Higher-gradient continua: The legacy of Piola, Mindlin, Sedov and Toupin and some future research perspectives

TL;DR: Mindlin and Toupin this article formulated a conceptual frame for continuum mechanics which is able to effectively model the complex behaviour of so-called architectured, advanced, multiscale or microstructured (meta)materials.

Abstract: Since the first studies dedicated to the mechanics of deformable bodies (by Euler, D’Alembert, Lagrange) the principle of virtual work (or virtual velocities) has been used to provide firm guidance to the formulation of novel theories. Gabrio Piola dedicated his scientific life to formulating a continuum theory in order to encompass a large class of deformation phenomena and was the first author to consider continua with non-local internal interactions and, as a particular case, higher-gradient continua. More recent followers of Piola (Mindlin, Sedov and then Richard Toupin) recognized the principle of virtual work (and its particular case, the principle of least action) as the (only!) firm foundation of continuum mechanics. Mindlin and Toupin managed to formulate a conceptual frame for continuum mechanics which is able to effectively model the complex behaviour of so-called architectured, advanced, multiscale or microstructured (meta)materials. Other postulation schemes, in contrast, do not seem able to ...

Summary

1. Historical perspective as a guide for future researches

• The research of the first sources of higher-gradient continua has its own scholarly interest.
• It can also be motivated by a more cogent aim: the search for the most effective tools for conceiving, finding and developing novel theories or models in physics and, in particular, in mechanics.
• The authors share Piola’s ideas and want to support his point of view by examining the historical evolution of the theory of higher-gradient continua since its first formulation by him.
• Exactly as happened with Piola (see his preface of the 1848 work in [1]) the authors are surprised that the principles of virtual work and least action, even though they have been fully supported by undisputed scientific authorities (for instance by D’Alembert, Lagrange, Hamilton, Landau [25, 26], Feynman [27, 28], Sedov [29]), still need to be advocated.
• It seems necessary to reaffirm, at least to the advantage of the community of specialists in continuum mechanics, that the continuum models which are needed when describing microscopically strongly inhomogeneous systems (see e.g. [30–33]) must consider internal work functionals (see Germain [24], Salençon [34]) involving second (and higher) gradients of virtual displacements.

1.3. Piola’s higher-gradient continua

• Piola never considers the particular case of linearized deformation measures (which is indeed physically rather unnatural).
• A late disciple of Piola9 In [42, 43], the reader will find a deep, erudite and original presentation of field theories, also known as 1.4. Sedov.
• The authors analysis will show that the Lagrange variational equation for material continua and physical fields can be employed as a basis for all physical models not only of reversible phenomena, but in cases of irreversible phenomena as well.
• Confusion is bred, on the one hand, by the fact that the mechanics of deformable bodies is usually concerned with linear problems in which one can assume that the observer’s system coincides with the comoving system.
• In many cases such papers have a particular character, sometimes they are connected essentially only with simplest particular problems and with empirical formal approach.

1.5. Truesdell’s difficulties with the principle of virtual works

• In his work ‘Essays in the history of mechanics’ (see [44]), Truesdell shows he has misunderstood the ideas of Lagrange and consequently those expressed by Piola, Mindlin and Toupin.
• Still it does not seem to be easy enough for most historians of science to penetrate the contents.
• Clearly Truesdell overestimates the role of Cauchy in the process of founding continuum mechanics.
• In the following statements Truesdell claims that Lagrange’s understanding of mechanics is limited.

2. State of the art: Higher-gradient continua theory in the language of functional analysis

• The pioneering works [13, 14, 45, 46] and especially those authored by Paul Germain [22–24] clarified the role of functional analysis in continuum mechanics.
• In [41] and in [47, 48], continuing Germain’s line of thought, it has been remarked that the new tools supplied by the theory of distributions developed by Laurent Schwartz (see the fundamental book [49]) are really adapted to frame generalized continuum theories.
• In some cases a suitable subset of D is to be considered: this circumstance can be accounted for via the Hahn–Banach prolongation theorem (for a reference see e.g. [49]) but for simplicity will not be treated here.
• By guest on January 14, 2016mms.sagepub.comDownloaded from.
• Therefore the representation theorems presented in Schwartz [49, pp. 82–104] can be fruitfully used to describe the structure of virtual work functionals.

