aa r X i v : . [ c ond - m a t . m t r l - s c i ] N ov Slow failure of quasi-brittle solids
Leonid S.Metlov ∗ Donetsk Institute of Physics and Engineering, Ukrainian Academy of Sciences,83114, R.Luxemburg str. 72, Donetsk, Ukraine † (Dated: November 22, 2018)A new mesoscopic non-equilibrium thermodynamic approach is developed. The approach is basedon the thermodynamic identity associated the first and second law of thermodynamics. In theframework of the approach different internal dissipative channels of energy are taken in account inan explicit form, namely, the thermal channel and channels of defect subsystems. The identity hasa perfect differential form what permits to introduce an extended non-equilibrium state and usethe good developed mathematical formalism of equilibrium and non-equilibrium thermodynamics.The evolution of non-equilibrium variables of a physical system are described by a Landau-basedequation set expressed through internal or different kinds of free energy connected by means of theLegendre transforms. The accordance between the different kinds of energy is possible owing tointroduction of some trends into the equation subset described the defect subsystems and havinga nature of structural viscosity. The possibilities of the approach are illustrated on the example ofquasibrittle solid damage and failure. Taking into account only one type of defects (viz., microcracks)and mechanical parameters in an expansion of free energy down to third powers in relative toaverage energy per microcrack, the description of destruction of quasi-brittle solids during long-term loading is considered. The consideration allows to find equilibrium and non-equilibrium valuesof the free energy. A qualitative behavior of the system on parameters of the theory is analyzed. Thedestruction of material is described from the uniform positions, both at uniform tension and uniaxialcompression. Origins of the high stability of mine workings at small depths and their instability atlarge depths are explained. PACS numbers: 05.70.Ce; 05.70.Ln; 61.72.Bb; 62.20.MkKeywords: mesoscopic non-equilibrium thermodynamics, internal energy, free energy, evolutin equations,stationary states
I. INTRODUCTION
A problem of solid fracture is one of the most diffi-cult challenges for modern science. The fracture is acompound multilevel process that depends on many fac-tors, such as physical properties of material, its struc-tural organization and environment and so on. Varietyof materials and environments predetermines many frac-ture mechanisms that are distributed in the range frombrittle to plastic failure. These are sudden dynamic de-struction during quick loading and a slow accumulationof microdefects with their subsequent coalescence duringa long-term static loading.Rocks are one of types of poorly studied solids of pe-culiar structural organization. Rock properties to resistto long-term destructive effects of rock pressure at largedepths are important component of stabilization of un-derground structures. At the same time rocks demon-strate a number of common for all solids features of be-havior that can be of certain interest for investigation ofprocesses that are initiated by a strong external effecton a matter and also for investigation of the resultingnon-equilibrium states. ∗ Electronic address: [email protected] † I thank all my colleagues who put me hard questions that essen-tially develop this approach
The governing relaxation mechanism in the case ofquasi-brittle solids is accumulation and growth of micro-cracks, its coalescence and branching and finally destruc-tion of a solid [1], [2]. Previously this mechanism for thecase of large deformation was considered of general ther-modynamic positions [3], [4], [5], [6], [7]. Applicability ofequilibrium thermodynamics was limited by intermediatetime scale less than that of necessary for diffusive heal-ing of microcracks. It was expected that a real solid hasalready natural weak points, non-continuities and so on,which became more intense due to external force action.Rock was considered usually as passive mechanicalcontinuum, properties of which are evolved just under theinfluence of active external stresses. Yet, it is clear fromgeneral considerations that in the process of metamor-phism at large depths the rocks reach state what is equi-librium in thermodynamic meaning for the given depths.In these conditions, the formation of an artificial cavityresults not only to mechanical disequilibrium, but, whatis much more essential, to thermodynamic disequilibriumof rocks. Actually rocks turn out in a non-equilibriumstate and relax by means of microcrack accumulation,subsequent coalescence of them and macroscopic destruc-tion. Depending on the level of accumulated internalenergy, the autowave processes of different kinds, whichare typical for an active media, are possible [8], includinggeneration of circular zonal destruction structures aroundmine workings [9], [10].In this work on the basis of the principles of thermo-dynamics an attempt to formulate a consistent theoryis made that takes into account both mentioned factorsenergy accumulated in the form of internal stresses andwork of external stresses. Areas of increased internalstresses make additional contribution to internal energy;generation of microcracks also results in growth of inter-nal energy of a solid of the external work which is equalto a total energy of break bonds.
