# A mathematical justification for nonlinear constitutive relations between
stress and linearized strain

K. R. Rajagopal and C. Rodriguez<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We present an asymptotic framework that rigorously generates nonlinear
constitutive relations between stress and linearized strain for elastic
bodies. Each of these relations arises as the leading order relationship
satisfied by a one-parameter family of nonlinear constitutive relations
between stress and nonlinear strain. The asymptotic parameter limits the
overall range of strains that satisfy the corresponding constitutive relation
in the one-parameter family while the stresses can remain large (relative to a
fixed stress scale). This differs from classical linearized elasticity where a
fixed constitutive relation is assumed, and the magnitude of the displacement
gradient serves as the asymptotic parameter. Also unlike classical approaches,
the constitutive relations in our framework are expressed as implicit
relationships between stress and strain rather than requiring stress
explicitly expressed as a function of strain, adding conceptual simplicity and
versatility. We demonstrate that our framework rigorously justifies nonlinear
constitutive relations between stress and linearized strain including those
with density-dependent Young’s moduli or derived from strain energies beyond
quadratic forms.

##  1\. Introduction

###  1.1. Nonlinear constitutive relations between stress and linearized
strain

The celebrated eponymous constitutive relation to describe the response of
elastic solids subject to “small strains” remains a cornerstone of solid
mechanics.111Referring to it as “Hooke’s law” is clearly unwarranted as it is
merely a constitutive relation that holds for a small subset of bodies in an
approximate sense while we expect a “law” to have more universal validity.
Unfortunately, most people that use the constitutive relation, including
experts in the field, are unaware that Hooke did not limit the use of the
relation to solids. Hooke did not have a clear grasp of the applicability of
the empirical constitutive relation he was proposing in his immortal treatise
“Lectures de potential restitutiva, or of Spring, Explaining the Power of
Springy Bodies”. He erroneously supposed that the constitutive relation was
equally applicable to gases and thereby confused his constitutive relation
with the one proposed by Boyle (see [12] for a discussion of the relevant
issues). Moyer [12] remarks:

> While it is a simple and straightforward matter to extract from the early
> pages of De potentia restitutiva the modern statement of ‘Hooke’s law’ for
> elastic solids, $F=-kx$, one must first suppress or ignore Hooke’s ‘trial’
> involving ‘a body of Air.’ That is, in modern terms, he inappropriately
> associated ‘Hooke’s law’-a direct proportionality between displacement and
> force-with ‘Boyle’s law’-an inverse proportionality between the volume of an
> ideal gas at constant temperature and the pressure to which it is subjected
> (or, for a gas in a cylinder with a uniform cross-sectional area, between
> the length of the column of gas and the force on it).

In fact, Moyer [12] points out that Hooke’s theoretical explanation of his
empirical relation $F=-kx$ is consistent with Boyle’s relation for gases but
is inconsistent with Hooke’s linear relation for solids. Indeed, assuming that
a spring has natural length of 1 unit, Hooke proposed that the force due to
the spring’s particles’ “endeavor of receding from each other” is inversely
proportional to its stretched length $d$ [12]. As a consequence of balance of
forces, Moyer [12] observes:

> $\ldots$realizing that the stretched length $d$ is equal to the original
> length 1 unit plus some increment distance $x$, we obtain a relation between
> the force on a spring due to an external macroscopic agent and the spring’s
> change in length:
>
> $\displaystyle F_{e}=k(1-[1/(1+x)]),$ (1.1)
>
> which is also equal to $k[x/(1+x)]\ldots$In other words, for slight changes
> in length, Hooke’s model does lead approximately to his empirical law.

In his authoritative article in the Handbuch der Physik, Bell [1] devotes a
significant amount of discussion to the work by illustrious contemporaries of
Hooke on nonlinear constitutive relations between stress and linearized
strain.222A great deal of care has to be exercised in analyzing the discussion
in Bell [1] as some of the works, while concerned with small strains, are
suffering inelastic response. As no unloading data is provided, it is possible
that during the reported experiments, some or many of the specimens might have
undergone inelastic response. According to Bell, many of them did not believe
that Hooke’s empirical constitutive relation was appropriate, even for elastic
solids undergoing small displacement gradients.333Of course, it depends on
what one means by small displacement gradients. Here, it means the use of
linearized strain wherein the nonlinear part of the strain is ignored and
merely the symmetric part of the displacement gradient is used, or put
differently, wherein the square of the norm of the displacement gradient is
negligible compared to the norm of the displacement gradient. For example, it
would be reasonable to say the displacement gradient is small if its Frobenius
norm is less than $10^{-2}$. See [24] for more on this point. Concerning the
data that James Bernoulli had sent him, Leibniz observed that it seems to be
fitted best by a hyperbola. Later, James Bernoulli himself proposed a
parabolic relation, namely

$\displaystyle t=kx^{m},$ (1.2)

where $t$ is the applied force and $x$ is the elongation. Bullfinger [2]
proposed the value $m=3/2$.

Wertheim [35] carried out extensive and systematic experiments on a miscellany
of biological materials. He carried out experiments on a variety of bones
(femur, fibula), tendons (big toe flexor, small plantar), sartorial muscles,
nerves, femoral arteries, veins, for both males and females. With the
exception of bones, all his experiments could be fitted to the nonlinear
constitutive relation

$\displaystyle\epsilon^{2}=A\sigma^{2}+B\sigma,$ (1.3)

where $\epsilon$ denotes the strain, $\sigma$ the stress, and $A$ and $B$ are
constants. This led Bell [1] to remark:

> The experiments of 280 years have demonstrated amply for every solid
> substance examined with sufficient care, that the strain resulting from
> applied stress is not a linear function thereof.

The above observation of Bell remains true today with regard to a large class
of solid bodies.

Numerous recent works study a variety of materials wherein one observes
nonlinear response even when the body is subject to sufficiently small
strains, regimes where the linearized constitutive relation due to Hooke seems
reasonable. It is now well established that the response of the trabecular
bone is nonlinear, even for small strains, and more importantly such bones
respond differently in tension and compression (see, e.g., [11] and [7]). It
is interesting to note that the title of [11] is _Nonlinear behavior of
trabecular bone at small strains_ , emphasizing that even in the small strain
range where the linearized constitutive relation due to Hooke is usually
thought to be valid, the response is nonlinear.444If the strain is
sufficiently small and one assumes that the stress as a function of strain is
sufficiently smooth that one can use a Taylor series expansion, there will be
a small interval containing the origin wherein the linear relationship will
hold. However, this might be a absurdly small range of strains. The point
being made here is that even in the range wherein one usually applies the
linearized approximation, many materials exhibit nonlinear response. There is
considerable experimental evidence for a large class of metallic alloys
wherein the response is nonlinear in the small strain regime. For the case of
Gum metal and titanium alloys, see [27, 28, 29, 5, 8, 32, 36, 37]. Once again,
we quote few of remarks from the papers which emphasize the fact that the
response is nonlinear even though the strain is small. Talling et al. [32]
remark: “As the elastic regions of the microscopic stress–strain curves are
nonlinear in Figure 3, the moduli were calculated from the lowest stress
portion of the curve, before deviation from linearity.”. The first sentence of
the abstract of Zhang et al. [37] reads “We report the fatigue endurance of a
multifunctional b type titanium alloy exhibiting nonlinear elastic deformation
behavior”. Finally, Withey et al. [36] demarcate the nonlinear response region
which is in the small strain regime. Interestingly, even traditional materials
such as concrete and rocks exhibit nonlinear behavior in the small strain
regime. Grasley et al. [4] show that concrete subject to simple uniaxial
compression test starts to respond nonlinearly even at a very low strain of
$2\times 10^{-3}$, where one would unhesitatingly apply the classical
linearized constitutive relation.

Thus, it is necessary to put into place a framework wherein the linearized
strain is a nonlinear function of the stress, or wherein the linearized strain
and the stress are implicitly related as an approximation. Moreover, such an
approximation should stem rigorously from a proper constitutive relation that
meets the basic requirements such as frame-indifference and appropriate
material symmetry.

###  1.2. Implicit constitutive theory

We now discuss modern advances in constitutive theory that inspired this
study’s proposed asymptotic foundation for rigorously generating nonlinear
constitutive relations between stress and linearized strain.

Cauchy’s classical theory of elasticity is founded upon a functional relation
between the Cauchy stress tensor $\boldsymbol{T}$ and left Cauchy-Green strain
tensor $\boldsymbol{B}=\boldsymbol{F}\boldsymbol{F}^{T}$ for a given
deformation,555Here,
$\boldsymbol{F}={\nabla}_{\boldsymbol{X}}\boldsymbol{\chi}$ is the gradient of
a smooth, invertible deformation
$\boldsymbol{\chi}:\mathcal{B}\rightarrow\boldsymbol{\chi}(\mathcal{B})\subseteq\mathbb{E}^{3}$
of a body with reference configuration $\mathcal{B}$ in three-dimensional
Euclidean space. See Section 2 for more on the definitions of the tensors
appearing in this discussion. Throughout this work, we use standard tensor
notation. In particular, if $\boldsymbol{A}$ is a tensor, we denote the
Frobenius norm
$|\boldsymbol{A}|=[\mathrm{tr}\,(\boldsymbol{A}\boldsymbol{A}^{T})]^{1/2}$.

$\displaystyle\boldsymbol{T}=\boldsymbol{g}(\boldsymbol{B}).$ (1.4)

Here and throughout our study, we omit the dependence of various quantities on
current points in the body, that is, we assume the body to be homogeneous.
Alternatively, a functional relationship between the symmetric Piola-Kirchhoff
stress tensor $\bar{\boldsymbol{S}}$ and the Green-St. Venant strain tensor
$\boldsymbol{E}=\frac{1}{2}(\boldsymbol{F}^{T}\boldsymbol{F}-\boldsymbol{I})$
is given,

$\displaystyle\bar{\boldsymbol{S}}=\boldsymbol{f}(\boldsymbol{E}).$ (1.5)

Classic linearized elasticity is obtained from (1.5) or (1.4) in the
asymptotic limit of infinitesimal strains. More precisely, if
$|\boldsymbol{F}-\boldsymbol{I}|=\delta_{0}\ll 1$,
$\boldsymbol{f}(\boldsymbol{0})=\boldsymbol{0}$, and $\boldsymbol{f}$ is twice
continuously differentiable, then (1.5) implies that

$\displaystyle\boldsymbol{\sigma}=\boldsymbol{\mathsf{C}}[\boldsymbol{\epsilon}]+\boldsymbol{O}(\delta_{0}^{2}),$
(1.6)

where
$\boldsymbol{\mathsf{C}}=D_{\boldsymbol{E}}\boldsymbol{f}(\boldsymbol{0})$ is
the elasticity tensor,
$\boldsymbol{\epsilon}=\frac{1}{2}(\boldsymbol{F}+\boldsymbol{F}^{T})-\boldsymbol{I}$
is the linearized strain, and $\boldsymbol{\sigma}$ has replaced
$\bar{\boldsymbol{S}}$ as the stress variable.666We use standard big-oh and
little-oh notation. In particular, a tensor $\boldsymbol{A}$ and scalar $B$
satisfy $\boldsymbol{A}=\boldsymbol{O}(B)$ if there exists a constant $C>0$
such that $|\boldsymbol{A}|\leq CB$. If $C$ depends on parameters $a$ and $b$,
we say that the big-oh term $\boldsymbol{O}(B)$ depends on $a$ and $b$.

