing
data sparsity in such responses is a challenge that merits consideration in
future works. In this proof-of-concept simulation, we predict the response up
to just before complete fracture, with a small ligament not fully broken.
Additionally, we find it beneficial to update the metric tensors
$\bm{\mathsf{C}}_{e}$, $\bm{\mathsf{D}}_{e}$, and $\bm{\mathsf{A}}_{e}$ during
this stage. To achieve this, optimal moduli may be considered [68]. For
simplicity, in this simulation, we employ the initial undamaged moduli until
the final propagation stage. At that point, the secant moduli, which may be
computed directly from the data, seem to yield better predictions.

[width=0.97]V_str_v2_

Figure 15: Reference simulation results for the V-notched specimen (top) and
mechanical (middle) and material (bottom) data-driven predictions, showing the
stress component $\sigma_{yy}+\tau_{yy}$ (left) and the double stress
components $\mu_{yyy}$ (middle) and $\mu_{yyx}$ (right). The snapshots
correspond to initial crack propagation at the peak load and crack merging at
failure.

Figure 13b compares the force-displacement response with the data-driven
predictions, considering both the material
($\mathbf{y}^{*}\in\mathrm{D}^{k+1}$) and mechanical
($\mathbf{z}^{*}\in\mathrm{E}^{k+1}$) states. The response shows a visually
close agreement, even during the final crack propagation stage. Moreover, the
proximity of the material and mechanical curves signals a relatively small
distance metric throughout the simulation. Figures 14 and 15 (bottom) show the
corresponding spatial distribution of nodal quantities and generalized stress
states. We observe a reasonable agreement between the reference response and
the data-driven predictions.

Note, however, that some noise is observed in the damage zone for the data-
driven simulations, particularly at the final load step. Nevertheless, recall
that the entire data set, consisting of over 1 million generalized stress-
strain states, was read by all the material points with no particular
assignment and in no particular order. Therefore, the deviations, mostly
limited to specific zones near fractured elements, are deemed reasonable.
Indeed, figure 16 presents the absolute error in generalized stresses between
the data-driven predictions and the reference simulation. The absolute error
maps highlight regions of agreement and discrepancy in stress and double
stress components, with most errors limited to regions near the crack tips.

[width=0.97]V_err_v2_

Figure 16: Deviation from the reference simulation for the V-notched specimen,
showing the mechanical (top) and material (bottom) absolute error maps for the
stress component $\sigma_{yy}+\tau_{yy}$ [MPa] (left) and the double stress
components $\mu_{yyy}$ [MPa/mm] (middle) and $\mu_{yyx}$ [MPa/mm] (right). The
snapshots correspond to initial crack propagation at the peak load and crack
merging at failure.

###  4.3 Biaxial compression test with plasticity

The final example applies data-driven micromorphic computing to a biaxial
compression test under plane strain conditions (figure 17), where the material
is assumed to obey the micromorphic frictional-dilatant plasticity model
presented in A.2. Vertical displacements are applied in 65 increments from
$\bar{u}=0$ to $\bar{u}=0.6$ mm, keeping the lateral confining stress
$\sigma_{0}=5$ MPa constant. We employ the following material parameters,
taken from Borja [85] for the local version of the model: Young’s modulus
$E=14$ GPa, Poisson’s ratio $\nu=0.3$, plastic yield strength
$\sigma^{\mathrm{p}}=14.85$ MPa, friction coefficient $A_{\varphi}=0.29$, and
dilation coefficient $A_{\theta}=0.25$. In addition, we consider a
micromorphic constant $c_{1}=1$ and length scale
$\ell_{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}=0.05$
mm. Finally, for the sake of simplicity, we take
$a_{1}=a_{2}=a_{3}=a_{4}=0$.333Hence, only the classical plastic strain
$\bm{\varepsilon}^{\mathrm{p}}$ is involved while
${\bm{\gamma}}^{\mathrm{p}}=\bm{0}$ and ${\bm{\zeta}}^{\mathrm{p}}=\bm{0}$.
The same assumption is made in Dillard et al. [70] in the context of metallic
foams. The 2D specimen is discretized into 3090 quadrilateral elements using
standard bilinear shape functions.