II. DEFINING RELATIONS
For infinitely small changes of external parameters ofa defect-free homogeneous system during its interactionwith external bodies a change of internal energy (first lawof thermodynamics) is equal to du = T δs + σ ij dε eij + σ ij dδε nij , (1)where u is density of internal energy, J/m ; T is temper-ature, K ; s is entropy density (here from external sourcesonly), J/K ∗ m ; σ ij is the stress tensor, H/m and ε eij , ε nij are the reversible and non- reversible parts of defor-mation tensor. Here increments of variables, which arestate functions, are perfect differentials. In this form thelaw expresses external forced and thermal actions on thestudied object and it fits for description of both equilib-rium and non-equilibrium processes.Irreversible part of the external work goes to internalheat at the cost of entropy production by internal sourcesand to generation and modification of defect subsystem,what may be written as σ ij δε nij = T δs ′′ + χδs ′ + N X k =1 ϕ k δh k ≥ . (2)Here s ′′ is that part of entropy produced by internalsources, which has time to go to equilibrium state dur-ing the external act, s ′ is that part of entropy pro-duced by internal sources, which remains yet into thenon-equilibrium state and it lies ahead to relax in thenext time, χ is the non-equilibrium temperature, sϕ k , h k are average k -type defect energy ( J ) and defect density( m − ), respectively, N is number of defect subsystems.The relation (2) is one of equivalent form of the secondlaw of thermodynamics. Substituting it to (1) and as-sociating equilibrium entropy produced by external andinternal sources one may deduce next identity du = T ds + σ ij dε eij + χds ′ + N X k =1 ϕ k dh k . (3)The relation (3) is combination of the first and secondlaw of thermodynamics in the form of identity, but notin the known inequality form. It turns out possible towrite it in this form as all internal channels of energydissipation are known for the given model of a solid. Thesound property of this identity is that all increments of the thermodynamic variables are perfect differentials inone or another sense. The first two terms are treatedin the sense of equilibrium thermodynamics; the secondtwo terms describe non-equilibrium processes passing ina solid. The temperature and stress tensor may be calcu-lated by means of differentiation of internal energy withrespect to full equilibrium entropy and elastic strain ten-sor as it does in the classical thermodynamics. The non-equilibrium temperature and defect energy are too calcu-lated by means of differentiation of internal energy withrespect to non-equilibrium entropy and defect densities χ = ∂u∂s ′ , ϕ k = ∂u∂h k , (4)but in the sense of the extended thermodynamic forcesin the Landau-based equations. Owing to this, the ex-pression (3) may be regarded as complete differentialwith respect to variables s , ε eij , s ′ , and h k and this vari-ables determines some generalized non-equilibrium stateof the system. What does it give? Now one can usethe simple mathematics of equilibrium thermodynamicsfor the non-equilibrium case. Time evolution of non-equilibrium thermodynamic parameters s ′ , and h k canbe calculated by means of set of equations each of it isLandau-Chalatnikov or Ginzburg-Landau equation type[7], [11]: ∂s ′ ∂t = γ s ′ ∂u∂s ′ = γ s ′ χ,∂h k ∂t = γ h k ( ∂u∂h k − ϕ k ) = γ h k ( ϕ k − ϕ k ) . (5)These equations differ from the classical prototype.First, here the internal energy is used as a more fun-damental quantity entered to the first law of thermody-namics. Second, in the right parts of equations the signplus stands it means that the stable stationary states areplaced near a maximum of the internal energy. This pe-culiarity is conditioned by energetic pumping in the costof irreversible work. The third difference is presence ofviscosity shift ϕ k in the equation describing defect sub-system. In consequence of the shift the stable stationarystates do not coincide exactly