In a series of papers [19, 20, 21, 22], the first author put into place an
implicit framework for the response of elastic bodies,

$\displaystyle\boldsymbol{g}(\boldsymbol{B},\boldsymbol{T})=\boldsymbol{0},$
(1.7)

with a simpler subclass taking the form

$\displaystyle\boldsymbol{B}=\boldsymbol{g}(\rho/\rho_{R},\boldsymbol{T}).$
(1.8)

Inspired by [19, 20, 21, 22], Mai and Walton [9, 10] studied constitutive
relations of the form

$\displaystyle\boldsymbol{E}=\boldsymbol{f}(\bar{\boldsymbol{S}}).$ (1.9)

They proved in [9] that for a popular class of isotropic forms of
$\boldsymbol{f}$ appearing in the literature, strong ellipticity for (1.9)
holds as long as $|\bar{\boldsymbol{S}}|$ is sufficiently small and fails for
extreme compression. Results of a similar spirit were obtained in [10] for a
form of monotonicity that is strictly weaker than strong ellipticity (and thus
applies when $\boldsymbol{f}$ is not Frechét differentiable). It was shown in
[18] that certain relations of the form (1.8) arise when describing the
response of nonlinear elastic solids specified in terms of Gibbs potentials.
For simple choices of the Gibbs potential (see Section 5 of [18]), (1.8)
becomes

$\displaystyle\boldsymbol{H}=\beta_{0}(\rho/\rho_{R})(\mathrm{tr}\,\boldsymbol{T})\boldsymbol{I}+\beta_{1}(\rho/\rho_{R})\boldsymbol{T},$
(1.10)

where $\boldsymbol{H}=\frac{1}{2}\log\boldsymbol{B}$ is the Hencky strain.
Based on the assumption that the Frobenius norm of the displacement gradient
is small, the works [19, 20, 21, 22] formally argued that the implicit
relation (1.7) implies
$\boldsymbol{g}(\boldsymbol{I}+2\boldsymbol{\epsilon},\boldsymbol{T})=\boldsymbol{0}$,
up to negligible errors of size
$|\boldsymbol{F}-\boldsymbol{I}|^{2}$.777Alternatively, the arguments in [19,
20, 21, 22] suggest that the implicit relation
$\boldsymbol{f}(\boldsymbol{E},\bar{\boldsymbol{S}})=\boldsymbol{0}$ implies
$\boldsymbol{f}(\boldsymbol{\epsilon},\bar{\boldsymbol{S}})=\boldsymbol{0}$,
up to negligible errors of size $|\boldsymbol{F}-\boldsymbol{I}|^{2}$.

However, one may show using the implicit function theorem that if
$\boldsymbol{g}$ is continuously differentiable,
$\boldsymbol{g}(\boldsymbol{I},\boldsymbol{0})=\boldsymbol{0}$, and
$D_{\boldsymbol{T}}\boldsymbol{g}(\boldsymbol{I},\boldsymbol{0})$ is
invertible on the linear space of symmetric tensors, then there exists a
function $\boldsymbol{h}$ such that (1.7) is satisfied if and only if
$\boldsymbol{T}=\boldsymbol{h}(\boldsymbol{B})$, provided
$|\boldsymbol{F}-\boldsymbol{I}|$ is sufficiently small. A similar statement
holds for implicit relations of the form

$\displaystyle\boldsymbol{f}(\boldsymbol{E},\bar{\boldsymbol{S}})=\boldsymbol{0}.$
(1.11)

Thus, under the previous assumptions, fixed constitutive relations of the form
(1.7) or (1.11) will always lead to (1.6) as the leading order relation
satisfied in the asymptotic limit,
$\delta_{0}=|\boldsymbol{F}-\boldsymbol{I}|\rightarrow 0$. Instead, our
approach is inspired by the theory discussed in [20, 23] wherein strains are
_limited by the constitutive relation_ to a range of size $\delta\ll 1$ (while
the stresses can remain large relative to a fixed stress scale). Rather than
$\delta_{0}$, it is the _limiting strain_ determined by the constitutive
relation, $\delta$, that serves as the asymptotic parameter in our framework.
The central thesis demonstrated in this work, is that “linearization” with
respect to $\delta$ rigorously leads to nonlinear relations between stress and
linearized strain $\boldsymbol{\epsilon}$ while, as discussed above,
“linearization” with respect to $\delta_{0}=|\boldsymbol{F}-\boldsymbol{I}|$
only leads to linear relations between stress and the linearized
strain.888Here, our use of “linearization” with respect to a small parameter
$a$ means to neglect terms of order $a^{2}$ appearing in either the strain
variable or constitutive relation.

We illustrate the above discussion and motivate our framework via a simple
$1d$ example. Consider the relations

$\displaystyle E=\delta
a(1+|a\bar{S}|^{p})^{-1/p}\bar{S},\quad\bar{S}\in\mathbb{R},$ (1.12)
$\displaystyle\epsilon:=-1+(1+2E)^{1/2},\quad\delta_{0}:=|\epsilon|,$ (1.13)

where $\epsilon$ is the linearized strain variable, $\bar{S}$ is the stress
variable, and

$E=\epsilon+\frac{1}{2}\epsilon^{2}$

is the nonlinear strain. Here, we assume that $\delta,\delta_{0}\ll 1$. By
(1.12), for all $\bar{S}\in\mathbb{R}$, $|E|\leq\delta$. By (1.13),
$\epsilon=E+O(\delta^{2})$, and (1.12) yields

$\displaystyle\epsilon+O(\delta^{2})=\delta
a(1+|a\bar{S}|^{p})^{-1/p}\bar{S}.$ (1.14)

Thus, linearization with respect to $\delta$ yields the nonlinear relation
between stress and linearized strain,

$\displaystyle\epsilon=\delta a(1+|a\bar{S}|^{p})^{-1/p}\bar{S}.$ (1.15)

On the other hand, we may invert (1.12), leading to

$\displaystyle a\bar{S}=(1-|E/\delta|^{p})^{-1/p}E/\delta.$ (1.16)

Since $\epsilon=E+O(\delta_{0}^{2})$, we conclude that
$a\bar{S}=\epsilon/\delta+O(\delta_{0}^{2}/\delta^{2}).$ Thus, linearization
with respect to $\delta_{0}$ yields the linear relation

$\displaystyle a\bar{S}=\epsilon/\delta.$ (1.17)

This interplay between inversion and linearization of constitutive relations
was first explored in [25]. We point out that it was quite simple to
rigorously arrive at (1.15) directly from (1.12) when linearizing with respect
to $\delta$. However, it is much less clear how one arrives at
$a\bar{S}=(1-|\epsilon/\delta|^{p})^{-1/p}\epsilon/\delta$ directly from
(1.16) when linearizing with respect to $\delta$.

###  1.3. Main results and outline

In this work, we consider implicit constitutive relations,

$\displaystyle\boldsymbol{E}=\boldsymbol{f}_{\delta}(\boldsymbol{E},\bar{\boldsymbol{S}}),$
(1.18)

with $\boldsymbol{f}_{\delta}$ bounded by $\delta\ll 1$ on its domain. In
Section 2 we briefly review the kinematics and balance laws necessary for this
study to be self-contained. Rather than consider each constitutive relation
(1.18) individually, it is mathematically desirable from the standpoint of
asymptotic analysis to view each $\boldsymbol{f}_{\delta}$ as a member of one-
parameter family of _strain-limiting functions with limiting small strain_
introduced in Section 3. See Definition 3.1.

We discuss several examples in Section 3.2, and we show in Section 3.3 that
within our framework, a nonlinear constitutive relation between stress and
linearized strain is the leading-order-in-$\delta$ relationship satisfied by
the family of strain-limiting constitutive relations represented by (1.18);
informally,

$\displaystyle\boldsymbol{\epsilon}=\boldsymbol{f}_{\delta}(\boldsymbol{\epsilon},\boldsymbol{\sigma})+\boldsymbol{O}(\delta^{2}).$
(1.19)

See Proposition 3.3 and Corollary 3.4 for the precise statements. Our results
are clearly distinct but in similar spirit to the situation in classical
linearized elasticity (1.6). As applications of Proposition 3.3 and Corollary
3.4, we first obtain nonlinear constitutive relations between stress and
linearized strain wherein the stress is derived from a strain energy depending
on the linearized strain; see (3.57). In contrast to classical linearized
elasticity, however, our framework allows for strain energies beyond quadratic
forms of the linearized strain. In addition, we also obtain certain popular
nonlinear constitutive relations between stress and linearized strain having
density dependent Young’s moduli; see (3.69), (3.72), (3.74), and (3.75). We
emphasize that the nonlinear constitutive relations between stress and
linearized strain obtained correspond to the leading order relations satisfied
by appropriate families of strain-limiting constitutive relations.

In the concluding Section 4, we speculate on the interesting open question
concerning the solvability of the fully nonlinear problem if one can prove the
solvability of the associated “linearized problem”, the body force being kept
the same.

### Acknowledgments

The authors gratefully acknowledge support provided by NSF Grant DMS-2307562.

##  2\. Preliminaries

In this brief section, we give necessary definitions for the ensuing
discussion and analysis presented in this work.

