preliminary framework

1

Preliminary framework

1.1 Configurations and displacement

At a given reference time, say t = to, a material continuum body occupies aregion of space denoted by Bo. Physically one can imagine that the particlesof the body are distributed continuously over the region Bo. Mathematicallyone can say that there is a one-to-one correspondence between the materialparticles and points in Bo, and one can identify each of the particles P bythe position vector X = Xiei, where ei (with i = 1, 2, 3) are the base unitvectors for a fixed rectangular Cartesian coordinate system (Fig. 1.1).

Imagine that at a typical subsequent time t, the body moves and occupiesanother region B in space. All particles of the body are now continuouslydistributed over the region B, and there is a one-to-one correspondence be-tween the material particles of the body and points in B. Similarly, the newlocation of the particle P of the body, denoted by p in the new configurationB, can be specified by the position vector x = xiei.

Since the particle P in the configuration Bo with position vector X occu-pies the position p in the configuration B with position vector x, the motioncan be mathematically described as

x = x(X, t) (1.1)

The motion is assumed to be smooth enough and thereby Eq. (1.1) canbe inverted to give:

X = X(x, t) (1.2)

To ensure that Eq. (1.1) is invertible, function x = x(X, t) should besingle valued and continuously differentiable at a given time t. Thus theJacobi determinant of x = x(X, t) must be non zero in Bo:

1

2 Preliminary framework

Bo

B

xX

u

u + du

dxdX

Referenceconfigura-on

Currentconfigura-on

e1

e2

e3

Pp

Figure 1.1: Body and configurations

det

(∂xi∂Xj

)≡ det (xi,j) 6= 0 (1.3)

previous relationship being usually referred to as axiom of continuity.Since previous considerations, a typical particle that moves in space ac-

cording to Eq. (1.1) must have its reference position X in Bo, X resultingits starting point or its reference point. At any subsequent time t, one canalways find this particle by identifying its reference point. Regardless ofhow it moves and the distance it travels, it can change its current positionand will always have the same original reference position. In this sense thereference position X and corresponding cooridnates Xi can be thought of asfixed labels of a typical particle and they will not change for a given particle.Thus these coordinates are called material coordinates or reference coordi-nates, or even mistakenly Lagrangian coordinates (as a matter of fact theywere introduce by Euler in 1762 [1]). Following the same terminology, theset of all particles of the body at t = to (namely, the region Bo) is termedreference configuration, where the word configuration refers to the space re-gion occupied by the body at any given time. It should be noted that thereference configuration does not have to be the region that the material ac-tually occupies. It can be an imaginary one. However, it is preferable toadopt any region occupied by the material at some previous time. Thus,the position X is usually the starting point (i.e., at an initial time to) of thetypical particle.

On the other hand, the position vector x specifies a spatial point thatcan be occupied by different particles at different times. The coordinatesxi are called spatial coordinates or Eulerian cooridnates, even though theywere introduced by d’Alembert in 1752 [1]. The region B occupied by thebody at time t is called current configuration.

University of Rome “Tor Vergata”- Giuseppe Vairo 3

In describing the motion (or deformation) of materials, either the ma-terial coordinates Xi or the spatial ones xi can be used as independentvariables to identify any particle of the material. This is true because themotion as denoted by Eq. (1.1) is invertible. Accordingly, one can determinethe current position x of the particle at any time if one knows its referencelabel X. Inversely, one can specify its reference position X by knowing itscurrent position x at time t. In detail, according to Eq. (1.2), the particlethat currently occupies position x originates from reference position X att = to.

It takes a material or Lagrangian description if one chooses the materialcoordinate Xi as independent variables. Alternatively, it takes the spatial orEulerian description if the spatial coordinates xi are chosen as independentvariables. In the material description the typical particle or particles arefollowed. In the spatial description the concern is with any phenomenon ata particular point in the space, with different particles occupying this pointat different times.

Particle motion can be described by introducing a displacement vectoru, such that:

x = X + u (1.4)

In material description u is represented as

u(X, t) = x(X, t)−X (1.5)

Alternatively, one can employ the spatial description:

u(x, t) = x−X(x, t) (1.6)

It should be noted that Eqs (1.5) and (1.6) have different physical mean-ings. Equation (1.5) expresses the displacement at the current time t ofthe particle identified by X in the reference configuration. Equation (1.6)is the displacement of any particle which at the current time t occupies theposition x.

It is worth observing that, although for the sake of simplicity the materialcoordinates Xi and the spatial coordinates xi are herein both referred tothe same Cartesian coordinate system, they generally are associated to twodifferent coordinate systems1.