[scale=0.70]biaxial_bvp_fd

Figure 17: BVP for the biaxial compression test with dimensions in mm (right)
and force-displacement response comparing the data-driven solution to the
reference simulation (right).

[width=0.95]biaxial_kin_v2

Figure 18: Reference simulation results for the biaxial compression test (top)
and data-driven predictions (bottom), showing the vertical displacement
component $-u_{y}$ [mm] (left), the micro-deformation
$\|\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}\|$
(middle), and the equivalent plastic strain field $\kappa$ (right). The
snapshots correspond to the initial yielding stage at $\bar{u}=0.25$ mm and
the final load step at $\bar{u}=0.6$ mm.

We proceed to reproduce the reference simulation using data-driven
micromorphic computing, taking the time-history data from 25% of all material
points in a random order. Figure 17b shows the global force-displacement curve
measured on the top edge of the sample, comparing the reference solution to
the material ($\mathbf{y}^{*}\in\mathrm{D}^{k+1}$) and mechanical
($\mathbf{z}^{*}\in\mathrm{E}^{k+1}$) data-driven predictions. A reasonable
agreement is observed throughout the simulation, with the material and
mechanical states virtually overlapping. Accordingly, the displacements
$u_{y}$, microstrains
$\|\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}\|$,
and equivalent plastic strains $\kappa$ in figure 18 show that the data-driven
predictions capture the reference shear-banding response. In particular, at
the initial yielding stage, where $\bar{u}\approx 0.25$ mm, the micro-
deformations show a slight concentration near the central hole in an inclined
pattern, coinciding with the equivalent plastic strains. As the material
points progressively yield, shear deformation bands develop symmetrically and
propagate toward the right and left sides. The data-driven predictions capture
this deformation profile accurately, with no prior data assignment and all
material points reading a single data set.

##  5 Conclusions and perspectives

We have presented a data-driven formulation for generalized micromorphic
continua. Its enhanced capabilities compared to the standard Cauchy-continuum
framework have been illustrated in problems with strain localization without
assigning data sets to specific material points a priori. The response of a 1D
softening bar, assumed to be governed by a gradient damage model, has been
studied systematically. It is shown that local minima found by the standard
data-driven framework may capture the global response but not the spatial
distribution of the state variables expected from the reference data. In
contrast, the proposed framework captures both the global response and the
generalized stress-strain profiles, effectively extracting the material length
scale through the solution of the data-driven BVP. The framework has been
further validated in 2D for (i) a V-notched sample undergoing fracture in
tension and (ii) a biaxial sample with shear banding, showing, in both cases,
accurate predictions.

The present study paves the way for subsequent developments on various fronts.
Firstly, given that a generalized continuum framework is seemingly required to
properly capture responses with strain localization, the natural question that
arises is whether suitable generalized stress-strain data can be measured in a
practical scenario and employed directly in data-driven computations. Focusing
on granular media, it has been shown previously [68] that this task is indeed
possible for micropolar continua, suitable for responses characterized by
shear bands with micro-rotations. For more general failure processes, e.g.,
compaction bands in granular materials or tensile failure in cohesive
materials, as considered in the present work, the full first-order
micromorphic variables must be sampled from experiments or lower-scale
simulations. This task is beyond the scope of this paper but appears feasible
in light of previous studies that bridge micromorphic quantities with results
from molecular dynamics [86] or discrete element methods [80, 81].

Finally, we emphasize that the present study has made no use of data
structures, using, at all times, a single data set for all material points.
This approach serves as a proof of concept but is by no means efficient. More
dedicated techniques that efficiently navigate through generalized stress-
strain data may be envisaged, perhaps adopting a clustering approach [87].
Furthermore, the high-dimensional space of generalized stress-strain states
may benefit from model order reduction. Applications of the present framework
to specific materials may consider such developments in the future.

## Acknowledgements

JU and JA acknowledge support for this research provided by US ARO funding
through the Multidisciplinary University Research Initiative (MURI) grant no.
W911NF- 19- 1- 0245. LS and MO gratefully acknowledge the financial support of
NExT ISite program of Nantes Université through International Research Project
(IRP) iDDrEAM.