Let $\mathcal{B}\subseteq\mathbb{E}^{3}$ be a smooth domain in three-
dimensional Euclidean space, the reference configuration of a body with
reference mass density $\rho_{R}$. Let
$\boldsymbol{\chi}:\mathcal{B}\rightarrow\boldsymbol{\chi}(\mathcal{B})\subseteq\mathbb{E}^{3}$
be a smooth invertible deformation of $\mathcal{B}$. For a given reference
element $\boldsymbol{X}\in\mathcal{B}$, we denote by
$\boldsymbol{x}=\boldsymbol{\chi}(\boldsymbol{X})$ its position in the current
configuration $\boldsymbol{\chi}(\mathcal{B})$. The deformation gradient of
$\boldsymbol{\chi}$ is defined by
$\boldsymbol{F}(\boldsymbol{X})=\frac{\partial\boldsymbol{\chi}}{\partial\boldsymbol{X}}(\boldsymbol{X})$.

The left and right Cauchy-Green stretch tensor fields are given respectively
by

$\displaystyle\boldsymbol{B}=\boldsymbol{F}\boldsymbol{F}^{T},\quad\boldsymbol{C}=\boldsymbol{F}^{T}\boldsymbol{F},$
(2.1)

and the _Green-St. Venant strain_ is the second order tensor

$\displaystyle\boldsymbol{E}=\frac{1}{2}(\boldsymbol{C}-\boldsymbol{I}).$
(2.2)

The displacement field is defined by

$\displaystyle\boldsymbol{u}=\boldsymbol{x}-\boldsymbol{X},$ (2.3)

and the displacement gradient is given by

$\displaystyle\nabla_{\boldsymbol{X}}\boldsymbol{u}=\boldsymbol{F}-\boldsymbol{I}.$
(2.4)

Then

$\displaystyle\boldsymbol{B}=I+\bigl{[}\nabla_{\boldsymbol{X}}\boldsymbol{u}+(\nabla_{\boldsymbol{X}}\boldsymbol{u})^{T}\bigr{]}+\nabla_{\boldsymbol{X}}\boldsymbol{u}(\nabla_{\boldsymbol{X}}\boldsymbol{u})^{T},$
(2.5)
$\displaystyle\boldsymbol{C}=I+\bigl{[}\nabla_{\boldsymbol{X}}\boldsymbol{u}+(\nabla_{\boldsymbol{X}}\boldsymbol{u})^{T}\bigr{]}+(\nabla_{\boldsymbol{X}}\boldsymbol{u})^{T}\nabla_{X}\boldsymbol{u}.$
(2.6)

Assuming that

$\displaystyle|\nabla_{\boldsymbol{X}}\boldsymbol{u}|=\delta_{0}\ll 1,$ (2.7)

we then have that

$\displaystyle\boldsymbol{B}=\boldsymbol{I}+2\boldsymbol{\epsilon}+\boldsymbol{O}(\delta_{0}^{2}),\quad\boldsymbol{E}=\boldsymbol{\epsilon}+\boldsymbol{O}(\delta_{0}^{2}),$
(2.8)

where $\boldsymbol{\epsilon}$ is the _linearized strain_

$\displaystyle\boldsymbol{\epsilon}=\frac{1}{2}\bigl{[}\nabla_{\boldsymbol{X}}\boldsymbol{u}+(\nabla_{\boldsymbol{X}}\boldsymbol{u})^{T}\bigr{]}=\frac{1}{2}(\boldsymbol{F}+\boldsymbol{F}^{T})-\boldsymbol{I}.$
(2.9)

In the purely mechanical setting and in the absence of external forces and
body couples, the classical equations expressing conservation of mass, balance
of linear momentum, and balance of angular momentum on the current
configuration are given respectively by:

$\displaystyle\dot{\rho}+\rho\operatorname{div}\boldsymbol{v}=\boldsymbol{0},$
(2.10)
$\displaystyle\rho\dot{\boldsymbol{v}}=\operatorname{div}\boldsymbol{T},\quad\boldsymbol{T}^{T}=\boldsymbol{T}.$
(2.11)

Here, $\dot{}=\partial_{t}+\sum_{j=1}^{3}v_{j}\partial_{x_{j}}$ is the
material time derivative, $\rho$ is the current mass density, $\boldsymbol{v}$
is the velocity field, and $\boldsymbol{T}$ is the (symmetric) Cauchy stress
tensor. On the reference configuration, the balance laws can be expressed by:

$\displaystyle\rho=(\det\boldsymbol{F})^{-1}\rho_{R},$ (2.12)
$\displaystyle\rho_{R}\partial_{t}^{2}\boldsymbol{\chi}=\mbox{Div}\,(\boldsymbol{S}),\quad\boldsymbol{S}\boldsymbol{F}^{T}=\boldsymbol{F}\boldsymbol{S}^{T},$
(2.13)

where $\boldsymbol{S}$ is the first Piola-Kirchhoff stress tensor. We denote
the second symmetric Piola-Kirchhoff stress tensor by
$\bar{\boldsymbol{S}}=\boldsymbol{F}^{-1}\boldsymbol{S}$. We note that if
$|\nabla_{\boldsymbol{X}}\boldsymbol{u}|=\delta_{0}\ll 1$, then

$\displaystyle\rho/\rho_{R}=(\det\boldsymbol{B})^{-1/2}=(\det\boldsymbol{C})^{-1/2}=[\det(\boldsymbol{I}+2\boldsymbol{E})]^{-1/2}=1-\mathrm{tr}\,\boldsymbol{\epsilon}+O(\delta_{0}^{2}).$
(2.14)

##  3\. An asymptotic framework for nonlinear relations between stress and
linearized strain

In this section, we introduce a novel asymptotic framework for constitutive
relations. Within this framework, a nonlinear constitutive relation between
stress and linearized strain is the leading-order-in-$\delta$ relationship
satisfied by a one-parameter family of strain-limiting constitutive relations
$\boldsymbol{E}=\boldsymbol{f}_{\delta}(\boldsymbol{E},\bar{\boldsymbol{S}})$.
These one-parameter families are given in terms of families of strain-limiting
functions with limiting small strains, $\boldsymbol{f}_{\delta}$, defined in
the following subsection.

###  3.1. Families of strain-limiting functions with limiting small strains

######  Definition 3.1.

For small $\tilde{\delta}>0$, and each $\delta\in(0,\tilde{\delta})$, let
$U_{\delta}\subseteq B(\boldsymbol{0},1/2)$ and $V$ be open subsets of
$\mathrm{Sym}$.999Here, $\mathrm{Sym}$ denotes the set of symmetric tensors on
$\mathbb{R}^{3}$, and
$B(\boldsymbol{0},r)=\bigl{\\{}\boldsymbol{E}\in\mathrm{Sym}\mid|\boldsymbol{E}|<r\bigr{\\}}$.
We note that if $\boldsymbol{E}\in B(\boldsymbol{0},1/2)$, then it
straightforward to see that $\boldsymbol{C}=\boldsymbol{I}+2\boldsymbol{E}$ is
positive definite. We say that a collection of bounded Lipschitz continuous
functions $\boldsymbol{f}_{\delta}:U_{\delta}\times V\rightarrow\mathrm{Sym}$
indexed by $\delta\in(0,\tilde{\delta})$ is a family of strain-limiting
functions with limiting small strains if there exist $C_{0},C_{1}>0$ and
$D_{0}>0$, independent of $\delta$, such that for all
$\delta\in(0,\tilde{\delta})$,

$\displaystyle\forall\boldsymbol{E},\bar{\boldsymbol{S}},\quad|\boldsymbol{f}_{\delta}(\boldsymbol{E},\bar{\boldsymbol{S}})|\leq
C_{0}\delta,\qquad\forall\boldsymbol{E}_{1}\not=\boldsymbol{E}_{2},\bar{\boldsymbol{S}},\quad\frac{|\boldsymbol{f}_{\delta}(\boldsymbol{E}_{2},\bar{\boldsymbol{S}})-\boldsymbol{f}_{\delta}(\boldsymbol{E}_{1},\bar{\boldsymbol{S}})|}{|\boldsymbol{E}_{2}-\boldsymbol{E}_{1}|}\leq
C_{1},$ (3.1)
$\displaystyle\forall\boldsymbol{E},\bar{\boldsymbol{S}}_{1}\not=\bar{\boldsymbol{S}}_{2},\quad\frac{|\boldsymbol{f}_{\delta}(\boldsymbol{E},\bar{\boldsymbol{S}}_{2})-\boldsymbol{f}_{\delta}(\boldsymbol{E},\bar{\boldsymbol{S}}_{1})|}{|\boldsymbol{S}_{2}-\boldsymbol{S}_{1}|}\leq
D_{0}\delta.$ (3.2)

We note that if $\boldsymbol{E}$ represents a strain variable,
$\bar{\boldsymbol{S}}$ represents a stress variable, and
$\boldsymbol{f}_{\delta}$ represents a dimensionless response function, then
$\delta,C_{0},$ and $C_{1}$ are dimensionless and $D_{0}$ has physical units
of $(\mbox{length})^{2}(\mbox{force})^{-1}$. Each individual function
$\boldsymbol{f}_{\delta}$ is referred to as a strain-limiting function with
limiting small strains on $U_{\delta}\times V$.

###  3.2. Examples of families of strain-limiting functions

We now discuss a series of example families of strain-limiting functions with
limiting small strains. Let $\boldsymbol{f}:U\times V\rightarrow\mathrm{Sym}$
be a bounded Lipschitz continuous function satisfying,

$\displaystyle\forall\boldsymbol{E},\bar{\boldsymbol{S}},\quad|\boldsymbol{f}(\boldsymbol{E},\bar{\boldsymbol{S}})|\leq\delta_{1},\qquad\forall\boldsymbol{E}_{1}\not=\boldsymbol{E}_{2},\bar{\boldsymbol{S}},\quad\frac{|\boldsymbol{f}(\boldsymbol{E}_{2},\bar{\boldsymbol{S}})-\boldsymbol{f}(\boldsymbol{E}_{1},\bar{\boldsymbol{S}})|}{|\boldsymbol{E}_{2}-\boldsymbol{E}_{1}|}\leq\tilde{C}_{1},$
(3.3)
$\displaystyle\forall\boldsymbol{E},\bar{\boldsymbol{S}}_{1}\not=\bar{\boldsymbol{S}}_{2},\quad\frac{|\boldsymbol{f}(\boldsymbol{E},\bar{\boldsymbol{S}}_{2})-\boldsymbol{f}(\boldsymbol{E},\bar{\boldsymbol{S}}_{1})|}{|\boldsymbol{S}_{2}-\boldsymbol{S}_{1}|}\leq\tilde{D}_{0}\delta_{1}.$
(3.4)

Then

$\displaystyle\boldsymbol{f}_{\delta}(\boldsymbol{E},\bar{\boldsymbol{S}}):=\frac{\delta}{\delta_{1}}\boldsymbol{f}\Bigl{(}\frac{\delta_{1}}{\delta}\boldsymbol{E},\bar{\boldsymbol{S}}\Bigr{)},$
(3.5)

for $\boldsymbol{E}\in
U_{\delta}:=\frac{\delta}{\delta_{1}}U=\bigl{\\{}\frac{\delta}{\delta_{1}}\tilde{\boldsymbol{E}}\mid\tilde{\boldsymbol{E}}\in
U\bigr{\\}}$ and $\bar{\boldsymbol{S}}\in V$, defines a family of strain-
limiting functions with limiting small strains satisfying
$\boldsymbol{f}_{\delta_{1}}=\boldsymbol{f}$. Moreover, the constants
appearing in (3.1) and (3.2) satisfy

$\displaystyle C_{0}=1,\quad C_{1}=\tilde{C}_{1},\quad D_{0}=\tilde{D}_{0}.$
(3.6)

Roughly speaking, this example shows that an arbitrary bounded Lipschitz
continuous function generates a family of strain-limiting functions with
limiting small strains in a natural way.