1For instance, to solve a problem of beam bending with initial rectangular configurationwhich is deformed to a circular shape, it is simpler to choose the Cartesian coordinatesystem to describe the initial reference configuration, and the curvilinear polar coordinatesto describe the current configuration.


1.2 Deformation gradient

The map described by Eq. (1.1) identifies the deformation process from thereference configuration Bo to the current one B. Because deformation, thepoints, lines, areas, and volumes in Bo are transformed into the correspond-ing ones in B.

An elementary line element dX at X in Bo becomes dx after the defor-mation. Since Eq. (1.1), the following relationship holds:

dx =∂x

∂XdX = ∇x dX ≡ F dX (1.7)

where F = ∂x/∂X = I + ∇u, or component-wise Fij = ∂xi/∂Xj = xi,j =δi,j +ui,j , is the deformation gradient tensor, I being the second-order iden-tity tensor2. Accordingly, F is a linear operator which relates every materialline element dX in Bo to the corresponding dx in B.

According to the axiom of continuity in Eq. (1.3), det(F) 6= 0. Thereby,Eq. (1.7) can be inverted, with F−1 component-wise resulting in: (Fij)

−1 =∂Xj/∂xi.

Let dΩo an elementary volume element in Bo be considered. After thedeformation it becomes dΩ in B. It is possible to show [3] that:

dΩ = det(F) dΩo (1.8)

As a physical consequence of such a relationship, it results: det(F) > 0.

1.3 Velocity, acceleration, and material derivatives

The velocity vector v of a particle is the rate of change of its displacement,and it can be expressed as:

v = lim∆t→0

x(X, t+ ∆t)− x(X, t)

∆t

= lim∆t→0

u(X, t+ ∆t)− u(X, t)

∆t=

(∂x

∂t

)X

=

(∂u

∂t

)X

(1.9)

where the subscript X denotes holding X constant (namely, referring to thesame material particle) while taking the partial derivative with respect totime.

Similarly, the acceleration vector a results in:

2The second-order identity tensor I can be component-wise expressed via the Kroneckerdelta: Iij = δij .


a = lim∆t→0

v(X, t+ ∆t)− v(X, t)

∆t=

(∂v

∂t

)X

(1.10)

As previously discussed, either material (Lagrangian) or spatial (Eule-rian) description can be used to describe the motion and deformation of amedium. This means that we can write the velocity vector v in either thefollowing forms

v = v(X, t) (1.11)

v = v(x, t) (1.12)

Accordingly, expression of acceleration of a particle X and as given inEq. (1.10) can be adopted by using the Lagrangian description of v, thatis Eq. (1.11). Time derivatives with Lagrangian or material coordinate Xheld constant (as in Eqs. (1.9) and (1.10)) are called material derivativeswith respect to t.

If the velocity vector is specified via an Eulerian description, that is viaEq. (1.12), then Eq. (1.10) is no longer be used to calculate the accelerationvector a. Nevertheless, by observing that v = v(x, t) = v[x(X, t), t], onecan calculate the material derivative as:

a =

(∂v

∂t

)X

= v,t +∂v(x, t)

∂x

(∂x

∂t

)X

(1.13)

or in components

ai = vi,t + vi,jxj,t (1.14)

where v,t = (∂v(x, t)/∂t)x and, by definition, (∂x/∂t)X = v. Therefore,the acceleration of a material particle, that is the material derivative withrespect to t of the velocity v, results in:

a = v,t + (gradv)v (1.15)

or in components

ai = vi,t + vi,jvj (1.16)

where the gradient operator ‘grad’ has to be intended as referred to thespatial coordinates xi.

In general, the material derivative of a quantity q expressed by an Eule-rian description, that is q = q(x, t), can be obtained similarly to Eq. (1.15).


As a notation rule, the material time derivative of q is usually denoted byq = dq/dt, and depending if q is a scalar (q = α), a vector (q = w), or atensor (q = A) field, then the following rules apply3:

α = α,t + (gradα) · v, α = α,t + α,ivi (1.17)

w = w,t + (gradw)v, wi = wi,t + wi,jvj (1.18)

A = A,t + (gradA)v, Aij = Aij,t +Aij,kvk (1.19)

1.4 Strain and strain rate

By adopting a Lagrangian description, many measures for finite deformationare based on the Green-Lagrange second-order strain tensor E(X, t):

E =1

2(C− I) (1.20)

where second-order tensor C = FTF denotes the so called right Cauchy-Green tensor [2]. Accordingly, since Fij = xi,j = δij + ui,j , E results in:

Eij =1

2(xk,ixk,j − δij) =

1

2(ui,j + uj,i + uk,iuk,j)

E =1

2[∇u + (∇u)T + (∇u)T∇u] (1.21)

where gradient expressed via the operator ∇ has to be intended with respectto the material coordinates Xi.