2.1. Work functionals

• Once the authors fix a generic subbody SB (i.e. a subset of material particles occupying, in a given configuration, an admissible domain) of a given continuous body B and consider the set A(SB) of all infinitesimal displacement fields admissible for SB, it is natural to admit that in A(SB) are included infinitely differentiable functions having compact support included in SB.
• It is also natural (as done e.g. in [22–24]) to assume that the work expended by the interactions between SB and its external world is a linear and continuous functional (with respect to the Fréchet topology) when restricted to D(SB) ⊂ A(SB).
• In other words the authors accept the following (fundamentally due to D’Alembert and Lagrange).

POSTULATE ON WORK FUNCTIONALS

• The work expended by all the interactions relative to a subbody SB are distributions (in the sense of Schwartz) concentrated on U(SB), where the authors denote by U(SB) the topological closure (in the sense of the natural topology on Rn) of an open set U(SB) including SB.
• It is clear that, once the previous postulate is accepted, theorems and definitions of the theory of distributions (see [49]) become really relevant in continuum mechanics.
• The authors can obviously exploit the Schwartz general representation theorems and, by taking into account the aforementioned definitions and theorems, they get that the postulate on work functionals can be rephrased into the following.

2.2. External and internal work functionals

• Having explained this nomenclature it is obvious what the authors mean by internal and external work functionals.
• When following the approach à la D’Alembert one will introduce the following.

POSTULATE ON WORK BALANCE OR D′ALEMBERT PRINCIPLE OF VIRTUAL WORK

• Indeed the external world interacts with a continuous body B and its subbodies exert internal interactions on each other.
• The authors call internal and external the work expended on any virtual displacement by internal and external interactions respectively: since the works by D’Alembert, inertial forces are included in external interactions.
• In these works, it is shown that this principle is the most suitable when dealing with more general systems than finite systems of material points: it is for example very effective in continuum mechanics.
• Piola, Mindlin and Toupin limited themselves to considering the following class of external interactions.

CONSTITUTIVE ASSUMPTION FOR EXTERNAL WORK

• The external interactions exerted on some subbody SB are described by a distribution Pext made of two parts.
• The first part corresponds to long-range external interactions exerted on SB.
• The inertial power, which D’Alembert included in Pext, is of this type.
• The second part corresponds to contact actions.

2.3. Contact interactions and stress states

• The theorems recalled in the previous section suggest that the expression for the work of contact interactions usually considered in continuum mechanics, when the classical format due to Cauchy is considered, is very restrictive.
• This is the case for Eringen’s microstructured continua.
• This tensorial nature of kinematical fields is irrelevant in the present context and therefore, for the sake by guest on January 14, 2016mms.sagepub.comDownloaded from of efficiency, the authors operate as if the kinematics were described by a real-valued function U .
• First, this is due to the fact that virtual work is not always the preferred tool for some mechanicians while, on the other hand, it gives the conceptual framework in which generalized contact interactions arise naturally.
• This inequality may be considered as a basis for a postulation for continuum mechanics when higher order continua are also considered, as proven in [47, 48].

3. Cauchy straightjacket and the consequent conceptual blockages

• The polemics between Poisson and Piola also involved Cauchy.
• By guest on January 14, 2016mms.sagepub.comDownloaded from Recall that the Piola’s schoolmaster is Lagrange and that Piola, while expressing himself against the views of Poisson, seems to be hesitating in starting a controversy against Cauchy.
• Noll [57] has proven that these two assumptions imply the so-called Cauchy postulate: contact surface density of forces depends only on the normal of the Cauchy cuts.
• If one wants to use N th-gradient theories the equilibrium equations involve (see [41]) exactly N stress tensors of increasing orders (from the second to the (N +1)th in the case of continua where the only kinematical field is placement).

4. Higher-gradient continua as models for microscopically complex systems

• Many papers in the literature try to deduce from the structure of microscopic models for mechanical systems the properties of their macroscopic ones.
• This contrast has relevant effects on their macroscopic behaviour and requires (yet to be accounted for), to be accounted for, that higher gradients of displacement or suitable microstructure fields or both must appear in the constitutive equations for deformation energy.
• Note that the heuristic procedure systematically presented in the literature can be tracked already in the works by Piola.
• By guest on January 14, 2016mms.sagepub.comDownloaded from Piola assumes that there exists a continuous macroscopic placement function which describes the global behaviour of the considered lattice of particles.
• Then he calculates the virtual work of the microscopic systems in the presence of a virtual displacement obtained by the variation of the macro-displacement function.