We now show that if $\delta_{n}\rightarrow 0$, then, up to passing to a
subsequence, an arbitrary family of strain-limiting functions with limiting
small strains takes the form (3.5) to leading order in $\delta_{n}$.

######  Proposition 3.2.

Let $U,V\subseteq\mathrm{Sym}$ be bounded open sets. For each $\delta>0$, let

$\displaystyle U_{\delta}=\delta
U=\\{\delta\tilde{\boldsymbol{E}}\mid\tilde{\boldsymbol{E}}\in U\\}.$ (3.7)

Suppose that $\\{\delta_{n}\\}_{n}$ is a sequence of positive numbers
converging to $0$, and $\boldsymbol{f}_{\delta}:U_{\delta}\times
V\rightarrow\mathrm{Sym}$ is a family of strain-limiting functions with
limiting small strains. Then there exists a subsequence
$\\{\delta_{n_{m}}\\}_{m}$ and a bounded Lipschitz continuous function
$\boldsymbol{g}:U\times V\rightarrow\mathrm{Sym}$ such that

$\displaystyle\sup_{(\boldsymbol{E},\bar{\boldsymbol{S}})\in
U_{\delta_{n_{m}}}\times
V}|\boldsymbol{f}_{\delta_{n_{m}}}(\boldsymbol{E},\bar{\boldsymbol{S}})-\delta^{-1}_{n_{m}}\boldsymbol{g}(\delta^{-1}_{n_{m}}\boldsymbol{E},\bar{\boldsymbol{S}})|=o({\delta_{n_{m}}}),$
(3.8)

as $m\rightarrow\infty$.

###### Proof.

Define $\boldsymbol{h}_{n}:U\times V\rightarrow\mathrm{Sym}$ by

$\displaystyle\boldsymbol{h}_{n}(\tilde{\boldsymbol{E}},\bar{\boldsymbol{S}})=\delta_{n}^{-1}\boldsymbol{f}_{\delta_{n}}(\delta_{n}\tilde{\boldsymbol{E}},\bar{\boldsymbol{S}}),\quad(\tilde{\boldsymbol{E}},\bar{\boldsymbol{S}})\in
U\times V.$ (3.9)

Then by (3.1) and (3.2), for all $n$,

$\displaystyle\forall\tilde{\boldsymbol{E}},\bar{\boldsymbol{S}},\quad|\boldsymbol{h}_{n}(\tilde{\boldsymbol{E}},\bar{\boldsymbol{S}})|\leq
C_{0},\qquad\forall\tilde{\boldsymbol{E}}_{1}\not=\tilde{\boldsymbol{E}}_{2},\bar{\boldsymbol{S}},\quad\frac{|\boldsymbol{h}_{n}(\tilde{\boldsymbol{E}}_{2},\bar{\boldsymbol{S}})-\boldsymbol{h}_{n}(\tilde{\boldsymbol{E}}_{1},\bar{\boldsymbol{S}})|}{|\tilde{\boldsymbol{E}}_{2}-\tilde{\boldsymbol{E}}_{1}|}\leq
C_{1},$ (3.10)
$\displaystyle\forall\tilde{\boldsymbol{E}},\bar{\boldsymbol{S}}_{1}\not=\bar{\boldsymbol{S}}_{2},\quad\frac{|\boldsymbol{h}_{n}(\tilde{\boldsymbol{E}},\bar{\boldsymbol{S}}_{2})-\boldsymbol{h}_{n}(\tilde{\boldsymbol{E}},\bar{\boldsymbol{S}}_{1})|}{|\boldsymbol{S}_{2}-\boldsymbol{S}_{1}|}\leq
D_{0}.$ (3.11)

By the Arzela-Ascoli theorem, there exists a subsequence
$\\{\boldsymbol{h}_{n_{m}}\\}_{m}$ and $\boldsymbol{g}:U\times
V\rightarrow\mathrm{Sym}$ such that

$\displaystyle\sup_{(\tilde{\boldsymbol{E}},\bar{\boldsymbol{S}})\in U\times
V}|\boldsymbol{h}_{n_{m}}(\tilde{\boldsymbol{E}},\bar{\boldsymbol{S}})-\boldsymbol{g}(\tilde{\boldsymbol{E}},\bar{\boldsymbol{S}})|=o(1),$
(3.12)

as $m\rightarrow\infty$, concluding the proof. ∎

A simple subclass of the example (3.5) is generated by a Lipschitz continuous
function $\boldsymbol{f}_{1}:V\rightarrow\mathrm{Sym}$ on an open set
$V\subseteq\mathrm{Sym}$ satisfying

$\displaystyle\forall\bar{\boldsymbol{S}},\quad|\boldsymbol{f}_{1}(\bar{\boldsymbol{S}})|\leq
1.$ (3.13)

Then

$\displaystyle\boldsymbol{f}_{\delta}(\bar{\boldsymbol{S}})=\delta\boldsymbol{f}_{1}(\bar{\boldsymbol{S}}),$
(3.14)

is a family of strain-limiting functions on $V$ with limiting small strains.
If there exists a scalar function $W^{*}:V\rightarrow\mathbb{R}$ such that
$\boldsymbol{f}_{1}=\partial_{\bar{\boldsymbol{S}}}W^{*}$, then the relation

$\displaystyle\boldsymbol{E}=\delta\boldsymbol{f}_{1}(\bar{\boldsymbol{S}})=\partial_{\bar{\boldsymbol{S}}}[\delta
W^{*}(\bar{\boldsymbol{S}})],\quad\bar{\boldsymbol{S}}\in V,$ (3.15)

models an elastic solid body with complementary energy $\delta W^{*}$. If, in
addition, $V$ is convex and $W^{*}$ is a twice continuously differentiable,
convex function on $V$, then the relation (3.15) can be inverted yielding

$\displaystyle\bar{\boldsymbol{S}}=\partial_{\boldsymbol{E}}[\delta
W\bigl{(}\delta^{-1}\boldsymbol{E}\bigr{)}],\quad\boldsymbol{E}\in\mbox{range}(\delta\boldsymbol{f}_{1}).$
(3.16)

In (3.16), $W$ is the Legendre transform of $W^{*}$, and (3.16) is the
classical constitutive relation for a Green elastic solid with strain energy
$\delta W(\delta^{-1}\cdot)$ (see [33]).

An explicit example of such a family of strain-limiting functions with
limiting small strains, inspired by [21, 9, 10], is generated by

$\displaystyle\boldsymbol{f}_{1}(\bar{\boldsymbol{S}})=a(1+a^{p}|\bar{\boldsymbol{S}}|^{p})^{-1/p}\bar{\boldsymbol{S}},\quad\bar{\boldsymbol{S}}\in\mathrm{Sym},$
(3.17)

where $a>0$ and $p\geq 1$. Then for all $\boldsymbol{E},\bar{\boldsymbol{S}}$,

$\displaystyle|\boldsymbol{f}_{1}(\boldsymbol{E},\bar{\boldsymbol{S}})|\leq
1.$ (3.18)

Using the product and chain rules, we compute the Frechét derivative,

$\displaystyle
D_{\bar{\boldsymbol{S}}}\boldsymbol{f}_{1}(\bar{\boldsymbol{S}})=a(1+b^{p}|\bar{\boldsymbol{S}}|^{p})^{-1/p}\boldsymbol{\mathsf{I}}-a(1+b^{p}|\bar{\boldsymbol{S}}|^{p})^{-1/p-1}b^{p}|\bar{\boldsymbol{S}}|^{p-2}\bar{\boldsymbol{S}}\otimes\bar{\boldsymbol{S}},$
(3.19)

where $\boldsymbol{\mathsf{I}}:\mathrm{Sym}\rightarrow\mathrm{Sym}$ is the
identity map. By the triangle inequality, the operator norm of
$D_{\bar{\boldsymbol{S}}}\boldsymbol{f}_{1}$ satisfies

$\displaystyle\|D_{\bar{\boldsymbol{S}}}\boldsymbol{f}_{1}(\bar{\boldsymbol{S}})\|\leq
a(1+b^{p}|\bar{\boldsymbol{S}}|^{p})^{-1/p}\Bigl{(}1+b^{p}(1+b^{p}|\bar{\boldsymbol{S}}|^{p})^{-1}|\bar{\boldsymbol{S}}|^{p}\Bigr{)}\leq
2a.$ (3.20)

The fundamental theorem of calculus and (3.20) then imply that

$\displaystyle|\boldsymbol{f}_{1}(\bar{\boldsymbol{S}}_{2})-\boldsymbol{f}_{1}(\bar{\boldsymbol{S}}_{1})|$
$\displaystyle=\Bigl{|}\int_{0}^{1}D_{\bar{\boldsymbol{S}}}\boldsymbol{f}_{\delta}(s\bar{\boldsymbol{S}}_{2}+(1-s)\bar{\boldsymbol{S}}_{1})[\bar{\boldsymbol{S}}_{2}-\bar{\boldsymbol{S}}_{1}]ds\Bigr{|}$
(3.21) $\displaystyle\leq 2\delta
a|\bar{\boldsymbol{S}}_{2}-\bar{\boldsymbol{S}}_{1}|.$ (3.22)

By (3.18) and (3.22), we conclude that
$\boldsymbol{f}_{\delta}=\delta\boldsymbol{f}_{1}$ is a family of strain-
limiting functions with limiting small strains. We leave it to the reader to
verify that $\boldsymbol{f}_{1}=\partial_{\bar{\boldsymbol{S}}}W^{*}$ for an
appropriately chosen twice continuously differentiable, convex function
$W^{*}$.