with the maximums of theinternal energy, but coincide with some points in whichthe curve of the internal energy has finite positive slopeof tangents. This shift has same physical sense as orderparameter in the case of a supercooled liquid and playsan important role for construction of a consistent set ofnon-equilibrium thermodynamic potentials connected bymeans of the Legendre transformations.So as number of conjugate pairs of thermodynamicvariables increases in comparison with equilibrium casethe number of the Legendre transformations increasestoo. For example, if one carries out the Legendre trans-formation f = u − T s the classic free energy is resulted.Difference of it from classic one is that this type of freeenergy has non-equilibrium nature. The stable station-ary states of a system in reference to defect subsystemare placed near the maximums of the free energy sameas for internal energy.If one carries out the Legendre transformation in theform ˜ f = u − T s − N X k =1 ϕ k h k (6)the complete free energy is resulted. Despite to the abovethe stable stationary states of the system in reference todefect subsystem are placed near the minimums of thisfree energy. Its total differential can be written as d ˜ f = − sdT + σ ij dε eij + χds ′ − N X k =1 h k dϕ k . (7)From this expression one notices that the complete freeenergy is function of such own arguments as T , ε eij , s ′ and ϕ k . Time evolution of non-equilibrium thermodynamicparameters s ′ and ϕ k can be calculated by means of setof equations which is similar to (5) ∂s ′ ∂t = γ s ′ ∂ ˜ f∂s ′ = γ s ′ χ,∂ϕ k ∂t = γ ϕ k ( ∂ ˜ f∂ϕ k − h k ) = γ ϕ k ( h k − h k ) . (8)The non-equilibrium temperature and defect densityare calculated by means of differentiation of the completefree energy with respect to non-equilibrium entropy anddefect energy χ = ∂ ˜ f∂s ′ , h k = − ∂ ˜ f∂ϕ k . (9)These relations and relations (4) can be interpretedas the state equations. Ignoring the thermal effects andsupposing that the drive parameter ε eij is a constant thestate equation can be written in two equivalent forms: ϕ k = ∂u∂h k , h k = − ∂ ˜ f∂ϕ k . (10) III. POWER SERIES ANALYSIS
Viewing only one type of defect power series expan-sions of the internal and free energies in terms of its ownarguments can be written in the above approximation as u = u + ϕ h − ϕ h + 13 ϕ h − ϕ h , (11)˜ f = ˜ f − h ϕ + 12 h ϕ − h ϕ + 14 h ϕ . (12)For simplicity sake, only one type of defects is consid-ered here. The coefficients ϕ , ϕ , ϕ and ϕ characterize energetic processes at microscopic levels, the coefficients h , h , h and h characterize dependence of defect den-sity from processes at microscopic levels. Different pow-ers in the expansions describe thermodynamic processesof different levels. The main process is described by thelowest power; the processes corrected to it are describedby higher powers of the expansions. Alternation of thesigns is due to the Le Chatelier principle the processesof each next level point in opposite direction to the pro-cesses of the previous level. The state equations for eachcase have the form: ϕ = ∂u∂h = ϕ − ϕ h + ϕ h − ϕ h , (13) h = − ∂ ˜ f∂ϕ = h − h ϕ + h ϕ − h ϕ . (14)Let us notice that in the linear (quadratic for energy)approximation the dependence between thermodynamicvariables ϕ and h is biunique in the both cases. Takingin account that in this approximation the both equationsmust lead to the same physical result the connection be-tween expansion coefficients is next: ϕ = h h , ϕ = 1 h , (15) h = ϕ ϕ , h = 1 ϕ . (16)Calculation of the internal and free energy and thestate equation at model parameters of u = 0 . · J/m , ϕ = 0 . · − J/m , ϕ = 0 . . · − J/m is presented in the fig. 1a. The dimensionof the variable h is m − . If in accordance with the stateequation (13), (14) written in the linear approximationone makes transition from variable h to variable ϕ . thensequence of extremes are changed, the maximum of theinternal energy (at zero value of ϕ ) is placed