##  Appendix A Reference models

###  A.1 Micromorphic gradient damage model

This appendix presents the reference continuum model used in the examples of
sections 4.1–4.2. The model is an extension of the conventional gradient-
damage/phase-field approach to fracture [30, 26] to micromorphic elasticity.
Thus, we introduce, in addition to the displacement field $\bm{u}$ and the
micro-deformation field $\textstyle\chi$, a damage variable
$\alpha(\bm{x},t)\in[0,1]$, with 0 corresponding to an undamaged material
point and 1 characterizing a completely damaged state. For the sake of
brevity, we consider displacement boundary conditions on
$\Gamma^{u}_{\mathrm{D}}$ and omit external double forces and applied micro-
deformations.

We derive the model in a variational form in terms of an internal energy
functional $\mathcal{E}$ and dissipation power functional $\mathcal{R}$:

$\mathcal{E}(\bm{u},\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}},\alpha)\coloneqq\int_{\Omega}\psi(\bm{\varepsilon},\bm{\gamma},\bm{\zeta},\alpha,\nabla\alpha)\,\mathrm{d}\bm{x}\,,\quad\mathcal{R}(\dot{\alpha};\alpha)\coloneqq\int_{\Omega}\phi(\dot{\alpha};\alpha)\,\mathrm{d}\bm{x}\,.$
(40)

The free energy density reads

$\psi(\bm{\varepsilon},\bm{\gamma},\bm{\zeta},\alpha,\nabla\alpha)\coloneqq\frac{1}{2}\Big{[}\hat{\bm{\mathsf{C}}}(\alpha)\bm{\varepsilon}:\bm{\varepsilon}+\hat{\bm{\mathsf{D}}}(\alpha)\big{(}\nabla\bm{u}-\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}\big{)}:\big{(}\nabla\bm{u}-\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}\big{)}+\hat{\bm{\mathsf{A}}}(\alpha)\nabla\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}\,\Shortstack{...}\,\nabla\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}\big{]}+w_{1}\,\ell^{2}\nabla\alpha\cdot\nabla\alpha\,,$
(41)

yielding the generalized stresses

$\bm{\sigma}=\frac{\partial\psi}{\partial\bm{\varepsilon}}=\hat{\bm{\mathsf{C}}}(\alpha)\bm{\varepsilon}\,,\qquad\bm{\tau}=\frac{\partial\psi}{\partial\bm{\gamma}}=\hat{\bm{\mathsf{D}}}(\alpha)\bm{\gamma}\,,\qquad\bm{\mu}=\frac{\partial\psi}{\partial\bm{\zeta}}=\hat{\bm{\mathsf{A}}}(\alpha)\bm{\zeta}\,.$
(42)

The dissipation potential is given by

$\phi(\dot{\alpha};\alpha)\coloneqq\begin{dcases}w^{\prime}(\alpha)\,\dot{\alpha}\quad&\text{if
}\dot{\alpha}\geq 0\,,\\\ +\infty\quad&\text{otherwise}\,,\end{dcases}$ (43)

with $w(\alpha)=w_{1}\alpha^{2}$ (AT-2 formulation). Here, $\ell$ is the
damage length scale, while the parameter $w_{1}=w(1)$ represents the energy
dissipated in a complete damage process for a homogeneous volume element.

One can show that the energy balance equation

$\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{E}(\bm{u},\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}},\alpha)+\mathcal{R}(\dot{\alpha};\alpha)-\int_{\Gamma^{u}_{\mathrm{N}}}\bar{\bm{t}}\cdot\dot{\bm{u}}\,\mathrm{d}S-\int_{\Gamma^{u}_{\mathrm{D}}}\bm{t}\cdot\dot{\bar{\bm{u}}}\,\mathrm{d}S=0$
(44)

and the first-order stability condition

$\displaystyle\delta\mathcal{E}(\bm{u},\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}},\alpha)(\tilde{\bm{u}},\tilde{\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}},\tilde{\alpha})+\mathcal{R}(\tilde{\alpha};\alpha)-\int_{\Gamma^{u}_{\mathrm{N}}}\bar{\bm{t}}\cdot\tilde{\bm{u}}\,\mathrm{d}S\geq
0\quad\forall\,\\{\tilde{\bm{u}},\tilde{\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}},\tilde{\alpha}\\}\in\tilde{\mathcal{U}}\times\tilde{\mathcal{X}}\times\tilde{\mathcal{A}},$
(45)

