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# The Method of Virtual Power in the Continuum Mechanics

Info: 7576 words (30 pages) Dissertation
Published: 16th Dec 2019

Tagged: Mechanics

# 1         Introduction

The method of virtual power, which is proposed by Paul Germain and distinguished by Gerad A.Maugin, is applied in continuum mechanics to extend the generalized continuum field, Non-classical and higher order theories [1]. The effective character of the method is explained in the state of gradient models. Then It is used to the introduction of concentration gradient effects in the fields of displacements, temperature representing the balance equation of motion and heat, for instance, and other coupled physical fields.

The two same way mathematically represent forces acting on continuum mechanical media: forces and couples in the balance of moment and moment of momentum equation, and the method of virtual power which is a variatioanl expression of dynamics of bodies [2]. The latter is presented by P. Germain and explained for the state of first and second gradient continuum theories.

Following Germain, Maugin made the advances in the method of virtual power and developed it to non-mechanical fields to build the continuum theories with the coupling of mechanics and electromagnetism [3]. He used continuum thermodynamics and the principle of virtual power to develop the fully dynamical theory of electromagnetic continua. The resultant equations are used by electronic engineers and theoretical mechanicians to study coupled electro-magneto-mechanical effects in electronic components. A full illustration of the application of this method is given all states of magnetism and dielectricity, the case of elastic dielectrics, liquid crystals and Ferro-fluids. In the end, this method is compared with other energy approaches used nowadays in continuum physics.

He also wrote a review paper in continuum physics on the fields of application, multi-dimensional continuum mechanics, which includes beam, plate and shell theories [4].

Following Germain’s idea, the method of virtual power, combined with the concepts of continuum thermodynamic, build a framework which includes the constitutive equations for the development of continuum theories [5-6].

By means of the method of virtual power and continuum thermodynamic, the extending of generalized continuum approaches like strain gradient and micromorphic models to plasticity and damage will be possible [7].

## 1.1         Historical notes

This part of the study doesn’t have the main purpose to set rules for using the principle of virtual power (PVP) as an efficient and safe conceptual tool to construct the governing equations of a complex theory of continuous media. It reminds the important role of the ancient Greek Aristotle, of the late sixteenth century John Bernoulli, Leonard Euler, Simon Stevin, Samuel Koenig and, Jean d’Alembert and others who played the essential role by the principle of virtual work.

This contribution is not planned to provide a history of the principle of virtual power in pre-d’Alembertian times or of its development by physicists or applied mathematicians.

The fact, that engineering mechanics soon recognized this efficient power in formulating the expression of forces, is to be distinguished with the enduring controversy between British mechanics (this means Newton’s and followers) and continental scientists (this meaning essentially the French, German and Italian “mechanicians”). For a long time, the scientific career of Paul Germain was solving difficult problems in theoretical fluid mechanics (transonic flows, flows around delta wings, the structure of shock waves, magnetohydrodynamics, perturbation methods, etc). He presented the general energy principle in elasticity and the theory of structural members appearing in continuum mechanics in the 1960s.

Paul Germain’s publication [2] was to propose a new framework for the education of continuum mechanics in engineering schools. Furthermore, the proposed continuum models led him to a research approach in the 1960s–1970s in the noted papers [8-12]. Paul Germain naturally engaged more deeply in the line of the principle of virtual power, when he changed his views toward the principles of mechanics.

## 1.2         The importance of this study

Here we denote the main Contents in continuum mechanics which make the necessity of using the method of virtual power.

The method of virtual power in the Continuum mechanics

### 1.2.1        Classical mechanics and classical continua:

The Theoretical framework

Remarks:

• The name classical mechanics generally denotes the common field of mechanics, as opposed to special disciplines, such as quantum mechanics, relativistic mechanics, or statistical mechanics. Continuum mechanics is the division of classical mechanics which have some relations with continuous deformable bodies. Its most traditional object of study, the classical continuum, is a continuum for which the external effects consists of two systems of forces, the body forces and the surface tractions. Larger varieties of external actions define classes of non-classical continua.
• Classical mechanics is based on Newton’s three laws of motion. Classical continua are governed by Euler’s balance laws of linear and angular momentum, which are direct consequences of Newton’s first two laws. The absence of the third law may look strange. In fact, as we shall see, this absence is a peculiarity of classical continua. In non-classical continua, the third law recovers its fundamental role. This is one of the reasons that make interesting the study of non-classical continua.
• A classical continuum is defined by two primitive elements: A continuous body, and a system of external actions, body force and surface traction. They are subject to Euler’s balance laws of linear and angular momentum.

