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TPCM2017 IOP Publishing
IOP Conf. Series: Journal of Physics: Conf. Series 991 (2018) 012003 doi :10.1088/17426596/991/1/012003
Fundamentals of continuum mechanics – classical
approaches and new trends1
H Altenbach
Institut f¨ur Mechanik, OttovonGuerickeUniversit¨at Magdeburg, Magdeburg, Germany
Email: holm.altenbach@ovgu.de
Abstract. Continuum mechanics is a branch of mechanics that deals with the analysis of
the mechanical behavior of materials modeled as a continuous manifold. Continuum mechanics
models begin mostly by introducing of threedimensional Euclidean space. The points within
this region are deﬁned as material points with prescribed properties. Each material point is
characterized by a position vector which is continuous in time. Thus, the body changes in
a way which is realistic, globally invertible at all times and orientationpreserving, so that
the body cannot intersect itself and as transformations which produce mirror reﬂections are
not possible in nature. For the mathematical formulation of the model it is also assumed to
be twice continuously diﬀerentiable, so that diﬀerential equations describing the motion may
be formulated. Finally, the kinematical relations, the balance equations, the constitutive and
evolution equations and the boundary and/or initial conditions should be deﬁned. If the physical
ﬁelds are nonsmooth jump conditions must be taken into account.
The basic equations of continuum mechanics are presented following a short introduction.
Additionally, some examples of solid deformable continua will be discussed within the
presentation. Finally, advanced models of continuum mechanics will be introduced.
1. Introduction
The basic ideas with respect to continuum mechanics are given in a German textbook [1] which
is written in the style of famous Russian school of profs’ A. I. Lurie (1901–1980), V. A. Pal’mov
and P. A. Zhilin (1942–2005). These professors presenting from the 60th of the last century up
to the beginning of 2000s various courses at the Leningrad Polytechnic Institute/Peter the Great
State Polytechnical University containing various elements of Continuum Mechanics. Lurie was
more focused on elasticity problems. Palmov’s idea was to combine continuum mechanics with
rheological modeling. Zhilin presented, together with the threedimensional approach, special
courses devoted to one and twodimensional continua. Unfortunately, a part of monographs,
related to various courses, have been published only in Russian [2–5] in which the following
topics were typically discussed. The remaining monograph were translated [6–8].
The common idea of their courses was
•the use of the direct tensor notation,
•the clear split of materialindependent and materialdependent equations,
•the introduction of ﬁve balance equations only,
1The paper is dedicated to Alexander Manzhirov’s 60th birthday.
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•the formulation of boundary conditions, and
•the formulation of initial conditions.
It should be noted that the courses were strongly based on mathematics. But the starting point
was always mechanics (or physics) and only after that mathematics. This is partly in contrast
to Cliﬀord Truesdell (1919–2000) and his followers.
These items are discussed brieﬂy as follows. At ﬁrst, the elements are presented for the case
of classical continua, sometimes named Cauchy (after AugustinLouis Cauchy, 1789–1857) or
Boltzmann (after Ludwig Boltzmann, 1844–1906) continua, and then some advanced models are
introduced. The latter step is not ﬁnal since many new models were introduced in recent years.
Continuum mechanics is a branch of mechanics that deals with the analysis of the mechanical
behavior of materials and structures modeled as a continuous mass rather than as discrete
particles. Maybe the French mathematician AugustinLouis Cauchy was the ﬁrst to formulate
such models in the 19th century. Up to now, one has to understand that the starting point of
continuum mechanics is not only the axioms of Isaac Newton (1643–1727), but also the ideas of
Leonhard Euler (1707–1783). With the help of Newton’s axiom, we will get only the simplest
theory based on force interactions. Euler mentioned that there is more: for the creation of the
beam theory it was necessary to assume the independence of forces and moments. Nowadays,
research on developing new theories, based on Newton’s axioms, is being carried out in several
directions.
Some elements of continuum mechanics should be introduced from the very beginning. Firstly,
one has to assume a space: let us consider here the Euclidean threedimensional space — in
special cases twodimensional or onedimensional spaces will be used. The fourdimensional
space, discussed, for example, by Minkowski (1864–1909) and mentioned in [9, 10], is not applied
here. Next step is the introduction of the (pseudo) time t≥0. After that, one has to deﬁne
the body, with/without mass, which has a volume and a surface with an outwardpointing
normal. Last but not least, homogeneity and isotropy should be assumed among other things.
A continuum theory must be valid for solids and ﬂuids. In addition, the theory should be scale
independent. Namely, not only the theory is applicable the the macroscale but also it should
provide proper results in the case of meso, micro or nano.