As it is well known, for infinitesimal deformation, namely for |ui,j | 1,the Green-Lagrange strain tensor reduces to the infinitesimal strain tensorε, component-wise expressed by

εij =1

2(ui,j + uj,i) (1.22)

In this case the difference between the material (Lagrangian) and spatial(Eulerian) descriptions is not significant. Thereby, ε = (1/2)[∇u+(∇u)T ] =(1/2)[gradu + (gradu)T ] = sym(∇u) = sym(gradu).

Recall that the velocities of all points in B constitute a vector fieldv(x, t) = x(X, t) = u(X, t). In general, this vector field varies with time.But for a fixed time t, vector v(x, t) varies only with the spatial coordinatesx. We can calculate the velocity variation dv caused by an infinitesimalvariation of the position coordinates dx as

3It is worth observing that gradA, component-wise expressed by Aij,k, is a third-ordertensor.


dv =∂v

∂xdx = gradv dx = Ldx (1.23)

where L = ∂v/∂x = ∂u/∂x, that is Lij = vi,j = xi,j = ui,j . It is worthobserving that dv can be considered as the velocity at point x+ dx relativeat point x.

Since the previous relationship, the material derivative of F results in

F =d

dt

∂x

∂X=∂x

∂X=∂v(x, t)

∂X=∂v

∂x

∂x

∂X= LF, Fij = LikFkj (1.24)

and FT = FTLT . Accordingly, the material derivative of C reads as

C = FTF + FT F = FT (L + LT )F = 2FTDF (1.25)

where

D = sym(L) = sym(gradv) (1.26)

is the strain rate tensor4. In detail, it is possible to show that, if C or E (fol-lowing a Lagrangian description) can completely determine the deformationstate of any material point, the strain rate tensor D allows to characterizethe deformation-rate state in the current configuration B [3, 5].

Due to Eq. (1.25), the material derivative of the strain tensor E is

E =1

2C = FTDF (1.27)

Accordingly, D and E are in general different. Nevertheless, if the dis-placement gradient ∇u is small compared to the identity tensor I (i.e., theinfinitesimal strain assumption), then F ' I and thereby

E ' ε = D (1.28)

Often in the following, we will implicitly assume to hold relationship(1.28).

4The skew-symmetric part of L, that is W = (L−LT )/2 = skew(gradv) is called spintensor and it is associated to a rigid rotation [3, 5].

2

General Principles

At a given time t, let a continuum body be in the current configuration B,characterized by the volume Ω and the boundary surface Σ. Let Ωo and Σo

the corresponding quantities associated to the reference configuration Bo,with x = x(X, t) the deformation map. Let ρ = ρ(x, t) be a continuousfunction of x and t, denoting the mass density in B, and let ρo(X, to) thedensity function in Bo. Moreover, let b the density of body force per unitmass be introduced in Ω, and p the density of surface force per unit areaacting on Σ.

As an assumption, the body does not interchange the mass with itssurroundings, that is we assume that the body is a closed system.

2.1 Conservation of mass

The principle of conservation of mass states that no mass can be destroyed orcreated. In order to obtain a local form of such a condition, let a portion ofthe body be considered, characterized by the volume ∆Ω and the surface ∆Σin B (associated to ∆Ωo and ∆Σo in Bo). At a given time t, the correspondingmass is

∫∆Ω ρ(x, t) dΩ. Since the conservation of mass, the following equality

holds:

∫∆Ω

ρ dΩ =

∫∆Ωo

ρodΩo (2.1)

Accordingly, since ρo and ∆Ωo are constant in time, the first time-derivative of Eq. (2.1) reads as:

d

dt

∫∆Ω(t)

ρ(x, t) dΩ = 0 (2.2)

the latter being equivalent to:

8


n

vt

d

d = tv · nd

(t)(t + t)

(t)

(t)

Figure 2.1: Volume variation.