where $\tilde{\mathcal{U}}\coloneqq\mathrm{H}^{1}_{0}(\Omega;\mathbb{R}^{n})$,
$\tilde{\mathcal{X}}\coloneqq\mathrm{H}^{1}_{0}(\Omega;\mathbb{R}^{n\times
n})$, and
$\tilde{\mathcal{A}}\coloneqq\mathrm{H}^{1}_{0}(\Omega;\mathbb{R}_{+})$, yield
the following governing equations in strong form:

Stress equilibrium
$\displaystyle\begin{dcases}\operatorname*{div}\,(\bm{\sigma}+\bm{\tau})=\bm{0}\
\ \text{in}\ \ \Omega\,,\\\ \bm{n}\cdot(\bm{\sigma}+\bm{\tau})=\bm{0}\ \
\text{on}\ \ \Gamma^{u}_{\mathrm{N}}\,;\end{dcases}$ (46) Double stress
equilibrium
$\displaystyle\begin{dcases}\operatorname*{div}\,\bm{\mu}+\bm{\tau}=\bm{0}\ \
\text{in}\ \ \Omega\,,\\\ \bm{\mu}\cdot\bm{n}=\bm{0}\ \ \text{on}\ \
\Gamma\,;\end{dcases}$ (47) Damage evolution
$\displaystyle\begin{dcases}-\frac{\partial\psi}{\partial\alpha}(\bm{\varepsilon},\bm{\gamma},\bm{\zeta},\alpha,\nabla\alpha)-w^{\prime}(\alpha)+w_{1}\ell^{2}\mathrm{div}[\nabla\alpha]\leq
0\ \ \text{in}\ \ \Omega\,,\\\
\bigg{[}-\frac{\partial\psi}{\partial\alpha}(\bm{\varepsilon},\bm{\gamma},\bm{\zeta},\alpha,\nabla\alpha)-w^{\prime}(\alpha)+w_{1}\ell^{2}\mathrm{div}[\nabla\alpha]\bigg{]}\dot{\alpha}=0\
\ \text{in}\ \ \Omega\,,\\\ \dot{\alpha}\geq 0\ \ \text{in}\ \ \Omega\,,\\\
\nabla\alpha\cdot\bm{n}={0}\ \ \text{on}\ \ \Gamma\,.\end{dcases}$ (48)

For the sake of simplicity, following remark 2.1, we consider a microstrain
formulation, with $\textstyle\chi$ purely symmetric, and the following damaged
elasticity tensors:

$\hat{\mathsf{C}}_{ijkl}(\alpha)=(1-\alpha)^{2}\mathsf{C}_{ijkl},\quad\hat{\mathsf{D}}_{ijkl}(\alpha)=c_{1}\hat{\mathsf{C}}_{ijkl}(\alpha),\quad\hat{\mathsf{A}}_{ijklmn}(\alpha)=c_{1}\ell_{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}^{2}\hat{\mathsf{C}}_{ijkl}(\alpha)\delta_{mn}\,,$

where $\mathsf{C}_{ijkl}$ is the standard isotopic elasticity tensor.

In sections 4.1–4.2, we apply this model to solve different BVPs for $\bm{u}$
and $\textstyle\chi$, from which the generalized stress/strain data is
extracted as

$\displaystyle\begin{dcases}(\bm{\varepsilon},\bm{\sigma})=\big{(}\operatorname*{sym}\nabla\bm{u},\,\hat{\bm{\mathsf{C}}}(\alpha)\operatorname*{sym}\nabla\bm{u}\big{)}\,,\\\
(\bm{\gamma},\bm{\tau})=\big{(}\nabla\bm{u}-\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}},\,\hat{\bm{\mathsf{D}}}(\alpha)(\nabla\bm{u}-\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}})\big{)}\,,\\\
(\bm{\zeta},\bm{\mu})=\big{(}\nabla\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}},\,\hat{\bm{\mathsf{A}}}(\alpha)\nabla\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}\big{)}\,.\end{dcases}$