In continuum mechanics, the balance equations work perfectly for classical continua, cannot be easily generalized to non-classical continua. This justifies a reconsideration of the position of the balance equations in the theory [13].

• The third object then is needed to add to the two primitive elements mentioned above to state the set of the virtual displacements. Virtual displacements are the initial velocities in a possible motion from the current configuration from the view of the observer. Because they describe infinitesimal changes of the current configuration, the virtual displacements are considered as infinitesimal and the defined external power is called a virtual power.

The Concept of the method of virtual power:

Remarks:

• The problems met in formulating balance equations for non-classical continua influenced several authors to converse the traditional approach, by assuming the equation of virtual power and deriving from it the balance equations.

This is the method of virtual power. For a classical continuum, this method is equivalent to the traditional approach. Some problems arise for non-classical continua. Definitely, because of the presence of additional internal actions, there are many possible expressions of the internal energy from which Euler’s laws can be derived.

• The energy is the sum of two terms for a classical continuum, the strain energy and the energy coming from the external loads. All energy approaches require the description of the functional dependence of the strain energy on the deformation, that is, the choice of a specific material. However, In the method of virtual power, this choice can be suspended, since the equation of virtual power is independent of the constitutive relations.

Non-classical continua, Continua with microstructure:

P. Germain [9] provided the benefits of a systematic application of the method of virtual power are represented by the study of continuous media with microstructure. His Results on micromorphic media of order one are found and the equations of motion for the general micromorphic medium are created for the first time. Also, He reported various interesting special cases, derived upon applying some convenient constraints.

Remarks:

• A continuum with microstructure is a continuum in which the deformation acts on two length scales of the different order of magnitude, macroscopic and microscopic.

The literature on this subject is very large. The first point was the theory of elastic bodies with couple stresses of the Cosserats [14], refreshed at the beginning of the 1960’s in papers of (Mindlin & Tiersten [15-16]). After that, the more general discussion of micropolar continua was started by (Eringen [18]), (Green [19]), and others.

• The displacement vector describes the macroscopic deformation, while the micro-deformation is described by a finite number of order parameters, also called internal variables.
• Models for continua with microstructure can be constructed by generalizing the approaches discussed in classical continua.
• In the approach based on the method of virtual power, a generalized expression of the internal power is assumed (Germain [9]). It consists of the products of the internal forces by the generalized deformations which mean the independence of the constitutive equations.

# 2         The formulation of the method of virtual power (PVP)

## 2.1         Defining the Principals and terms in PVP

The principle of virtual power: for a Newtonian history, the virtual power of the inertial forces of a mechanical system balances the virtual power of all other forces, internal, or external, influenced on the system, for any virtual velocity field.

Objectivity and Virtual Power of Internal Forces: In continuum physics, the basically new concept is that of internal forces or that of the virtual power of internal forces. In the simple ease of classical pure continuum mechanics, the internal forces are stresses, which represent the short area interaction of a matter with others. Regarding this concept of internal force, we can have access to the principle of the virtual power of internal forces. we can establish this principle in the following general form.

The principle of Objectivity: The virtual power of internal forces to a system disappears for all rigidifying virtual motions of a system at any time t.

Virtual Power of External Forces: In the first place, it needs to be comprehended the concept of external force within the framework of continuum physics. We suppose that these forces are of two different kinds. The first represents those forces which are applied to the system S by other systems put on S. These are provided by physical theories such as gravitation and electromagnetism. For instance, the action of gravity and of the Maxwellian electromagnetic fields is supposed to give such forces. The second kind of external forces represent those forces which are applied to the parts of S. we assume in continuum mechanics; these are contact forces. The boundary forces, bounded by the regular surface, and if the bonding surface is not regular enough, such forces may also be defined along singularity lines (edges) or at singular points (apexes).