lim∆t→0

∫∆Ω(t+∆t) ρ(t+ ∆t) dΩ−

∫∆Ω(t) ρ(t) dΩ

∆t= 0 (2.3)

where ρ(t+∆t) has to be intended as ρ(x, t+∆t) and similarly ρ(t) = ρ(x, t),with x being the spatial coordinates which describe the current configurationof the body portion. Accordingly, denoting by δΩ(∆t) the volume variationattributed to the volume element ∆Ω(t) (the latter having ∆Σ(t) as bound-ary surface) after the time interval ∆t, that is ∆Ω(t+∆t) = ∆Ω(t)+δΩ(∆t)(see Fig. 2.1), Eq. (2.3) can be recast in:

lim∆t→0

∫∆Ω(t) ρ(t+ ∆t) dΩ +

∫δΩ(∆t) ρ(t+ ∆t) dΩ−

∫∆Ω(t) ρ(t) dΩ

∆t

=

∫∆Ω(t)

lim∆t→0

ρ(t+ ∆t)− ρ(t)

∆tdΩ + lim

∆t→0

∫δΩ(∆t) ρ(t+ ∆t) dΩ

∆t

=

∫∆Ω(t)

∂ρ

∂tdΩ + lim

∆t→0


∆t= 0 (2.4)

Therefore, in agreement with the sketch in Fig. 2.1 and in the limit ofvanishing values for ∆t, it is immediate to verify that the following relation-ship holds:

∫δΩ(∆t)

(·) dΩ =

∫∆Σ(t)

(·) [∆tv(t)] · n(t) dΣ (2.5)

where n(t) is the outward normal unit vector to ∆Σ(t), v is the velocityvector, and ∆tv is the displacement vector of the elementary surface dΣ of∆Σ(t) during ∆t, so that the elementary volume dΩ of δΩ(∆t) is ∆tv · n.

10 General Principles

Accordingly, by using Eq. (2.5), the second term in Eq. (2.4) results in:

lim∆t→0


∆t= lim

∆t→0

∫∆Σ(t) ρ(t+ ∆t) [∆tv(t)] · n(t) dΣ

∆t

=∆t

∆t

∫∆Σ(t)

lim∆t→0

ρ(t+ ∆t)v(t) · n(t) dΣ

=

∫∆Σ

ρv · n dΣ =

∫∆Ω

div(ρv) dΩ (2.6)

Therefore, by combining Eqs. (2.4) and (2.6), relationship in Eq. (2.2)reads as:

∫∆Ω

∂ρ

∂t+ div(ρv) dΩ = 0 (2.7)

Since the volume ∆Ω is arbitrary, by applying the localization lemma1,the integrand must be zero everywhere:

∂ρ

∂t+ div(ρv) = 0 in Ω (2.8)

or equivalently2

ρ+ ρdivv = 0 in Ω (2.9)

Equation (2.8) (or Eq. (2.9)) is the differential form of the continuity orconservation mass condition expressed by Eq. (2.1). Alternative deductionsof Eqs. (2.8)-(2.9) can be found in [3, 4, 5, 6].

If the material is incompressible, so that mass density ρ remains thesame when the material moves or deforms, we have ρ = 0. Thereby, thecontinuity equation (2.8) reduces to (see Eq. (1.26))

divv = vi,i = tr(D) = 0 (2.10)

The conservation of mass discussed above is in the current configuration.The resulting continuity equation can be also derived in the reference con-figuration. In detail, in the limit of an infinitesimal volume and due to Eq.(1.8), Eq. (2.1) prescribes that:

1Let f(x) be a smooth function in Ω. If∫

∆Ωf(x) dΩ = 0 for any arbitrary neighbour-

hood ∆Ω ⊂ Ω of the position x, then f(x) = 0.2It is worth observing that the material derivative ρ results in: ρ = dρ(x,t)

dt= ∂ρ

∂t+

∂ρ∂xi

dxidt

= ∂ρ∂t

+ v · gradρ. Moreover, div(ρv) = (ρvi),i = ρ,ivi + ρvi,i = v · gradρ+ ρ divv.


ρ dΩ = ρo dΩo ⇒ ρ(x(X, t), t) det[F(X, t)] = ρo(X, to) (2.11)

the latter being generally called the material description of the continuityrelationship.

For incompressible materials, since ρ = ρo, Eq. (2.11) leads to

detF = 1 (2.12)

corresponding to prescribe that the deformation process is isochoric (namely,at constant volume).

Almost all plasticity theories for metals assume that the plastic deforma-tion is isochoric, namely that it takes place with negligible volume change,based on Bridgeman’s experimental observations. In these cases, if we fur-ther assume that elastic deformation is small compared with plastic deforma-tion (thereby, neglecting possible volume changes associated with the elasticregime), Eqs. (2.10) and (2.12) can be used as a first-order approximationin formulating plasticity models.