###  A.2 Micromorphic plasticity model

This appendix presents the reference continuum model used in the examples of
section 4.3. The model is an extension of conventional non-associative
Drucker-Prager plasticity to the micromorphic setting. As in the previous
section, we consider a microstrain continuum with $\textstyle\chi$ purely
symmetric. Moreover, similar to models presented in previous works [69, 72,
46], we may assume an additive decomposition of the micro-deformations into
elastic and plastic components, such that
$\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}=\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}^{\mathrm{e}}+\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}^{\mathrm{p}}$.
The generalized strain measures are decomposed accordingly as

$\bm{\varepsilon}=\bm{\varepsilon}^{\mathrm{e}}+\bm{\varepsilon}^{\mathrm{p}},\qquad\bm{\gamma}=\bm{\gamma}^{\mathrm{e}}+\bm{\gamma}^{\mathrm{p}},\qquad\bm{\zeta}=\bm{\zeta}^{\mathrm{e}}+\bm{\zeta}^{\mathrm{p}}\,.$
(49)

The free energy density reads

$\psi(\bm{\varepsilon},\bm{\gamma},\bm{\zeta},\bm{\varepsilon}^{\mathrm{p}},\bm{\gamma}^{\mathrm{p}},\bm{\zeta}^{\mathrm{p}})\coloneqq\frac{1}{2}\Big{[}\bm{\mathsf{C}}\bm{\varepsilon}^{\mathrm{e}}:\bm{\varepsilon}^{\mathrm{e}}+\bm{\mathsf{D}}\bm{\gamma}^{\mathrm{e}}:\bm{\gamma}^{\mathrm{e}}+\bm{\mathsf{A}}\bm{\zeta}^{\mathrm{e}}\,\Shortstack{...}\,\bm{\zeta}^{\mathrm{e}}\Big{]}\,.$
(50)

The conjugate stresses follow as

$\bm{\sigma}^{\mathrm{p}}=-\frac{\partial\psi}{\partial\bm{\varepsilon}^{\mathrm{p}}}=\bm{\mathsf{C}}\bm{\varepsilon}^{\mathrm{e}}\equiv\bm{\sigma}\,,\qquad\bm{\tau}^{\mathrm{p}}=-\frac{\partial\psi}{\partial\bm{\gamma}^{\mathrm{p}}}=\bm{\mathsf{D}}\bm{\gamma}^{\mathrm{e}}\equiv\bm{\tau}\,,\qquad\bm{\mu}^{\mathrm{p}}=-\frac{\partial\psi}{\partial\bm{\zeta}^{\mathrm{p}}}=\bm{\mathsf{A}}\bm{\zeta}^{\mathrm{e}}\equiv\bm{\mu}\,.$
(51)

At this point, different approaches may be taken to establish evolution laws
for the three plastic internal variables
$(\bm{\varepsilon}^{\mathrm{p}},\bm{\gamma}^{\mathrm{p}},\bm{\zeta}^{\mathrm{p}})$.
In particular, as done in Forest and Lorentz [69] for compressible plasticity,
we may consider ether a combined yield function for all variables or an
independent yield function for each variable. Focusing on non-associative
Drucker-Prager plasticity, similar to Regueiro [46] and Isbuga and Regueiro
[88], we may employ a combined yield function $f$ and plastic potential $g$:

$\displaystyle
f(\bm{\sigma},\bm{\tau},\bm{\zeta})\coloneqq\sqrt{\frac{3}{2}\big{(}\|\mathrm{dev}\,\bm{\sigma}\|+a_{1}\|\mathrm{dev}\,\bm{\tau}\|+a_{3}\|\mathrm{dev}\,\bm{\mu}\|\big{)}}+A_{\varphi}\big{(}\mathrm{tr}\,\bm{\sigma}+a_{2}\mathrm{tr}\,\bm{\tau}+a_{4}\|\mathrm{tr}\,\bm{\mu}\|\big{)}-\sigma^{\mathrm{p}}\,,$
(52) $\displaystyle
g(\bm{\sigma},\bm{\tau},\bm{\zeta})\coloneqq\sqrt{\frac{3}{2}\big{(}\|\mathrm{dev}\,\bm{\sigma}\|+a_{1}\|\mathrm{dev}\,\bm{\tau}\|+a_{3}\|\mathrm{dev}\,\bm{\mu}\|\big{)}}+A_{\theta}\big{(}\mathrm{tr}\,\bm{\sigma}+a_{2}\mathrm{tr}\,\bm{\tau}+a_{4}\|\mathrm{tr}\,\bm{\mu}\|\big{)}-\sigma^{\mathrm{p}}\,,$

