Non-classical solutions of a continuum model for rock descriptions

Mikhail A.Guzev

Institute for Applied Mathematics,Far Eastern Branch of the Russian Academy of Sciences,Vladivostok 690041,Russia

Non-classical solutions of a continuum model for rock descriptions

Mikhail A.Guzev*

Institute for Applied Mathematics,Far Eastern Branch of the Russian Academy of Sciences,Vladivostok 690041,Russia

A R T I C L E I N F O

Article history:

Received 18 January 2014

Received in revised form

12 March 2014

Accepted 21 March 2014

Available online 1 April 2014

Zonal disintegration

The strain-gradient and non-Euclidean continuum theories are employed for construction of nonclassical solutions of continuum models.The linear approximation of both models’results in identical structures in terms of their kinematic and stress characteristics.The solutions obtained in this study exhibit a critical behaviour with respect to the external loading parameter.The conclusions are obtained based on an investigation of the solution for the scalar curvature in the non-Euclidean continuum theory. The proposed analysis enables us to use different theoretical approaches for description of rock critical behaviour under different loading conditions.

?2014 Institute of Rock and Soil Mechanics,Chinese Academy of Sciences.Production and hosting by Elsevier B.V.All rights reserved.

1.Introduction

Development of underground rock engineering structures at great depths is a challenging issue for underground engineers. Deep rock masses are characterised by zonal disintegration that is known as a zonal periodic structure around excavated rocks.

The zonal structure of fracturing has been discovered in the drilling-and-blasting processes of openings and in experiments. The zonal disintegration phenomenon around the tunnels in the Witwatersrand gold mines in South Africa at depths of 2000-3000 m was initially observed with a periscope(Cloete and Jager, 1972).Subsequently,the zonal disintegration phenomenon was widely observed in the surrounding rock mass in deep gold mines in South Africa(Adams and Jager,1980).Similarly,the zonal disintegration phenomena have also been discovered in the Taimyrskii and Mayak mines in Russia(Shemyakin et al.,1986a). Shemyakin,who f i rst studied the zonal disintegration phenomenon of deep rock masses in 1986,obtained satisfactory results via in situ observations(Shemyakin et al.,1986b).After that,great efforts have been continuously made to understand the zonal disintegration phenomena as a type of failure mode of deep rock masses.

The complex development of a quantitative theory regarding the phenomenon of zonal disintegration is determined by the necessity to simulate the behaviour of a medium that has the properties of elastic deformation(non-fractured zones around a working zone)and of fracture.In the physical perspective,the formation of fracture zones(macrocracks)depends on the presence of microdefects in a medium.Rock masses without pre-existing macrocracks can basically be regarded as the materials with only microcracks.During tunnel excavation,microcracks may nucleate, grow and propagate through the rock matrix.Then,secondary microcracks may appear,and discontinuous and incompatible rock mass deformation may occur.The classical continuum elastoplastic theory is not suitable for analysing the discontinuous and incompatible deformation of rock masses because this theory does not consider the defects included in rock materials.Thus,novel ideas and methods must be introduced to gain a better understanding of the zonal disintegration mechanism in deep rock mass engineering.

At present,different failure mechanisms and effects of zonal disintegration phenomena around deep tunnels are obtained.The hot issue concerned is the experimental observation that macroscopically uniform rock masses with defects develop localised(i.e. non-uniform)deformation f i elds under uniform surface tractions. The phenomenon of the localisation of macroscopic deformations is the motivating force for the developmentof continuum theory with“higher gradients”.In the early 1960s,Toupin(1962)and Mindlin (1964)proposed the strain-gradient theory,suggesting that energy depends on both deformations and the gradient of deformations.For the past 20 years,there has been a growing interest in this theory in mechanics because of the successful applications of this theory to the mechanics of materials.Traditional continuum theories are notcapable of solving such problems withoutthe use of length parameters that determine the scale of microstructures. Numerous extensions of this theory have subsequently been developed and used for various applications to investigate theproblems such as strain localisation and size effects in materials,as well as challenging micro-and nano-scale issues.There are many other gradient theories that have also received considerable attention in many engineering f i elds.However,in this paper,we do not attempt to make a comprehensive comparison of the various gradient theories;for such a comparison,the readers can refer to other recent papers,e.g.the work of Chambon et al.(2004).