The details of the direct tensor notation will be not given here. However, a very good
presentation of the tensor calculus is given in the appendices of [6, 7] and in the recent published
textbook [11]. All these references followed the ideas of Josiah Willard Gibbs (1839–1903) and
Max Lagally (1881–1945), see for example [12, 13].
2. Materialindependent equations
In this section, the materialindependent equations will be introduced, i.e., the equations are
independent from the individual material response. The kinematical relations are based only on
geometrical considerations and contain the changes of the geometry. Forces and stresses are the
basic reasons for the geometrical changes in the classical continuum mechanics. The balances
are physical principles formulated here in the general sense for the continuum volume. We have
to distinguish between balance equation and inequality. There are four balance equations, the
ﬁfth one is an inequality necessary for the process direction estimation. Boundary and initial
conditions allowing to solve initialboundary problems. Jump conditions are necessary if the
physical ﬁelds are nonsmooth.
2.1. Kinematics
The starting point for any kinematical relation is the introduction of conﬁgurations. Generally, a
minimum of two conﬁgurations is employed: the reference conﬁguration at the initial moment t0
and the actual conﬁguration at the moment t. It is obvious that instead of time we have a
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(pseudo)time. It is our decision to ﬁx the moment t0(the real moment of the beginning is
unknown). In addition, it should be underlined that in many textbooks the initial conﬁguration
is related to the Lagrangian (named after JosephLouis Lagrange, 1736–1813) description and
the reference to the Eulerian one. However historically, this is not correct: the Langrangian
description was introduced by Leonhard Euler (1707–1783) in 1762 and the Eulerian by Jean
Baptiste le Rond d’Alambert (1717–1783) in 1752 [14, 15]. It should be noted that many other
conﬁgurations may also exist: stressfree conﬁguration, damaged conﬁguration, etc. For solids,
the Lagrangian description is preferred whereas the Eulerian description is used for ﬂuids. But
this restrictive split is under question if we deal, for example, with plasticity problems.
Any conﬁguration can be introduced with the help of a reference point and a position vector.
A coordinate system at the beginning is not necessary, but may be helpful in solving problems.
At the same time, one has to select either a ﬁxed or a moving reference point. The latter allows
to increase the number of distinct conﬁgurations.
In mechanics, geometrical changes are related to the motion and to the strains. At this
stage, it is necessary to make a distinction by introducing a property of the medium: rigidity or
deformability. In the ﬁrst case, all material points of the body have the same motion (translations
and rotations), while the distance between each two arbitrary points remains the same at all
times. In the second case, we disregard the rigid motions and focus on the geometrical changes
in the inﬁnitesimal surrounding of the material points. It can be shown that these changes are
related to extension/shorting and angle changes. In the case of extension/shortening, we look
at the length of inﬁnitesimal small radiusvector whereas in the second case, the focus is on
the angle between two inﬁnitesimal small radiusvectors. Note that in some applications, both
approaches should be adopted concurrently.
The radiusvector approach for description of the material point positions results in two
strain measures in the literature. After introducing the radiusvector in both conﬁgurations
that means R
R
Rfor the reference conﬁguration and r
r
rfor the actual one, the displacement vector
can be expressed as:
u
u
u(R
R
R, t) = r
r
r(R
R
R, t)−R
R
R
with r
r
r(t=t0) = R
R
R. This approach is the common way to present the Cauchy’s strains in
the geometricallylinear theory. The second approach is related to the deformation gradient —
better gradient of the radiusvector, see [7]:
F
F
F= (∇
∇
∇R
R
Rr
r
r)T
This property is more appropriate for the case of large strains since after some manipulation a
more clear representation of the strains and rigid body motions is obtained. Note that linearizing
each one of the relations coincides with the other one. More details about diﬀerent strain
measures are presented, for example, in [2, 7]. Concerning the linearization procedure, a excellent
method is given in [16, 17].
With the strain measures and strain tensors one has only one part of the kinematics. In
addition, the velocity vector, the acceleration vector, the strain rate tensor, and the strain
gradient rate should be introduced. The calculation of time derivatives can be readily done in
the case of geometricallylinear theory while it is more complicated in geometrically nonlinear
problems. One problem arises from the objectivity of these derivatives for instance. Diﬀerent
approaches are presented in the literature, see [5] and [6] among others. Some actual statements
are presented in [18, 19].
2.2. Forces and stresses
The following classiﬁcation of the external loading can be given:
(i) Natural loading models
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Figure 1. Force stress vector.
•body / mass / volume loading (forces, moments), and
•surface / contact loading (forces, moments).
(ii) Artiﬁcial loading models
•line loading (forces, moments), and
•single point loading (forces, moments).
Note that as in the Course of General Mechanics forces and moments are introduced. In addition,
it should be mentioned that a moment is not only a moment of the force (torque) or a pair of
forces — it can deﬁned as an independent quantity. Dimensional analysis provides the following
results for the single loads:
•dimension of force [F] = N and
•dimension of moment [M] = Nm.