in the leftside, the minimum of free energy is placed in the rightside.From the fig. 1a one notices that the maximum of theinternal energy does not coincide with the minimum ofthe free energy. The standard procedure of minimizationof free energy has led to the zero value of the variable h as in the same time the analogous procedure of max-imization of internal energy has led to the zero valueof the variable ϕ suggesting that there is some conflictbetween descriptions in terms of the internal and free en-ergy. It is possible to eliminate this conflict offsetting theextremums of potentials (energies) to meet one anotherin such away that they both turn out in the same pointsatisfying to the state equation. In the terms of the ini-tial potentials (energies) it means the stable point is thatin which the slopes of tangents to the internal and freeenergy curves are not equal to zero and together they aresatisfied to the state equation. In fact, an arbitrary point FIG. 1: Dependence of the thermodynamic energies versus defect density: a - initial ones; b - shifted ones; 1 internal energy;2 free energy; 3 state equation ϕ = ϕ ( h ) between zero and h max satisfies to this condition, conse-quently the position of the point must be determinedfrom experimental data. The plots of shifted thermody-namic potentials for value h = 0 . m − are presented inthe fig. 1b. The value of ϕ determined from the stateequation is 0 . · Jm − .From the figure we notice that the curve of internalenergy is shifted a little to the left side, and the curveof free energy is essentially shifted in the right side so asthe extremums of both energies are coincided. Thus, asystem during its evolution tends to common stable stateindependently of what kind of energy is used for descrip-tion. In consequently of it the evolution of a system canbe described by two kinds of Landau-Khalatnikov equa-tions expressed in terms of internal energy (5) or free one(8).All along, it is possible to determine the kinetic co-efficients γ h and γ ϕ so that the equations (5) and (8)at a arbitrary time will describe one and the same stateassociated with thermodynamic pair h and ϕ connectedby the state equation (13) or (14). The evolution is fin-ished with achievement of one and the same system stateassociated with pair h and ϕ . IV. FAILURE OF QUASIBRITTLE MATERIALSA. General
In a special case of isothermal processes, which is justconsidered here, valid is d ˜ f = σ ij dε eij − N X k =1 h k dϕ k , (17)so ˜ f = ˜ f ( ε eij , ϕ k ) If explicit expression of the free energydensity dependence of its arguments is known, then vari-ables σ ij and h k can be determined by simple differentia-tion. As determination of the explicit dependence for thedensity of the free energy is intractable problem, we usea standard method for such cases, expansion of the freeenergy into a series up to cubic power of its arguments:˜ f ( ε eij , ϕ ) = ˜ f − h ϕ + 12 h ϕ − h ϕ ++ 12 λ ( ε eii ) + µ ( ε eij ) − gϕε eii + (18)+ 12 λϕ ( ε eii ) + µϕ ( ε eij ) + eϕ ε eii . Here we consider only one type of defect, which is mainfor quasibrittle failure, namely, micro-cracks. Then h is micro-crack density and ϕ is average energy per onecrack. During an external mechanical action the micro-cracks are produced, collected and merged into macro-scopic clusters such as main-cracks and so on.Corresponding state equations have the form: h = h − h ϕ + h ϕ + gε eii ++ 12 λ ( ε eii ) + µ ( ε eij ) − eϕε eii ,σ ij = λε ii δ ij + 2 µε ij − gϕδ ij (19) − λϕε ii δ ij − µϕε ij + eϕ δ ij . The evolution equation in an explicit form now is ∂ϕ∂t = χ ϕ ( h − h − h ϕ + h ϕ + gε eii ++ 12 λ ( ε eii ) + µ ( ε eij ) − eϕε eii ) . (20)The positive terms represent microscopic processes asa contributory factor for increasing of average energy ofseparated micro-cracks; the negative terms are same fordecreasing energy. The constants h and h are constantsink and source for average micro-crack energy. Micro-scopic sense of the constant h is