6. Research perspectives

• Notwithstanding all the efforts made all his life by Gabrio Piola, and despite the fact that he was the beginner of a strong school of mathematical physics (see [3]), Cauchy’s particular kind of continuum has been considered the most general one which can be logically formulated.
• This principle has been exploited in the investigations presented in [76] where it has been shown how the microstructural response of a hypothetical bioresorbable material may positively influence the remodelling process in a reconstructed bone tissue.
• Such a modelling procedure shows some limits: it requires huge calculating devices even for very simple situations and does not allow for any effective analytical or semi-analytical optimization process.
• The use of higher-gradient continuum models can play a relevant role in modelling the physical behaviour of many complex mechanical systems and structures [91–95].
• The models for complex materials and structures may require the introduction of micro-structured continua endowed with additional kinematical fields, also accounting for the activation of internal degrees of freedom.

Notes

• This quotation was repeated many times by R. Toupin during the symposium in his honour held at the 4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013).
• Già vedemmo nella precedente Memoria copia di risultati che se ne deducono, e toccammo di molte teoriche che potrebbero rannodarsi alle varie parti di essa.
• In §205 he asserts that mutual forces are central and are analogous to forces of constraint, and in §217 he derives the integral of moment of momentum for a system subject to steady holonomic constraints.
• These tensor fields are, by definition, orthogonal to the manifold where they are concentrated.
• Having reached agreement that the authors should base the classical field theories on a set of axioms, they must now admit, ruefully, their inability to do so.

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Higher-gradient continua: The legacy of Piola, Mindlin,
Sedov and Toupin and some future research perspectives
Francesco Dell’Isola, Alessandro Della Corte, Ivan Giorgio
To cite this version:
Francesco Dell’Isola, Alessandro Della Corte, Ivan Giorgio. Higher-gradient continua: The legacy of
Piola, Mindlin, Sedov and Toupin and some future research perspectives. Mathematics and Mechanics
of Solids, SAGE Publications, 2016, 21 p. �10.1177/1081286515616034�. �hal-01256929�

2 Mathematics and Mechanics of Solids
1. Historical perspective as a guide for future researches
The research of the first sources of higher-gradient continua has its own scholarly interest. However, it can
also be motivated by a more cogent aim: the search for the most effective tools for conceiving, finding and
developing novel theories or models in physics and, in particular, in mechanics. Gabrio Piola (see [1–3]) spent
all his scientific activity and his intellectual efforts in proving that the principle of virtual work (or its particular
case, the principle of least action) is the most effective conceptual tool for use by a scientist who wants to create
new models able to successfully predict the observed experimental evidence and to forecast the existence of
unknown phenomena in Continuum Mechanics.
We share Piola’s ideas and want to support his point of view by examining the historical evolution of the
theory of higher-gradient continua since its first formulation by him. We will show how and why these models
were abandoned (or considered logically inconsistent) by those scholars who refused to accept the Lagrangian
postulation of mechanics and we will see that they, instead, could be successfully developed only by those
scientists (e.g. Rivlin [4], Green and Naghdi [5–8], Pipkin [9, 10], Mindlin [11–16], Toupin [17], Casal [18–
21], Germain [22–24]) who could manage to accept the use of the powerful abstract concepts given to us by
the genius of Lagrange (in this sentence we are paraphrasing Piola [1, 3]). Exactly as happened with Piola
(see his preface of the 1848 work in [1]) we are surprised that the principles of virtual work and least action,
even though they have been fully supported by undisputed scientific authorities (for instance by D’Alembert,
Lagrange, Hamilton, Landau [25, 26], Feynman [27, 28], Sedov [29]), still need to be advocated. It seems
necessary to reaffirm, at least to the advantage of the community of specialists in continuum mechanics, that
the continuum models which are needed when describing microscopically strongly inhomogeneous systems
(see e.g. [30–33]) must consider internal work functionals (see Germain [24], Salençon [34]) involving second
(and higher) gradients of virtual displacements. We want to stress that such a statement can already be found
in the work by Piola and we will see why it has sometimes been overlooked: we claim that in the theory of
higher-gradient continua one can observe the processes of erasure, loss or removal and rediscovery of scientific
knowledge which happened to many other scientific theories (see [35, 36]).
1.1. Gabrio Piola advocates the importance of variational principles in mechanics
We reproduce here some excerptions from the work published in 1848 by Piola and translated in [1].
Piola advocates the use of variational principles in mechanics: he claims that this way of thinking has been
proven by Lagrange to be the most effective. To support this statement he uses a simile by establishing a parallel
between the theory of differential curves and rational mechanics: he also explicitly states that the synthetic
analysis allowed by variational methods greatly reduces the possibility of being misled.
Since Lagrange managed to reduce all the questions of Rational Mechanics to the Calculus of Variations, the decision to insist
on avoiding its use is similar to wishing to behave as those who, being involved in the researches in higher geometry, instead of
flying to use the formulas taken from integral and differential calculus, stubbornly persist in using, in a pedestrian way, the synthetic
methods. Proceeding in this way one does not manage to get many results and one highly risks being wrong. It is instead convenient
to persuade oneself that the greater is the part in which the demonstrations are based on simple reasoning, the more they are likely
to be wrong, as the intuitive grasp of our reason is very limited and we are very easily misled as soon as the elements of the question
increase to a great number and are interconnected in a complex way.
2
Piola then proceeds by answering a usual objection of those opposing variational principles. Indeed the op-
posers of variational methods, in every historical period, always use the same argument: they cannot understand
the ‘intrinsic evidence’ of the consequences to which one can arrive by using the variational principles. They
usually ask ‘why are you using this expression for the action functional?’ or ‘why are you using this particular
virtual work functional?’ They also add: ‘Lag r angian postulation is abstract and too mathematical and cannot be
justified on physical grounds’. The opposers of Lagrangian methods declare that they need to g r asp the physical
content of every statement in the theories they use. They refuse to accept a unique basic principle (least action
or virtual work principles) and calculate from it all relevant logical consequences and, after this mathematical
process, to check if the consequences are acceptable from a phenomenological point of view. In our opinion
there is, in this kind of criticism, a certain degree of epistemological misunderstanding. If one thinks to the
principles of a theory as to its ‘true’ foundation, it is legitimate to ask for such things as the ‘intrinsic evidence’
of the principles themselves. If, on the other hand, one judges the cor rectness of a given set of postulates just by
their capability to generate a theory which accurately describes and foresees observable phenomena, it tur n s out
that there is no point in investigating such questions as the ‘truth content’ of a principle besides the relationship
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dell’Isola et al. 3
between its logical consequences and experimental evidence, other questions being purely metaphysical. In our
opinion, supported by the classic works by Popper (see e.g. [37]), this last approach is the wiser one.
It seems to us that the opposers of Piola’s Lagrangian postulation of continuum mechanics (through all ages
from 1824 until now) show the same attitude beautifully (and ironically) described by Galileo Galilei in the
following excerption from the Assayer (Il Saggiatore) (see [38, 39]).
[...]Your Excellency must consider that, for somebody who wants to prove a statement which, if not false, is at least very dubious, it is
really advantageous the possibility of using arguments which are probable, conjectures, examples, comparisons and even sophisms,
and then of fortifying and entrenching himself by means of influential texts and the authority of other philosophers, rhetors and
historians: while the genuine appeal to the severity of the geometrical demonstrations is too dangerous a challenge for those who
are not able to handle them correctly; indeed exactly as ex parte rei there is no alternative between the true and the false, also in the
necessary demonstrations either one can indubitably conclude or one inexcusably paralogizes, and there is not any other possibility
to keep oneself standing by means of limitations, distinctions, words twists and other [logical] whirligigs, and it is instead necessary,
in few words and at the first assault, to stay ‘either Caesar or nothing’. This geometrical strictness will lead me, shortly and with
less tedium for Your Illustrious Excellency, to be able to disentangle from the following demonstrations; which I will call optical or
geometrical more with the aim of seconding Mr. Sarsi rather than because I can really find in them, except for the used figures, some
perspective or geometry.
3
They declare that it is preferable to accept many different principles ‘on physical grounds’, hoping that the
whole set of assumptions will not reveal itself to be logically inconsistent. In other words: instead of accepting
only one, clearly formulated, principle, they prefer to accept many (and one by one!) principles, too often risking
being deceived; they prefer to be obliged to find the intrinsic (‘physical’?) evidence of many and different prin-
ciples instead of using a logically correct procedure leading to clear results from a well-specified assumption;
in conclusion, they prefer to indefinitely multiply the number of not-well-grounded assumptions. Let us leave
Piola to express his ideas again.
We need to use powerful methods which, representing the simultaneous and compendious expression of many principles, are able
to act by simultaneously gathering the power of all of them and are not using each of them separately and one by one, as usually
happens in the logical reasoning: [we need] methods which, once reduced to well-determined and immutable processes, do not allow
us to be deceived.
Of course, even when it is using this kind of tools our reason still keeps its rights, as it is able to recognize as true their fundaments
and correct their applications, although our reason it is not allowed, most of the time, to reach the intrinsic evidence relatively to the
consequences to which it managed to arrive.
4
Piola declares then that variational principles are one of the most poderosi (formidable, mighty) conceptual tools
to be used in mechanics.
It is in this way that in our search for the truth we manage to accomplish those great explorations in which direct reasoning is
absolutely insufficient, while it becomes advantageous again when, having reached certain destinations, we want to extend the
benefits of the obtained knowledge. One of the most formidable among the indicated tools for mechanicians is precisely the calculus
of variations. However, I deeply feel that all the present work is also very far from exhausting the fecundity of the Lagrangian
methods: I believe I can assure that with these same methods one can conquer all the various parts of mathematical physics. In the
previous Memoir we have already seen the panoply of results which can be deduced by means of its use and we treated the many
theories which could be connected with the various parts which constitute it.
5
Piola knew very well how bitter was the opposition to the Lagrangian principles and methods in mechanics
and could forecast that future opposers also would tr y to confute his reasoning. Therefore he concludes his
Memoir published in 1848 with the following words, whose correctness we strongly believe in.
I have another work ready, which is not short and which will continue the present one, and I strongly desire to be able to produce
ulterior factual evidence of the stated assertion: but whenever I am able to conclude my work and no matter how successful my own
efforts will be I am strongly persuaded that time will prove right the words with which I started this Memoir.
6
1.2. Piola’s foundation of continuum mechanics
Gabrio Piola intended to generalize to the theory of deformable bodies the methods introduced by Lagrange for
studying mechanical systems. Referring to [3] for an accurate textual analysis of his contribution we resume his
ideas here. It has to be recognized that they are topical even nowadays.
7
Piola considers a deformable body and chooses a reference (Lagrangian) configuration C
∗
for this body.
The placement function χ maps every material particle X belonging to C
∗
to the position occupied in the
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4 Mathematics and Mechanics of Solids
actual configuration by X .Denotingby
¯
X another material particle, the actual distance between the considered
particles (X ,
¯
X )isgivenby
ρ(X ,
¯
X ) =