Virtual Power of Inertial Forces: The presentation of inertial forces converts the statement of the principle of virtual power to the statement of d’Alembert’s principle. These forces, in general, is considered as the controversial part of the PVP method.

The virtual power of inertial (or acceleration) forces experienced by a continuous medium occupying the domain at time t is a continuous linear functional on body which (whose elements are never objective).

## 2.2         Coupled Fields

The Concept of Interaction in Electromagnetic Continua

Problem statement: Simple theories of magneto-elasticity [19,20], magneto-hydrodynamics [21], electro-elasticity [22] and electro-hydrodynamics [23] consist mostly in putting together without presenting new concepts the subject of continuum mechanics and Maxwellian electromagnetism, contributing in the terms of known expression in the usual mechanical and energetic balance laws. However, if the magnetic or dielectric behavior of the continuous material is more involved, the case in magnetic materials, or when one intends to consider dielectric effects at a finer scale, for example, in ferroelectrics, then one must present new fields to account for the new interactions (see Fig. 1) which described in a phenomenological manner. We now deal with a wider subject matter referred to us as the electrodynamics of continua with interactions.

Figure 1 Electrodynamics of continua with interactions [2].

The new interactions (B) of Fig. 1 will be stated by constitutive equations in the general nonlinear problem like stresses and heat flux in thermos-elasticity. Moreover, the interactions (B) will contribute to new field equations such as the spin-precession equation in ferromagnetism or the equations of intramolecular “force balance” in dielectrics. This formulation can be made via creative models of interactions whose construction requires some insight into physical phenomena [24-27]. It can also be made by selecting a particular thermodynamically reversible behaviour, a specific internal energy dependence, and using a Hamiltonian principle.

The method of virtual power then demonstrates its full power in constructing complex physical models of continua within the framework of finite-deformation theory and thermodynamical processes.

Previous studies propose, within the framework of modern continuum physics, a combined approach to the fully dynamical theory of electromagnetic continua including interactions. The theory maintains all fields and constitutive equations that govern, at the Galilean approximation, dielectric continua (including polarization gradients), ferromagnetic continua and nonlinear paramagnetic continua. It incorporates all presently known physical effects (which show up at the humane scale of experience) in moving (at a speed much less than that of light in vacuum) magnetized dielectrics solids or fluids. A memory dependent type of behavior can also be considered. The electromagnetic description of matter is limited to electric and magnetic dipoles because of simplicity.

## 2.3         The requirements and hypotheses

In modern view of forces, standard vectors or complicated objects from the tensorial view are presented as factors of displacements, velocities or generalized such parametric quantities.

We must consider two hypotheses for this modern view to use them in the following examples [4]:

H1: Internal forces should not expend any power in a rigidified deformation field.

They must be introduced as factors of kinematic quantities that are objective.

H2: The virtual power of internal forces of a mechanical system must be written as a continuous linear form on a set of objective virtual velocities.

## 2.4         The statement of the PVP

### 2.4.1        Example.First-order gradient theory of deformable bodies

The basic kinematic field is the physical velocity

v(X, t),where is

tNewtonian’s time and

Xthe selected space parametrization (three material coordinates)[4]

We have to consider the extended set:

V = {v,∇v}  1

The second argument decomposes into a symmetric part and a skew-symmetric part, respectively, the rate of strain D and the rate of rotation Ω. According to H2, both the rotation Ω and the velocity v are not objective, so the only set of symmetric tensors D remains such that,

Vobj = D=DT  2

Vobj

is the space of V represented by the set called distributor of rigid body motions. Accordingly, the power expended by internal forces for a starting first-order gradient velocity of type 1 is written, for a 3D continuous material body of volume Bin a Cartesian tensorial notation,

Pint*V=-∫Bσji Dij*dB,  3

where a star denotes a virtual field, and the symmetric tensor of components

σjiis called the intrinsic stress tensor. For a rigidifying virtual motion,

v*satisfies both H1 and H2.