where $[\mathrm{dev}\,\bm{\mu}]_{ijk}=\mu_{ijk}-(1/3)\delta_{ij}\mu_{aak}$ and
$[\mathrm{tr}\,\bm{\mu}]_{i}=\mu_{aai}$, or independent functions of the form

$\displaystyle
f^{\varepsilon}(\bm{\sigma})\coloneqq\sqrt{\frac{3}{2}}\|\mathrm{dev}\,\bm{\sigma}\|+A_{\varphi}\mathrm{tr}\,\bm{\sigma}-\sigma^{\mathrm{p}}\,,\qquad$
$\displaystyle
g^{\varepsilon}(\bm{\sigma})\coloneqq\sqrt{\frac{3}{2}}\|\mathrm{dev}\,\bm{\sigma}\|+A_{\theta}\mathrm{tr}\,\bm{\sigma}-\sigma^{\mathrm{p}}\,;$
(53) $\displaystyle
f^{\gamma}(\bm{\tau})\coloneqq\sqrt{\frac{3}{2}}a_{1}\|\mathrm{dev}\,\bm{\tau}\|+a_{2}A_{\varphi}\mathrm{tr}\,\bm{\tau}-a_{5}\sigma^{\mathrm{p}}\,,\qquad$
$\displaystyle
g^{\gamma}(\bm{\tau})\coloneqq\sqrt{\frac{3}{2}}a_{1}\|\mathrm{dev}\,\bm{\tau}\|+a_{2}A_{\theta}\mathrm{tr}\,\bm{\tau}-a_{5}\sigma^{\mathrm{p}}\,;$
$\displaystyle
f^{\zeta}(\bm{\mu})\coloneqq\sqrt{\frac{3}{2}}a_{3}\|\mathrm{dev}\,\bm{\mu}\|+a_{4}A_{\varphi}\|\mathrm{tr}\,\bm{\mu}\|-a_{6}\sigma^{\mathrm{p}}\,,\qquad$
$\displaystyle
g^{\zeta}(\bm{\mu})\coloneqq\sqrt{\frac{3}{2}}a_{3}\|\mathrm{dev}\,\bm{\mu}\|+a_{4}A_{\theta}\|\mathrm{tr}\,\bm{\mu}\|-a_{6}\sigma^{\mathrm{p}}\,.$

Here, as in classical plasticity, $\sigma^{\mathrm{p}}\geq 0$ is the yield
strength, $A_{\varphi}\geq 0$ is the friction coefficient,
$A_{\theta}\in[0,A_{\varphi}]$ is the dilation coefficient reflecting the non-
associativity of the model, and $a_{1},\dots,a_{6}$ are additional material
parameters for the micromorphic case.

A stress-dependent dissipation potential may now be evaluated following a
generalization of the principle of maximum dissipation to non-associative
models. For the case of independent functions (53):

$\phi(\dot{\bm{\varepsilon}}^{\mathrm{p}},\dot{\bm{\gamma}}^{\mathrm{p}},\dot{\bm{\zeta}}^{\mathrm{p}};\bm{\sigma},\bm{\tau},\bm{\mu})=\phi^{\varepsilon}(\dot{\bm{\varepsilon}}^{\mathrm{p}};\bm{\sigma})+\phi^{\gamma}(\dot{\bm{\gamma}}^{\mathrm{p}};\bm{\tau})+\phi^{\zeta}(\dot{\bm{\zeta}}^{\mathrm{p}};\bm{\mu}),$
(54)

where (cf. [78])

