2. Continuum Mechanics
2.3. Physical Balance Principles
We now specify the canonical balance principle introduced in 2.2.3 to the balances of mass, linear momentum, angular momentum, energy and entropy. To be able to specify the related supplies and fluxes, we first apply Euler’s cut principle to a subdomain BP and specify the relevant quantities acting onBP, specifically the stress and heat flux.
2.3.1. Isothermal and Quasistatic Conditions
Throughout this work, we will assume that all occurring processes are isothermal and quasistatic. The assumption ofisothermal conditionsis based on the idea that the bodyB is surrounded by an infinite heat bath of constant temperature θ=θo such that the body will continuously try to reach the resulting equilibrium state θ(X, t) = θo through heat exchange with the heat bath. If this heat exchange and the heat conduction within the body occur much faster and the heat production much slower than all other relevant
2.3 Physical Balance Principles 15
X X
x x
N dA
dA n
da da
B S
BP BP
∂BP
∂BP
SP SP
∂SP
∂SP γo
γo
t T q
ϕt(X)
ϕt(X)
Figure 2.5: Euler’s cut principle: The material part BP ⊆ B cut out from the reference domain B and the corresponding deformed spatial part SP ⊆ S cut out from the current domainS. The mechanical action of the rest of the body is replaced by the tractiont(x, t) acting on∂SP, the thermal action is replaced by the heat fluxQ(X, t) acting on∂BP.
processes (e.g. mechanic deformation), we can assume a constant temperature distribution in the body, leading to the relations
θ(X, t)≈ θo = const. ⇒ θ(X, t)˙ ≈0 and Gradθ(X, t)≈0 inB. (2.18) The assumption of quasistatic conditions used here is based on the idea that the me-chanical loading is applied extremely slowly such that the body is always in mechanical equilibrium. As a consequence, the energy and the forces related to inertial effects will be much smaller than e.g. the energy and the forces related to the deformation of the body.
We can hence neglect all contributions related to inertia, leading to
kϕ˙tk ≈0 and kϕ¨tk ≈0 inB. (2.19) It should be noted here that (2.19) does not rule out gradients of ˙ϕt, such that e.g.
F˙t 6=0 also in quasistatic processes. Furthermore, since we will account for dissipation where it occurs (i.e. during phase transformations or plastic deformation), the assumption of quasistatic processes does not imply reversibility (even though the reverse holds).
To guarantee consistency, all derivations in this section are carried out taking into account both temperature and inertia and then specialized using (2.18) and (2.19).
2.3.2. Mechanical and Thermal Fluxes
To introduce the notion of stress and heat flux, we consider a body B whose particles are mapped from referential positions X ∈ B to spatial positions x ∈ S by the point mapϕt(X). ApplyingEuler’s cut principle, we “cut out” a referential subdomainBP ⊆ B and the associated deformed spatial subdomain SP = ϕt(BP) ⊆ S and consider the mechanical and thermal fluxes over the referential and spatial boundaries ∂BP and ∂SP. 2.3.2.1. Stress Tensors. To replace the mechanical action of the remainder of the body S \ SP on the cut out part SP, we introduce a spatial traction vector t(x, t) acting on its surface ∂SP, see Figure 2.5. The Cauchy theorem then states that at every point
on the surface x ∈ ∂SP, the traction vector is a linear function of the spatial outward unit normaln(x, t) to ∂SP,
t(x, t) =σ(x, t)n(x, t), (2.20) where σ(x, t) is called the Cauchy stress. While t measures the current force per unit spatial area da, we can also introduce a nominal traction vector T parallel to t that measures the current force per unit reference area dA. For this nominal traction vector, we postulate a Cauchy-like theorem similar to (2.20),
T(X, t) =P(X, t)N(X), (2.21)
where P(X, t) is called the first Piola-Kirchhoff stress and where N(X) is the outward reference unit normal to ∂BP. The link between the two stress tensors is established by noting thatP N dA=σnda. Recalling (2.11) this yields
PdA=σda ⇔ P dA=σcof(Ft)dA ⇔ P =σcof(Ft). (2.22) Note that the Cauchy stress is often referred to as true stress, whereas the first Piola-Kirchhoff stress is denoted as thenominal stress.