A family of strain-limiting functions with limiting small strains outside of
the scope of our discussion thus far is the following. Let $E_{0}$ and $a$ be
fixed positive constants. For $\delta>0$ small, define

$\displaystyle
E_{\delta}(\boldsymbol{E})=\delta^{-1}E_{0}\Bigl{[}1+a\delta^{-1}([\det(\boldsymbol{I}+2\boldsymbol{E})]^{-1/2}-1)\Bigr{]},$
(3.23)
$\displaystyle\boldsymbol{f}_{\delta}(\boldsymbol{E},\bar{\boldsymbol{S}})=\frac{1+\nu}{E_{\delta}}\bar{\boldsymbol{S}}-\frac{\nu}{E_{\delta}}(\mathrm{tr}\,\bar{\boldsymbol{S}})\boldsymbol{I},$
(3.24)

One may interpret the constitutive relation
$\boldsymbol{E}=\boldsymbol{f}_{\delta}(\boldsymbol{E},\bar{\boldsymbol{S}})$
as a generalization of the classical linear constitutive relation for an
isotropic solid with a generalized Young’s modulus $E_{\delta}$ depending on
the density via
$E_{\delta}=\delta^{-1}E_{0}\bigl{[}1+a\delta^{-1}(\rho/\rho_{R}-1)\bigr{]}$
(see (2.14)).

Suppose that $b>0$, $c>0$ and $ab<1/2$. We claim that for all $\delta$
sufficiently small (depending on $a$ and $b$), (3.24) is a family of strain-
limiting functions with limiting small strains on domains
$B(\boldsymbol{0},b\delta)\times B(\boldsymbol{0},c)$. Towards proving our
claim, we observe that if $\boldsymbol{E}\in B(\boldsymbol{0},b\delta)$, then
for all $\delta$ sufficiently small,

$\displaystyle\Bigl{|}[\det(\boldsymbol{I}+2\boldsymbol{E})]^{-1/2}-1\Bigr{|}=|-\mathrm{tr}\,\boldsymbol{E}+o(|\boldsymbol{E}|)|\leq\sqrt{3}|\boldsymbol{E}|+o(|\boldsymbol{E}|)\leq
2|\boldsymbol{E}|.$ (3.25)

Above we used the fact that by the Cauchy-Schwarz inequality, for any tensor
$\boldsymbol{A}$ we have
$|\mathrm{tr}\,\boldsymbol{A}|=|\boldsymbol{I}\cdot\boldsymbol{A}|\leq\sqrt{3}|\boldsymbol{A}|$.
By the reverse triangle inequality, we have

$\displaystyle
E_{\delta}(\boldsymbol{E})\geq\delta^{-1}E_{0}(1-2a\delta^{-1}|\boldsymbol{E}|)\geq\delta^{-1}E_{0}(1-2ab)>0.$
(3.26)

We conclude that

$\displaystyle|\boldsymbol{f}_{\delta}(\boldsymbol{E},\bar{\boldsymbol{S}})|\leq\frac{1+\nu}{E_{\delta}}|\bar{\boldsymbol{S}}|+\frac{\nu}{E_{\delta}}|\mathrm{tr}\,\bar{\boldsymbol{S}}||\boldsymbol{I}|\leq\frac{(1+4\nu)c}{E_{0}(1-2ab)}\delta.$
(3.27)

Let $\boldsymbol{E}_{1},\boldsymbol{E}_{2}\in B(\boldsymbol{0},b\delta)$ and
$\bar{\boldsymbol{S}}_{1},\bar{\boldsymbol{S}}_{2}\in B(\boldsymbol{0},c)$.
Then

$\displaystyle\boldsymbol{f}_{\delta}(\boldsymbol{E}_{2},\bar{\boldsymbol{S}}_{2})-\boldsymbol{f}_{\delta}(\boldsymbol{E}_{1},\bar{\boldsymbol{S}}_{1})=\frac{E_{\delta}(\boldsymbol{E}_{1})-E_{\delta}(\boldsymbol{E}_{2})}{E_{\delta}(\boldsymbol{E}_{1})E_{\delta}(\boldsymbol{E}_{2})}[(1+\nu)\bar{\boldsymbol{S}}_{2}-\nu(\mathrm{tr}\,\bar{\boldsymbol{S}}_{2})\boldsymbol{I}]$
(3.28)
$\displaystyle+\frac{1}{E_{\delta}(\boldsymbol{E}_{1})}[(1+\nu)(\bar{\boldsymbol{S}}_{2}-\bar{\boldsymbol{S}}_{1})-\nu(\mathrm{tr}\,[\bar{\boldsymbol{S}}_{2}-\bar{\boldsymbol{S}}_{1}])\boldsymbol{I}].$
(3.29)

For all $\delta$ sufficiently small, we have,

$\displaystyle\forall
s\in[0,1],\quad|(\boldsymbol{I}+s\boldsymbol{E}_{1}+(1-s)\boldsymbol{E}_{2})^{-1}|\leq(1-b\delta)^{-1}\leq
2,$ (3.30) $\displaystyle\forall
s\in[0,1],\quad|\det(\boldsymbol{I}+s\boldsymbol{E}_{1}+(1-s)\boldsymbol{E}_{2})|\geq
1/2.$ (3.31)

By the fundamental theorem of calculus and the chain rule, we have

$\displaystyle|E_{\delta}(\boldsymbol{E}_{1})-E_{\delta}(\boldsymbol{E}_{2})|$
(3.32)
$\displaystyle=\delta^{-2}E_{0}a\Big{|}\int_{0}^{1}-\frac{1}{2}[\det(\boldsymbol{I}+s\boldsymbol{E}_{1}+(1-s)\boldsymbol{E}_{2})]^{-1/2}$
(3.33)
$\displaystyle\qquad\times(\boldsymbol{I}+s\boldsymbol{E}_{1}+(1-s)\boldsymbol{E}_{2})^{-1}\cdot(\boldsymbol{E}_{1}-\boldsymbol{E}_{2})ds\Bigr{|},$
(3.34)

and thus,

$\displaystyle|E_{\delta}(\boldsymbol{E}_{1})-E_{\delta}(\boldsymbol{E}_{2})|\leq\delta^{-2}E_{0}a\sqrt{2}.$
(3.35)

We also have

$\displaystyle|(1+\nu)\bar{\boldsymbol{S}}_{2}-\nu(\mathrm{tr}\,\bar{\boldsymbol{S}}_{2})\boldsymbol{I}|\leq(1+4\nu)|\bar{\boldsymbol{S}}_{2}|\leq(1+4\nu)c,$
(3.36)
$\displaystyle|(1+\nu)(\bar{\boldsymbol{S}}_{2}-\bar{\boldsymbol{S}}_{1})-\nu(\mathrm{tr}\,[\bar{\boldsymbol{S}}_{2}-\bar{\boldsymbol{S}}_{1}])\boldsymbol{I}|\leq(1+4\nu)|\bar{\boldsymbol{S}}_{1}-\bar{\boldsymbol{S}}_{2}|.$
(3.37)

Using the previous two estimates along with (3.35) and (3.26), we conclude
from (3.29) and repeated use of the triangle inequality that

$\displaystyle|\boldsymbol{f}_{\delta}(\boldsymbol{E}_{2},\bar{\boldsymbol{S}}_{2})-\boldsymbol{f}_{\delta}(\boldsymbol{E}_{1},\bar{\boldsymbol{S}}_{1})|$
(3.38)
$\displaystyle\leq\frac{(1+4\nu)ac\sqrt{2}}{E_{0}(1-2ab)^{2}}|\boldsymbol{E}_{2}-\boldsymbol{E}_{1}|+\frac{1+4\nu}{E_{0}(1-2ab)}\delta|\bar{\boldsymbol{S}}_{2}-\bar{\boldsymbol{S}}_{1}|.$
(3.39)

By (3.27) and (3.39), we conclude that for all $\delta$ sufficiently small
(depending on $a$ and $b$), (3.24) is a family of strain-limiting functions
with limiting small strains.

A final example of a family of strain-limiting functions with limiting small
strains is given in terms of a slightly different form of a density dependent
generalized Young’s modulus:

$\displaystyle
E_{\delta}=\delta^{-1}E_{0}\Bigl{[}1+a\delta^{-1}([\det(\boldsymbol{I}+2\boldsymbol{E})]^{1/2}-1)\Bigr{]}^{-1},$
(3.40)
$\displaystyle\boldsymbol{f}_{\delta}(\boldsymbol{E},\bar{\boldsymbol{S}})=\frac{1+\nu}{E_{\delta}}\bar{\boldsymbol{S}}-\frac{\nu}{E_{\delta}}(\mathrm{tr}\,\bar{\boldsymbol{S}})\boldsymbol{I},$
(3.41)

Again, by (2.14), one may interpret the generalized Young’s modulus as
depending on the density via
$E_{\delta}=\delta^{-1}E_{0}\bigl{[}1+a\delta^{-1}(\rho_{R}/\rho-1)\bigr{]}^{-1}$.
We note that both (3.24) and (3.41) have the physical property that an
increase in density, $\rho$, yields an increase in the Young’s modulus,
$E_{\delta}$. Arguing as above, one may conclude that (3.41) also yields a
family of strain-limiting functions with limiting small strains on domains
$U_{\delta}\times V=B(\boldsymbol{0},b)\times B(\boldsymbol{0},c)$ for
appropriate choices of $b$ and $c$.