Investigations of the microcharacteristics of various materials by physical methods are responsible for the introduction of concepts such as dislocations,disclinations,vacancies,and other characteristics of crystal structure defects in literature.From the perspective of classical mechanics of continuous medium and non-equilibrium thermodynamics,the characteristics for internal structure descriptions introduced by different investigators clearly require an extension of the kinematic basis of the theory.In the 1950s,the natural kinematic variables for the models of defects observed in the crystal structure of materials were demonstrated to be non-Euclidean objects(Bilby et al.,1955;Myasnikov and Guzev,2000). The idea is introduced by abandoning the kinematics hypothesis of the classical continuum model and suggests that the internal geometry of a material does not coincide with the geometry of the observer’s Euclidean space.

The f i rst non-Euclidean continuum model(Myasnikov and Guzev,2000)to describe the stress-f i eld distribution around underground workings with a round cross-section was formulated in the framework of non-equilibrium thermodynamics.A modi f i cation of this model was presented by Guzev(2008).Variants of the generalised elasticity classical theory in the framework of various non-Euclidean models of continuous medium have been proposed by many authors(Kadic and Edelen,1983;Valanis,1995;Ciancio and Francaviglia,2003;Guzev,2010).

From a physical perspective,the geometric characteristics of af f i ne-metric spaces are the internal variables and cannot be measured directly.Therefore,the results of the non-Euclidean theory can be veri f i ed in comparison with the results of other approaches.For this purpose,we use the strain-gradient theory formulated by Toupin(1962)and Mindlin(1964).The non-Euclidean theory and the strain-gradient theory de f i ne the inhomogeneous structures of the displacement and stress f i elds that are different from the classical structures.The solutions also have a critical behaviour with respect to the value of the external force. When the external force does not reach the critical value,the solution is provided by a monotonic function;when the critical value is reached,the solution is calculated through oscillatory functions.

2.Strain-gradient theory

The Toupin-Mindlin theory assumes that the bulk Helmholtz free energyΨTMdepends on the strain tensorAij=(?Ui/?xj+?Uj/?xi)/2 and on its gradients.For simplicity,we assume that the free energy is equal to

whereA=Akkis the f i rst strain invariant.

The free energy can then be written as

whereλ,μare the Lamé constants;andGis an elastic constant associated with the gradient term.

The constitutive equations of the strain-gradient theory are obtained from the condition under which the deformation of the body minimises the total energy∫ΨTMdV.This condition results in the equilibrium equation:

whereσij=λδijA+2μAij+GδijΔA.

Let us write the displacement f i eld in the form:

where the function Uclassiccoincides with the elastic strain f i eld.

Then,the equilibrium equations are reduced to

wherew=div W=Δφ.

Now we haveA=Aclassic+wandwis an additional f i rst invariant.Thus,the invariant satis f i es(λ+2μ)w+GΔw=0.

For a deep-level tunnel with a circular cross-section,the cylindrical coordinates(r,φ)can be used for convenience.For the problem under consideration,the dependence on the polar angle φ is absent.Then,the equation for the additional f i rst invariantw=w(r) can be written in the following form:

Eq.(6)is Bessel’s differential equation.The behaviour of the solutionwdepends on the sign ofG.IfGis negative,i.e.G＜0,we can obtain:

whereK0(·)is the zeroth-order MacDonald function,andcis an amplitude coef f i cient.

The MacDonald function behaves monotonically.IfGis positive, i.e.G＞0,the solution is given by

whereJ0(·)andY0(·)are the zeroth-order Bessel and Neumann functions,respectively,aandbare the amplitude coef f i cients.The Bessel and Neumann functions have an oscillatory behaviour. Hence,the sign ofGde f i nes the character of the solutionw.

LetZn(γr)denote any Bessel function or a linear combination with constant coef f i cients of Bessel functions of ordern,we have

There is a relation between different Bessel functions in the following form:

wheres=γris considered.

This function allows us to construct solutions for non-classical displacement,deformations and stresses.There is the unique displacement f i eldUrin cylindrical coordinates as follows:

whered1andd2are constants.

The non-classical deformations are given by

The non-classical stresses are calculated by

Eqs.(12)-(14)indicate that the non-classical solutions for the kinematics and the force characteristics have a generalstructure.This structure is the sum of the classical solution and an additional term, which is underlined in Eqs.(12)-(14).This term exhibits a critical behaviour with respect to the phenomenological parameterG.