And the virtual powers of inertial forces, body and surface forces are given by:

Pinertia*B=-∫Ba.v* dB,  4

Pvol*B=-∫Bρf. v*dB,  5

Psurf*B=-∫∂BTd.v* ds,  6

Where

a=ρv̇is the acceleration parameter,

fis the body force, and

Tdis the surface traction. Then, the principle of virtual power (PVP) is stated in this form:

Pinertia*B=Pvol*B+Psurf*B+Pint*B,  7

valid for any element of volume and surface (the continuity of continuum mechanics) and any virtual field

v*(the PVP). this yields

ρv̇=divσ+ρf

, at any

X∈B;  8

n.σ=Td, at X∈∂B,  9

After application of the divergence theorem and localization.

Remarks:

1. The natural boundary conditions associated with the field equation 9 always follow from the application of  7.
2. The symmetry of the stress tensor

σcan be identified with the Cauchy stress.

3. The application of 9 replaces the application of the two balance laws of linear momentum and moment of momentum.

The following procedure has been followed:

Step 1: Selecting the kinematic space 1, selecting the gradient order of the theory.

Step 2: Applying the hypotheses of H1 or H2 to illustrate 2 in order to write 3.

Step 3: Using the acceleration term in classical mechanics in order to write 4 according to the knowledge of physics. This term is never objective.

Step 4: Writing 5 on V as a continuous linear form; there is no limitation of objectivity, as the force contribution is external and known by physical theory (e.g. gravity force, electromagnetic interactions).

Step 5: Its clear 6 can be written on a set of velocities corresponding to gradient order due to restriction in divergence theorem.

Remarks:

• The notable fact is that the equation 7 potentially contains the formulation of all modern theories of complex and micro-structured continua.

We can follow the above-given procedure when the space generalizing 1 is chosen.

• Now, we are able to propose a complete description of a thermomechanical model using first and second laws of thermodynamics for continua.

The combination of equation 7 and the thermodynamic laws yields the theorem of internal energy in the global form[4]:

ddtEB+PintB=Q̇(B,∂B)  10

Where

Eis internal energy and

Q̇is the derivative of the influx of heat in the body,

B, and it’s boundary

∂B.

Then, this concludes the overview of the first-order gradient theory of continua.

# 3         The Generalization of PVP

## 3.1         Formulation of PVP for different form of Generalization

There is the main interest will be found in the formulation of theories of special continua in which the velocity field 1 is particularly for some applicable geometrical bodies (plates, shells, beams).

First, we indicate that the formulation 1 through 7 mentions in the different forms of generalization and develop them in next sections:

1. Higher-order gradients with extending set 1 and with defining the other powers than those obtained by the internal forces.
2. Describing the primary kinematics by a point (as in 1) butequipped itself with a microstructure, like a rigid or deformable micro-body, so with internal degrees of freedom which describesthis extra structure are required and then it’s called microstructured continua;
3. Presenting otherphysical fields added in 2 and then it’s called theories of coupled fields. The idea of gradient order can also be represented for these new fields.

## 3.2         Higher-order gradient theories

In typical local theories of continuous media, the concept of “gradient theory” is presented by that of the gradient of strains. However, this concept is mainly approached in terms of the velocity fields (or time-rates of some fundamental fields) and not in terms of the strains fields.

The definition of the higher order-gradient theory of continua can be given in the following. We first examine the purely mechanical case, and then we extend the description of phenomena of a more physical nature.

The generalization of 1 consists of the set of velocities, for instance, for the second gradient of

v:

V = {v,∇v, ∇∇v,… }  11

The following are essential remarks about 11:

• An objective set derived from 11 is:

Vobj = D=∇vs=∇∇v

,  12

With this equation, we can add a power like

Pinternal*. Also dealing with the term

∇∇v, the transformation of the integral on lines with using Stokes’ theorem would be possible over the surface

∂B.

