Strength properties of reinforced porous medium

Transcription

1 ème Congrès Français de Mécanique Bordeaux, 6 au 30 août 03 Strength properties of reinforced porous medium Z. He a, L. Dormieux a, D. Kondo b a. E.N.P.C, Institut Navier, Marne-la-Vallee, France b. U.M.P.C, Institut d Alembert, 755 PARIS CEDEX 05, France Résumé : Nous étudions les résistances des matériaux poreux ductiles renforcés par des particules rigides(argilite Callovo-Oxfordian). Nous considérons un milieu microporeux constitué d une phase solide de Drucker- Prager contenant des vides sphériques; le comportement du milieu est décrit par un critère elliptique dont la fonction d appui est ensuite déterminée. Celle-ci est mise en oeuvre dans une démarche d analyse limite dans laquelle une attention particulière est accordée à la contribution de l interface inclusion/matrice. Ceci aboutit à la détermination des résistances du matériau poreux renforcé par des particules rigides sous la forme d équations paramétriques. Les prédictions sont comparées aux résultats obtenus par des procédés d homogénéisation variationnelle successivement appliqués aux transitions micro-méso, et puis méso-macro. De plus, nous analysons en détail les prédictions de la résistance des matériaux sous sollicitations isotropes. A cette fin, des solutions statiques sont dérivées et comparées aux solutions cinématiques. Enfin, une interprétation dans des cas limites est fournie. Abstract : We investigate the strength properties of ductile porous materials reinforced by rigid particles(callovo- Oxfordien argillite). The microporous medium is constituted of a Drucker-Prager solid phase containing spherical voids; its behaviour is described by means of an elliptic criterion whose corresponding support function is determined. The latter is then implemented in a limit analysis approach in which a careful attention is paid for the contribution of the inclusion matrix-interface. This delivers parametric equations of the effective strength properties of the porous material reinforced by rigid particles. The predictions are compared to results obtained by means of variationnal methods successively applied for micro-to-meso and then for meso-to-macro scales transitions. Moreover, we discuss in detail the predictions of the material strength under isotropic mechanical loadings. To this end, additional static solutions are derived and compared to the kinematics limit analysis ones. Finally, an interpretation in limit cases is provided. Mots clefs : résistance; inclusion-matrice; matrix poreux ductiles Introduction Being hard clayey rocks, Callovo-Oxfordien (COx) Argillite is a porous clay matrix in which quartz or silica inclusions are embedded. In the present study, we mainly aim to derive new closed-form results for the strength of the COx argillite, under the assumption that the solid phase of the clay is a Drucker-Prager perfectly plastic material. Therefore, the behaviour of the microporous clay is described by means of an elliptic criterion [] whose corresponding support function is determined in this paper. Then by using this support function, we explore an alternative approach which can be viewed as an extension of the original Gurson model. Instead of a spherical cavity surrounded by a matrix, the proposed rigid core sphere model consists of a rigid spherical core surrounded by the homogeneous porous material. The failure criterion of this rigid core sphere model is derived in the framework of the kinematic approach of limit analysis (LA). It is worth noting that from the LA point of view the failure mechanism can include a strain concentration at the core-matrix interface which can be described mathematically by a velocity discontinuity.