###  3.3. Nonlinear relations between stress and linearized strain

We now show that a family of constitutive relations of the form
$\boldsymbol{E}=\boldsymbol{f}_{\delta}(\boldsymbol{E},\bar{\boldsymbol{S}})$,
with $\boldsymbol{E}$ interpreted as the Green-St. Venant strain tensor
associated to a deformation and $\bar{\boldsymbol{S}}$ as the symmetric Piola-
Kirchhoff stress, asymptotically yield nonlinear relations between stress and
the associated linearized strain $\boldsymbol{\epsilon}$, up to a
$\boldsymbol{O}(\delta^{2})$ error. Moreover, Corollary 3.4 and the examples
discussed thereafter show that in general, the resulting asymptotic relations
between stress and linearized strain can be genuinely nonlinear to leading
order in $\delta$. This stands in contrast to the setting of a fixed
constitutive relation, which under mild differentiability assumptions, always
can be asymptotically reduced to a linear relation between stress and
linearized strain to leading order in the displacement gradient
$|\boldsymbol{F}-\boldsymbol{I}|$.

The preliminary set-up for our result is as follows. For an element
$\boldsymbol{E}_{\delta}\in B(\boldsymbol{0},1/2)$ and
$\bar{\boldsymbol{S}}\in\mathrm{Sym}$, we associate a deformation
$\boldsymbol{F}_{\delta}$ via

$\displaystyle\boldsymbol{C}_{\delta}:=(\boldsymbol{I}+2\boldsymbol{E}_{\delta})^{1/2},\quad\boldsymbol{F}_{\delta}:=\boldsymbol{C}_{\delta}^{1/2},\quad\epsilon_{\delta}:=\frac{1}{2}(\boldsymbol{F}_{\delta}+\boldsymbol{F}_{\delta}^{T})-\boldsymbol{I},$
(3.42)

and a stress tensor via

$\displaystyle\boldsymbol{\sigma}_{\delta}:=\frac{1}{2}\bigl{(}\boldsymbol{F}_{\delta}^{-1}\bar{\boldsymbol{S}}+\bar{\boldsymbol{S}}\boldsymbol{F}_{\delta}^{-T}\bigr{)}.$
(3.43)

The tensor $\boldsymbol{\sigma}_{\delta}$ can be interpreted as the symmetric
part of the associated first Piola-Kirchhoff stress tensor.

######  Proposition 3.3.

Let $\boldsymbol{f}_{\delta}:U_{\delta}\times V\rightarrow\mathrm{Sym}$ be a
family of strain-limiting functions with limiting small strains. Let
$\bar{\boldsymbol{S}}\in V$. Assume that there exists $r>0$ such that for each
$\delta>0$ sufficiently small, there exists $\boldsymbol{E}_{\delta}\in
U_{\delta}$ such that $B(\boldsymbol{E}_{\delta},r\delta)\subseteq U_{\delta}$
and

$\displaystyle\boldsymbol{E}_{\delta}=\boldsymbol{f}_{\delta}(\boldsymbol{E}_{\delta},\bar{\boldsymbol{S}}).$
(3.44)

Then for all $\delta$ sufficiently small,
$\boldsymbol{(}\boldsymbol{\epsilon}_{\delta},\boldsymbol{\sigma}_{\delta})\in
U_{\delta}\times V$, and as $\delta\rightarrow 0$,

$\displaystyle\boldsymbol{\epsilon}_{\delta}=\boldsymbol{f}_{\delta}(\boldsymbol{\epsilon}_{\delta},\boldsymbol{\sigma}_{\delta})+\boldsymbol{O}(\delta^{2}),$
(3.45)

where the big-oh term depends on $C_{0}$, $C_{1}$, and
$D_{0}|\bar{\boldsymbol{S}}|$.

###### Proof.

For all $\delta>0$ sufficiently small, we have by (3.44) and (3.1) that

$\displaystyle|\boldsymbol{F}_{\delta}-\boldsymbol{I}|=|(\boldsymbol{I}+2\boldsymbol{E}_{\delta})^{1/2}-\boldsymbol{I}|\leq(1-2C_{0}\delta)^{-1/2}|\boldsymbol{E}_{\delta}|\leq(1-2C_{0}\delta)^{-1/2}C_{0}\delta.$
(3.46)

Since
$\boldsymbol{C}_{\delta}=\boldsymbol{F}_{\delta}^{T}\boldsymbol{F}_{\delta}$
and
$\boldsymbol{E}_{\delta}=\frac{1}{2}(\boldsymbol{C}_{\delta}-\boldsymbol{I})$,
we conclude that

$\displaystyle\boldsymbol{C}_{\delta}=\boldsymbol{I}+2\boldsymbol{\epsilon}_{\delta}+\boldsymbol{O}(\delta^{2}),\quad\boldsymbol{E}_{\delta}=\boldsymbol{\epsilon}_{\delta}+\boldsymbol{O}(\delta^{2}),$
(3.47)

and

$\displaystyle\boldsymbol{\sigma}_{\delta}=\frac{1}{2}(\boldsymbol{I}+\boldsymbol{O}(\delta))\bar{\boldsymbol{S}}+\frac{1}{2}\bar{\boldsymbol{S}}(\boldsymbol{I}+\boldsymbol{O}(\delta)),$
(3.48)

where the big-oh terms depend only on $C_{0}$.

Since $\bar{\boldsymbol{S}}\in V$ and $V$ is open, there exists $r_{0}>0$ such
that $B(\bar{\boldsymbol{S}},r_{0})\subseteq V$. The relations (3.47) and
(3.48) imply that there exists a constant $C$ depending on $C_{0}$ such that
for all $\delta$ sufficiently small,

$\displaystyle|\boldsymbol{E}_{\delta}-\boldsymbol{\epsilon}_{\delta}|\leq
C\delta^{2}<r\delta,\quad|\boldsymbol{\sigma}_{\delta}-\bar{\boldsymbol{S}}|\leq
C\delta|\bar{\boldsymbol{S}}|<r_{0},$ (3.49)

and thus, $(\boldsymbol{\epsilon}_{\delta},\bar{\boldsymbol{S}})\in
B(\boldsymbol{E}_{\delta},r\delta)\times
B(\bar{\boldsymbol{S}},r_{0})\subseteq U_{\delta}\times V$. Then by (3.44),
(3.47), (3.1), and (3.2),

$\displaystyle\boldsymbol{\epsilon}_{\delta}$
$\displaystyle=\boldsymbol{f}_{\delta}(\boldsymbol{E}_{\delta},\bar{\boldsymbol{S}})+\boldsymbol{\epsilon}_{\delta}-\boldsymbol{E}_{\delta}$
(3.50)
$\displaystyle=\boldsymbol{f}_{\delta}(\boldsymbol{\epsilon}_{\delta},\boldsymbol{\sigma}_{\delta})+[\boldsymbol{f}_{\delta}(\boldsymbol{\epsilon}_{\delta},\bar{\boldsymbol{S}})-\boldsymbol{f}_{\delta}(\boldsymbol{\epsilon}_{\delta},\boldsymbol{\sigma}_{\delta})+\boldsymbol{\epsilon}_{\delta}-\boldsymbol{E}_{\delta}+\boldsymbol{f}_{\delta}(\boldsymbol{E}_{\delta},\bar{\boldsymbol{S}})-\boldsymbol{f}_{\delta}(\boldsymbol{\epsilon}_{\delta},\bar{\boldsymbol{S}})]$
(3.51)
$\displaystyle=\boldsymbol{f}_{\delta}(\boldsymbol{\epsilon}_{\delta},\boldsymbol{\sigma}_{\delta})+\boldsymbol{O}(\delta^{2}),$
(3.52)

as $\delta\rightarrow 0$, where the big-oh term depends only on $C_{0}$,
$C_{1}$, and $|D_{0}||\bar{\boldsymbol{S}}|$. This concludes the proof. ∎

######  Corollary 3.4.

In addition to the assumptions of Proposition 3.3, suppose that there exist an
open set $U\subseteq\mathrm{Sym}$, $\boldsymbol{f}_{1}:U\times
V\rightarrow\mathrm{Sym}$, and a constant $C_{2}>0$ such that

$\displaystyle U_{\delta}=\delta
U=\\{\delta\tilde{\boldsymbol{E}}\mid\tilde{\boldsymbol{E}}\in U\\},$ (3.53)

and

$\displaystyle\forall(\boldsymbol{E},\bar{\boldsymbol{S}})\in U_{\delta}\times
V,\quad\bigl{|}\boldsymbol{f}_{\delta}(\boldsymbol{E},\bar{\boldsymbol{S}})-\delta\boldsymbol{f}_{1}\bigl{(}\delta^{-1}\boldsymbol{E},\bar{\boldsymbol{S}})\bigr{|}\leq
C_{2}\delta^{2}.$ (3.54)

Then, in the notation of Proposition 3.3, we have as $\delta\rightarrow 0$,

$\displaystyle\boldsymbol{\epsilon}_{\delta}=\delta\boldsymbol{f}_{1}(\delta^{-1}\boldsymbol{\epsilon}_{\delta},\boldsymbol{\sigma}_{\delta})+\boldsymbol{O}(\delta^{2}),$
(3.55)

where the big-oh term depends on $C_{0},C_{1},C_{2}$ and
$D_{0}|\bar{\boldsymbol{S}}|$.

We now discuss the examples from Section 3.2 within the context of Proposition
3.3 and Corollary 3.4. It is clear that a general family of strain-limiting
functions with limiting small strains given by (3.14) satisfies the hypotheses
of both Proposition 3.3 and Corollary 3.4. In particular, if (3.15) holds with
a twice continuously differentiable, convex function $W^{*}$, then Corollary
3.4 implies in the notation therein that

$\displaystyle\boldsymbol{\epsilon}_{\delta}=\partial_{\boldsymbol{\sigma}_{\delta}}[\delta
W^{*}(\boldsymbol{\sigma}_{\delta})]+\boldsymbol{O}(\delta^{2}),\quad\boldsymbol{\sigma}_{\delta}\in
V.$ (3.56)

Omitting the $\boldsymbol{O}(\delta^{2})$ term yields
$\boldsymbol{\epsilon}_{\delta}=\partial_{\boldsymbol{\sigma}_{\delta}}[\delta
W^{*}(\boldsymbol{\sigma}_{\delta})]$ or, equivalently,

$\displaystyle\boldsymbol{\sigma}_{\delta}=\partial_{\boldsymbol{\epsilon}_{\delta}}[\delta
W\bigl{(}\delta^{-1}\boldsymbol{\epsilon}_{\delta}\bigr{)}],$ (3.57)

where $W$ is the Legendre transform of $W^{*}$. The relation (3.57) models a
Green elastic solid, i.e., the stress is derived from a strain energy
depending on the linearized strain. In contrast to classical linearized
elasticity, however, our framework allows for strain energies $W$ beyond
quadratic forms of the linearized strain.