3.Riemann non-Euclidean theory

If a rock mass contains defects,then the set of additional parameters 3ijmust be introduced to describe its mechanical state. The structure of the components of 3ijmay be different.In the classical theory of elasticity,the functionsAijsatisfy the Saint-Venant compatibility conditions.These functions include six identities in three-dimensional(3D)space.These identities are violated in the presence of defects.Let us illustrate this situation for the case of a plane-strain approximation,which corresponds to a model of a deep-level tunnel with a circular cross-section.There is a unique function:

Eq.(15)characterises the incompatibility of the components3ij. The theory of elasticity proves that the vanishing ofRin a region is a necessary and suf f i cient condition for the equality 3ij=Aij.If the quantity in Eq.(15)does not vanish,i.e.R≠0,there is a source of elastic incompatibility(Eshelby,1956).This quantity is critically important when the state of the internal stress of a body is investigated becauseRis a natural source of the internal stress.

In a physical point of view,defects lead to changes in the internal stress and in the appearance of an incompatible component 3ijof deformation.From a mathematical perspective,Ris the scalar curvature of the Riemann tensorRijkl.IfRijkl=0,the internal geometry of a body is Euclidean.WhenRijkl≠0,the geometry of the body is non-Euclidean.For the plane-strain approximation,the scalar curvature is the unique invariant of the Riemann tensor.The minimal complete set of kinematic variables for the description of a material with structural defects includesAij,3ijandR.The corresponding extension of the elastic continuum model by introducing 3ijandRas additional parameters leads to the Riemann non-Euclidean continuum model.

Considering an elastic body with defects and neglecting the thermal effects,the bulk Helmholtz free energy is equal to ΨNE=ΨNE(Aij,3ij,R).We write the bulk free energy in the form:

whereΨAandΨ3are parameters dependent onAijand 3ij,respectively;ΨA3characterises interaction of the f i eldsAijand 3ij;andΨRis a function of the scalar parameterR.

Because small deformations are considered in comparison of the classical theory of elasticity,we assume that the functionsΨAand Ψ3are de f i ned by the f i rst and second invariants of their tensor arguments and thatΨA3depends on the invariantsAkkand 3kk.

The constitutive equations of the strain-gradient theory and those of the non-Euclidean theory can then be obtained by minimising the total energy∫ΨNEdV.The structure of the irrotational displacement f i eld U and stress f i eldσijwas analysed in Guzev (2011).The displacement f i eld has the following form:

whereτis a constant.

We omit the calculation procedures and formulate the results for the problem of the stress f i eld distribution around an excavation with a circular cross-section in the plane-strain approximation with the stressP∞speci f i ed at in f i nity.In this model,the problem is given in a stationary formulation under the condition thatσr=0 at the boundaryr=r0.In polar coordinates,we can obtain the following equation:

whereα,β,γare constant coef f i cients.

It follows from Eq.(18)that the stress f i eld is the sum of the elastic stress f i eld and an additional f i eld.The former is determined in accordance with Hooke’s law,and the latter is calculated through the parameter of incompatibilityR.

4.Equation for the parameter of incompatibility

The parameter of incompatibilityRhas a clear geometric meaning and coincides with the scalar curvature(Weinberg,1972). In the 3D case,this parameter is calculated in accordance with the following relations:

The standard formalism of non-equilibrium thermodynamics is used to obtain an equation for the non-Euclidean parameterR(Guzev,2010).For the Riemann non-Euclidean continuum model, the state equation can be written as follows:

whereρ0is the density,Uis the material internal energy,andDis the dissipative function.It is assumed that the dissipative function is a homogeneous function of some order in regards to thermodynamic forces(Kondepudi and Prigogine,1998).To provide the function non-negativity,we accept the linear relations between the source and thermodynamic forces as follows:

whereξis an non-negative phenomenological coef f i cient,Δis the Laplace operator.

The exact equation for the scalar curvature,which was obtained in formula(4.10)(Guzev,2010),is shown as

whereEijis given in Eq.(23).

Due to a small value of deformations,in the right-hand part of the Eq.(24),we assumeg js=δjs,whereas in the functionFlijq,we leave only the contribution containing derivatives ofDij,l.In this approximation,Eqs.(19)and(20)can be written as follows:

Then,the componentsRijare calculated(Weinberg,1972)as follows:

As a result,Eq.(24)is written in the shortened form:

5.Stationary equation for the scalar curvature

From Eq.(28),we can see that the stationary equation for the scalar curvature can be written as

With the work of Guzev and Paroshin(2001),we obtain the following equation forRin the following form:Δ2R-γ2R=0.From a mathematical point of view,its structure is similar to Eq.(29). However,Eq.(29)clari fi es the meaning of phenomenological parameters that are linked with the external load.A modi fi ed plasticity model incorporating Eq.(28)has been developed in Galanin et al.(2008).The model was numerically investigated in a simplifi ed plain-strain case and was found to show some threshold behaviour.Below,we will construct an analytical solution of Eq. (29).