Hence, one can conclude that:
•body loads are expressed per unit volume,
•surface loads are expressed per unit area, and
•line loads are expressed per unit length.
Let us introduce body loadings:
•body force ρ(r
r
r, t)k
k
k(r
r
r, t) = k
k
kV(r
r
r, t), and by analogues
•ρ(r
r
r, t)l
l
l(r
r
r, t) = l
l
lV(r
r
r, t) the body moment.
Here ρ=ρ(r
r
r, t) is the density. Examples of body forces are the weight force, the inertia force
and potential force. The body moments in the classical theory will be ignored.
After introduction of forces and moments, we can deﬁne various stress vectors
•the force stress vector
t
t
t= lim
∆A→0
∆F
∆A,
•and the couple stress vector
m
m
m= lim
∆A→0
∆M
∆A.
As usual, the second one is ignored in the textbooks of strength of materials. There are several
arguments why the couple stresses can be ignored among which one is presented in [20].
Let us introduce an inﬁnitesimal small surface element dAwith the unit normal n
n
n(ﬁgure 1).
The acting force stress vector does not have the same value over the surface but the diﬀerence
of any two neighbouring magnitudes is negligible. From the deﬁnition of moment, it follows
that the moment of the couple is inﬁnitesimal since a small diﬀerence is multiplied by a small
distance. Therefore, the couple stress vector can be ignored in this case.
A counterexample is the case of stress vectors on the crack tip: even over an inﬁnitesimal small
distance, the change in the force stress vector can be huge. Such cases initiate the arguments over
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introducing couple stresses. Maybe this was the reason for extending the appendix in [20] and
to publish another book on elasticity taking into account asymmetric stress tensors in addition
to the symmetric ones [21].
Note Similar derivations can be established for both the actual and the reference conﬁguration.
Below we are performing the procedures for the actual conﬁguration. That means writing ∇
∇
∇
implies ∇
∇
∇R
R
R, also t
t
tand T
T
Tare the Cauchy stress vector and tensor in the actual conﬁguration,
respectively.
The components of the stress vector can be presented in Cartesian coordinates as
t
t
t=tn
n
nn
n
n+tt
t
te
e
et=tn
n
nn
n
n+tt1e
e
et1+tt2e
e
et2,
where e
e
et1and e
e
et2are two arbitrary tangential directions on the surface dA. Since we have a
Cartesian coordinate system n
n
n⊥e
e
et1,n
n
n⊥e
e
et2, and e
e
et1⊥e
e
et2, so n
n
n,e
e
et1, and e
e
et2form an arbitrary
orthonormal unit base.
Following Cauchy, a lemma can be written down
t
t
t(r
r
r, n
n
n, t) = n
n
n·T
T
T(r
r
r, t)
connecting the stress vector with the stress tensor. Cauchy got his lemma from equilibrium
considerations [22]. Note that in many textbooks [23, for example] the lemma is presented in a
diﬀerent way
t
t
t(r
r
r, n
n
n, t) = T
T
T(r
r
r, t)·n
n
n.
This is equivalent to the linear mapping of two vector spaces with the help of a second rank
tensor. Both formulations coincide if the stress tensor is symmetrical.
As in General Mechanics now the resultant force
F
F
FR=ZV
ρk
k
kdV+ZA
t
t
tdA
and the resultant moment
M
M
MR=ZV
(r
r
r×ρk
k
k+ρl
l
l) dV+ZA
(r
r
r×t
t
t+m
m
m) dA
can be introduced. Taking into account only force actions the static equilibrium can be
formulated for the continuum with F
F
FR= 0
0
0 and M
M
MR= 0
0
0
ZV
ρk
k
kdV+ZA
t
t
tdA= 0
0
0,ZV
(r
r
r×ρk
k
k) dV+ZA
(r
r
r×t
t
t) dA= 0
0
0.
With the divergence theorem (formulated independently by Carl Friedrich Gauß, 1777–1855,
and Michail Wassiljewitsch Ostrogradski, 1801–1862) for the ﬁrst equilibrium equation
ZA
t
t
tdA=ZA
n
n
n·T
T
TdA=ZV
∇
∇
∇ · T
T
TdA
one gets ZV
(ρk
k
k+∇
∇
∇ · T
T
T) dV= 0
0
0
and the local form can be obtained (if all ﬁelds are smooth)
∇
∇
∇ · T
T
T+ρk
k
k= 0
0
0
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Figure 2. Formulation of the general balance equation.
