Abstract
In this chapter, we review the fundamental concepts and formulation of the thermomechanics of continuous media. First, we revise the expressions of the two first laws of thermodynamics for thermomechanical processes, that is, those with changes in temperature and strains as state-independent variables. These equations constitute the generalization of the energy balance equation that was presented, particularized for the isothermal case, in chapter “Basic Equations of Continuum Mechanics” [10]. Next, we introduce different thermodynamic potentials such as the internal energy density, the Helmholtz free energy density, and the dissipation density. This latter is expressed in terms of additional independent internal state variables that take into account history-dependent changes in the material’s internal microstructure. The associated thermodynamic fluxes are then defined as derivatives of the dissipation density with respect to the corresponding thermodynamic drivers (internal variables). The next section introduces the fundamental principles for simple (local, nongraded) materials. These allow establishing the general constitutive equations for the rest of the state- dependent variables, stress and entropy, from the expression of the chosen thermodynamic potential and the fulfillment of the second law of thermodynamics. These expressions are finally applied to several examples: two types of non-dissipative materials, such as ideal fluids and elastic solids in thermomechanical processes, and two dissipative cases: damage mechanics and nonlinear viscoelastic solids. We finish with the formulation and results of a complex thermomechanical process, the extrusion forming process of an aluminum billet under high temperature and viscoplastic behavior.
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Notes
- 1.
From now on, we shall consider only Euclidean reference frames, without any restriction, but allowing a simpler notation.
- 2.
Note that the material’s microstructure is associated with an intrinsic length scale, leading most times to size effects in the constitutive response.
- 3.
Other possible energy sources (chemical, electrical, metabolical) are considered only in an indirect form through changes in the internal state variables.
- 4.
Each variable \(\varvec{\kappa }_i\) may have a different tensorial order (scalar, vector, or tensor) according to its definition.
- 5.
In the following, we shall consider the usual case of body forces \(\mathbf {b}\) independent of the displacements.
- 6.
When \(r=0\) and \(j=0\), that is, in adiabatic thermally isolated processes (no heat sources or sinks are present), this equation reduces to
$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}\int _{\mathrm {R}_t} \rho \eta \, \mathrm {d}v\ge 0, \end{aligned}$$meaning that the total entropy of the system cannot decrease.
- 7.
Remember that balance of angular momentum, in standard non-micropolar continuum mechanics, is equivalent to the symmetry of the Cauchy stress tensor, so only six independent functions will be considered for this tensor.
- 8.
This means that such initial stored energy in the reference state is provided by an energy provision mechanism not considered explicitly in the equations.
- 9.
For example, when internal growth is considered, the associated deformation gradient \(\mathbf {F}^\mathrm {g}\) or its rate \(\dot{\mathbf {F}}^\mathrm {g}\) are described by a convenient expression.
- 10.
For example, the need to remain on the surface of a certain (elastic) state domain, like in plasticity or damage, and/or additional thermodynamic assumptions like the fulfillment of the maximum dissipation principle.
- 11.
This principle, although usually assumed, has been controversial and even some authors [15] ridiculed it, so it has to be taken with care.
- 12.
Although some additional variables will appear in the following in the expressions of the state functions, e.g., internal state variables, their value at each time, according to this principle, must be obtained from the histories of motion and temperature at all points of the body up to such time t.
- 13.
It is noteworthy that the fulfillment of locality is sometimes excessive. There exist materials whose constitutive behavior at one point also depends on the values of the configuration and temperature in a finite neighborhood of such a point, leading to a nonlocal theory that falls out the limits of this chapter (see, for example, [18]).
- 14.
Again, this principle does not always hold, so it should be considered as an additional assumption.
- 15.
This principle, although not directly derived from any fundamental universal law, and therefore, at a different level than the previous ones, is a direct consequence of the trend of any physical system to dissipate as much energy as possible and therefore to maximize entropy production.
- 16.
