Constitutive Equations
6.1 Constitutive Axioms
It has already been stated in Chaps. 4 and 5 that balance equations are general relations whose validity does not depend on body properties. However, we also know from experience that two bodies with the same dimensions and shape may react differently when subjected to the same load and thermal conditions.
Thus we intuitively conclude that, in contrast to rigid body dynamics, the evolution of a continuous deformable body cannot be completely predicted by knowing the equations of motion, the mass distribution, and the external forces acting on the body. We will now use the results of Chap.5to express this statement in a formal way.
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A. Romano, A. Marasco,Continuum Mechanics using MathematicaR, Modeling and Simulation in Science, Engineering and Technology, DOI 10.1007/978-1-4939-1604-7__6
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In the Eulerian description,f;v;T; ;h; grepresent the unknown fields, and the balance equations for mass conservation, momentum balance, and energy balance are the equations we use to determine these fields. Similarly, in the Lagrangian description, the unknowns becomefv;T; ;h; gand the balance equations are the momentum balance and energy balance.
In both cases the balance equations do not form a closed set offield equations for the above-mentioned fields, so that we must add relations that connect thestress tensor, theinternal energy, and theheat fluxwith the basic fields.
These relations are called constitutive equations because they describe the material constitution of the system from a macroscopic view point.
In order to obtain such relationships, we presume that the macroscopic response of a body, as well as any macroscopic property of it, depends on its molecular structure, so that response functions could in principle be obtained from statistical mechanics, in terms of the average of microscopic quantities.
As a matter of fact, such an approach, although promising from a theoretical point of view, is not straightforward if applied to the complex materials that interest us in continuum mechanics.
In addition, the basic assumption of continuum mechanics consists of erasing the discrete structure of the matter, so that constitutive equations are essentially based on experimental evidence. Again, this is not an easy task, but continuum mechanics greatly simplifies this task through the introduction of general rules, calledconstitutive axioms. In fact, they represent constraints for the structure of constitutive equations.
In order to discuss these axioms, we introduce some definitions.
The history of a thermokinetic process up to time t is defined by the two functions
xtS Dx.Y; /; St D .Y; / .Y; /2C.1; t : (6.1) According to this definition, two processes arelocally equivalentatXif there is at least a neighborhood ofXin which the two histories coincide.
By dynamic process we mean the set of fields x.Y; t /, .Y; t /, T.Y; t /, .Y; t /,h.Y; t /,.Y; t /, where TFT D FTT, which are solutions of equations (5.69)1(5.80)1, under a given body densityband a sourcer.
Taking into account all the previous remarks, it turns out that the material responseis formally expressed by the set of fields A D fT; ; h; g, which depend on the history of the thermokinetic process. Constitutive axioms, listed below, deal with such functional dependence.
1. Principle of Determinism(Noll, 1958).
At any instantt,the value of AatX2 Cdepends on the whole history of the thermokinetic processxtS, St up to the timet.
This is equivalent to stating that the material response atX2 Cand at time tis influenced by the history ofSthrough a functional
AD‡.xtS; St;X/; (6.2)
6.1 Constitutive Axioms 165 which depends on X 2 C (nonhomogeneity) as well as on the reference configuration.
If the Eq. (6.2) are substituted into (5.69)1, (5.80)1, withTFT DFTT, then a system of 4 scalar equations for the 4 unknown functionsx.X; t /; .X; t /is obtained. Ifband the history of the process are given up to the instantt0under suitable boundary and initial conditions, then this system allows us to predict, at least in principle, the process fromt0on.