Applying the divergence theorem to the second equilibrium equation the symmetry condition
for the stress tensor (T
T
T=T
T
TT) can be established. Finally, with the d’Alambert’s Principle one
gets the dynamic equations (Euler’s equation of motion)
ZV
ρk
k
kdV+ZA
t
t
tdA−ZV
ρ¨
r
r
rdV= 0
0
0.
Applying again the divergence theorem we obtain
ZV
(ρk
k
k+∇
∇
∇ · T
T
T−ρ¨
R
R
R) dV= 0
0
0
and ﬁnally, if all ﬁelds are smooth, the local form is obtained
∇
∇
∇ · T
T
T+ρk
k
k=ρ¨
r
r
r.
2.3. Balances
2.3.1. General balance equation. Let us introduce some quantity named balance variable of the
continuum Ψ(r
r
r, t) and deﬁned in the actual conﬁguration. The integration over all body points
results in Y(t)
Y(t) = ZV
Ψ(r
r
r, t) dV.
With dV= detF
F
FdV0the recalculation of the properties for the reference conﬁguration is
possible.
The rate of Y(t) is inﬂuenced by actions onto the volume and through the surface (ﬂux)
D
DtY(t) = D
DtZV
Ψ(r
r
r, t) dV=ZA
Φ(r
r
r, t) dA+ZV
Ξ(r
r
r, t) dV,
where D/Dtis the total time derivative, Φ and Ξ are named ﬂux and supply, respectively. The
visualization of the general balance equation is given in ﬁgure 2.
It should be underlined that the general balance equations can be introduced in a diﬀerent
form with three terms on the righthand side [23]. Instead of one surface and one volume integral
for the ﬂux and the supply one surface and two volume integrals for the ﬂux, the supply and
the production are introduced. But there are only few applications for production. If Cauchy’s
lemma is valid for the general form after application of the divergence theorem one gets again
the local form.
Note If the righthand side of the general balance is equal to zero we have a conservation law. In
other words, the ﬁrst integral is a constant.
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2.3.2. Balance of mass. In solid mechanics applications, the formulation of the balance
equation for mass starts with
m=ZV
ρ(r
r
r, t) dV= const,
where mis the mass. Then the integral form of the balance of mass is
Dm
Dt=D
DtZV
ρ(r
r
r, t) dV= 0
and the local form is D
Dt(dm) = D
Dt(ρdV) = 0.
From this equation the continuity equation can be deduced
Dρ
Dt+ρ∇
∇
∇ · v
v
v= 0
with the velocity v
v
v=˙
r
r
r. If ρ= const with respect to the time the incompressibility condition is
following as
∇
∇
∇ · v
v
v= 0.
2.3.3. Balance of momentum. By introducing the linear momentum of the continuum, the
integral form can be written down as
D
DtZV
ρ(r
r
r, t)v
v
v(r
r
r, t) dV=ZA
t
t
t(r
r
r, t)dA+ZV
ρ(r
r
r, t)k
k
k(r
r
r, t) dV.
The local form is D
Dtρ(r
r
r, t)v
v
v(r
r
r, t) = ∇
∇
∇ ·
·
·T
T
T(r
r
r, t) + ρ(r
r
r, t)k
k
k(r
r
r, t).
2.3.4. Balance of moment of momentum. By introducing the angular momentum of the
continuum, the integral form can be written down as
D
DtZV
r
r
r×ρ(r
r
r, t)v
v
v(r
r
r, t)dV=ZA
r
r
r×t
t
t(r
r
r, t)dA+ZV
r
r
r×ρ(r
r
r, t)k
k
k(r
r
r, t) dV.
If the balance of momentum is valid as given in the previous section, the local form results again
in the symmetry condition for the stress tensor.
2.3.5. Balance of energy Let us assume at the beginning that we have only mechanical actions.
In this case, the balance of energy, which is equivalent to the ﬁrst law of thermodynamics, can
be presented as
D
DtZV1
2v
v
v·v
v
v+uρdV=ZA
t
t
t·v
v
vdA+ZV
k
k
k·v
v
vρ dV.
The kinetic and potential energy is inﬂuenced only by surface and volume forces. If the balance
of moment The local form can be expressed as
ρ˙u=T
T
T··(∇
∇
∇v
v
v)T=T
T
T··D
D
D.
The general ﬁrst law of thermodynamics states that:
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The changes in time of the total energy Wwithin the volume is equal to the heat ﬂux Q
and the power of all external loadings Pext
D
DtW=Pext +Q,
where W=U+Kis the sum of the inner energy Uand the kinetic energy K.
Let us introduce the four terms
K=1
2ZV
v
v
v·v
v
vρ dV,
U=Zm
udm=ZV
ρu dV,
Pext =ZA
t
t
t·v
v
vdA+ZV
k
k
k·v
v
vρ dV,
Q=ZV
ρr dV−ZA
n
n
n·h
h
hdA.