2 ème Congrès Français de Mécanique Bordeaux, 6 au 30 août 03 The micro-to-meso transition : support function of porous matrix The first homogenization step approach starts at the microscopic scale. At this scale, the porous clay matrix is described as a heterogeneous material being made up of a Drucker-Prager perfectly plastic solid in which pores are embedded. The scalar deviatoric stress σ d and mean stress σ m at the microscopic scale are defined as σ d = σ : K : σ and σ m = ( : J : σ)/3 and the Drucker-Prager criterion reads : σ d +T( σ m h) 0. The parameters T and h respectively characterize the friction coefficient and the tensile strength of the solid phase of the clay matrix. The result of the first homogenization step is the derivation of the strength properties of the porous clay matrix at the mesoscopic scale where it is described as a homogeneous material. This homogeneous material can be described by an elliptic criterion (see []). In the framework of limit analysis theory, a dual characterization of the strength criterion F(σ) 0 is the support function π F (d) = sup(σ : d,f(σ) 0} of the convex set of admissible stress states. The support function π(d) associated with the elliptic criterion of the porous matrix finally takes the form π F (d) = σ 0 d EQ λd v with d EQ = 3 d : H : d () where H = αj+k is a fourth order tensor, with α = 3f/ T 3 3+f ;σ 0 = ( f)h 3f 3f T T +f/3 ; λ = ( f)h T 3f T () 3 Overall dissipation at the mesoscopic scale We now focus on the transition from the mesoscopic scale to the macroscopic scale which constitutes the second homogenization step and is the main subject of the present paper. We seek the macroscopic criterion by means of a Gurson-type approach. As already stated, the microstructure at the mesoscopic scale is described by a composite sphere Ω with a rigid core surrounded by the homogenized clay resulting from the micro-to-meso transition. The external (resp. internal) radius is denoted by r e (resp. r i ). The shell Ω m (r i r r e ) around the core represents the clay. The volume fraction ρ = (r i /r e ) 3 of the core in the composite sphere is equal to the volume fraction of the rigid inclusions in a representative volume element of argillite. For geomaterials, we define here a family of kinematically admissible (k.a.) velocity fields at the mesoscopic scale with D, depending on one compressible parameter A : r r i : v A = Ax+(D m A) r3 e r e r +D d x (3) The strain rate in the clay (r i r r e ) can be determined from (3) : d A = A+D d +(D m A) r3 e r 3 ( 3e r e r ) (4) The velocity is v O = 0 in the rigid core. Note that the condition v A = 0 on the core boundary r = r i cannot be fulfilled by the velocity field defined in (3). This implies that the dissipation associated with a discontinuity of velocity must be considered at the boundary I (r = r i ). 3. Macroscopic support function and Macroscopic criterion Defining the macroscopic strength domain G hom as the set of admissible macroscopic stress states Σ, the macroscopic support function reads Π hom (D) = sup(σ : D,Σ G hom ). Considering the set K of k.a. velocity fields with D, Π hom (D) is characterized as [] : Π hom (D) = Ω inf v K ( π F (d A )dv + Ω m I ) π( v A )ds where Ω = 4πr 3 e/3. In the surface integral, v denotes the velocity discontinuity at the core boundary I and π( v ) represents the associated surface density of dissipation. In the line of reasoning of Gurson (5)

3 ème Congrès Français de Mécanique Bordeaux, 6 au 30 août 03 approach, Π hom (D) is approximated by the minimal dissipation obtained among the velocity fields v A defined in (3) : Π hom (D) = ( ) Ω inf π F (d A )dv + π( v A )ds (6) A R Ω m I For further use, let us introduce the following notation : π Ω F (d A )dv = Π m (D,A) ; Ω m Ω Accordingly : I π( v A )ds = Π I (D,A) (7) Π hom (D) = inf Π(D,A) ; Π(D,A) = Πm (D,A)+ Π I (D,A) (8) A R Once Π hom (D) is determined, the macroscopic admissible stress states on the boundary G hom are derived according to : Σ = Πhom (D) (9) D The stress state of (9) lies on the boundary of G hom at the location where the normal is parallel to D. The overall dissipation of (8) proves to read in the following form : with the notations : Y = Π(D,A) = σ 0 [Narcsinh P +Q 5α ( ) un M M +u N u ] ρ +σ 0 3 Y 3λD m (0) M = A α + D d 3 ; N = 4(D m A) () ( ) 5 ; Q = α+6 ρ Dd ; P = 45(+α)[D m A( ρ)] () The macroscopic support function can be determinated by minimizing the sum Π(D,A) = Π m (D,A)+ Π I (D,A) with respect to the parameter A. Accordingly, the boundary of G hom is determined according to (9)[3] : Σ = Π(D,A) D, with Π(D,A) = 0 (3) A Figure Comparison between the results predicted by (LA) and the (VA) for different volume fractions of inclusions. (line,point,diamond denote LA results; cross,box,circle denote VA results) 3