followed; it representsthat part of defectiveness of a deeper structural levelwhich provokes activation of new micro-cracks. Follow-ing from it increasing of micro-crack density reduces theelastic property of the material and, as consequence, theaverage energy of a micro-crack. Constant h representsthat part of defectiveness of the deeper structural levelwhich provokes micro-crack healing, what leads to de-creasing of defect density and increasing of the averageenergy of the remaining micro-cracks. Regeneration ofdefects of the deeper structural level is supported by ir-reversible work. It is notice that in a particular problemthe constants h and h may be integrated in one effectiveconstant h = h − h .The term h ϕ also represents micro-crack healing butconnected with surplus energy relaxation of the selfmicro-cracks (self-action). The term h ϕ representsmicro-crack merge. At small ϕ this process is weak and itis easily compensated by the above described processes.At large ϕ micro-crack density increases sharply, in thesame time the spatial distribution of the micro-cracks isnot totally independent, but forms chains of macroscopiclength from merged micro-cracks, that is, new objects ofa higher structural level. Although energy of the newobject (macro-crack) is high the part of energy per onemicro-crack included in it becomes less, what is repre-sented by the minus sign before the term.In the given approximation free energy curve is a cu-bic parabola (Fig. 2). In accordance with a sign of thedeterminant D = ( h + 2 eε eii ) ) + 4 h ( h − λ ( ε eii ) − µ ( ε eij ) ) (21)three kinds of cubic parabola are possible: with two ex-tremes at D > D = 0 (curve 3), and monotonic without pointof inflexion at D < ε eij )The equation of state calculated according to (19) isrepresented by the curve 1 in Fig. 2. It is seen thatat such consideration, the extremes of free energy corre-spond to zero values of defect density, i.e. from the pointof view of mechanics, to a continuous defect-free solid.Defect density between extremes has a negative value,and it seemingly has no physical meaning. Because ofthis there is something like a gap between the right andleft branches of the free energy, which prohibits a spon-taneous transition from one branch to another. The plotof the state equation for the free energy described by thecurve 3 just touches the abscissa axis and for that de-scribed by the curve 4 it is above the abscissa axis. Inboth cases the defect density decreases at first and thenincreases with increase in the average defect energy. Itimplies that due to internal processes the material in thegiven stressed state (at predetermined ε eij ) first tends tobe consolidated and only then loosened and collapsed.Presence of the free energy gap results in that the transi-tion from the consolidation stage to the destruction onecannot be realized exclusively through the internal pro-cesses. In this case, material is stable and will not bedestroyed during unlimited time.The left extreme of the free energy is the minimumand, corresponding to it, state is stable. The right ex-treme is the maximum and, corresponding to it, stateis unstable and provokes destruction of a material. Thefree energy difference of these two states represents anenergy barrier. Existence of such barrier is a prerequisitefor sustained stability of a material. At large stress thestate is described by the free energy in the forms 3, 4.In these cases, the energy barrier vanishes, and materialwill be in a state of long-time creep-like destruction thatwill finally result in its microscopic destruction. B. Numerical calculations
Let us assume for numerical estimates the followingset of reference model parameters: f = 2 · Jm − , h = 10 m − , h = 1 . · J − m − , h = 0 . · − J − m − , λ = µ = 0 . · P a , g = 1 . · m − , λ = 0, µ = 3 . · m − , e = 5 . · J − m − . Inall calculations, the main components of the deforma-tion tensor are assumed to be constant and equal to ε e = ε e = ε e = 0 .