Finally, we have
D
DtZVu+1
2v
v
v·v
v
vρdV=ZA
t
t
t·v
v
vdA+ZV
k
k
k·v
v
vρ dV−ZA
n
n
n·h
h
hdA+ZV
ρr dV.
With the help of the Cauchy lemma and after some manipulations the local form can be deduced
(it is assumed that the balance of momentum is valid):
Du
Dt+T
T
T··D
D
D− ∇
∇
∇ · h
h
h+ρr = 0.
2.3.6. Balance of entropy. The last balance cannot be formulated in the standard form since
now we have two formulations: as an equation for nondissipative processes or as an inequality
for dissipative processes. The decision on the type of process can only be made after introducing
statements concerning the material behavior.
Let us introduce the second law of thermodynamics in the integral form
D
DtZV
ρs dV≥ZV
r
ΘρdV−ZA
n
n
n·h
h
h
ΘdA,
where sis the speciﬁc entropy, ris a energy source, Θ denotes the absolute temperature, and h
h
his
the heat ﬂux vector. The changes in time of the entropy within the volume under consideration
is not smaller then the rate of the outer entropy ﬂux. Again after some manipulations
ZA
n
n
n·h
h
h
ΘdA=ZV
∇
∇
∇ · h
h
h
ΘdV=ZV∇
∇
∇ · h
h
h
Θ−h
h
h· ∇
∇
∇Θ
Θ2dV
and taking into account 1
Θh
h
h· ∇
∇
∇Θ = h
h
h· ∇
∇
∇ln Θ
we get the local form
ρθ Ds
Dt−ρr − ∇
∇
∇ ·
·
·h
h
h−h
h
h·
·
· ∇
∇
∇ln θ≥0.
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Figure 3. Temperature ﬁeld Θ1<Θ2<Θ3.
The underlined terms are also in the ﬁrst law of thermodynamics. Combining both equations
we get
ρθ Ds
Dt−Du
Dt+T
T
T··D
D
D−h
h
h·
·
· ∇
∇
∇ln θ≥0.
With the Helmholtz free energy
u−Θs=f
we can deﬁne the dissipation function
T
T
T··D
D
D−ρ(˙
f+s˙
Θ) = Φ ≥0.
Finally, we obtain
Φ−h
h
h·
·
· ∇
∇
∇ln θ≥0.
Since Φ is positive, the remaining part can be written
h
h
h·
·
· ∇
∇
∇ln θ≤0.
The limiting cases are deﬁned by the following two processes:
•h
h
h= 0
0
0 adiabatic process and
• ∇
∇
∇Θ = 0
0
0 isothermal process.
Assuming a nondissipative process Φ = 0 with temperature ﬁeld (ﬁgure 3).
The following solution is valid only for
∠(h
h
h,∇
∇
∇Θ) >π
2.
Note that the perpendicular case is excepted.
Going back to the ﬁrst law
ρΘDs
Dt=T
T
T··D
D
D−ρDf
Dt+sDΘ
Dt+ρr − ∇
∇
∇ · h
h
h= Φ + ρr − ∇
∇
∇ · h
h
h
the following situations can be considered:
•nondissipative process: Φ = 0
ρΘDs
Dt=ρr − ∇
∇
∇ · h
h
h,
which is a heat transfer equation,
•isothermal process: no heat transfer, mechanical and thermal processes are decoupled, and
•adiabatic process with h
h
h= 0
0
0
ρΘDs
Dt=ρr.
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2.4. Boundary and initial conditions
The balance equations in the local form are a system of coupled diﬀerential equations in terms of
the positionvector and time. That means for solving the system one should formulate boundary
and initial equations.
There are diﬀerent types of boundary conditions:
•A boundary condition which speciﬁes the value of the function itself is a Dirichlet boundary
condition (Peter Gustav Lejeune Dirichlet, 1805–1859).
•A boundary condition which speciﬁes the value of the normal derivative of the function is
a Neumann boundary condition (Carl Gottfried Neumann, 1832–1925).
•A boundary condition which speciﬁes the sum of the value of the function itself multiplied
by a constant and the value of the normal derivative of the function multiplied by another
constant is a Robin boundary condition (Victor Gustave Robin, 1855–1897).
•If the boundary has the form of a curve or surface that gives a value to the normal derivative
and the variable itself then it is a Cauchy boundary condition.
•Last but not least there are mixed boundary conditions that on one part of the boundary
is given one condition, on another part — another condition.
If we have a pure mechanical problem, the simplest boundary conditions are related to the
displacements or the stress vector on the boundary. In more complicated problems, the
conditions should also be formulated for the temperature or the heat ﬂux vector, etc.