Similar equations can be written for the rest of constitutive equations (\(\mathbf {Q}, \varvec{\varSigma }^{\varvec{\kappa }_i}\)), since E and \(\varUpsilon \), as scalar fields, do not change under rotations.
- 17.
Observe that in purely mechanical processes, where (23) holds, \(\varvec{\varSigma }^{\varvec{\kappa }_i} = \partial \mathcal {D}/\partial \varvec{\kappa }_i, \ \forall \ i=1,..,N\), which implies that \(\dot{\mathcal {D}} = (\partial \mathcal {D}/\partial \varvec{\kappa }_i ) \varvec{\dot{\kappa }}_i \ge 0\).
- 18.
Note that constitutive functions for \(\mathbf {T}^{(2)}, \varUpsilon , \varvec{\varSigma }^{\varvec{\kappa }_i}\) are determined by a single scalar function \(\varPsi (\mathbf {F}^\mathrm {e},\varTheta , \varvec{\kappa }_i)\) (equivalently \(E(\mathbf {F}^\mathrm {e},\varTheta , \varvec{\kappa }_i)\)) or, for purely mechanical problems, by the two functions \(\mathcal {V}(\mathbf {F}^\mathrm {e})\), \(\mathcal {D}(\varvec{\kappa }_i)\), while the constitutive function for the heat flux satisfying (28) has to be added explicitly.
- 19.
Observe that the dependence on motion here is only through changes in density or, equivalently, through changes in volume.
- 20.
Note that, in this particular case, all constitutive functions are determined by a single scalar function \(\varPsi =\hat{\varPsi }(\mathbf {F},\varTheta )\) (\(E=\hat{E}(\mathbf {F},\varUpsilon )\)), except the heat flux, which satisfies the first inequality of (28).
- 21.
Observe that the condition of polyconvexity usually requested for hyperelastic strain energy functions is not necessary, but, on the other hand, is a sufficient condition for quasiconvexity [2].
- 22.
For an isotropic material, for example,
$$ \mathbf {F}^\theta = \lambda ^\theta \mathbf {I}, \quad \lambda ^\theta = \exp \left( \int _{\theta ^0}^\theta \alpha (\theta )\,\mathrm {d}\theta \right) , $$with \(\lambda ^\theta \) the isotropic thermal stretch and \(\alpha \) the coefficient of thermal expansion of the material.
- 23.
These stresses are null if and only if \(\mathbf {F}^{\theta }\) fulfills internal and boundary compatibility, but, in general, this is not the case. This means that, except for such a compatible situation, the above stresses are not, in general, in equilibrium. An intermediate step is usually solved to get equilibrium under zero external forces, leading to a different set of strains, that may not be fully compatible yet. This is a problem intrinsically associated to this phenomenological decomposition of the deformation gradient.
- 24.
In the presence of an adiabatic process (\(r=0, \mathbf {q} = \mathbf {0} \)), the heat transfer equation yields
$$ \dot{\theta } =\frac{\varvec{\sigma }: \mathbf {D}^\theta }{\rho C_p}, $$which defines the temperature rate in an isolated material due to the strain rate, and constitutes the so-called Kelvin formula.
- 25.
We consider again the isothermal case without restriction, working with the appropriate strain-energy density function when required.
- 26.
Normally, it is assumed that around 90% of the energy mechanically generated becomes heat while the remaining 10% goes for generating plastic dislocations, generation of grain boundaries and phase changes [23]. Therefore, we can also write \(\varvec{\sigma } : \mathbf {D}^{\mathrm {ne}} \approx 0.9 \varvec{\sigma } : \mathbf {D}\).
- 27.