Note that (6.2) assumes that the material has a memory of the whole history of the motion. We make two remarks about this assumption: first, it is unrealistic to have experience of the whole history of the thermokinetic process; moreover, the results are quite reliable if we presume that the response of the systemSis mainly influenced by itsrecent history. Depending on the kind of memory selected, there are different classes of materials (e.g., materials with fading memory,plastic memory, and so on). Furthermore, if only the recent history (how recent has to be specified) is supposed to be relevant, then for suitably regular motions inC .1; t , the history of the motion can be expressed, at least in a neighborhood of .X; t /, in terms of a Taylor expansion at the initial point.X; t /of ordern. Such an assumption allows us to consider a class of materials for whichAdepends on Fas well as on its spatial and time derivatives evaluated at.X; t /:
ADA.F;F;P rXF; : : : ;X/: (6.3) The materials described by a constitutive relation (6.3) are said to be ofgraden ifnis the maximum order of the derivatives of motion and temperature.
It is worthwhile to note that this principle includes, as special cases, both classical elasticityand thehistory-independent Newtonian fluids(see Chap.7).
2. Principle of Local Action(Noll, 1958).
FieldsA.X; t /depend on the history of the thermokinetic process through a local class of equivalence atX.
According to the notion of contact forces, this principle states that the thermokinetic process of material points at a finite distance from X can be disregarded in computing the fieldsA.X; t /atX.
The previous two principles (i.e., determinism and local action), when combined, imply that the response at a point depends on the history of the thermokinetic process relative to an arbitrary small neighborhood of the particle.
Materials satisfying these two principles are calledsimple materials.
3. Principle of Material Frame-Indifference.
Constitutive equations (6.2) must be invariant under changes of frame of reference.
We first remark that the principle ofmaterial frame-indifferenceormaterial objectivity is not to be confused with the term objectivity in the sense of transformation behavior as discussed in Chap.4.
In fact, the term objectivity denotes transformation properties of given quantities, whereas the principle ofmaterial objectivitydiscussed here postulates the complete independence of the material response from the frame of reference.
In other words, since constitutive equations represent a mathematical model of material behavior, they are supposed to be independent of the observer. To make this principle clear, consider a pointP with massmmoving in an inertial frameI under the action of an elastic forcek.xxO/, whereO is the force center. The motion equation ofP inI is written as
mRxD k.xxO/:
If a frameI0 is considered to be in rigid motion with respect to I, so that the transformationx0 D x.t /CQ.t/x holds, where x.t / is the position vector fixing the origin ofI with respect toI0andQ.t /is an orthogonal matrix, then the motion equation assumes the form
mRx0D k.x0x0O/ma2m!Px0;
whereaand!are the acceleration and the angular velocity ofI0with respect to I. The motion of P in I0 can be obtained by integrating this equation or, alternatively, by integrating the previous one and by applying the rigid transformation rule to the result. In this example it is relevant to observe that, when writing the motion equation inI0, it has been assumed that the elastic force is an invariant vector and theconstitutive elastic lawis invariant when passing fromItoI0.
In summary, the principle of material frame-indifference states that consti-tutive quantities transform according to their nature (e.g.,is invariant, Tis a tensor, and so on), and, in addition, their dependence on the thermokinetic process is invariant under changes of frame of reference.
4. Principle of Dissipation(Coleman e Noll, 1963).
Constitutive equations satisfy the reduced dissipation inequality(5.83)1 in any thermokinetic process compatible with momentum and energy balance(5.69)1, (5.80)1.
To investigate the relevance of this principle, we first observe that, given (6.2) withTFT DFTT, the momentum and energy balance equations can always be satisfied by conveniently selectingbandr. This remark implies that momentum and energy balance equations (5.69)1, (5.80)1do not,as a matter of principle, play any role of constraint for the thermokinetic process; in other words, any thermokinetic process is a solution of the balance equations provided thatband rare conveniently selected.
Moreover, we note that the reduced dissipation inequality is equivalent to the entropy principle, presuming that the energy balance is satisfied. Historically, the entropy inequality has been considered to be aconstraint for the processes:
i.e., only those processes that are solutions of the balance equations and satisfy the second law of thermodynamics for given body forcesb, energy sources, and prescribed boundary conditions are admissible. It is apparent how complex such a requirement is, since explicit solutions can only be obtained in a few cases.