Remark 1 Due to the conservation of mass expressed by ρ dΩ = ρo dΩo

(Eq. (2.11)), the following equality holds

d

dt

∫ΩρA dΩ =

d

dt

∫Ωo

ρoA dΩo =

∫Ωo

ρodAdt

dΩo =

∫ΩρdAdt

dΩ (2.13)

where A can be a scalar, vector, or tensor of any order.

2.2 Conservation of momentum

Let an internal portion of the body be considered in B. The correspondingequilibrium (in the sense of the Newton’s second law of motion, that is byincluding also inertial forces) results in

∫∆Ω

ρv dΩ =

∫∆Ω

ρb dΩ +

∫∆Σ

tn dΣ (2.14)

where v = dv/dt denotes the acceleration and tn = σn is the stress vectoron the surface ∆Σ, whose outward normal unit vector is n, σ being theCauchy second-order stress tensor.

By using the divergence theorem, that is

∫∆Σ

tn dΣ =

∫∆Σσn dΣ =

∫∆Ω

divσ dΩ (2.15)


or in components∫

∆Σ tnidΣ =∫

∆Σ σijnj dΣ =∫

∆Ω σij,j dΩ, Eq. (2.14) canbe transformed in: ∫

∆Ω[divσ + ρb− ρv] dΩ = 0 (2.16)

that, since the volume ∆Ω is arbitrary, leads to

divσ + ρb = ρv on Ω (2.17)

representing the differential form of the conservation of momentum (namely,the generalization of the Newton’s second law of motion for continuum me-chanics).

Previous relationship is also known as Cauchy’s equation of motion, andin components it reads as

σij,j + ρ bi = ρ vi on Ω (2.18)

For material particles on the boundary Σ the conservation of momentum(proof is omitted for the sake of brevity) reduces to:

σn = p on Σ (2.19)

or component-wise σijnj = pi on Σ.

2.3 Conservation of moment of momentum

As a further generalization of the Newton’s second law of motion for con-tinuum mechanics, it is possible to identify the conservation of moment ofmomentum. It states that the rate of change of the moment of momentumfor any given set of material particles equals the vectorial sum of momentsacting on it. If it is assumed that there are no distribute couples and theaction between the internal material particles of the body is the tractionforce only (namely, if reference is made to the Cauchy’s continuum modelor nonpolar case), we can write the conservation of moment of momentumfor a body portion as:

d

dt

∫∆Ω

(r× ρv) dΩ =

∫∆Σ

(r× tn dΣ) +

∫∆Ω

(r× ρb) dΩ (2.20)

where r is the position vector of the particle under consideration with re-spect to a given pole. By using the Cauchy’s equation of motion and byapplying the localization lemma, the conservation of moment of momentumprescribes3 the symmetry of the Cauchy’s stress tensor, that is σ = σT orσij = σji.

3Proof is herein omitted for the sake of compactness; reader can refer to [3].


2.4 Principle of virtual powers

Let w = w(x, t) a generic smooth vector field defined in B be considered.Since Eq. (2.16), the following relationship holds on a body portion:

∫∆Ω

[divσ + ρb− ρv] ·w dΩ = 0 (2.21)

By observing that, due the symmetry of the Cauchy stress (i.e., σij =σji), it results4:

div(σw) = (σijwj),i = σij,iwj + σijwj,i = σij,jwi + σijwi,j

= div(σ) ·w + σ : grad(w)

= div(σ) ·w + σ : sym(gradw) (2.22)

then Eq. (2.21) becomes

∫∆Ω

[div(σw)− σ : sym(gradw) + ρb ·w] dΩ =

∫∆Ω

ρv ·w dΩ (2.23)

where the first term can be transformed in a surface integral via the diver-gence theorem:

∫∆Ω

div(σw) dΩ =

∫∆Σσw · n dΣ =

∫∆Σσn ·w dΣ =

∫∆Σ

tn ·w dΣ

(2.24)

As a result, Eq. (2.23) reads as

(∫∆Σ

tn ·w dΣ +

∫∆Ω

ρb ·w dΩ

)−∫

∆Ωσ : sym(gradw) dΩ

=

∫∆Ω

ρv ·w dΩ (2.25)

By regarding the vector field w as a virtual velocity field5, and by spe-cializing Eq. (2.25) to the overall body, the following condition holds:

Pve − Pvi = Pva (2.26)

4Since Cauchy stress tensor σ is symmetric, then σ : gradw = σ : [sym(gradw) +skew(gradw)] = σ : sym(gradw).