With respect to the time, we have diﬀerential equations of second order in the case of a
pure mechanical problem. That means one can formulate initial conditions for the function
itself or the ﬁrst time derivative. Examples are a prescribed displacement vector and/or velocity
velocity at the beginning t0. If both values are zero, we get the static solution only (no motion
or constant velocity for all points of the continuum). In more complex problems, we have more
initial conditions. The temperature problem, for example, is a ﬁrst order diﬀerential equation
with respect to time. That means the initial temperature can be prescribed.
2.5. Jump conditions
Up to now, it was assumed that the physical ﬁelds are continuously diﬀerentiable within the
volume in both conﬁgurations. This assumption was also introduced for the Reynolds transport
theorem (Osborne Reynolds, 1842–1912) necessary for the recalculation of the actual balances
into balances in the reference conﬁguration and the divergence theorem. If a surface within a
material body exists with a discontinuous physical quantity, it is called a singular surface [23].
Examples are shock front in supersonic ﬂows, surface between two diﬀerent bodies (skisnow,
atmosphereocean) or the front between two phases. In all these cases, the balance can be
formulated in the integral form but they should incorporate additional terms expressing the
jump conditions. Details concerning the jump conditions are presented in [23–27] among others.
3. Materialdependent equations
Now our focus is on the speciﬁc (individual) response of the given material under an arbitrary
load. We should include the information on material behavior since the number of governing
equation is not equal to the number of unknowns in the set of governing equations. For example,
consider a homogeneous threedimensional solid which is only mechanically loaded. In this case,
we have the following governing equations:
•balance of mass — one scalar equation,
•balance of momentum — one vectorial or three scalar equation,
•balance of moment of momentum — one vectorial or three scalar equation,
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•balance of energy — one scalar equation, and
•balance of entropy — one scalar expression.
The unknown are:
•the density (one scalar),
•the displacement vector (one vector or three scalars),
•the stress tensor (one second rank tensor or 9 scalars), and
•the inner energy (one scalar).
It is obvious that we have only 8 governing equations, but 14 unknowns. To close the gap, we
need more equations which can be established only with the help of information on the behavior
of the materials. The material behavior itself is speciﬁc to each material, and thus a universal
constitutive equation cannot be established.
3.1. Constitutive equations
The formulation of constitutive equations cannot be based on some physical principles (exception
is that the requirements of the second of thermodynamics/balance of entropy should be fulﬁlled).
The following modeling principles are established in the literature:
•inductive approach that is moving from the simplest to more complex models,
•deductive approach that is moving from a general frame to more speciﬁc cases, and
•rheological modeling which is a combination of both approaches.
The constitutive equations contain parameter, parameter functions, etc. which should be
identiﬁed with the help of
•experimental observations (real or virtual),
•mathematical analysis, and
•theory of symmetry [28].
The last one should be used together with the CurieNeumann’s principle (Pierre Curie, 1859–
1906, Fritz Ernst Neumann, 1798–1895). In addition, sometimes the models are restricted by
some constraints (for example, the incompressibility condition). There are a lot of references
in this matter among which is [17]. Herein, the deductive approach and the problems of
identiﬁcation are discussed. In addition, a classiﬁcation of the material behavior can be done as
follows:
•spontaneous material behavior with the subclasses of elastic and plastic behavior, and
•timedependent material behavior with the subclasses of viscoelastic and viscoplastic
behavior.
Formulation of suitable constitutive and evolution equations ﬁnally should be proved by
•checking the correctness of the formulation and the adequateness in terms of thermodynamic
considerations and
•experimental validation or falsiﬁcation, etc.
Let us introduce the following deﬁnition: Constitutive equations connect all macroscopic
phenomenological variables describing the behavior of the continuum [29]. It is clear that this
deﬁnition is very general and we need some restrictions with respect to the mathematical form
of the constitutive equations. Such restrictions are introduced in [30] among others. One basic
deﬁnition in this paper is: Simple materials of the rank 1 are materials which are described
by constitutive equations connecting local variables, e.g. the local strain tensor and the local
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heat ﬂux vector with the local stress tensor and the local temperature gradient, respectively. All
statements are related to the same material point and its diﬀerential neighborhood of rank 1. As a
consequence only the ﬁrst gradients are in the constitutive equations, higher order gradients are
ignored. The next important deﬁnition is: Process is the change of the constitutive parameters
with respect of time [2]. In addition, one can assume: The behavior of the continuum in each
material point is given as a set of constitutive variables which are operators with respect to
time [2].
If we want to distinguish solids and ﬂuids, several deﬁnitions are presented in the literature:
•Under a given load, a solid will have nonzero components in the stress deviator, i.e., it
shows resistance against any shape change.
•In contrast, under a given, a ﬂuid will have only zero components in the stress deviator,
i.e., it does not preserve its shape.
