Fundamentals of continuum mechanics – classical approaches and new trends

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DOI: 10.1088/1742-6596/991/1/012003
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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 three-dimensional 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 orientation-preserving, 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 non-smooth 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.
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Fundamentals of continuum mechanics – classical
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TPCM-2017 IOP Publishing
IOP Conf. Series: Journal of Physics: Conf. Series 991 (2018) 012003 doi :10.1088/1742-6596/991/1/012003
Fundamentals of continuum mechanics – classical
approaches and new trends1
H Altenbach
Institut f¨ur Mechanik, Otto-von-Guericke-Universit¨at Magdeburg, Magdeburg, Germany
E-mail: 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 three-dimensional 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 orientation-preserving, 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 non-smooth 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 three-dimensional approach, special
courses devoted to one- and two-dimensional 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 material-independent and material-dependent 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 Augustin-Louis 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 Augustin-Louis 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 three-dimensional space — in
special cases two-dimensional or one-dimensional spaces will be used. The four-dimensional
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 t0. After that, one has to define
the body, with/without mass, which has a volume and a surface with an outward-pointing
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 macro-scale 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. Material-independent equations
In this section, the material-independent 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 initial-boundary problems. Jump conditions are necessary if the
physical fields are non-smooth.
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 Joseph-Louis 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: stress-free 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 radius-vector whereas in the second case, the focus is on
the angle between two infinitesimal small radius-vectors. Note that in some applications, both
approaches should be adopted concurrently.
The radius-vector approach for description of the material point positions results in two
strain measures in the literature. After introducing the radius-vector 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 geometrically-linear theory. The second approach is related to the deformation gradient —
better gradient of the radius-vector, 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 geometrically-linear theory while it is more complicated in geometrically non-linear
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
A0
F
A,
and the couple stress vector
m
m
m= lim
A0
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
ne
e
et1,n
n
ne
e
et2, and e
e
et1e
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
tdAZV
ρ¨
r
r
rdV= 0
0
0.
Applying again the divergence theorem we obtain
ZV
(ρk
k
k+
∇ · T
T
Tρ¨
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 right-hand 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 right-hand 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
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
dV,
U=Zm
udm=ZV
ρu dV,
Pext =ZA
t
t
t·v
v
vdA+ZV
k
k
k·v
v
dV,
Q=ZV
ρr dVZA
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
dVZA
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 non-dissipative 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 dVZV
r
ΘρdVZA
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
Θh
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
Dtρr − ∇
∇ ·
·
·h
h
hh
h
h·
·
· ∇
ln θ0.
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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
DtDu
Dt+T
T
T··D
D
Dh
h
h·
·
· ∇
ln θ0.
With the Helmholtz free energy
uΘs=f
we can define the dissipation function
T
T
T··D
D
Dρ(˙
f+s˙
Θ) = Φ 0.
Finally, we obtain
Φh
h
h·
·
· ∇
ln θ0.
Since Φ is positive, the remaining part can be written
h
h
h·
·
· ∇
ln θ0.