We now consider the explicit family (3.24). We write
$\boldsymbol{f}_{\delta}(\boldsymbol{E},\bar{\boldsymbol{S}})=\delta\boldsymbol{g}_{\delta}(\boldsymbol{E}/\delta,\bar{\boldsymbol{S}})$
where

$\displaystyle\boldsymbol{g}_{\delta}(\tilde{\boldsymbol{E}},\bar{\boldsymbol{S}})=\frac{1+\nu}{\delta
E_{\delta}(\delta\tilde{\boldsymbol{E}})}\bar{\boldsymbol{S}}-\frac{\nu}{\delta
E_{\delta}(\delta\tilde{\boldsymbol{E}})}(\mathrm{tr}\,\bar{\boldsymbol{S}})\boldsymbol{I},\quad(\tilde{\boldsymbol{E}},\bar{\boldsymbol{S}})\in
B(\boldsymbol{0},b)\times B(\boldsymbol{0},c),$ (3.58)

and we note that for all $\tilde{\delta}$ and $b$ sufficiently small,

$\displaystyle(0,\tilde{\delta})\times
B(\boldsymbol{0},b)\ni(\delta,\tilde{\boldsymbol{E}})\mapsto\delta
E_{\delta}(\delta\tilde{\boldsymbol{E}})$
$\displaystyle=E_{0}\Bigl{[}1+a\delta^{-1}([\det(\boldsymbol{I}+2\delta\tilde{\boldsymbol{E}})]^{-1/2}-1)\Bigr{]},$
(3.59)

extends smoothly to a function of
$(\delta,\tilde{\boldsymbol{E}})\in(-\tilde{\delta},\tilde{\delta})\times
B(\boldsymbol{0},b)$. Moreover, we have for all $\tilde{\boldsymbol{E}}\in
B(\boldsymbol{0},b)$,

$\displaystyle\delta
E_{\delta}(\delta\tilde{\boldsymbol{E}})=E_{0}[1-a\mathrm{tr}\,\tilde{\boldsymbol{E}}+O(ab^{2}\delta)].$
(3.60)

We observe that
$\boldsymbol{E}=\boldsymbol{f}_{\delta}(\boldsymbol{E},\bar{\boldsymbol{S}})$
if and only if

$\displaystyle\tilde{\boldsymbol{E}}=\boldsymbol{g}_{\delta}(\tilde{\boldsymbol{E}},\bar{\boldsymbol{S}}),$
(3.61)

where $\boldsymbol{E}=\delta\cdot\tilde{\boldsymbol{E}}$,
$\boldsymbol{g}_{0}(\boldsymbol{0},\boldsymbol{0})=\boldsymbol{0}$, and

$\displaystyle
D_{\tilde{\boldsymbol{E}}}\bigl{(}\tilde{\boldsymbol{E}}-\boldsymbol{g}_{\delta}(\tilde{\boldsymbol{E}},\bar{\boldsymbol{S}})\bigr{)}\Big{|}_{(\delta,\tilde{\boldsymbol{E}},\bar{\boldsymbol{S}})=(0,\boldsymbol{0},\boldsymbol{0})}=\boldsymbol{\mathsf{I}}.$
(3.62)

Thus, by the implicit function theorem and choosing $\tilde{\delta}$, $b$, and
$c$ sufficiently small, we conclude that for each $0<\delta<\tilde{\delta}$
and $\bar{\boldsymbol{S}}\in B(\boldsymbol{0},c)$, there exists a unique
$\tilde{\boldsymbol{E}}_{\delta}\in B(\boldsymbol{0},b)$ satisfying (3.61).
Written differently, we have shown that for each $0<\delta<\tilde{\delta}$ and
$\bar{\boldsymbol{S}}\in B(\boldsymbol{0},c)$, there exists a unique
$\boldsymbol{E}_{\delta}\in B(\boldsymbol{0},\delta b)$ satisfying
$\boldsymbol{E}_{\delta}=\boldsymbol{f}_{\delta}(\boldsymbol{E}_{\delta},\bar{\boldsymbol{S}})$,
and the hypotheses of Proposition 3.3 are satisfied on $U_{\delta}\times
V=B(\boldsymbol{0},b\delta)\times B(\boldsymbol{0},c)$.

Finally, define

$\displaystyle\boldsymbol{f}_{1}(\tilde{\boldsymbol{E}},\bar{\boldsymbol{S}})=\boldsymbol{g}_{0}(\tilde{\boldsymbol{E}},\bar{\boldsymbol{S}})=E_{0}^{-1}(1-a\mathrm{tr}\,\tilde{\boldsymbol{E}})^{-1}\bigl{[}(1+\nu)\bar{\boldsymbol{S}}-\nu(\mathrm{tr}\,\bar{\boldsymbol{S}})\boldsymbol{I}\bigr{]}.$
(3.63)

By (3.60), it follows that for all $\boldsymbol{E}\in B(\boldsymbol{0},\delta
b)$ (with $ab<1/2$),

$\displaystyle\Bigl{|}\frac{1}{E_{\delta}(\boldsymbol{E})}-\frac{\delta}{E_{0}(1-a\mathrm{tr}\,(\delta^{-1}\boldsymbol{E}))}\Bigr{|}$
$\displaystyle=\frac{\bigl{|}\delta^{-1}E_{0}(1-a\mathrm{tr}\,(\delta^{-1}\boldsymbol{E}))-E_{\delta}(\boldsymbol{E})\bigr{|}}{\delta^{-1}E_{0}(1-a\mathrm{tr}\,(\delta^{-1}\boldsymbol{E}))E_{\delta}(\boldsymbol{E})}$
(3.64)
$\displaystyle\leq\frac{a\bigl{|}1-[\det(\boldsymbol{I}+2\boldsymbol{E})]^{-1/2}-\mathrm{tr}\,\boldsymbol{E}\bigr{|}}{E_{0}(1-2ab)^{2}}$
(3.65) $\displaystyle\leq\frac{Ca|\boldsymbol{E}|^{2}}{E_{0}(1-2ab)^{2}}$
(3.66) $\displaystyle\leq\frac{Cab^{2}\delta^{2}}{E_{0}(1-2ab)^{2}},$ (3.67)

where $C$ is an absolute constant. Thus, for all
$(\boldsymbol{E},\bar{\boldsymbol{S}})\in B(\boldsymbol{0},b\delta)\times
B(\boldsymbol{0},c)$,

$\displaystyle\bigl{|}\boldsymbol{f}_{\delta}(\boldsymbol{E},\bar{\boldsymbol{S}})-\delta\boldsymbol{f}_{1}\bigl{(}\delta^{-1}\boldsymbol{E},\bar{\boldsymbol{S}})\bigr{|}\leq\frac{Cab^{2}(1+4\nu)c\delta^{2}}{E_{0}(1-2ab)^{2}}.$
(3.68)

The estimate (3.68) proves that the hypotheses of Corollary 3.4 are satisfied.
In the notation therein,

$\displaystyle\boldsymbol{\epsilon}_{\delta}=\delta
E_{0}^{-1}\bigl{[}1-a\delta^{-1}\mathrm{tr}\,(\boldsymbol{\epsilon}_{\delta})\bigr{]}^{-1}\bigl{[}(1+\nu)\boldsymbol{\sigma}_{\delta}-\nu(\mathrm{tr}\,\boldsymbol{\sigma}_{\delta})\boldsymbol{I}\bigr{]}+\boldsymbol{O}(\delta^{2}),$
(3.69)

as $\delta\rightarrow 0$, and omitting the $\boldsymbol{O}(\delta^{2})$ term
yields

$\displaystyle\boldsymbol{\epsilon}_{\delta}=\delta
E_{0}^{-1}\bigl{[}1-a\delta^{-1}\mathrm{tr}\,(\boldsymbol{\epsilon}_{\delta})\bigr{]}^{-1}\bigl{[}(1+\nu)\boldsymbol{\sigma}_{\delta}-\nu(\mathrm{tr}\,\boldsymbol{\sigma}_{\delta})\boldsymbol{I}\bigr{]}.$
(3.70)

One may similarly show that the hypotheses of Proposition 3.3 and Corollary
3.4 are satisfied for (3.41) on a family of nontrivial open sets
$U_{\delta}\times V$ with

$\displaystyle\boldsymbol{f}_{1}(\tilde{\boldsymbol{E}},\bar{\boldsymbol{S}})=E_{0}^{-1}(1+a\mathrm{tr}\,\tilde{\boldsymbol{E}})\bigl{[}(1+\nu)\bar{\boldsymbol{S}}-\nu(\mathrm{tr}\,\bar{\boldsymbol{S}})\boldsymbol{I}\bigr{]}.$
(3.71)

In the notation of Proposition 3.3 and Corollary 3.4, we conclude that

$\displaystyle\boldsymbol{\epsilon}_{\delta}=\delta
E_{0}^{-1}\bigl{[}1+a\delta^{-1}\mathrm{tr}\,(\boldsymbol{\epsilon}_{\delta})\bigr{]}\bigl{[}(1+\nu)\boldsymbol{\sigma}_{\delta}-\nu(\mathrm{tr}\,\boldsymbol{\sigma}_{\delta})\boldsymbol{I}\bigr{]}+\boldsymbol{O}(\delta^{2}),$
(3.72)

as $\delta\rightarrow 0$, and omitting the $\boldsymbol{O}(\delta^{2})$ term
yields

$\displaystyle\boldsymbol{\epsilon}_{\delta}=\delta
E_{0}^{-1}\bigl{[}1+a\delta^{-1}\mathrm{tr}\,(\boldsymbol{\epsilon}_{\delta})\bigr{]}\bigl{[}(1+\nu)\boldsymbol{\sigma}_{\delta}-\nu(\mathrm{tr}\,\boldsymbol{\sigma}_{\delta})\boldsymbol{I}\bigr{]}.$
(3.73)

Constitutive relations of the form

$\displaystyle\boldsymbol{\epsilon}=E_{\mathsf{ref}}^{-1}\bigl{[}1-\Gamma\mathrm{tr}\,\boldsymbol{\epsilon}\bigr{]}^{-1}\bigl{[}(1+\nu)\boldsymbol{\sigma}-\nu(\mathrm{tr}\,\boldsymbol{\sigma})\boldsymbol{I}\bigr{]},$
(3.74)

or

$\displaystyle\boldsymbol{\epsilon}=E_{\mathsf{ref}}^{-1}\bigl{[}1+\Gamma\mathrm{tr}\,\boldsymbol{\epsilon}\bigr{]}\bigl{[}(1+\nu)\boldsymbol{\sigma}-\nu(\mathrm{tr}\,\boldsymbol{\sigma})\boldsymbol{I}\bigr{]},$
(3.75)

where $\boldsymbol{\sigma}$ is the stress variable, have been appeared
extensively in the literature; see e.g. [26, 15, 16, 14, 13, 6, 17, 34]. Our
framework and the expansions (3.69) and (3.72) place (3.74) and (3.75) on firm
mathematical footing as the asymptotic leading order relations (in $\delta$)
arising from a constitutive theory for finite deformations.