$\displaystyle\phi^{\varepsilon}(\dot{\bm{\varepsilon}}^{\mathrm{p}};\bm{\sigma})$
$\displaystyle=\sup\big{\\{}\tilde{\bm{\sigma}}:\dot{\bm{\varepsilon}}^{\mathrm{p}}\
\ \colon\ \
g^{\varepsilon}(\tilde{\bm{\sigma}})\leq(A_{\theta}-A_{\varphi})\mathrm{tr}\,\bm{\sigma}\big{\\}}$
(55)
$\displaystyle=\begin{dcases}\frac{\mathrm{tr}\,\dot{\bm{\varepsilon}}^{\mathrm{p}}}{3A_{\theta}}\big{[}(A_{\theta}-A_{\varphi})\mathrm{tr}\,\bm{\sigma}+\sigma^{\mathrm{p}}\big{]}&\text{if}\quad\mathrm{tr}\,\dot{\bm{\varepsilon}}^{\mathrm{p}}\geq\sqrt{6}A_{\theta}\|\mathrm{dev}\,\dot{\bm{\varepsilon}}^{\mathrm{p}}\|\,,\\\
+\infty&\text{otherwise};\end{dcases}$
$\displaystyle\phi^{\gamma}(\dot{\bm{\gamma}}^{\mathrm{p}};\bm{\tau})$
$\displaystyle=\sup\big{\\{}\tilde{\bm{\tau}}:\dot{\bm{\gamma}}^{\mathrm{p}}\
\ \colon\ \ g^{\gamma}(\tilde{\bm{\tau}})\leq
a_{2}(A_{\theta}-A_{\varphi})\mathrm{tr}\,\bm{\tau}\big{\\}}$
$\displaystyle=\begin{dcases}\frac{\mathrm{tr}\,\dot{\bm{\gamma}}^{\mathrm{p}}}{3a_{2}A_{\theta}}\big{[}a_{2}(A_{\theta}-A_{\varphi})\mathrm{tr}\,\bm{\tau}+a_{5}\sigma^{\mathrm{p}}\big{]}&\text{if}\quad\mathrm{tr}\,\dot{\bm{\gamma}}^{\mathrm{p}}\geq\frac{a_{2}}{a_{1}}\sqrt{6}A_{\theta}\|\mathrm{dev}\,\dot{\bm{\gamma}}^{\mathrm{p}}\|\,,\\\
+\infty&\text{otherwise};\end{dcases}$
$\displaystyle\phi^{\zeta}(\dot{\bm{\zeta}}^{\mathrm{p}};\bm{\mu})$
$\displaystyle=\sup\big{\\{}\tilde{\bm{\mu}}\,\Shortstack{...}\,\dot{\bm{\zeta}}^{\mathrm{p}}\
\ \colon\ \ g^{\zeta}(\tilde{\bm{\mu}})\leq
a_{4}(A_{\theta}-A_{\varphi})\|\mathrm{tr}\,\bm{\mu}\|\big{\\}}$
$\displaystyle=\begin{dcases}\frac{\|\mathrm{tr}\,\dot{\bm{\zeta}}^{\mathrm{p}}\|}{3a_{4}A_{\theta}}\big{[}a_{4}(A_{\theta}-A_{\varphi})\|\mathrm{tr}\,\bm{\mu}\|+a_{6}\sigma^{\mathrm{p}}\big{]}&\text{if}\quad\|\mathrm{tr}\,\dot{\bm{\zeta}}^{\mathrm{p}}\|\geq\frac{a_{4}}{a_{3}}\sqrt{6}A_{\theta}\|\mathrm{dev}\,\dot{\bm{\zeta}}^{\mathrm{p}}\|\,,\\\
+\infty&\text{otherwise}.\end{dcases}$