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 non-dissipative 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
DρDf
Dt+s
Dt+ρr − ∇
∇ · h
h
h= Φ + ρr − ∇
∇ · h
h
h
the following situations can be considered:
non-dissipative 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 position-vector 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 (ski-snow,
atmosphere-ocean) 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. Material-dependent 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 three-dimensional 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 Curie-Neumann’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
time-dependent material behavior with the subclasses of visco-elastic and visco-plastic
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 non-zero 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 Cauchy-Green 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
FTpI
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=
or torsion with τ— shear stress, G— shear modulus and γ— shear strain
τ=
and generalize towards the following cases:
three-dimensional isotropic case,
three-dimensional 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,
visco-elastic = elastic + viscous
visco-plastic = 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 three-dimensional equation by means of hypotheses, mathematical
simplification, etc. Examples of such hypotheses are:
Euler-Bernoulli 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 mid-plane 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
well-known 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 world-wide 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 light-weight structural material
is designed. The classical case is the are uni-directional fibre-reinforced 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 particle-reinforced 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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  • Book
    Das Buch führt in möglichst einfacher Weise in die Grundlagen der Kontinuumsmechanik ein, wobei der Schwerpunkt bei festen deformierbaren Körpern liegt. Zahlreiche Beispiele mit vollständigen Lösungen illustrieren den theoretischen Teil und erleichtern so das Verständnis. In der 4. Auflage wurden zahlreiche Abschnitte überarbeitet und präzisiert, wobei auch die unterschiedlichen Konzepte der Kontinuumsmechanik noch deutlicher gemacht werden. Zahlreiche Fehler wurden beseitigt. Gleichzeitig wurde die Referenzliteratur erweitert sowie die Liste der weiterführenden Literatur ergänzt und aktualisiert. Der Inhalt Teil I Grundbegriffe und mathematische Grundlagen.- Einführung.- Mathematische Grundlagen der Tensoralgebra und Tensoranalysis.- Teil II Materialunabhängige Gleichungen.- Kinematik des Kontinuums.- Kinetische Größen und Gleichungen.- Bilanzgleichungen.- Teil III Materialabhängige Gleichungen.- Materialverhalten und Konstitutivgleichungen.- Deduktiv abgeleitete Konstitutivgleichungen.- Induktiv abgeleitete Konstitutivgleichungen.- Methode der rheologischen Modelle.- Teil IV Anfangs-Randwertprobleme der Kontinuumsmechanik.- Grundgleichungen der linearen Elastizitätstheorie.- Grundgleichungen linearer viskoser Fluide.- Teil V Anhang.- Elastizitäts- und Nachgiebigkeitsmatrizen. Die Zielgruppen Das Buch richtet sich an Studierende im Bereich Maschinenbau und Bauingenieurwesen, Physik und Technomathematik sowie an Wissenschaftler und Praktiker in der Industrie. Der Autor Holm Altenbach studierte am Leningrader Polytechnischen Institut (heute St. Petersburger Staatliche Polytechnische Universität „Peter der Große“), wo er auch promoviert wurde und sich habilitierte. Sein beruflicher Weg begann an der TH Magdeburg, ab 1996 war er Professor für Technische Mechanik an der Martin‐Luther‐Universität Halle‐Wittenberg, seit 2011 vertritt er dieses Gebiet an der Otto‐von‐Guericke‐Universität Magdeburg.
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    This textbook is written for use not only in engineering curricula of aerospace, civil and mechanical engineering, but also for materials science and applied mechanics. Furthermore, it addresses practicing engineers and researchers. No prior knowledge of composite materials and structures is required for the understanding of its content. The structure and the level of presentation is close to classical courses of "Strength of Materials" or "Theory of Beams, Plates and Shells". Yet two extensions have been included: the linear elastic material behavior of isotropic and non-isotropic structural elements, and inhomogeneous material properties in the thickness direction. The Finite Element Analysis of laminate and sandwich structures is briefly presented. Many solved examples illustrate the application of the techniques learned.
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    This book summarizes the actual state of the art and future trends of surface effects in solid mechanics. Surface effects are more and more important in the precise description of the behavior of advanced materials. One of the reasons for this is the well-known from the experiments fact that the mechanical properties are significantly influenced if the structural size is very small like, for example, nanostructures. In this book, various authors study the influence of surface effects in the elasticity, plasticity, viscoelasticity. In addition, the authors discuss all important different approaches to model such effects. These are based on various theoretical frameworks such as continuum theories or molecular modeling. The book also presents applications of the modeling approaches.
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    The book presents mathematical and mechanical aspects of the theory of plates and shells, applications in civil, aero-space and mechanical engineering, as well in other areas. The focus relates to the following problems: • comprehensive review of the most popular theories of plates and shells, • relations between three-dimensional theories and two-dimensional ones, • presentation of recently developed new refined plates and shells theories (for example, the micropolar theory or gradient-type theories), • modeling of coupled effects in shells and plates related to electromagnetic and temperature fields, phase transitions, diffusion, etc., • applications in modeling of non-classical objects like, for example, nanostructures, • presentation of actual numerical tools based on the finite element approach.