5If displacement constraints are assigned as boundary conditions on some surface por-tion Σu of Σ, there w has to respect homogeneous conditions, namely w = 0 at Σu.


where Pve denotes the external virtual power

Pve =

∫Σp ·w dΣ +

∫Ωρb ·w dΩ (2.27)

Pvi is the internal virtual power

Pvi =

∫Σσ : Dw dΣ (2.28)

and Pva is the virtual power of the acceleration (associated to virtual inertialforces)

Pva =

∫Ωρv ·w dΩ (2.29)

In previous relationships, Eq. (2.19) has been enforced and Dw =sym(gradw) denotes the virtual strain rate field associated to the virtualvelocity field w.

Equality (2.26) corresponds to the principle of virtual powers6.

2.5 Theorem of kinetic energy

Starting from Eq. (2.26), if w = v, that is if the vector field w coincides withthe physical velocity field v, then quantities introduced in Eqs. (2.27) to(2.29) acquire the meaning of real powers and thereby Eq. (2.26) describesa balance of real powers. In detail, since Eq. (2.13), it results

Pa =

∫Ωρv · v dΩ =

1

2

∫Ωρd

dt(v · v) dΩ =

d

dt

∫Ω

1

2ρv · v dΩ = K (2.30)

that is, the power of acceleration corresponds to the rate of the kinetic energyK =

∫Ω ρv · v/2 dΩ of the system7. Accordingly, the power balance for the

system states that the difference between external and internal powers hasto be equal to the rate of the kinetic energy

Pe − Pi = K (2.31)

with

6Often, as a consequence of different notations, the internal virtual power in Eq. (2.28)is defined with an opposite sign. In this case, Eq. (2.26) results characterized by a signplus at the left side.

7It is worth observing that K is not an objective quantity.


Pe =

∫Ωρb · vdΩ +

∫Σp · vdΣ (2.32)

Pi =

∫Ωσ : DdΩ (2.33)

where D = sym(gradv). Equation (2.31), referred to a body portion, leadsto:

∆Pe −∆Pi =

∫∆Ω

ρb · vdΩ +

∫∆Σ

tn · vdΣ−∫

∆Ωσ : DdΩ

= ∆K =d

dt

∫∆Ω

1

2ρv · vdΩ (2.34)

Equation (2.31) (or equivalently Eq. (2.34)) express the so-called theo-rem of the kinetic energy: for a closed8 system the external power is equalto the internal power plus the rate of the kinetic energy.

8In thermodynamics it is usual to introduce the following definition. Open system: asystem that can exchange with the surrounding both energy and mass. Closed system:a system that can exchange with the surrounding only energy but not mass. Isolatedsystem: a system that can not exchange with the surrounding energy and mass.

3

Thermodynamics principlesfor continuous media

3.1 Thermodynamic state and state variables

Continuous media have, on a macroscopic scale, geometric and physico-chemical features (position, temperature, density, stresses, electric charge,chemical potential, etc.) represented via tensor quantities (scalars, vectors,second order tensor, and so on). Depending on the problem studied andthe fineness of the sought modelling, only the properties representative ofthe dominant phenomena are generally retained. Their fields define, at anymoment, the thermodynamic state of the continuum. It is assumed that itis possible to completely characterize the thermodynamic state of the con-tinuous medium in an equilibrium condition by a finite number N of fieldsof independent physical quantities. In other words, if one knows these N in-dependent fields, any other physical quantity of interest for the continuumcan be uniquely determined. These N physical variables (χi = 1, ..., N),called state variables, depend only on the point considered and its immedi-ate neighbourhood. They describe not only the equilibrium of the medium,but also its successive states in any non-equilibrium evolution. In detail,this means that for any particle of the evolving system we can associate alocal equilibrium state described precisely by the state variables in question.It is worth pointing out the choice of state variables strictly depends on thephenomena that one wishes to model as well as the fineness of the modellingin question. Moreover, such a choice is in general not unique a priori.

Two of the primitive variables generally adopted to describe the equi-librium of a continuous medium are the displacement relative to a refer-ence configuration and the temperature. At the level of a material point(namely, a material particle), the relevant kinematic variable, derived fromthe displacement, reflects the variations of distance between particles; it isthe strain (symmetrical part of the gradient of the displacement field un-

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der infinitesimal-strain assumption) for a solid continuum, and the specificvolume (related to the spacing of the molecules) for the a fluid continuum1.Thus the state of a perfect fluid in equilibrium is totally defined by itstemperature and its specific volume, or more generally by two of the threeindependent state variables (temperature, pressure and specific volume). Itis possible to prove that, a continuous medium in infinitesimal thermoe-lasticity is completely characterized by temperature and strain; stress, andentropy, and other characteristics of the state, are necessarily connected tothem.