With these functions in hand, one can show that the energy balance equation

$\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{E}(\bm{u},\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}},\bm{\varepsilon}^{\mathrm{p}},\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}^{\mathrm{p}})+\mathcal{R}(\dot{\bm{\varepsilon}}^{\mathrm{p}},\dot{\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}}^{\mathrm{p}};\bm{\sigma})-\int_{\Gamma^{u}_{\mathrm{N}}}\bar{\bm{t}}\cdot\dot{\bm{u}}\,\mathrm{d}S-\int_{\Gamma^{u}_{\mathrm{D}}}\bm{t}\cdot\dot{\bar{\bm{u}}}\,\mathrm{d}S=0$
(56)

and the first-order stability condition

$\displaystyle\delta\mathcal{E}(\bm{u},\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}},\bm{\varepsilon}^{\mathrm{p}},\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}^{\mathrm{p}})(\tilde{\bm{u}},\tilde{\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}},\tilde{\bm{\varepsilon}}^{\mathrm{p}},\tilde{\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}}^{\mathrm{p}})+\mathcal{R}(\tilde{\bm{\varepsilon}}^{\mathrm{p}},\tilde{\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}}^{\mathrm{p}};\bm{\sigma},\bm{\tau},\bm{\mu})-\int_{\Gamma^{u}_{\mathrm{N}}}\bar{\bm{t}}\cdot\tilde{\bm{u}}\,\mathrm{d}S\geq
0\quad\forall\,\\{\tilde{\bm{u}},\tilde{\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}},\tilde{\bm{\varepsilon}}^{\mathrm{p}},\tilde{\bm{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}}^{\mathrm{p}}\\}\in\tilde{\mathcal{U}}\times\tilde{\mathcal{X}}\times\tilde{\mathcal{P}}\times\tilde{\mathcal{Y}},$
(57)

where $\tilde{\mathcal{P}}\coloneqq\mathrm{L}^{2}(\Omega;\mathbb{R}^{n})$ and
$\tilde{\mathcal{Y}}\coloneqq\mathrm{H}^{1}_{0}(\Omega;\mathbb{R}^{n\times
n})$, yield the following governing equations in strong form:

Stress equilibrium
$\displaystyle\begin{dcases}\operatorname*{div}\,(\bm{\sigma}+\bm{\tau})=\bm{0}\
\ \text{in}\ \ \Omega\,,\\\ \bm{n}\cdot(\bm{\sigma}+\bm{\tau})=\bm{0}\ \
\text{on}\ \ \Gamma^{u}_{\mathrm{N}}\,;\end{dcases}$ (58) Double stress
equilibrium
$\displaystyle\begin{dcases}\operatorname*{div}\,\bm{\mu}+\bm{\tau}=\bm{0}\ \
\text{in}\ \ \Omega\,,\\\ \bm{\mu}\cdot\bm{n}=\bm{0}\ \ \text{on}\ \
\Gamma\,;\end{dcases}$ (59) Plasticity evolution
$\displaystyle\begin{dcases}f^{\varepsilon}(\bm{\sigma})\leq 0,\quad
f^{\gamma}(\bm{\tau})\leq 0,\quad f^{\zeta}(\bm{\mu})\leq 0\ \ \text{in}\ \
\Omega\,,\\\
\dot{\lambda}^{\varepsilon}\,f^{\varepsilon}(\bm{\sigma})=0,\quad\dot{\lambda}^{\gamma}\,f^{\gamma}(\bm{\tau})=0,\quad\dot{\lambda}^{\zeta}\,f^{\zeta}(\bm{\mu})=0\
\ \text{in}\ \ \Omega\,,\\\
\\{\dot{\bm{\varepsilon}}^{\mathrm{p}},\dot{\bm{\gamma}}^{\mathrm{p}},\dot{\bm{\zeta}}^{\mathrm{p}}\\}\in\\{\dot{\lambda}^{\varepsilon}\partial
g^{\varepsilon}(\bm{\sigma}),\dot{\lambda}^{\gamma}\partial
g^{\gamma}(\bm{\tau}),\dot{\lambda}^{\zeta}\partial g^{\zeta}(\bm{\mu})\\}\ \
\text{in}\ \ \Omega\,.\end{dcases}$ (60)

The evolution equations for a model with the combined functions (52) can be
derived similarly. Note further that a Cosserat-type plasticity model can be
derived from the general formulation, taking $\textstyle\chi$ purely skew-
symmetric. Recent studies [89] have shown numerically that a model of this
type resolves ill-posedness in BVPs caused by non-associativity.

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