The internal force for the term

∇∇vis a third-order tensor called the hyperstress m. It was presented by Mindlin, Eshel [28,29] accounting for the gradient of strain in second order theory.

• For inertia term, the expression 4 remains unchanged.
• The higher-order gradient theories in 3D physical space may be practically useless, as all new internal forces will have to satisfy homogeneous boundary conditions.

## 3.3         Mechanically structured media

Generalized continuum theories consider each material point X in continua, and this point has internal degrees of freedom within itself and equipped by a structure. If this structure can only rotate rigidly, we say that the medium is micropolar or a Cosserat continuum (Cosserat brothers [30]). And if this structure can deform, the medium is called a micromorphic continuum. (Eringen [31-33])

We have

χ=(X,t): the geometric object representing the relevant microstructure at material point X and Newtonian time t. And the time derivative of

χis:

χ̇=ω , 13

The set generalizing 2 is

V = {D, Ω-ω,∇ω}

.  14

Accordingly, all the powers generalizing 3, 4, and 5 are respectively,

Pint*B=-∫BσijDji*+σij̃(Ωji*-ωji*+μkjiωji,k*) dB

,

Pvol*B=∫Bρf.v*+ρcijωji*dB,

Psurf*B=∫∂BTd.v*+Mijdωji*ds,  15

Where

μkji,

cijand

Mijare, respectively, couple-stress tensor, applied couples in the bulk and at surface of B.

Then, Application of the PVP 7 yields the following dynamical equations and natural boundary conditions:

ρv̇=divσ+ρf

,

ρSij̇=μkij,k+σij+ρcij  at

X∈B,  16

n.σ=Td, n.μ=Md at X∈∂B

,  17

These equations are the basic field equations of the mechanics of 3D micropolar.

## 3.4         Physically structured media

Electro-magneto–mechanical interactions in continua generally reveal themselves through forces and couples. There exist electric and magnetic conditions of matter, like ferroelectricity, ferromagnetism, where a real nonmechanical microstructure exists even in the absence of the electric or magnetic fields.

The idea of the PVP formulation about these cases shows that this is, in fact, the best method for their definite continuum formulation.

• Ferroelectric:

The set generalizing 2 ,for ferroelectric body exhibiting a density of permanent electric dipoles P, is[4]:

V = {v,D, Ω,Ṗ,∇P}

,   18

Pint*B=-∫BσijDji*-ρEL.P*+EL∇P* dB

,  19

And the result of the application of PVP for a classical deformable medium to this theory yields:

ρv̇=divσ+ρf+ρfem

,

ρdep̈=EC+EL+ρ-1divE̅L  at

X∈B,  20

And natural boundary conditions:

n.σ=Td+Tem,  n.E̅L=Ed at X∈∂B

,  21

where

femand

Temare expressions from electromagnetism [34],

Td,

Ed,

EC,

ELcome from electrical field that they are in Maxwell’s equations.

• Ferromagnetism:

The magnetization is different from an electric polarization.

The generalization of 2  with considering nonzero virtual velocitiesfor magnetization is:

V = {v,D, Ω,μ̇=ω×μ,∇μ̇}

,  22

Pint*B=-∫BσijDji*-ρBL.μ̇*+EL∇μ* dB

,  23

where

μis the magnetization per unit mass.

And the result of the application of the PVP to this theory yields:

ρv̇=divσ+ρf+ρfem

,

μ̇=ω×μ  at

X∈B,  24

n.σ=Td+Tem,  (n.BL)×μ=Bd at X∈∂B

,  25

Where

BLis local magnetic induction.

• Other theories: dielectrics, liquid crystals

In particular, we denote the cases of elastic and also the theory of liquid crystals in which the extrainternal degree of freedom is represented by a vector field whose gradientsneed to be introduced.

The physical field adds an extra internal degree of freedom has a kind of interaction that resonates with the material microstructure. This means that the gradient order for the deformation field may be changed, thus the new formulation of PVP required. This is the case of the refined theory of micro-magnetism, happened mainly in the neighbourhood of boundary layers.