The fi rst step is to rewrite Eq.(29)in the linear approximation with respect to the parameterR.The source of incompatibilityEijis calculated throughσij,J.From Eq.(23),we see that?2Eij/?xi?xj=0. Then,the quantityEkkcan be written as follows:

In expression of the state in Eq.(22),we assume that the internal energyUis the function of the entropys,the internal deformation tensor 3ijand the scalar curvatureRasU=U(s,3ij,R).Because of neglecting thermal effects,we can replaceU=U(s,3ij,R)with the Helmholtz free energyΨNE=ΨNE(Aij,3ij,R)as follows:

We set the free energy internal energy as the following equation:

whereλA,μ3,g,μRare the phenomenological parameters.Let us introduce the internal deformation tensor 3ij,which is de f i ned by the following relation:

From Eqs.(15)and(33),we obtain the following equation for determiningfif the scalar curvatureRis given:

Let us transform the expression for the f i rst and second invariants of the tensorAij,i.e.

The substitution of Eq.(35)into Eq.(32)results in the following equation:

where

We choose

and write Eq.(36)in the following form:

From Eqs.(31)and(39),we obtain the following expression for the stress tensorσijandJ:

The invariant 3kkis equal to

Eq.(40)allows us to f i nd

The combination of Eqs.(41)and(42)with Eq.(15)results in the following equation:

The functionJis presented in the following form:

Hence,the linear approximation of the expressionEkkRwith respect to the parameterRis written as follows:

The combination of Eqs.(43)and(44)with Eq.(30)results in the following equation:

The further reduction of the Eq.(46)is linked with choosing the parameterμ2.Supposingμ2=6ν2/(3λ1+2μ1),we can obtain the following equation:

The usage of Eqs.(45)and(47)allows us to write Eq.(30)in the following f i nal form:

The accuracy of obtaining Eq.(48)allows us to replaceσkkby the following classical solutionσclassic:

whereν1=λ1/[2(λ1+μ1)]is an effective Poisson coef f i cient.

6.Stationary solution for the scalar curvature

From Eqs.(48)and(49),we see thatRsatis f i es the following equation:

Let us introduceR±as solutions of the following system:

7.Conclusions

We used the non-Euclidean theory and the strain-gradient theory to describe the inhomogeneous structure of the stress fi elds in a material.It is demonstrated that the constructed solutionis the sum of the classical solution and an additional term(see Eqs. (14)and(18)).This term has a critical behaviour with respect to the phenomenological parameters.In the strain-gradient theory,the constructed solution behaves monotonically when the phenomenological parameterG＜0.If the parameterG＞0,the solution contains wavy terms.In the non-Euclidean theory,the critical behaviour is de f i ned by the value of the external loading force. When the external load does not reach a critical value,the constructed solution behaves monotonically.When the critical value is reached,the solution contains wavy and monotonic terms.Hence, both theories were found to result in identical conclusions.

From a physical perspective,we simulated the behaviour of a medium that has classical elastic properties and non-classical properties.This simulation provides us with an opportunity to use models to describe different rock phenomena,including zonal disintegration and the anomalous deformation of rock samples.The corresponding application was proposed in Guzev(2010).

Con f l ict of interest

The author wishes to con f i rm that there are no known con f l icts of interest associated with this publication and there has been no signi f i cant f i nancial support for this work that could have in f l uenced its outcome.

Acknowledgements

The author is grateful to the academician Qihu Qian for constructive discussions.

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Mikhail A.Guzev,born in 1962,studied mathematical and theoretical physics at Saint-Petersburg(Leningrad)University.He earned Master of Sciences(theoretical and mathematical physics)in 1987,and Doctor of Sciences(solid mechanics)in 1999.He is Professor at The Information Department,Institute of Mathematics and Computer Science,Far Eastern National University,Vladivostok,Russia in 2000.Since 2003,he is the Corresponding Member of Russian Academy of Sciences,Director of Institute for Applied Mathematics Far Eastern Branch RAS,Russia,Vladivostok.His research interests cover asymptotic methods, nonlinear mechanics,semiclassical approximation in quantum mechanics,statistical theory of wave propagation,a nd mathematical modelling of elastoplastic materials.

*Tel.:+7 4232311856.

E-mail address:guzev@iam.dvo.ru.

Peer review under responsibility of Institute of Rock and Soil Mechanics,Chinese Academy of Sciences.

Strain-gradient theory

Non-Euclidean continuum model