3.2 Conservation of energy: The first law of Ther-modynamics

The conservation of energy principle is one of the most important generalprinciple in the universe. It comes from the extensive experimental obser-vation that energy con never be created or destroyed in the universe; it canonly be transformed from one form into another.

For a closed system the total rate of the work done on the system byall the external agencies must be equal the rate of the increase of the totalenergy of the system. This principle is also called the first law of thermody-namics.

There are various types of energies (e.g., mechanical, chemical, electrical,magnetic, thermal energy). The principle of conservation of energy providesus a relationship between the mechanical work done, the heat transferred tothe system, and the change in the internal energy of the system.

In particular, a closed system can exchange energy with its surroundingsby the work of mechanical external forces or by thermal way (namely, viaheat transfer). The rate of the work of external forces is clearly defined bythe external power Pe in Eq. (2.32). Moreover, denote the heat transferinto the system by Q. It consists of two parts: the heat flow through theboundary surface and the heat generated by internal sources. Therefore, therate Q of the heat transfer is

Q = −∫

Σq · n dΣ +

∫Ωr dΩ (3.1)

where q is the heat flux vector per unit time and per unit surface, and r isthe the heat source per unit time and per unit volume. The negative sign inthe previous equation is introduced because q, by convention, is the outwardheat flux.

1The specific volume of a substance is the ratio of the substance’s volume to its mass.It is the reciprocal of density.

18 Thermodynamics principles for continuous media

As a result of energy exchanges via external forces and heat transfer, amodification of the total energy of the system is expected. The total energyU of the system is composed of two parts: the kinetic energy K and theinternal energy E . Denote the specific internal energy per unit mass by e.We can write the rate of the total energy of the system as (see Eq. (2.13)):

U = K +d

dt

∫Ωρ e dΩ = K +

∫Ωρe dΩ = K + E (3.2)

Accordingly, the first law of thermodynamics can be expressed in integral(or global) form as:

Pe + Q = K + E (3.3)

or equivalently, due to the theorem of the kinetic energy in Eq. (2.31), as

E = Pi + Q (3.4)

Thereby, the rate of the internal energy of a closed system has to be equalto the internal power plus the rate of the heat transfer into the system.

In order to derive a local form of the first law of thermodynamics, letEq. (3.4) be referred to a body portion:

∆E =

∫∆Ω

ρedΩ = ∆Pi + ∆Q

=

∫∆Ωσ : D dΩ +

∫∆Ω

r dΩ−∫

∆Σq · ndΣ (3.5)

Observing that∫

∆Σ q · n dΣ =∫

∆Ω divq dΩ and by invoking the local-ization lemma (since ∆Ω is arbitrary), Eq. (3.5) leads to the following local(differential) form of the first law of thermodynamics:

ρe = σ : D + r − divq (3.6)

Equation (3.6) shows that the increase of the internal energy per unitvolume (ρe) consists of three parts: the stress power (σ : D) -representingthe mechanical work done by external forces not converted in kinetic energy-,the heat supplied by internally distributed sources (r), and the heat providedby the flow of thermal energy into the system (divq).

Finally, the first principle of thermodynamics can be enunciated as: thereexists a state function2 E(t) =

∫Ω ρ edΩ, called internal energy, such that

Eqs. (3.4) and (3.6) hold.

2A state function or function of state is a function defined for a system relating severalstate variables or state quantities that depends only on the current equilibrium state ofthe system. State functions do not depend on the path by which the system arrived at itspresent state. A state function describes the equilibrium state of a system.


3.3 Clausius-Duhem inequality: The second lawof Thermodynamics

The second law of thermodynamics limits the direction of the energy trans-formation. As noted previously, energy can be neither created nor destroyed;it can be only transformed from one form into another. Whenever this trans-formation happens, the energies involved in the process must obey the firstlaw of thermodynamics. Nevertheless, it has been experimentally observedthat while some energies transform from one to another, there are othertypes of transformations that are impossible. For instance, heat alwaysflows spontaneously from hotter to colder bodies, and never the reverse,unless external work is performed on the system. The kinetic energy ofa moving body can be converted into heat by friction and the body willstop due to the frictional resistance, but the heat caused by friction canbe never converted completely into the kinetic energy of the body causingit to move as in the initial stage. These observations cannot be describedby the first law of thermodynamics, since it simply relates the quantitiesof energies involved in a transformation process. It is the second law ofthermodynamics that governs this directional phenomenon observed in theenergy transformation processes.

