A new family of Constitutive Artiﬁcial Neural Networks towards automated model discovery Kevin Linka Ellen Kuhl

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A new family of Constitutive Artiﬁcial Neural Networks

towards automated model discovery

Kevin Linka & Ellen Kuhl

Department of Mechanical Engineering

Stanford University, Stanford, California, United States

We dedicate this manuscript to our Continuum Mechanics teachers

whose insights and passion for Continuum Mechanics have stimulated the ideas of this work,

Wolfgang Ehlers, Mikhail Itskov, Christian Miehe, Michael Ortiz,

Jörg Schröder, Erwin Stein, and Paul Steinmann

Abstract. For more than 100 years, chemical, physical, and material scientists have proposed

competing constitutive models to best characterize the behavior of natural and man-made ma-

terials in response to mechanical loading. Now, computer science offers a universal solution:

Neural Networks. Neural Networks are powerful function approximators that can learn con-

stitutive relations from large data without any knowledge of the underlying physics. However,

classical Neural Networks entirely ignore a century of research in constitutive modeling, violate

thermodynamic considerations, and fail to predict the behavior outside the training regime. Here

we design a new family of Constitutive Artiﬁcial Neural Networks that inherently satisfy com-

mon kinematical, thermodynamical, and physical constraints and, at the same time, constrain

the design space of admissible functions to create robust approximators, even in the presence of

sparse data. Towards this goal we revisit the non-linear ﬁeld theories of mechanics and reverse-

engineer the network input to account for material objectivity, material symmetry and incom-

pressibility; the network output to enforce thermodynamic consistency; the activation functions

to implement physically reasonable restrictions; and the network architecture to ensure polycon-

vexity. We demonstrate that this new class of models is a generalization of the classical neo Hooke,

Blatz Ko, Mooney Rivlin, Yeoh, and Demiray models and that the network weights have a clear

physical interpretation in the form of shear moduli, stiffness-like parameters, and exponential

coefﬁcients. When trained with classical benchmark data for rubber under uniaxial tension, biax-

ial extension, and pure shear, our network autonomously selects the best constitutive model and

learns its set of parameters. Our ﬁndings suggests that Constitutive Artiﬁcial Neural Networks

have the potential to induce a paradigm shift in constitutive modeling, from user-deﬁned model

selection to automated model discovery. Our source code, data, and examples are available at

https://github.com/LivingMatterLab/CANN.

Keywords. constitutive modeling; machine learning; Neural Networks; Constitutive Artiﬁcial

Neural Networks; theroodynamics; automated science

arXiv:2210.02202v2 [cs.LG] 21 Oct 2022

1 Motivation

“What can your Neural Network tell you about the underlying physics?” is the most common question

when we apply Neural Networks to study the behavior of materials and “Nothing.” is the honest

and disappointing answer.

This manuscript challenges the notion that Neural Networks can teach us nothing about the

physics of a material. It seeks to integrate more than a century of knowledge in continuum me-

chanics [3, 4, 22, 38, 40, 47, 50, 51] and modern machine learning [24, 29, 41] to create a new family

of Constitutive Artiﬁcial Neural Networks that inherently satisfy kinematical, thermodynamical,

and physical constraints, and constrain the space of admissible functions to train robustly, even

when data are space. While this general idea is by no means new and builds on several important

recent discoveries [2,27, 28, 32], the true novelty of our Constitutive Artiﬁcial Neural Networks is

that they autonomously discover a constitutive model, and, at the same time, learn a set of physically

meaningful parameters associated with it.

Interestingly, the ﬁrst Neural Network for constitutive modeling approximates the incremental

principal strains in concrete from known principal strains, stresses, and stress increments and is

more than three decades old [17]. In the early days, Neural Networks served merely as regres-

sion operators and were commonly viewed as a black box. The lack of transparency is probably

the main reason why these early approaches never really generated momentum in the constitu-

tive modeling community. More than 20 years later, data-driven constitutive modeling gained

new traction, in part powered by a new computing paradigm, which directly uses experimental

data and bypasses constitutive modeling altogether [26]. While data-driven elasticity builds on

a transparent and rigorous mathematical foundation [9], it can also become fairly complex, espe-

cially when expanding the theory to anisotropic [13] or history-dependent [14] materials. Rather

than following this path and eliminate the constitutive model entirely, here we attempt to build

our prior physical knowledge into the Neural Network and learn something about the constitu-

tive response [1].