001 that corresponds to uniformtension.Changes of a form and height of potential barrier withvariations of the parameters are illustrated in Fig. 3.The decrease in the parameter h in comparison withthe reference value increases the height of the potentialbarrier (curve 1, Fig. 3a) that is equivalent to increasein material strength. This does not contradict commonsense and satisfies our intuition, namely, the smaller isthe damage of a material on a deeper structural level, and FIG. 2: Schematic presentation of density of defects and free energy versus average energy of defects: 0 onset of coordinatesystem; 1 equation of state h = h ( ϕ ); charts of free energy: 2 with two extremes; 3 with point of inflexion; 4 declining curveFIG. 3: Dependence of free energy versus average energy per microcrack with variations of parameters: a - h , b - h , c - h .Numbers mean the following: 2 reference parameters; 1 decrease ones; 3 increase ones the more stable it is. The increase in the parameter h changes the height of the potential barrier sharper thanin the decrease case (curve 3, Fig. 3a). It is necessary tonote, that the free energy minimum at large damage ona deeper structural level (curve 3, Fig. 3a) has shiftednoticeably towards large values of ϕ . It means that in amicroscopically more damaged material the states withlarge average energy per a microcrack are much morepreferential from the point of view of thermodynamics.Inasmuch as defects of larger sizes have larger energies, the instability of material can be connected with achieve-ment by microcracks of some critical sizes or critical av-erage energy.Smaller values of the parameter h (curve 1, Fig. 3b),on the contrary, result in decrease in the potential barrierand describe materials with less strength, and its largervalues represent materials with higher strength (curve 3,Fig. 3b). This can be explained by the fact that decreasein efficient parameters of material with increase in theparameter h that taking into account interaction of mi-crocracks leads to the growth of adaptation capabilities ofa material, the reduction of internal stress concentrationthat results in growth of its stability.Influence of the parameter h on stability is outwardlythe same as that of the parameter h (compare Fig. 3 andFig. 3c). Namely, if it decreases the stability raises (curve1, Fig. 3c) and if it growths the stability reduces (curve3, Fig. 3c). However, this influence is realized accordingto the other mechanism in comparison with case for theparameter h . Namely, with growth h the processes ofcoalescence of old microcracks and generation of new onesare stimulated to a greater extent that does result in areduction of material strength.The dependence on the other parameters of the freeenergy expansion is determined by what power of theparameter ϕ they stand at. All items having the samepower with respect to ϕ , , can be united through theeffective constants of the theory: h ∗ = h + gε eii + 12 λ ( ε eii ) + µ ( ε eij ) ,h ∗ = h + 2 eε eii ) . (22)The second term in the expression h ∗ is linear withrespect to the reversible deformation. At the tension ε eii > ε eii < h ∗ arequadratic with respect to deformation. It means thatat positive parameters λ and µ stability of a material isequally reduced, both as in case of the tension and as incase of the compression. Further, we will assume that λ << µ , i.e. shear deformations play a major role in thedestruction of a material.In the above examples, the deformation remained aconstant value. It is necessary to clarify how a systemdescribed by the theory behaves in a thermodynamic cy-cle. Fig. 4 gives results of calculations of uniform tensionand three-axial unequal-component compression. In thefirst case, the main components of elastic deformationtensor are equal to ε e = ε e = ε e = ε where ε is thevalue given in Fig. 4. In the latter case, they are equalto ε e = 0, ε e = ε , ε e = 2 ε . Such conditions arespecially chosen to describe a situation in which rocksare in the vicinity of a free surface of a mine workingas in a real situation. It is seen that in both cases theheight of potential barrier decreases during increase inload, and at a definite deformation it becomes equal tozero. Material loses stability both in the first and in thesecond case. However, instability mechanisms are differ-ent. In the first case it is due with long-term ultimatetensile strength, in the second one it is due with shearstrength. However, critical values for a quasibrittle ma-terial in the given theory are not introduced explicitlyas the fundamental strength constants, they are derivedfrom the theory. V. EXPLANATION OF EXPERIMENTALFACTS AND APPLIED ASPECTS OF THEPROBLEM