• Ferrofluids

The theory of the Ferro-fluid scheme can be assumed, from the general equations of the ferromagnetic framework developed above, for a fluidic behaviour. But, the standard theory of Ferro-fluids (suspensions of ferromagnetic particles in an organometallic carrier) depends on simplifying some hypotheses. In particular, all electrical polarization effects being discarded, (a) the exchange forces are supposed to be minimal (b) the time characteristics of the flow of Ferro-fluids and inertia can be ignored.

## 3.5         Discontinuous fields, mixtures

Discontinuity surfaces:

In some problems of the propagation of shock waves, there are fields that are not appropriately continuous. These fields suffer finite discontinuities at interfaces.  In this case, the solution will be found in the PVP formulation is given by Daher [35-37] considering a virtual motion of the discontinuity surface along with a virtual motion of material points.

## 3.6         Mixtures of continuous media

The construction of a continuum theory of mixtures has modelled fundamental questions (Truesdell [37]) respect to thermodynamical concepts such as temperature and kinetic quantities such as the kinetic energy. The construction of the power of inertial forces in the PVP formulation solve these problems.

## 3.7         Dissipative media with internal variables of state

According to the definition of internal variables of state presented to describe phenomenologically complex dissipative processes [38,39], because of their dissipative existence, they can be identified experimentally, and cannot be measured directly by means of local body or surface forces. They appear only in the Clausius–Duhem inequality for the free energy density. Hence, they do not modify the general statements (PVP and laws of thermodynamics); in particular, they cannot appear in the virtual power of both internal and external forces.

## 3.8         Approximate kinematic fields: refined theories of structural members

Considering the previous sections results in that the PVP is an efficient way to make theories of definite structural members such as beams, plates and shells at different degrees of approximation.

We can specialize the basic field 1 to achieve this, rather than adding new generalized kinematic fields by expending the set of relevant virtual velocities [40]. For instance, in order to formulate the theory of plates, the 3D field of virtual velocities can be defined by the expressions:

vα*=uα*x1,x2+x3lα*x1,x2

,

v3*=ω*x1,x2.  26

Accordingly, the virtual power of internal forces of the plate theory is written as a continuous linear form:

Pint*S=-∫SNαβdαβ*+∏αβKαβ*+Qαβα* dS

,

Where S denotes the surface of the plate, and

Nαβ,

∏αβand

Qαare, respectively, the in-plane stress tensor, the tensor of bending torsion and the normal shear force.

## 3.9         Particular and more general cases

An example of a complicated representation of the basic 3D field is given by [41]:

vα*=uα*x1,x2-x3ωα*(x1,x2)+f(x3)γα*

,

v3*=ω*x1,x2,  27

Where

f = (h/π)sin(π x3/h)and his the thickness of the plate.

• In case of shells, unfortunately, the differential geometry of non-flat 2D objects implanted in the 3D physical space needs to be considered, and the basic velocity field must be considered the degree of approximation [40]. However, the method of the PVP seems to be the safest way to formulate the basic equations, andlinked natural boundary conditions, in these advanced theories.

From the previous examples, it provides the best framework to place in evidence the relations between the various theories and to offer a real classification of the many formulated theories of structural members.

# 4         Comparison with Other Energetic Methods

General Remark:

It is expected that the different involved cases briefly examined in the previous sections within the framework of the electrodynamics of continua have demonstrated the power, the class and the simplicity of the general the method of virtual power sketched out in section ‎2 for obtaining the local and global field equations that govern both the motion and the interactions in electromagnetic continua. Among the advantageous features of this method, we note the following outlines.

The formation of conservation laws for involved interaction phenomena does not require the use of any model of interactions, apart from the classical physical models supplied from fields of physical theory like gravitation, electromagnetism, which are used in construction the expression of the virtual power of volume and inertial forces. Physically, the virtual power of internal forces in agreement with the objectivity requirement allows us to uncouple the different interactions.

The Green-Rivlin Energy Method:

The Green-Rivlin energy method used to obtain the local field equations [42,43] in continua and its application to the electrodynamics of continua consists in considering the first principle of thermodynamics [44,45].