Two successful but fundamentally different strategies have emerged to integrate physical knowl-

edge into network modeling, Physics-Informed Neural Networks that add physics equations as ad-

ditional terms to the loss function [24] and Constitutive Artiﬁcial Neural Networks that explicitly

modify the network input, output, and architecture to hardwire physical constraints into the net-

work design [28]. The former approach is more general and typically works well for incorporating

ordinary [29] or partial [41] differential equations, while the latter is speciﬁcally tailored towards

constitutive equations [30]. In fact, one such Neural Network, with strain invariants as input,

free energy functions as output, and a single hidden layer with logistic activation functions in be-

tween, has been proposed for rubber materials almost two decades ago [46] and recently regained

attention in the constitutive modeling community [55]. While these Constitutive Artiﬁcial Neural

Networks generally provide excellent ﬁts to experimental data [6,36,52], exactly how they should

integrate thermodynamic constraints remains a question of ongoing debate.

Thermodynamics-based Artiﬁcial Neural Networks a priori build the ﬁrst and second law of ther-

modynamics into the network architecture and select speciﬁc activation functions to ensure com-

pliance with thermodynamic constraints [32]. Recent studies suggest that this approach can suc-

cessfully reproduce the constitutive behavior of rubber-like materials [18]. Alternative approaches

use a regular Artiﬁcial Neural Network and ensure thermodynamic consistency a posteriori via a

pseudo-potential based correction in a post processing step [25]. To demonstrate the versatility of

these different approaches, several recent studies have successfully embedded Neural Networks

within a Finite Element Analysis, for example, to model plane rubber sheets [28] or entire tires [46],

the numerical homogenization of discrete lattice structures [33], the deployment of parachutes [2],

or the anisotropic response of skin in reconstructive surgery [49]. Regardless of all these success

stories, one limitation remains: the lack of an intuitive interpretation of the model and its param-

eters [27].

The general idea of this manuscript is to reverse-engineer a new family of Constitutive Artiﬁcial

Neural Networks that are, by design, a generalization of widely used and commonly accepted

constitutive models [6, 12, 36, 43, 53, 54] with well-deﬁned physical parameters [31, 48]. Towards

this goal, we review the underlying kinematics in Section 2 and discuss constitutive constraints

in Section 3. We then introduce classical Neural Networks in Section 4 and our new family of

Constitutive Artiﬁcial Neural Networks in Section 5. In Section 6, we brieﬂy review the three spe-

cial homogeneous deformation modes that we use to train our model in Section 7. We discuss our

results, limitations, and future directions in Section 8 and close with a brief conclusion in Section 9.

2 Kinematics

We begin by characterizing the motion of a body and introduce the deformation map ϕthat, at

any point in time t, maps material particles Xfrom the undeformed conﬁguration to particles, x=

ϕ(X,t), in the deformed conﬁguration [3]. To characterize relative deformations within the body,

we introduce the deformation gradient F, the gradient of the deformation map ϕwith respect to

the undeformed coordinates X, and its Jacobian J,

F=∇Xϕwith J=det(F)>0 . (1)

Multiplying Fwith its transpose Ft, either from the left or the right, introduces the right and left

Cauchy Green deformation tensors Cand b,

C=Ft·Fand b=F·Ft. (2)

In the undeformed state, all three tensors are identical to the unit tensor, F=I,C=I, and b=I,

and the Jacobian is one, J=1. A Jacobian smaller than one, 0 <J<1, denotes compression and

a Jacobian larger than one, 1 <J, denotes extension.