The theory presented here can be applied to qualitativeexplanation of phenomenon of rock failure around mineworkings. It is well known that at small depths mineworkings have a rather high degree of stability. Aroundidentical workings at large depths one can observe phe-nomena of anomalously large displacements of rocks in-side the free working space that give large reduction inthe mine working section area and finally lead to the com-plete collapse of it [9]. According to the theory, at smallvalues of deformation tensor ε eij , i.e. at small depths (es-timates are given below) for rocks surrounding a mineworking the condition of existence of equilibrium state isfulfilled (the case of D <
0) which is described by freeenergy with one minimum (curve 2 in Fig. 2). At largevalues of shear deformations (in elastic approach the vol-umetric deformation is equal to zero), i.e. at large depths,kind of the free energy dependence is changed, and it isalready described by the curve without extremes (curve3 in Fig. 2a). In this case, a system has no equilibriumstates (at
D >
0) and therefore is permanently collapsed.The phenomenon described above suggests an elegantmethod to determine an additional estimate couplingamong the parameters of the theory. Let us assume thatmine depth H , , starting from which said phenomenonbegins to manifest itself, equals to 200 m . Correspond-ing to it hydrostatic pressure is γH and the tangentialcomponent of the stress tensor on the boundary of mineworking is 2 γH (where γ = ρg , ρ is rock density equal,in the mean, to 2 . · kg · m , g is the gravity con-stant). From here σ τ ≃ . M P a . Assuming that shearhardness of rocks µ = 30 GP a , we get a value for theshear component of the deformation tensor of the orderof ε τ = σ τ τ /µ = 0 . ε eij ) that appears in free energy expansion (18). Taking intoaccount that on the boundary of the mine working thefirst invariant of the deformation tensor is equal to zero(this follows from the exact problem solution in the elas-tic formulation), the zero condition for the determinant(21) can be written as h − h ( h − . µ ) = 0 . (23)This coupling of the theory parameters, parallel withattraction of the special experiment data, may be of usefor numerical estimates. At present a complete complexof such experiments aimed at measurement of parameters h , h , h , µ and other parameters of the theory is notavailable. The consideration suggested here may serve asan impetus to exploratory developments in this line. FIG. 4: Dependence of free energy versus average energy per microcrack during: a - a uniform tension; b - a three-axialunequal-component compression.
VI. CONCLUSION
Thus, a new mesoscopic non-equilibrium thermody-namic approach is developed. The approach is basedon the thermodynamic identity associated the first andsecond law of thermodynamics and having a perfect dif-ferential form. The introduction into the basic relation(2) of such internal parameters as average defect ener-gies and defect densities has allowed to give the phys-ical meaning of seeming purely mechanical problem ofmaterial destruction and to apply the standard methodsclosed to the equilibrium thermodynamics to the analy-sis of nonequilibrium processes. The identity permits tointroduce an extended non-equilibrium state and use thegood developed mathematical formalism of equilibriumand non-equilibrium thermodynamics.The evolution of non-equilibrium variables are de-scribed by a Landau-based equation set expressedthrough internal or different kinds of free energy con-nected by means of the Legendre transforms. The intro-duction of some structural viscosity-like shifts ϕ k and h k into the evolution equations (5) and (8) make possiblethe accordance between the different kinds of energy.Taking into account only one type of defects (viz., mi- crocracks) the description of destruction of quasi-brittlesolids is considered by means of an expansion of free en-ergy. Different powers in the expansions describe thermo-dynamic processes of different levels. The main processis described by the lowest power; the processes correctedto it are described by higher powers of the expansions.Alternation of the signs is due to the Le Chatelier prin-ciple, namely, the processes of each next level point inopposite direction to the processes of the previous level.The consideration allows to find stable and unstablestationary points of free energy in dependence of sign ofthe determinant (21). At D >
D <
D >
D <