2.3.2.2. Heat Flux. To replace thethermal action of the remainder of the bodyB \ BP on the cut out part BP, we introduce a heat flux Q(X, t) per unit area of the reference surface ∂BP. We then assume that at every point on the surface X ∈ ∂BP, the heat supply is a linear function of the referential outward unit normal N(X) to ∂BP,
Q(X, t) =q(X, t)·N(X), (2.23)
whereq(X, t) is called thereferential heat flux vector. In analogy to (2.21),qis sometimes also called the Piola-Kirchhoff heat flux vector, see e.g. Silhav´ˇ y [146].
2.3.3. Balance of Mass
The mass MBP of the material subdomain BP ⊆ B is defined in terms of the referential mass density ρo(X) as MBP = R
BP ρo(X)dV. In the absence of mass supply, produc-tion and transport, the mass of a material subdomain is conserved during deformaproduc-tion.
Recalling the canonical balance principle introduced in Section 2.2.3, we can state ξVM =ρo – density of mass Min BP ,
sMV = 0 – supply of mass MinBP , pMV = 0 – production of mass MinBP , fM =0 – flux of mass Mover ∂BP,
(2.24)
where we confirm from pMV = 0 that the mass is a conserved quantity. From (2.16) and (2.17), we can deduce the global and local forms of the balance of mass,
d
dtMBP = d dt
Z
BP
ρo(X, t)dV = 0 ⇒ ρ˙o = 0. (2.25) Equation (2.25) tells us that the mass MBP of any material subdomain as well as the referential mass density ρo(X) are constant. Considering the mass MSP of the related deformed spatial domainSP =ϕt(BP) with spatial mass density ρ(x, t), we can write
MSP = Z
SP
ρ(x, t)dv= Z
BP
ρ ϕt(X), t
det(Ft)dV = Z
BP
ρo(X)dV =MBP , (2.26)
2.3 Physical Balance Principles 17
where the volume map (2.12) has been used. Localization of (2.26) yields the relation ρo(X) =ρ ϕt(X), t
det Ft(X)
⇔ ρ(x, t) =ρo ϕ−t1(x)
/det Ft ϕ−t1(x)
, (2.27) which parametrizes the spatial mass density in terms of the deformation gradient.
2.3.4. Balance of Linear Momentum
The linear momentum IBP of a material subdomain BP ⊆ B is defined in terms of the velocity ˙ϕt as IBP = R
BP ρoϕ˙tdV. The resulting force FBP acting on the domain BP is given by FBP = R
BP γodV +R
∂BP T dA in terms of the body force γo per unit reference volume. The balance of linear momentum postulates the relation
d
dtIBP =FBP ⇔ d dt
Z
BP
ρoϕ˙tdV = Z
BP
γodV + Z
∂BP
T dA . (2.28) Comparison of (2.28) with the canonical balance principle (2.16) allows identification of
ξVI =ρoϕ˙t – density of linear momentum I in BP, sIV =γo – supply of linear momentum I in BP , pIV =0 – production of linear momentum I in BP , fI =P – flux of linear momentum I over ∂BP ,
(2.29)
where we note from pIV =0 that the linear momentum is a conserved quantity and where we have deduced fI = P from the combination of fI ·N = T with (2.21). Using the canonical local form (2.17) together with (2.25), we can directly write
ρoϕ¨t =γo+ Div(P). (2.30)
Finally, since we assume quasistatic conditions, we can make use of (2.19) and obtain
Div(P) +γo =0. (2.31)
Equation (2.31) is also referred to as the static mechanical equilibrium condition.