Isotropy. To characterize an isotropic material, we introduce the three principal invariants I1,I2,

I3, either in terms of the deformation gradient F,

I1=F:F∂FI1=2F

I2=1

2[I2

1−[Ft·F]:[Ft·F]] with ∂FI2=2[I1F−F·Ft·F]

I3=det (Ft·F) = J2∂FI3=2I3F−t,

(3)

or, equivalently, in terms of the right or left Cauchy Green deformation tensors Cor b,

I1=tr (C) = C:I∂CI1=II1=tr (b) = b:I∂bI1=I

I2=1

2[I2

1−C:C]∂CI2=I1I−Cor I2=1

2[I2

1−b:b]∂bI2=I1I−b

I3=det (C) = J2∂CI3=I3C−tI3=det (b) = J2∂bI3=I3b−t.

(4)

In the undeformed state, F=I, and the three invariants are equal to three and one, I1=3, I2=3,

and I3=1.

Near incompressibility. To characterize an isotropic,nearly incompressible material, we perform a

multiplicative decomposition of deformation gradient, F=J1/3 I·¯

F, into a volumetric part, J1/3 I,

and an isochoric part, ¯

F[15],

F=J−1/3 Fand ¯

J=det(¯

F) = 1 , (5)

and introduce the isochoric right and left Cauchy Green deformation tensors ¯

Cand ¯

C=¯

Ft·¯

F=J−2/3 Cand ¯

b=¯

F·¯

Ft=J−2/3 b. (6)

We can then introduce an alternative set of invariants for nearly incompressible materials, ¯

I1,¯

I2,J,

in terms of the deformation gradient ¯

I1=I1/J2/3 =F:F/J2/3 ∂F¯

I1=2/J2/3 F−2

3¯

I1F−t

I2=I2/J4/3 =1

2[¯

I1−[Ft·F]:[Ft·F]/J4/3]with ∂F¯

I2=2/J2/3 ¯

I1F−2/J4/3 F·Ft·F−4

3¯

I2F−t

J=det(F)∂FJ=JF−t,

(7)

or, equivalently, in terms of the right and left Cauchy Green deformation tenors Cor b,

I1=I1/J2/3 =C:I/J2/3 ¯

I1=I1/J2/3 =b:I/J2/3

I2=I2/J4/3 =1

2[¯

I1−C:C/J4/3]or ¯

I2=I2/J4/3 =1

2[¯

I1−b:b/J4/3]

J=det1/2(C)J=det1/2(b).

(8)

Perfect incompressibility. To characterize an isotropic, perfectly incompressible material, we recall

that the third invariant always remains identical to one, I3=J2=1. This implies that the princi-

pal and isochoric invariants are identical, I1=¯

I1and I2=¯

I2, and that the set of invariants reduces

to only these two.

Transverse isotropy. To characterize a transversely isotropic material with one pronounced direc-

tion with unit normal vector n, we introduce a fourth invariant [47],

I4=n·Ft·F·n=C:N=λ2

nwith ∂CI4=n⊗n=N. (9)

Here N=n⊗ndenotes the structural tensor associated with the pronounced direction n, with

a unit length of ||n|| =1 in the reference conﬁguration and a stretch of λn=||F·n|| in the

deformed conﬁguration. In the undeformed state, F=I, and the stretch and the fourth invariant

are one, λn=1 and I4=1.

3 Constitutive equations

In the most general form, constitutive equations in solid mechanics are tensor-valued tensor

functions that deﬁne the relation between a stress, for example the Piola or nominal stress,

P=limdA→0(df/dA), as the force d fper undeformed area dA, and a deformation measure,

for example the deformation gradient F[22, 50],

P=P(F). (10)

Conceptually, we could use any Neural Network as a function approximator to simply learn the

functional relation between Pand Fand many approaches in the literature actually do exactly

that [17, 32, 45]. However, the functions P(F)that we learn through this approach might be too

generic and violate well-known thermodynamical arguments and widely-accepted physical con-

straints [18]. Also, for limited amounts of data, the tensor-valued tensor function P(F)can be

difﬁcult to learn and there is a high risk of overﬁtting [27]. Our objective is therefore to design

a Constitutive Artiﬁcial Neural Network that a priori guarantees thermodynamic consistency of

the function P(F), and, at the same time, conveniently limits the space of admissible functions to

ensure robustness and prevent overﬁtting when available data are sparse.