4 ème Congrès Français de Mécanique Bordeaux, 6 au 30 août 03 4 Comparison with the result obtained by a variational approach Numerical result derived in the framework of the kinematic approach of (LA) are now compared with the result obtained by (VA) [4]. For the derivation of their criterion, these authors consider a (VA) in the two homogenization steps. Applying the parameters f = 0.5 and T = 0.55, the comparison between the results predicted by the two methods is shown in Fig. As it can be seen in Fig., the analytical estimate obtained by the VA and the present numerical result based on the rigid core sphere model show a very good agreement for purely isotropic stress states, both in traction and compression. It is noteworthy that the strength under purely isotropic stress seems surprisingly almost unaffected by the volume fraction of inclusions ρ. In order to gain a deeper understanding of the effect of the parameter ρ, the hydrostatic strength will be compared with the exact solution predicted by a static approach (sections 5). On the other hand, we note that the strength predicted by LA (upper bounds) overestimates that predicted by VA under shear loading.(we focus on the strength proprieties under isotropic loading. The strength under shear loading has not been discussed in the present paper.) 5 Strength under isotropic loading 5. Static approach of the limit analysis problem The theory of LA teaches that a kinematic approach like the Gurson one provides an upper bound of the true strength. In order to check the accuracy of a kinematic estimate, it is therefore highly desirable to derive a static approach which in turn will deliver a lower bound of the true strength. We therefore seek the stress field solution to an isotropic loading (traction or compression) in the framework of the rigid core sphere model. The macroscopic stress state is of the form Σ m. Accordingly the external boundary (r = r e ) is subjected to a radial surface force σ n = Σ m n. We implement the so-called static approach of limit analysis. It consists in deriving a mesoscopic stress field σ which must be statically admissible with these boundary conditions and meet the criterion F meso (σ) = 0 (see []). Owing to the spherical symmetry, this statically admissible stress field can be sought in the form (spherical coordinates) as σ = σ rr (r)e r e r +σ θθ (r) ( e θ e θ +e φ e φ ). The boundary condition on the surface r = r e reads σ rr (r e ) = Σ m. The momentum balance equation divσ = 0 reduces to dσ rr dr = σ θθ σ rr r (4) With the notation X = σ θθ σ rr (note that X = 3 σ d ), it is found that σ m = σ rr 3X, so that the elliptic criterion (see []) yields 3 (+ 3 f ) X T The values of X solutions of (5) read : + 3f T T ( σrr + 3 X) +( f)h ( σrr + 3 X) ( f) h = 0 (5) X = 3 (3f T )σ rr ht (f )± T, with (6) 5f 3 = (f +3)(T 3f)σ rr +4hT (f +3)(f )σ rr +h T (5f +3)(f ) (7) Recalling (4), an ordinary differential equation with respect to σ rr is obtained in the form : X(σ rr ) = r dσ rr dr Introducing (6) into X(σ rr ) and integrating over the interval [r i,r e ], one obtains 3 ln(ρ) = Σm c T 5f 3 3(T 3f)σ rr +ht (f )+ǫ dσ rr (8) where the notation c = σ rr (r i ) and the boundary condition at r = r e have been used. Note that no boundary condition is available at r = r i. The physical meaning of (8) is the following : Whenever there exists a constant c such that (8) is fulfilled (with ǫ = + or ), then Σ m is an admissible 4

5 ème Congrès Français de Mécanique Bordeaux, 6 au 30 août 03 loading for the value of ρ at stake. We seek the highest possible value Σ + m > 0 of Σ m (isotropic strength in traction) and the lowest one, denoted by Σ m < 0 (hydrostatic strength on compression). For the simplification of the following discussion, the denominator in the integral of (8) is denoted by D ǫ : D ǫ = (T 3f)σ rr +ht (f )+ǫ (9) In order for this integral to be defined, two mathematical conditions are to be met, namely 0 and D ǫ 0. This remark leads to introduce the solutions to the equations of = 0 and of D ǫ = 0. First, let Σ ± m denote the solutions to = 0, which read : [ ] Σ + m = T(3+f) 6f(3+f)(5f T +3) ( f)ht (3+f)(T 3f) [ ] Σ m = T(3+f)+ 6f(3+f)(5f T +3) ( f)ht (3+f)(T 3f) Secondly, let Σ + m (resp. Σ m ) denote the solution to D + = 0 (resp. D = 0) : (0) Σ m = (T+ 6 f)(f )Th 3f T ; Σ + m = (T 6 f)(f )Th 3f T () After some reasoning, the static solution can be finally determined by numerial interation. Due to the complexity of the integrals in (8), these equations can hardly be solved analytically. However,weobservethatΣ ± m remainintheneighborhoodofσ ± m.wethereforeproposetoapproximate the functions D + and D by series expansions in the neigborhood of Σ + m and Σ m respectively. In the case of of series expansion to the second order, The analytical solutions at order read : Σ ± m = η +ρ c η κ(c η)( ρ)+ () with the following parameters for isotropic compression or traction : η = Σ m κ = c = Σ m 3+f 6 (f ) fht or η = Σ + m κ = c = Σ + m 3+f 6 ( f) fht (3) The comparison between the static solution and the kinematic solution as functions of the rigid core volume fraction ρ (f = 0.5, T = 0.55) has been performed. It is found that the two solutions can hardly be differentiated. it can be concluded that they can be regarded as the exact strength of the composite material, within the rigid core model.. 5. Interpretation of similar hydrostatic strength in limit cases To well understand this phenomenon of similar strengths under isotropic loading for both porous matrix and inclusions reinforced porous matrix, the reasoning is that even in the limit case where ρ, we can demonstrate that the difference of the strengths under isotropic loading between a reinforced matrix cell (ρ ) and an homogenous and isotropic matrix sphere (ρ = 0) is insignificant (noting that, the porous matrix is considered as homogeneous and isotropic one in the mesolevel). In the case of the reinforced matrix (ρ ), H denotes the thickness of the spherical wall of matrix (H 0 when ρ ), r e is the radius of the exterior of matrix sphere. To clarify the reasoning, a simple velocity field in the matrix phase is chosen as : v = v r e r = Dr e H [r (r e H)]e r (4) The strain rate in the matrix can be determined from (4), the components of the matrix read : d = Dr e /H, d = d 33 = v r /r, d ij = 0(i j), where v r /r 0, when H 0. Then we have 5