The second law of thermodynamics postulates that there exists a statefunction called entropy S(t), expressed as

S =

∫Ωρs dΩ (3.7)

s(x, t) being the specific entropy per unit mass, such that the total entropyS(t) can never decrease over time for an isolated system, meaning a systemwhich neither energy nor matter can enter or leave3. The total entropycan remain constant in ideal cases where the system is in a steady state(equilibrium), or undergoing a reversible process. In all other real cases, thetotal entropy always increases and the process is irreversible.

With reference to closed systems, entropy changes are associated to heattransfer and temperature. In particular, for an infinitesimal region of aclosed system that passes from the starting state 1 to the ending state 2,the change in the elementary entropy is given by:

3An isolated system is a thermodynamic system enclosed by rigid immovable wallsthrough which neither matter nor energy can pass. This can be contrasted with what (inthe more common terminology used in thermodynamics) is called a closed system, beingenclosed by selective walls through which can pass energy as heat or work, but not matter;and with an open system, which both matter and energy can enter or exit, though it mayhave variously impermeable walls in parts of its boundaries.

20 Thermodynamics principles for continuous media

(dS)2 − (dS)1 = (ρsdΩ)2 − (ρsdΩ)1 ≥∫ 2

1

δ(dQ)

T(3.8)

where T = T (x, t) is the absolute temperature, δ(dQ) is an infinitesimalamount of heat transferred to the elementary portion of the system duringthe process4, and the greater than and equal to signs are for irreversible andreversible processes, respectively.

Let the starting state 1 be considered as corresponding at the time t andthe ending state 2 as associated at the time t+ dt. Thereby, in the limit ofvanishing dt, Eq. (3.8) leads to

dS(t+ dt)− dS(t) ≥ dQ(t+ dt)− dQ(t)

T⇒ dS ≥ dQ

T(3.9)

Accordingly, since (see Eq. (3.1)) dQ = r dΩ − q · n dΣ, the followingrelationship holds for the entire body:

S =d

dt

∫Ωρ s dΩ ≥

∫Ω

r

TdΩ−

∫Σ

q · nT

dΣ (3.10)

or equivalently

S −(∫

Ω

r

TdΩ−

∫Σ

q · nT

dΣ

)≥ 0 (3.11)

where the left-side term is generally called rate of entropy production5.

4Different notations are used for infinitesimal amounts of heat (δ) and infinitesimalamounts of entropy (d) because entropy is a function of state, while heat, like work, is notin general.

5Notice that if the system is isolated (thereby, process is an adiabatic process), thenδ(dQ) = dQ = 0 in Eqs. (3.8) and (3.9), so the relationship (3.11) reduces to S ≥ 0.Let the scenario of an isolated system (called the total system or universe) be consideredand let it made up of two parts: a sub-system of interest (labelled as ss, and the sub-system’s surroundings (sr). Whatever changes occur in the entropies of the sub-systemSss and the surroundings Ssr individually, according to the second law of thermodynamicsthe entropy of the universe Suniv must not decrease. Therefore Suniv = Sss + Ssr ≥ 0.It is worth observing that the increase of the universe entropy for irreversible processesdoes not imply that entropy has to increase in each region of the sub-system or of thesurroundings. As a matter of fact, local entropy reductions are possible provided that incomplementary regions entropy increases such that the condition Suniv ≥ 0 is satisfied.As a further observation, the increase of universe entropy determines an asymmetry be-tween future and past, thereby leads to the concept of the arrow of time, developed in1927 by the British astronomer Arthur Eddington, and involving the idea of the one-waydirection of time. As a matter of fact, all motion laws, as well as the first principle ofthermodynamics, can contemplate an inversion in time direction. On the contrary, thesecond law of thermodynamics prescribes that all natural processes (always irreversible)are associated to an entropy increase, and thereby that they can not be inverted, leadingto an asymmetry between future and past.


Accordingly, the second principle of thermodynamics can be stated as:there exists a state function S such that the rate of entropy production isalways greater than zero for irreversible processes (and equal to zero forideal reversible processes).

Bibliography

[1] Truesdell, C. (1952). The mechanical foundations of elasticity and fluiddynamics. Journal of Rational Mechanics and Analysis 1:125–300.

[2] Truesdell, C.A., Noll, W. (1965). Handbuch der physik. Springer.

[3] Gurtin, M.E. (1981). An introduction to continuum mechanics. Aca-demic Press.

[4] Khan, A.S., Huang, S. (1995). Continuum theory of plasticity. JohnWiley & Sons.

[5] Podio Guidugli, P. (2000). A primer in Elasticity. Journal of Elasticity58:1–104.

[6] Bigoni, D. (2012). Nonlinear solid mechanics: bifurcation theory andmaterial instability. Cambridge University Press.

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preliminary framework

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