χ(
¯
X ) − χ(X )


.(1)
Piola assumes that the internal virtual work corresponding to a virtual displacement δχ, that is, the virtual
work relative to the ‘internal’ interactions between the material particles constituting the body, can be given by
the double integral
ˆ
C
∗
ˆ
C
∗
1
2
K(X ,
¯
X , ρ) δρ (2)
where the variation δρ corresponds to the variation δχ, the scalar quantity K is introduced as the intensity of the
force (see page 309 in [1], or page 147 in the original work by Piola [40]) exerted by the particle
¯
X on the particle
X and the factor
1
2
is present as the action–reaction principle holds. The quantity K is assumed to depend on
¯
X , X and ρ and is manifestly measured in Nm
−6
(SI units). In number 72 starting on page 150 in [40] (translated
completely in [1]), Piola discusses the physical meaning of this scalar quantity and consequently establishes the
properties to be verified by the constitutive equations which have to be assigned to it. He refrains moreover from
any effort to obtain for it an expression in terms of microscopic quantities relative to the interactions between the
microscopic particles which he believes constitute the deformable body considered. On the contrary, he limits
himself to requiring that K is objective by assuming that it depends, among all possible Eulerian quantities, only
on ρ: this is an assumption which, in the sequel of his considerations, will have some important consequences.
Moreover, he argues that if one wants to deal with continua more general than fluids (for a discussion of this
point one can have a look at the recent paper [2]) then it may depend (in a symmetric way) also on the Lagrangian
coordinates of both
¯
X and X : therefore he assumes that
K(
¯
X , X , ρ) = K(X ,
¯
X , ρ).
Piola formulates the principle of virtual work (or virtual velocities, as it is called by Lag range and Piola
himself) for a deformable body as follows, where we use more modern notation than in his original formulation:
(
∀δχ
)