The starting point of the deductive approach in formulation of the constitutive equations is
introducing of several axioms of the material theory [17]:
•causality,
•determinism,
•equipresence,
•material objectivity,
•local action,
•memory, and
•physical consistency.
A typical example of the set of constitutive equations is
T
T
T(R
R
R, τ ) = P
P
P{R
R
R, θ(R
R
R, τ ),˙
θ(R
R
R, τ ),∇
∇
∇r
r
rθ(R
R
R, τ ),∇
∇
∇r
r
r˙
θ(R
R
R, τ ),Γ
Γ
Γ(R
R
R, τ )},
h
h
h(R
R
R, τ ) = h
h
h{R
R
R, θ(R
R
R, τ ),˙
θ(R
R
R, τ ),∇
∇
∇R
R
Rθ(R
R
R, τ ),∇
∇
∇r
r
r˙
θ(R
R
R, τ ),Γ
Γ
Γ(R
R
R, τ )},
f(R
R
R, τ ) = f{R
R
R, θ(R
R
R, τ ),˙
θ(R
R
R, τ ),∇
∇
∇R
R
Rθ(R
R
R, τ ),∇
∇
∇r
r
r˙
θ(R
R
R, τ ),Γ
Γ
Γ(R
R
R, τ )},
s(R
R
R, τ ) = s{R
R
R, θ(R
R
R, τ ),˙
θ(R
R
R, τ ),∇
∇
∇R
R
Rθ(R
R
R, τ ),∇
∇
∇r
r
r˙
θ(R
R
R, τ ),Γ
Γ
Γ(R
R
R, τ )},
describing simple thermomechanical material behavior. The constitutive parameters are: the
temperature, the temperature rate, the temperature gradient, the temperature rate gradient and
a strain measure. Since 0 < τ < t, the constitutive equations are functionals. Other examples
are given, for example, in [2, 14, 15].
Several simpliﬁcations can be made such as speciﬁcation of the strain measure, neglecting
aging eﬀects and viscositywhich is equivalent to removing the explicit time dependency. Among
such cases, the following simpliﬁed ones are worth mentioning:
•nonlinear elastic anisotropic material behavior
T
T
T(F
F
F) = 2ρF
F
F·u,C
C
C(C
C
C)·F
F
FT
with C
C
Cas the right CauchyGreen strain tensor (C
C
C=F
F
FT·F
F
F).
•nonlinear elastic isotropic material behavior
T
T
T(F
F
F) = 2ρF
F
F·(φ0I
I
I+φ1C
C
C+φ2C
C
C2)·F
F
FT
with the functions φi(i= 0,1,2) depending on the ﬁrst, second and third invariants of the
tensor C
C
C.
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•nonlinear elastic isotropic incompressible material behavior
T
T
T= 2ρF
F
F·∂u
∂I1
+I1
∂u
∂I2I
I
I−∂u
∂I2
C
C
C·F
F
FT−pI
I
I
with the hydrostatic pressure p, and
•simple thermoelastic material behavior.
The inductive approach is the more engineering way of formulation constitutive equations.
For example, we start we the Hooke’s law
•tension with σ— normal stress, E— Young’s modulus (named after Thomas Young,
1773–1829) and εnormal strain
σ=Eε
•or torsion with τ— shear stress, G— shear modulus and γ— shear strain
τ=Gγ
and generalize towards the following cases:
•threedimensional isotropic case,
•threedimensional anisotropic case,
•nonlinear behavior,
•. . .
In each case, the thermodynamical consistency, which is guaranteed in the deductive approach,
should be checked separately.
The last possibility to formulate constitutive equations is the rheological modeling [2, 8, 29].
This approach is founded on
•the introduction of some basic models, for example, related to
–the elastic behavior,
–the plastic behavior, and
–the viscous behavior,
•and the assumption that the connection of basic models can be realized only
–in parallel or
–in series.
Then any complex behavior can be represented by these connection, for example,
•viscoelastic = elastic + viscous
•viscoplastic = plastic + viscous
The rheological modeling was introduced in [31] and discussed in [2, 8] for isotropic and
anisotropic materials.
3.2. Evolution equations
In many cases it is enough to present the material behavior by constitutive equations discussed in
the previous subsection. Sometimes the changes of the material behavior are related to evolution
processes. Examples are
•the establishment of plastic zones,
•the development of damage processes,
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•creep processes,
•the deformation induced anisotropy,
•the hardening and softening processes, or
•phase transformation processes.
The description of the evolution process is based on ordinary diﬀerential equations with respect
to time derivatives of ﬁrst order. Special initial conditions should be introduced. For example,
if we have creep or plastic evolution the initial state is assumed to be elastic.
4. Advanced models
The last part of this paper presents some new directions in continuum mechanics. This overview
is not comprehensive and only few comments are given. Nevertheless, the required references
are given for further reading.
