We shall outline a simple model of porous media by introducing the volume fraction, pore fluid pressure, effective solid stress, etc. As an example of an equilibrium solution in a linear theory for incompressible porous media, porosity distribution in saturated soil is considered. Remark The specific form of the relations 5 can be derived provided that the ClausiusDuhem assumptions on the entropy flux and the entropy supply are valid for each constituent.
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However, only the entropy production of the mixture as a whole is postulated to be non-negative. It has been shown that such assumptions are appropriate but not necessarily general enough to account for the behavior of a mixture within 22 I.
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Liu the framework of continuum mechanics. In , it has been shown by virtue of the jump condition 14 in equilibrium, that the fluid pressure measured from the manometer tube attached to the porous body is the pore fluid pressure. Moreover, since the resistive force vanishes in equilibrium, i. It agrees with the observation in soil mechanics from experimental measurements that the manometer pressure in the soil is the pressure as if the medium were bulk fluid, unaffected by the presence of the solid constituent in the medium. Mixture theory of porous media 27 We remark that this result is sometimes overlooked in the mixture theory of porous media.
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It is mainly due to the fact that in the theory of simple mixture, which omits the second gradients of deformations as independent constitutive variables, the equilibrium interactive force m0 is identically zero by constitutive hypothesis. This remark, we shall regard as a strong evidence that porous media must be treated as non-simple mixtures even for a linear equilibrium theory.
This is the effective stress principle in soil mechanics first introduced by Terzaghi see . Both are uninteresting cases. It offers a check for the applicability of the present theory based on such a simple model.
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References 1. Bowen, R. In: Eringen, A. Continuum Mechanics, vol. Academic Press, London 2. Liu, I.
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Springer, Heidelberg 4. Terzaghi, K. Truesdell, C. Evolutionary genetics requires both: The randomness of mutations and the determination of selection. In that sense genetics is analogous to thermodynamics which requires the randomness of thermal motion and the purpose of energetic interaction. The analogy can be made explicit by defining an entropy of a population and a selective energy of interaction of the population with the environment, or a breeder.
Thermodynamicists deplore the simplicity of the mathematics that is involved while, on the other hand, the sociologists say complain about the use of mathematics — such as it is —, because according to them no mathematics is permissible that goes beyond the four basic elements of calculus, e. And yet, in most fields of knowledge — upon reflection — we can detect a random influence on events that competes with the strife toward a definite and desirable goal, and prevents the full attainment of that goal. That competition between opposing tendencies was first identified in thermodynamics and it led to the doctrine of entropy and energy: Randomization under the influence of the thermal motion tends to increase the entropy, and energy tends to decrease, because it attracts the elements of a body toward the depth of definite potential wells.
That theme dictates — again and again — physical properties and it has been identified for many circumstances in elasticity, chemistry, metallurgy, and meteorology. Extrapolation of that competition into other fields requires some thought, and some insight. It is not a trivial exercise of turning the crank.
Indeed, mutation and selection may act as opposing tendencies in the development of a population. Mutation is random so that — in the simplest case — it tends to be impartial to different genotypes in the population in the sense that they all have an equal chance of appearing. The intensity of mutation may be controlled — accelerated or slowed down — by radiation. Selection on the other hand is determined by the extant environment — possibly shaped by a breeder — and may prefer one phenotype in a population over others in the sense that the fittest has a more numerous progeny.
In a manner of speaking mutation and selection compete and lead the population to a distribution of genotypes to which both tendencies contribute: Selection more when the intensity of mutation is weak, and mutation more, when its intensity is strong. Thus mutational chance may be balanced by selective order.
The situation is not unlike one frequently encountered in thermodynamics of a gas say , where entropy and energy are often opposing each other in determining the shape of the gas. Entropy S measures the effect of the random thermal motion whose intensity is controlled by temperature T ; therefore entropy is biggest for a homogeneous distribution of the atoms in the gas.
The energy E on the other hand depends on the environmental situation, e. Thus the temperature determines to what extent energetic order dominates entropic chance. The theme was exploited and illustrated for many circumstances in . The perceived analogy of thermodynamics and genetics was expounded in my paper  for simple cases of haploid and diploid populations. The present paper reviews the previous arguments — albeit only for haploids — and it emphasises the observation that evolutionary genetics can be interpreted in terms of the notions governing the thermodynamics of mixtures of reacting fluid constituents; that aspect was only touched on in  at the very end.
Mixtures of fluids may be seen as fluids with an internal structure, which is represented by the concentrations of the constituents.
Deformation waves in microstructured solids and dimensionless parameters; pp. 109–115
That interpretation is not a common one but it is all I have to offer for arguing that my contribution fits — loosely — into the scheme of the articles in this book. The cells are equal in all aspects except on one locus of one chromosome, which may carry alleles A or a. The A-cells and the a-cells are supposed to have the same molecular energy.
In  I have discussed haploid and diploid cells. The latter contain pairs of chromosomes. Cells with alleles a may be selectively privileged by the environment. But, if there is no selective bias, a and A occur with equal probability, because they are energetically equal. And by the rules of statistical thermodynamics the entropy of the population is defined by 2N! By 2 through 4 this means that R and S will grow to a maximum. The speed of this growth is small in a physical sense, because time in genetics is measured in generations. Thus the growth of entropy is purely probabilistic.
It is entirely possible that the growth-process may be interrupted occasionally by a decrease of R and S and even in equilibrium there may be fluctuations away from the maxima of R and S.
Entropy as a function of q, b. Selective energy as a function of q. S q is plotted in Fig. The bonus may consist of being able to find more adequate food and therefore have a more numerous progeny than the A-cells. In the end, — over many generations —, this may lead to dominance of the a-cells and the elimination of A-cells.
Thus we imagine that the environment, or the breeder, offers a potential well for the population. In the biological literature, e. This is entirely equivalent, but we choose wells — instead of peaks — in order to emphasize the analogy with physics. Their minima determine the values of q in constrained equilibrium for three given mutational intensities T. Suppose we start at some value of q close to one. Such a distribution has a small number of realizations and the stochastic tendency for the number of realizations to grow under the mutational activity will lead the population to smaller values of q.
On the other hand, according to 5 and Fig. The slope of E q on the other hand remains constant as q decreases. If that happens, S q — or the number R of realizations — has reached the maximum that is compatible with the constraint E q. This is the state of constrained equilibrium. Their effect may be drastic. The temperature thus corresponds to the mutational intensity. The requirements 6 and 9 are equivalent. The preference for 9 in thermodynamics is entirely traditional; we follow it here, because we wish to emphasize the analogy of the reasoning in thermodynamics and genetics.
For the present case with a linear function E q , cf. This is a plausible ansatz and it may often be true. However, more complex alternatives may be discussed. Thus both the a-cells and the A-cells may be privileged, if they are rare. Thus, according to Haldane  parasites will evolve so as to attack most effectively the common cells of the host population; therefore the rare host cells will enjoy a selective advantage.
Similarly hosts will acquire immunity to the common type of parasites, so that the rare types of parasites will be at an advantage. In such cases the potential well may be represented graphically by a double well, cf. Double well potential for q-dependent bonus, b. Actually it develops into a malus for q 11 A simple calculation shows that for T 0.
Inspection of Fig. The significance of the flatness of the selective free energy is discussed in .
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