ˆ

(
b
m
(X ) − a(X )
)
δχ(X ) +

ˆ
C
∗
(X ,
¯
X , ρ)δρ
2
d
¯
X

dX + δW (δχ, ∂C
∗
) = 0

(3)
Here the following definition has been conveniently introduced (in order to avoid dealing with the variations of
expressions involving square roots):
 =
1
4
K
ρ
(4)
by means of which it will be possible to introduce the quantity δρ
2
instead of the quantity
1
2
Kδρ in the integral
(2). In (3), moreover, ∂C
∗
is the set of boundary material particles in the reference configuration C
∗
, b
m
(X )isthe
externally applied (volumic) mass force density, a(X ) is the acceleration of material point X ,andδW(δχ, ∂C
∗
)
is the work expended on the virtual displacement δχ because of the interactions active through the boundary
∂C
∗
plus (possibly) the first variations of the equations expressing the applied constraints on that boundary
multiplied by the corresponding Lagrange multipliers.
The particular form of the principle of virtual work presented in (3) was reformulated many years later in
an interesting effort to find an effective way to study the onset and growth of cracks in continuous bodies (for
more details see [3]).
Some attention is required regarding the choice of the external interactions functional used, δW (δχ, ∂C
∗
)
(this point is carefully discussed in [2], [3] and [41]), as it is clear that a given class of bodies, as characterized
by a class of internal work functionals, can only sustain certain classes of external interactions. The fact that,
when trying to consider non-appropriate external interactions acting on a given class of bodies, one gets some
seeming paradoxes (e.g. diverging displacements or deformation energies) should be simply considered as the
manifestation of an inconsistency intrinsic in the introduced model, of the same kind, methodologically speak-
ing, as the choice of an inconsistent set of postulates in a purely mathematical theory. Indeed, when the model
for a given class of phenomena has to be chosen one has to determine, simultaneously, the most appropriate
functional for both the external and internal work functionals.
As modern calculation tools are available to us, the presentation of the theory could, nowadays, stop here. To
solve one ‘exercise’ of mechanics one has s imply to introduce the appropriate finite element s cheme for solving
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Frequently asked questions

Q1. What have the authors contributed in "Higher-gradient continua: the legacy of piola, mindlin, sedov and toupin and some future research perspectives" ?

HAL this paper is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. 

Q2. What is the effect of the inextensibility conditions on the formation of such layers?

The formation of such layers is induced in fibre reinforcements by the inextensibility conditions which induce high gradients of stress in very narrow material regions. 

Q3. What is the role of continuum models in the design of complex materials?

The models for complex materials and structures may require the introduction of micro-structured continua endowed with additional kinematical fields, also accounting for the activation of internal degrees of freedom. 

Q4. What is the proof of the existence of stress tensor?

Cauchy’s proof of the existence of stress tensor is based on the equilibrium of contact forces with a force which is assumed to be absolutely continuous with respect to volume. 

Q5. What is the main reason Sedov believes that the mechanics of deformable bodies is concerned?

Confusion is bred, on the one hand, by the fact that the mechanics of deformable bodies is usually concerned with linear problems in which one can assume that the observer’s system coincides with the comoving system. 

Q6. How does Piola calculate the virtual work of the microscopic systems?

Then he assumes that he knows the law of interaction between any couple of particles in the lattice and therefore knows the expression of virtual work for any virtual displacement. 

Q7. What is the metric of the comoving Lagrangian coordinate system?

On the other hand, it is encouraged by the fact that the metric of the comoving Lagrangian coordinate system in the theory of liquids and gases is manifested only by way of density. 

Q8. What is the way to restructure a living tissue?

When a living tissue is to be reconstructed by the addition of an artificial, although biocompatible and eventually bioresorbable, material, it is desirable that the added material has the closest possible behaviour to the natural living tissue. 

Q9. What is the purpose of the additional boundary and other conditions?

The additional boundary and other conditions just mentioned are a means of rendering specific (‘concretizing’) individual models and particular formulations of problems. 

Q10. What is the common feature of the Cauchy simplified version of continuum mechanics?

The common feature which is shared by all systems to which the Cauchy simplified version of continuum mechanics does not apply is clear: these systems show, at the microscopic level, high contrast in geometrical and mechanical properties. 

Q11. How can the authors forecast the localization of the interfacial zones?

by means of higher-gradient models it is possible, without any further assumptions added to the choice of deformation energy, to forecast the localization and the varying mechanical properties of interface regions, which are assumed to be three-dimensionally extended.