4.1. Lower dimensional continuum models
These models are interesting from the theoretical point of view. Commonly, structural models
(beam, rod, plate, shell, etc.) are presented in the literature as sets of equations which are
reduced from a general threedimensional equation by means of hypotheses, mathematical
simpliﬁcation, etc. Examples of such hypotheses are:
•EulerBernoulli beam theory, for example, is based on the assumption that the beam cross
sections remain planar and perpendicular to the neutral axis before and after deformation,
and
•Kirchhoﬀ plate theory, for example, assumes that the line elements of the beam remain
straight and perpendicular to the midplane before and after deformation.
Mathematical approaches are based on power series or asymptotic integration (in both case a
small parameter is assumed).
A third way is more elegant and natural. At the beginning it is assumed that we have a
deformable surface or line. Then, an exact continuum theory can be deduced. The disadvantage
of this approach is the identiﬁcation of the parameters in the constitutive equations.
Recently overviews concerning plate and shell theories and diﬀerent approaches were
published in [32, 33]. A micropolar plate model was presented in [34] and extended to micropolar
shells in [35]. A direct theory of rods was suggested in [36, 37]. Further discussion concerning
actual trends are given in [38, 39] among others.
4.2. Nanostructures
Nanostructures are a new class of structures with special properties. In many cases they are
stiﬀer which is a result of high speciﬁc surface to volume ratio. In the case of classical structures
the bulk behavior is dominant — in the case of nanostructures the stiﬀness properties are under
the inﬂuence of the surface behavior.
A large number of publications were released in recent years which makes it almost impossible
to provide a comprehensive overview. But within a more speciﬁc framework, the pioneering
works of [40, 41] suggest considering the surface eﬀects in the continuum mechanics of the bulk
material — where it is relevant. Some publications in this context are [42–45]. References for
further reading are given within these publications. An overview concerning surface eﬀects is
given also in [46].
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4.3. Growing Solids
In June 2015, Alexander Manzhirov has organized an IUTAM2Symposium on growing solids
in Moscow. From the ﬁrst announcement, one can comprehend this class of solids by means
of a nice description: The vast majority of objects or solids which surround us arise from
some growth processes. As an example, one can present such natural phenomena as growth
of biological tissues, glaciers, blocks of sedimentary and volcanic rocks, as well as space objects.
Similar processes determine the speciﬁc features of many technologies in industry, including
wellknown technologies of crystal growth, laser deposition, solidiﬁcation of melts, electrolytic
formation, pyrolytic deposition, polymerization and concreting. Recent research has shown that
solids, which were formed due to the processes of growth, diﬀer essentially in their properties
from solids in the traditional view. Moreover, the classical approaches of solid mechanics to the
modeling of growing solids behavior fail. They have to be replaced by new ideas and methods
of modern mechanics, mathematics, physics, and engineering sciences. Thus, at present, a new
area of solid mechanics, which deals with the construction of adequate models for solids growth
processes is forming.
There are worldwide diﬀerent schools investigating growing solids. One of them was
founded by N. Kh. Arutyunyan (1912–1993), who published a monograph in Russian on this
topic [47]. Actual problems of the mechanics are the general theory of growth processes of
solids, the numerical modeling of growth, moving boundaries and interfaces, surface eﬀect,
phase transitions, dislocations and disclinations in solids, crystal growth, growth of bone and
soft tissues, laser deposition, solidiﬁcation of melts, and electrolytic formation. Some of them
are discussed, for example, in [48–50].
4.4. Reinforced materials
The ﬁnal example of advanced materials is related to reinforced materials. By combining two
or more diﬀerent materials, one gets a new material with new properties. Considering the
speciﬁc eﬀective properties, it is obvious that, as a result a lightweight structural material
is designed. The classical case is the are unidirectional ﬁbrereinforced layers. Finally, the
layers can be combined and a laminate can be established. The classical ideas are published in
various textbooks and monographs, for example in [51]. One of the actual problems is related
to particlereinforced matrix materials while taking into account the interphases [52–54].
5. Final Remarks
There are much more advanced models. At the moment we have a renaissance of the
Cosserat theory (Eug`ıne Cosserat, 1866–1931, Fran¸cois Cosserat, 1852–1914) with applications
to foams [34] or bones [55]. Further developments are related to micropolar continua [56].
Another direction is related to the application of higher gradients [57–60].
Summarizing up to now Continuum Mechanics is a actual branch of Mechanics with new
directions. The limits are not clear since the continuum mechanics approach is applied to
problems with smaller and smaller sizes. There is no need for new theories — only the classical
continuum mechanics should be improved. There is only one important item: the requirements
of the continuum deﬁnition should be fulﬁlled.
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