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    This invaluable treatise belongs to the cultural heritage of mechanics. It is an encyclopaedia of the classic and analytic approaches of continuum mechanics and of many domains of natural science. The book is unique also because an impressive number of methods and approaches it displays have been worked out by the author himself. In particular, this implies a full consistency of notation, ideas and mathematical apparatus which results in a unified approach to a broad class of problems. The book is of great interest for engineers who will find a lot of analytical formulae for very different problems covering nearly all aspects of the elastic behavior of materials. In particular, it fills the gap between the well-developed numerical methods and sophisticated methods of elasticity theory. It is also intended for researchers and students taking their first steps in continuum mechanics as it offers a carefully written and logically substantiated basis of both linear and nonlinear continuum mechanics.
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    The concept of equivalent inhomogeneity has been introduced to analyze the effective properties of composites with interphases using techniques devised for problems without interphases. The basic idea is to replace the inhomogeneity and the surrounding it interphase by a single equivalent inhomogeneity with constant stiffness tensor, combining properties of both, which is then perfectly boned to the matrix. In this presentation a new definition of equivalent inhomogeneity is discussed. It is based on Hill's energy equivalence principle, applied to the problem consisting only of the original inhomogeneity and its interphase. It is more general than the definitions proposed in the past in that, conceptually and practically, it allows to consider inhomogeneities of various shapes and various models of interphases. This is illustrated considering spherical and cylindrical particles with two models of interphases, Gurtin-Murdoch material surface model and spring layer model. The resulting equivalent inhomogeneities are subsequently used to determine effective properties of randomly distributed unidirectional particulate composites. Properties of the equivalent cylindrical inhomogeneities are transversely isotropic, thus the method of conditional moments, which is a statistical method capable of handling anisotropy and randomness, has been employed for that purpose. Closed-form formulas for the effective stiffness tensor have been developed in all cases considered here. Comparisons with solutions available in the literature are made and other possible applications are discussed.
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    Springer Nature SharedIt: http://rdcu.be/tfs6 In continuum mechanics, there exists a unique theory for elasticity, which includes the first gradient of displacement. The corresponding generalization of elasticity is referred to as strain gradient elasticity or higher gradient theories, where the second and higher gradients of displacement are involved. Unfortunately, there is a lack of consensus among scientists how to achieve the generalization. Various suggestions were made, in order to compare or even verify these, we need a generic computational tool. In this paper, we follow an unusual but quite convenient way of formulation based on action principles. First, in order to present its benefits, we start with the action principle leading to the well-known form of elasticity theory and present a variational formulation in order to obtain a weak form. Second, we generalize elasticity and point out, in which term the suggested formalism differs. By using the same approach, we obtain a weak form for strain gradient elasticity. The weak forms for elasticity and for strain gradient elasticity are solved numerically by using open-source packages---by using the finite element method in space and finite difference method in time. We present some applications from elasticity as well as strain gradient elasticity and simulate the so-called size effect.
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    1 Foundations and equations of continuum mechanics.- 2 Plasticity theory and internal friction in materials.- 3 Three-dimensional cyclic deformations of elastoplastic materials.- 4 Single-frequency vibrations in elastoplastic bodies.- 5 Random deformation of elastoplastic materials.- 6 Random vibrations of elastoplastic bodies.- 7 Propagation of vibration in a nonlinear dissipative medium.- 8 Propagation of vibration in media with complex structure.- References.
  • Chapter
    This paper presents an in-depth discussion of the transformation properties of total and material (substantial) time derivative operators during a change of the Frame of Reference (FoR). For this purpose it is first necessary to establish the transformation properties of gradients of scalar fields, which then leads to the transformation properties of the gradient itself, a.k.a. nabla operator. The analysis is based on the notion of so-called tensor and vector image transfer from one frame to another, as originally introduced in the scientific work of Zhilin. Emphasis is put on several issues, namely, first, the observation point considered in context with all operators must be the same for all FoRs at all times. Second, the effect of all operators on an invariant scalar must be investigated. This will then consecutively result in transformation properties of the operators themselves. Third, the arguments of the scalar field must be the same for all FoRs in order to guarantee a meaningful comparison. This in mind it will be shown (a) that the nabla operator in the current configuration is invariant, (b) that the nabla operator in the reference configuration is not, and (c) that the aforementioned time derivatives in material and in spatial description are invariant.