However, by applying only a seven parameters from the group of Galilean invariance, this method is different from the case of non-dissipative processes for which the first principle can be used as an identity for virtual velocities [43], and is not applicable, for instance, to the cases described in dielectrics and ferroelectrics, because, in this limited case, the local balance of moment of momentum is not like the case in ferromagnetism and ferroelectricity.

Sedov’s Variational Principle:

The generalized variational principle considered by L. I. Sedov and his coworkers [46-48] appears to be very close to the method of virtual power designed in section ‎2. Clearly, this general formulation offers two differences with the formulation of section ‎2. Firstly, the functional dependence of strain must be specified, which means that the exact constitutive behaviour is fixed and, secondly, there is no attempt at applying the objectivity requirement.

# 5         Conclusion

In conclusion we note that the mathematical way of thought that underlies the method of virtual power, belongs to the “continental” tradition in mechanics, which has often been opposed to the British Newtonian tradition, However, it is expected that, in the present work, this opposition is resolved by the combination of both traditions in a harmonious frame.

This PVP tool, deal with modern mathematical concepts such as variational, weak formulation, test functions, distributions, and inequalities used in the previous parts. Students and engineers easily learn this tool which has a difficult form, with using the techniques of the finite element methods in calculations. It has proved its efficiency as a supportive tool in the construction of complex theories of continua, which is higher than the usual Euler–Cauchy standard modeling. The collection of examples explained in the previous parts is evidence of this effectiveness and productivity.

# 6         References

[1]. Altenbach, Holm, et al., eds. Generalized Models and Non-Classical Approaches in Complex Materials. Vol. 89. Springer, 2018.

[2]. Germain, Paul. “The method of virtual power in continuum mechanics. Part 2: Microstructure.” SIAM Journal on Applied Mathematics 25.3 (1973): 556-575.

[3]. Maugin, G. A. “The method of virtual power in continuum mechanics: Application to coupled fields.” Acta Mechanica 35.1-2 (1980): 1-70

[4]. Maugin, Gérard A. “The principle of virtual power: from eliminating metaphysical forces to providing an efficient modelling tool.” Continuum Mechanics and Thermodynamics25.2-4 (2013): 127-146.

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[6]. Maugin, Gérard A. The thermomechanics of nonlinear irreversible behaviors: an introduction. 1999.

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[9]. Germain, P.: La méthode des puissances virtuelles en mécanique des milieux continus, Première partie : théorie du second gradient. J. de Mécanique (Paris) 12, 235–274 (1973)

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[11]. Germain, P. ontribution à l’étude des milieux micropolaires et micromorphiques. In: Omaggio a Carlo Ferrari, Levretto e Bella, Torino, pp. 273–297 (1974)

[12]. Germain, P.: Duality and convection in continuum mechanics. In: Fichera, D. Trends in Applications of Pure Mathematics to Mechanics, pp. 107–128. Pitman, London (1976)

[13]. Del Piero, Gianpietro. “On the method of virtual power in continuum mechanics.” Journal of Mechanics of Materials and Structures 4.2 (2009): 281-292.

[14]. F. Cosserat. Theorie des corps d´eformables. Hermann, Paris 1909

[15]. R.D. Mindlin. Micro-structure in linear elasticity. Arch. Ration. Mech. Analysis 16: 51-78, 1964

[16]. R.D. Mindlin, H.F. Tiersten. Effects of couple-stresses in linear elasticity. Arch. Ration. Mech. Analysis 11: 415-448, 1962

[17]. A.C. Eringen. Mechanics of micromorphic media. Applied Mechanics, Proc. XI Congress on Applied Mechanics, M¨unchen, pp. 132-138, 1964

[18]. A.E. Green. Micro-materials and multipolar continuum mechanics. Int. J. Engng. Sciences 3: 533-537, 1965

[19] Nowacki, W.: Dynamic problems in thermoelasticity (Translation from the Polish), Chap. VI. Leyden: Noordhoff; Warsaw: P. W. N. 1975.

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