We conclude this section by noting that our results are not restricted solely
to constitutive relations involving the Green-St. Venant strain tensor and the
symmetric Piola-Kirchhoff stress tensor. Indeed, analogues of Proposition 3.3
and Corollary 3.4 can be proved in terms of variables $\boldsymbol{H}$ and
$\boldsymbol{T}$ representing the Hencky strain and Cauchy stress. The
preliminary set-up is as follows. For an element
$(\boldsymbol{H}_{\delta},\boldsymbol{T})\in\mathrm{Sym}\times\mathrm{Sym}$,
we associate a deformation via

$\displaystyle\boldsymbol{F}_{\delta}:=e^{\boldsymbol{H}_{\delta}},\quad\boldsymbol{\epsilon}_{\delta}:=\frac{1}{2}(\boldsymbol{F}_{\delta}+\boldsymbol{F}_{\delta}^{T})-\boldsymbol{I},$
(3.76)

and a stress tensor via

$\displaystyle\boldsymbol{\sigma}_{\delta}=(\det\boldsymbol{F}_{\delta})\frac{1}{2}\bigl{(}\boldsymbol{T}\boldsymbol{F}_{\delta}^{-T}+\boldsymbol{F}_{\delta}^{-1}\boldsymbol{T}\bigr{)}.$
(3.77)

As before, the tensor $\boldsymbol{\sigma}_{\delta}$ can be interpreted as the
symmetric part of the associated first Piola-Kirchhoff stress tensor.

######  Proposition 3.5.

Let $\boldsymbol{f}_{\delta}:U_{\delta}\times V\rightarrow\mathrm{Sym}$ be a
family of strain-limiting functions with limiting small strains. Let
$\boldsymbol{T}\in V$. Assume that there exists $r>0$ such that for each
$\delta>0$ sufficiently small, there exists $\boldsymbol{H}_{\delta}\in
U_{\delta}$ such that $B(\boldsymbol{H}_{\delta},r\delta)\subseteq U_{\delta}$
and

$\displaystyle\boldsymbol{H}_{\delta}=\boldsymbol{f}_{\delta}(\boldsymbol{H}_{\delta},\boldsymbol{T}).$
(3.78)

Then for all $\delta$ sufficiently small,
$\boldsymbol{(}\epsilon_{\delta},\boldsymbol{\sigma}_{\delta})\in
U_{\delta}\times V$, and as $\delta\rightarrow 0$,

$\displaystyle\boldsymbol{\epsilon}_{\delta}=\boldsymbol{f}_{\delta}(\boldsymbol{\epsilon}_{\delta},\boldsymbol{\sigma}_{\delta})+\boldsymbol{O}(\delta^{2}),$
(3.79)

where the big-oh term depends only on $C_{0}$, $C_{1}$, and
$D_{0}|\boldsymbol{T}|$.

###### Proof.

For all $\delta>0$ sufficiently small, we have by (3.78) and (3.1) that

$\displaystyle|\boldsymbol{F}_{\delta}-\boldsymbol{I}|=|e^{\boldsymbol{H}_{\delta}}-\boldsymbol{I}|\leq|\boldsymbol{H}_{\delta}|e^{|\boldsymbol{H}_{\delta}|}\leq
C_{0}\delta e^{C_{0}\delta}\leq eC_{0}\delta.$ (3.80)

Since
$\boldsymbol{H}_{\delta}=\frac{1}{2}\log\boldsymbol{F}_{\delta}\boldsymbol{F}^{T}_{\delta}$,
we conclude that

$\displaystyle\boldsymbol{H}_{\delta}=\boldsymbol{\epsilon}_{\delta}+\boldsymbol{O}(\delta^{2}),\quad\boldsymbol{\sigma}_{\delta}=\frac{1}{2}\boldsymbol{T}(\boldsymbol{I}+\boldsymbol{O}(\delta)+\frac{1}{2}(\boldsymbol{I}+\boldsymbol{O}(\delta))\boldsymbol{T},$
(3.81)

where the big-oh terms depend only on $C_{0}$ and $C_{1}$.

Arguing as in Proposition 3.3, we conclude that for all $\delta$ sufficiently
small, $(\boldsymbol{\epsilon}_{\delta},\boldsymbol{\sigma}_{\delta})\in
U_{\delta}\times V$, and

$\displaystyle\boldsymbol{\epsilon}_{\delta}$
$\displaystyle=\boldsymbol{f}_{\delta}(\boldsymbol{H}_{\delta},\boldsymbol{T})+\boldsymbol{\epsilon}_{\delta}-\boldsymbol{H}_{\delta}$
(3.82)
$\displaystyle=\boldsymbol{f}_{\delta}(\boldsymbol{\epsilon}_{\delta},\boldsymbol{\sigma}_{\delta})+[\boldsymbol{f}_{\delta}(\boldsymbol{\epsilon}_{\delta},\boldsymbol{T})-\boldsymbol{f}_{\delta}(\boldsymbol{\epsilon}_{\delta},\boldsymbol{\sigma}_{\delta})+\boldsymbol{\epsilon}_{\delta}-\boldsymbol{H}_{\delta}+\boldsymbol{f}_{\delta}(\boldsymbol{H}_{\delta},\boldsymbol{T})-\boldsymbol{f}_{\delta}(\boldsymbol{\epsilon}_{\delta},\boldsymbol{T})]$
(3.83)
$\displaystyle=\boldsymbol{f}_{\delta}(\boldsymbol{\epsilon}_{\delta},\boldsymbol{\sigma}_{\delta})+\boldsymbol{O}(\delta^{2}),$
(3.84)

as $\delta\rightarrow 0$, where the big-oh term depends only on $C_{0}$,
$C_{1}$, and $D_{0}|\boldsymbol{T}|$. This concludes the proof. ∎

##  4\. Conclusion

In this study, we have presented a rigorous asymptotic framework that provides
a mathematical foundation for nonlinear constitutive relations between stress
and linearized strain. Within our framework, we have demonstrated that a
nonlinear constitutive relation between stress and linearized strain emerges
as the primary term of an asymptotic expansion in $\delta$ of a family of
strain-limiting, nonlinear constitutive relations,

$\displaystyle\boldsymbol{E}=\boldsymbol{f}_{\delta}(\boldsymbol{E},\bar{\boldsymbol{S}})$
(4.1)

between stress and nonlinear strain. Here, the dimensionless parameter
$\delta$ determining the leading order term is (up to a fixed constant) the
limiting small strain of $\boldsymbol{f}_{\delta}$:
$\forall\boldsymbol{E},\bar{\boldsymbol{S}},$
$|\boldsymbol{f}_{\delta}(\boldsymbol{E},\bar{\boldsymbol{S}})|\leq\delta$.
Our approach diverges from classical linearized elasticity where a
constitutive relation is fixed and the asymptotic parameter is the size of the
displacement gradient $\delta_{0}=|\boldsymbol{F}-\boldsymbol{I}|$.

As our results show, strain-limiting constitutive relations
$\boldsymbol{E}=\boldsymbol{f}_{\delta}(\boldsymbol{E},\bar{\boldsymbol{S}})$
are approximated by
$\boldsymbol{\epsilon}=\boldsymbol{f}_{\delta}(\boldsymbol{\epsilon},\boldsymbol{\sigma})$
up to a quadratic error in $\delta$ with $\boldsymbol{\sigma}$ interpreted as
the symmetric part of the associated first Piola-Kirchhoff stress tensor;
however, a fundamental question remains. In particular, we conjecture that for
a fixed external body force $\boldsymbol{b}$ and for all $\delta$ sufficiently
small, solvability of the “linearized” equilibrium equations,

$\displaystyle\boldsymbol{0}=\mbox{Div}\,\boldsymbol{\sigma}_{L,\delta}+\boldsymbol{b},\quad\boldsymbol{\sigma}_{L,\delta}^{T}=\boldsymbol{\sigma}_{L,\delta},$
(4.2)
$\displaystyle\boldsymbol{\epsilon}_{L,\delta}=\boldsymbol{f}_{\delta}(\boldsymbol{\epsilon}_{L,\delta},\boldsymbol{\sigma}_{L,\delta}),$
(4.3)

implies solvability of the fully nonlinear equilibrium equations,

$\displaystyle\boldsymbol{0}=\mbox{Div}\,\boldsymbol{S}_{\delta}+\boldsymbol{b},\quad\boldsymbol{S}_{\delta}\boldsymbol{F}_{\delta}^{T}=\boldsymbol{F}_{\delta}\boldsymbol{S}_{\delta}^{T},$
(4.4)
$\displaystyle\boldsymbol{E}_{\delta}=\boldsymbol{f}_{\delta}(\boldsymbol{E}_{\delta},\bar{\boldsymbol{S}}_{\delta}),$
(4.5)

and

$\displaystyle\boldsymbol{E}_{\delta}=\boldsymbol{\epsilon}_{L,\delta}+\boldsymbol{O}(\delta^{2}),\quad\bar{\boldsymbol{S}}_{\delta}=\boldsymbol{\sigma}_{L,\delta}+\boldsymbol{O}(\delta),\quad\boldsymbol{S}_{\delta}=\boldsymbol{\sigma}_{L,\delta}+\boldsymbol{O}(\delta),$
(4.6)

as $\delta\rightarrow 0$.101010Of course, on a bounded domain $\mathcal{B}$, a
fixed set of boundary conditions must also be given. Such a fundamental result
would be unique but in line with analogous results established for classical
linearized elasticity; see, e.g., the classic works of Stoppelli [30, 31] and
the discussion of more recent results by Ciarlet in Chapter 6 of [3].

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K. R. Rajagopal

Department of Mechanical Engineering, Texas A&M University

College Station, TX 77843, USA

C. Rodriguez

Department of Mathematics, University of North Carolina

Chapel Hill, NC 27599, USA