Thermodynamic consistency. As a ﬁrst step towards this goal, we ensure thermodynamically

consistency and guarantee that the Piola stress Pinherently satisﬁes the second law of thermo-

dynamics, the entropy or Clausius-Duhem inequality [40], D=P:˙

F−˙

ψ≥0. It states that,

for any thermodynamic process, the total change in entropy, the dissipation D, should always

remain greater than or equal to zero, D ≥ 0. To a priori satisfy the dissipation inequality, we

introduce the Helmholtz free energy as a function of the deformation gradient, ψ=ψ(F)such

that ˙

ψ=∂ψ(F)/∂F:˙

F, and rewrite the dissipation inequality following the Coleman-Noll en-

tropy principle [50] as D= [ P−∂ψ/∂F]:˙

F≥0. For the hyperelastic case with D.

=0, for all

possible ˙

F, the entropy inequality reduces to P−∂ψ/∂F.

=0. The condition of thermodynam-

ically consistency implies that the Piola stress Pof a hyperelastic or Green-elastic material is a

thermodynamically conjugate function of the deformation gradient F[51],

P=∂ψ(F)

∂F. (11)

For our Neural Network, this implies that, rather than approximating the nine stress components

P(F)as nine generic functions of the nine components of the deformation gradient F, we train

the network to learn the free energy function ψ(F)and derive the stress Pin a post-processing

step to a priori satisfy the second law of thermodynamics. As such, satisfying thermodynamic

consistency according to equation (11) directly affects the output of the Neural Network.

Material objectivity and frame indifference. Second, we further constrain the choice of the free

energy function ψto satisfy material objectivity or frame indifference to ensure that the constitutive

laws do not depend on the external frame of reference [37]. Mathematically speaking, the consti-

tutive equations have to be invariant under rigid body motions, ψ(F) = ψ(Q·F), for all proper

orthogonal tensors Q∈SO(3). The condition of objectivity implies that the stress response func-

tions are independent of rotations and must be functions of the right Cauchy Green deformation

tensor C[50],

P=∂ψ(C)

∂F=∂ψ(C)

∂C:∂C

∂F=2F·∂ψ(C)

∂C. (12)

For our Neural Network, this implies that rather than using the nine independent components of

the deformation gradient Fas input, we constrain the input to the six independent components

of the symmetric right Cauchy Green deformation tensor, C=Ft·F. As such, satisfying material

objectivity according to equation (12) directly affects the input of the Neural Network.

Material symmetry and isotropy. Third, we further constrain the choice of the free energy func-

tion ψto include constraints of material symmetry, which implies that the material response re-

mains unchanged under transformations of the reference conﬁguration, ψ(F) = ψ(F·Q). Here

we consider the special case of isotropy for which the material response remains unchanged under

proper orthogonal transformations of the reference conﬁguration, ψ(Ft·F) = ψ(Qt·Ft·F·Q),

for all proper orthogonal tensors Q∈SO(3)[3]. The condition of isotropy implies that the stress

response functions, ψ(C) = ψ(b), must be functions of the left Cauchy Green deformation tensor,

b=F·Ft, and, together with the condition of objectivity, ψ(b) = ψ(Qt·b·Q), that the stress

response functions must be functions of the invariants of Cand b, for example ψ(I1,I2,I3)using

the set of invariants from equation (3). The Piola stress for hyperelastic isotropic materials then

becomes

P=∂ψ(I1,I2,I3)

∂F=∂ψ

∂I1

F+∂ψ

∂I2

F+∂ψ

∂I3

F=2∂ψ

∂I1

+I1

∂ψ

∂I2F−2∂ψ

∂I2

F·Ft·F+2I3

∂ψ

∂I3

F−t.

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AnewfamilyofConstitutiveArticialNeuralNetworkstowardsautomatedmodeldiscoveryKevinLinka&EllenKuhlDepartmentofMechanicalEngineeringStanfordUniversity,Stanford,California,UnitedStatesWededicatethismanuscripttoourContinuumMechanicsteacherswhoseinsightsandpassionforContinuumMechanicshavestimulatedtheide...

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