6 ème Congrès Français de Mécanique Bordeaux, 6 au 30 août 03 d v = Dr e /H and d d = 3 Dr e/h. Recalling the formula (), the support function of matrix reads : ( π(d) = Dr ) e k L H σ c (5) Then the macroscopic dissipation of the reinforced matrix (ρ ) is characterized as : ( Π(D) ρ= = 4πr e H π(d) = 4πr ) e 3 D k L Ω Ω σ c (6) On the other hand, in the case of the matrix (ρ = 0) under isotropic loading boundary condition D = D. The macroscopic dissipation is written as : Π(D) ρ=0 = 4πr e 3 D(kL σ c ) (7) Ω Dividing (6) by (7), the ratio χ of the strengths under isotropic loading between reinforced matrix and the pure matrix : χ = k L σ c ; σ c = T h( f) kl σ c T 3f ; L = 4(+f/3) 3f T ; k = ( f) T h +σc(3f/+t ) (+f/3) (8) Applying the paramaters T = 0.55, f = 0.5, h = 30, the numerical application of (8) yields χ =.05 which means that similar strengths under isotropic loading for both inclusions reinforced matrix (ρ ) and pure matrix (ρ = 0)have been found. 6 Conclusion On the basis of a limit analysis approach, we have proposed an extension of available models. This extension concern porous materials with a Drucker-Prager solid phase, reinforced by rigid particles. The proposed model concerns a composite material (Callovo-Oxfordien argillite) made up of rigid inclusions embedded in a porous clay matrix. The obtained results has been compared to the estimate of the strength recently derived by [4] on the basis of a variational approach. A good accuracy of the estimate of the strength under isotropic loadings has been shown by a comparison with the results of a static (stress-based) approach of the limit analysis problem. An interesting observation is that the estimates of the hydrostatic strength in traction or in compression do not depend on the homogenization method (limit analysis, variational method) and proves to be only slightly affected by the rigid core volume fraction. The practical implication is that the hydrostatic strength properties of the clay matrix and of the Callovo Oxfordien argillite are very close, irrespective of the quartz/calcite content. Also, an interpretation in limit cases has been provided. Acknowledgements The work presented in this paper was partly funded by ANDRA, the French national Agency for the management of radioactive wastes, which is gratefully acknowledged. Références [] J.-B. Leblond, G. Perrin, P. Suquet., 994. Exact results and approximate models for porous viscoplastic solids. International Journal of Plasticity [] S. Maghous, L. Dormieux, J. Barthelemy, 009. Micromechanical approach to the strength properties of frictional geomaterials. European Journal of Mechanics A/Solids [3] V. Monchiet, E. Charkaluk, D. Kondo, 007. An improvement of Gurson-type models of porous materials by using Eshelby-like trial velocity fields. Comptes Rendus Mecanique 335 Pages 3 4 [4] W.Q. Shen, L. Dormieux, D. Kondo, J.F. Shao, 03. A closed-form three scale model for ductile rocks with a plastically compressible porous matrix, Mechanics of Materials