2.3.5. Balance of Angular Momentum
Theangular momentum L0BP of a material subdomainBP ⊆ Bwith respect to the origin of the coordinate system 0 is defined by L0BP =R
BP ϕt×ρoϕ˙tdV. Accordingly, the resulting torque TB0P with respect to the origin of the coordinate system 0 acting on it is given by the expression TB0P =R
BP ϕt×γodV +R
∂BP ϕt×T dA. Thebalance of angular momentum postulates a relation between these two quantities, given by
d
dtL0BP =TB0P ⇔ d dt
Z
BP
ϕt×ρoϕ˙tdV = Z
BP
ϕt×γodV + Z
∂BP
ϕt×T dA . (2.32) Comparing (2.32) with the canonical balance principle (2.16), we obtain
ξLV0 =ϕt×ρoϕ˙t – density of angular momentum L0 in BP, sLV0 =ϕt×γo – supply of angular momentum L0 in BP , pLV0 =0 – production of angular momentum L0 in BP , fL0 =ϕt×P – flux of angular momentum L0 over ∂BP ,
(2.33)
where we note from pLV0 = 0 that the angular momentum is also a conserved quantity and where we have derived fL0 =ϕt×P using fL0 ·N =ϕt×T and (2.21). Recalling the canonical local form (2.17) together with (2.25), we can write
˙
ϕt×ρoϕ˙t+ϕt×ρoϕ¨t =ϕt×γo+ Div(ϕt×P). (2.34) Rewriting the divergence term Div(ϕt×P) in index notation using the permutation sym-bolεabc yields (εabcϕbPcD),D =εabcϕb,DPcD+εabcϕbPcD,D which translates to the symbolic relation Div(ϕt×P) = ε: (FtPT) +ϕt×Div(P). Insertion into (2.34) yields
[ ˙ϕt×ρoϕ˙t] +ϕt×[ρoϕ¨t−γo−Div(P)]−ε : (FtPT), (2.35) whereε =εabcea⊗eb⊗ec is the third order permutation tensor. Making use ofb×b=0
∀b ∈ R3, recalling the balance of linear momentum (2.30) and noting thatεabcϕb,DPcD= 0 impliesϕb,DPcD =ϕc,DPbD due to the skew symmetry εabc=−εacb, we finally obtain
ε : (FtPT) =0 ⇔ FtPT =P FTt . (2.36) Note that (2.36) does not depend on the choice of the reference point of the angular momentum and the torque. However, the fact that (2.32) holds for arbitrary reference points induces the balance of linear momentum (2.28), see e.g. Silhav´ˇ y [146]. Further note that reformulation of (2.36) by use of (2.22) yields the well-known conditionσ =σT. 2.3.6. Balance of Energy
The energy EBP of a material subdomain BP ⊆ B is defined in terms of the internal energy density e(X, t) per unit reference volume as EBP = R
BP(e+ 12ρoϕ˙2t)dV, where
1
2ρoϕ˙2t represents the kinetic energy. Denoting the heat supply per unit reference volume by r(X, t), we can identify the supply, production and flux for thebalance of energy as
ξVE =e+ 12ρoϕ˙2t – density of energy E inBP , sEV = ˙ϕt·γo+r – supply of energy E inBP, pEV = 0 – production of energy E inBP , fE = ˙ϕt·P −q – flux of energy E over ∂BP ,
(2.37)
where we note from pEV = 0 that the energy is a conserved quantity and where fE =
˙
ϕt·P −q in combination with (2.21) and (2.23) gives fE ·N = ˙ϕt·T −Q. Relations (2.37) can now be inserted into the canonical global balance principle (2.16), yielding
d
dtEBP = d dt
Z
BP
(e+ 12 ρoϕ˙2t)dV = Z
BP
( ˙ϕt·γo+r)dV + Z
∂BP
( ˙ϕt·T −q·N)dA . (2.38) Using the canonical local form (2.17) together with (2.25), we can write
˙
e+ρoϕ˙t·ϕ¨t= ˙ϕt·γo+r+ Div( ˙ϕt·P −q). (2.39) Making use of the identity Div( ˙ϕt·P) = Grad( ˙ϕt) :P+ ˙ϕt·Div(P) =P : ˙Ft+ ˙ϕt·Div(P), and rearranging terms, we can finally reformulate equation (2.39) as
˙ ϕt·
ρoϕ¨t−γo−Div(P)
+ ˙e=P : ˙Ft+r−Div(q) ⇒ e˙ =P : ˙Ft+r−Div(q), (2.40) where (2.30) has been used to eliminate the term in square brackets. The conservation of energy as stated in (2.39) and (2.40) is also referred to as thefirst law of thermodynamics.