Note that this fluidity parameter corresponds to the viscosity, if we reduce this model to a generalized viscous one. On the contrary, if this fluidity parameter is negligible (taken as null), this model identifies with a purely plastic behavior, reducing to
$$ s_{ij}=\frac{2\sigma _f(\bar{D},\theta )}{3\bar{D}}D_{ij}. $$
References
Alfaro I, Yvonnet J, Cueto E, Chinesta F, Doblaré M (2006) Meshless methods with application to metal forming. Comput Meth Appl Mech Eng 195:6661–6675
Ball J (1977) Convexity conditions and existence theorems in nonlinear elasticity. Arch Ration Mech Anal 61:317–401
Coleman BD, Noll W (1963) The thermodynamics of elastic materials with heat conduction and viscosity. Arch Ration Mech Anal 13:167–178
Cueto E, Sukumar N, Calvo B, Martínez-Barca MA, Cegonino J, Doblaré M (2003) Overview and recent advances in natural neighbour Galerkin methods. Arch Comput Meth Eng 10:307–384
Duvaut G, Lions JL (1972) Les Inéquations en Mécanique et en Physiques. Dunod, Paris
Lemaitre J, Chaboche JL (1990) Mechanics of solid materials. Cambridge University Press, Cambridge
Lof J, Blokhuis Y (2002) FEM simulations of the extrusion of complex thin-walled aluminium sections. J Mater Process Technol 122:344–354
Lubliner J (1984) A maximum-dissipation principle in generalized plasticity. Acta Mech 52:225–237
Marsden JE, Hughes TJR (1994) Mathematical foundations of elasticity. Dover Publications, New York
Merodio J, Ogden RW (2019) Basic equations of continuum mechanics. In: Merodio J, Ogden RW (eds) Constitutive modeling of solid continua. Series in Solids Mechanics and its Applications (In Press). Springer
Miehe C, Keck J (2000) Superimposed finite elastic-viscoelastic-plastoelastic stress response with damage in filled rubbery polymers. Experiments, modelling and algorithmic implementation. J Mech Phys Solids 48:323–365
Montagut E, Kachanov M (1988) On modeling a microcracked zone by weakened elastic material and on statistical aspects of crack-microcrack interaction. Int J Fract 37:R55–R62
Mullins L (1947) Effect of stretching on the properties of rubber. J Rubber Res 16:275–289
Perzyna P (1966) Fundamental problems in visco-plasticity. Adv Appl Mech 9:243–377
Rivlin RS (1997) Red herrings and sundry unidentified fish in nonlinear continuum mechanics. In: Barenblatt GI, Joseph DD (eds) Collected papers of R.S. Rivlin. Springer, New York
Sellars CM, Tegart WJ McG (1972) Hot workability. Int Metall Rev 17:1–24
Serrin J (1959) Mathematical principles of classical fluid mechanics. In: Flügge S (ed) Encyclopedia of physics, vol VIII/1. Springer, Berlin, Heidelberg, pp 125–263
Silling SA, Epton M, Weckner O, Xu J, Askari E (2007) Peridynamic states and constitutive modeling. J Elast 88:151–184
Simo JC (1987) On a fully three-dimensional finite-strain viscoelastic damage model: formulation and computational aspects. Comput Meth Appl Mech Eng 60:153–173
Simo JC, Hughes TJR (1998) Computational inelasticity. Springer, New York
Temam R (1985) Mathematical problems in plasticity. Gauthier-Villars, Paris
Truesdell C, Toupin RA (1960) The classical field theories. In: Flügge S (ed) Handbuch der Physik, III(1). Springer, Berlin, Heidelberg, pp 225–793
Zhou J, Li L, Duszczyk J (2003) 3D FEM simulation of the whole cycle of aluminium extrusion throughout the transient state and the steady state using the updated Lagrangian approach. J Mater Process Technol 134:282–297
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Doblaré, M., Doweidar, M.H. (2020). Thermomechanics. In: Merodio, J., Ogden, R. (eds) Constitutive Modelling of Solid Continua. Solid Mechanics and Its Applications, vol 262. Springer, Cham. https://doi.org/10.1007/978-3-030-31547-4_3
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