MICROSTRUCTURAL ANALYSIS OF CONCRETE FRACTURE
USING INTERFACE ELEMENTS
Carlos M. López†, Ignacio Carol††, and Antonio Aguado††
†
ICMAB (CSIC Institute of Material Science, Barcelona) Campus UAB, E-08193 Bellaterra, Spain
e-mail: [email protected]
††
ETSECCPB (School of Civil Engineering)- UPC (Technical Univ. of Catalonia) Jordi Girona 1-3, E-08034 Barcelona, Spain
e-mail: [email protected] e-mail: [email protected]
Key words: Fracture Mechanics, Interface Elements, Microstructural Analysis, Concrete. Abstract. A microstructural model for the mechanical behavior of concrete is developed and
verified for concrete specimens subject to several load scenarios. The model is based on interface elements equipped with a constitutive law representing non-linear fracture, which may be considered a mixed-mode generalization of Hillerborg’s “Fictitious Crack Model”1. Specimens with 4x4 and 6x6 arrays of aggregates are discretized into finite elements. Interface elements are inserted along the main lines in the mesh, representing potential crack lines. The continuum elements themselves are assumed to remain linear elastic. In this paper, some numerical results obtained in meshes under various loading cases, such as pure tension and compression, as well as Brazilian test are presented. The results show a good qualitative agreement with experimental observations and illustrate the capabilities of the model.
1 INTRODUCTION
The behavior of heterogeneous materials such as concrete, is clearly determined by the microstructural geometry and properties. Macroscopic models of the continuum type, which have been used traditionally in structural analysis, may be realistic for simple loading scenarios. However, as loading cases become more complicated and, especially, as other phenomena such as diffusion of coupled behavior are important, accurate understanding of the material response can be drastically improved via microstructural analysis.
In this paper, some of the work developed at ETSECCP-UPC along this line during recent years is summarized. The microstructural model is based on the use of “zero-thickness” interface elements equipped with the fracture-based constitutive law described in Sect. 2. The interface elements are inserted into the general-purpose FE code DRAC, as described in Sect. 3.
Meshes used and results obtained in the analysis of concrete specimens are described in Sect. 4. Loading cases considered are uniaxial tension (4.1), uniaxial compression (4.2) and Brazilian tests (4.3). Conclusions of the work developed so far are finally presented in Sect. 5. More details about the work presented, as well as additional results of application may be found in2.
2 INTERFACE CONSTITUTIVE MODEL
As recognized by several authors3,4,5, zero-thickness interface elements with appropriate constitutive laws formulated in terms of shear and normal stress and the corresponding relative displacements provide the means of extend the FCM concept1 into a modern
numerical analysis of fracture.
Interface behavior is formulated in terms of the normal and shear components of stresses (tractions) on the interface plane, σσσσ = [σN, σT] t, and corresponding relative displacements u =
[uN, uT] t (t = transposed). The constitutive model is analogous to that used for each potential
crack plane in the multicrack model6,7,8,9,10; it has been recently described in detail and compared to other existing interface models2,11, and its main features and verification are summarized in the following.
The constitutive formulation conforms to work-softening elasto-plasticity, in which plastic relative displacements can be identified with crack openings. The main features of the plastic model are represented in Figure 1. The initial loading (failure) surface F = 0 is given as a three-parameter hyperbola (tensile strength χ, asymptotic “cohesion” c, and asymptotic friction angle tanφ, Figure 1a). Classic Mode I fracture occurs in pure tension. A second Mode IIa is defined under shear and high compression, with no dilatancy allowed (Figure 1b). The fracture energies GIf and GIIaf are two model parameters. After initial cracking, c and χ
decrease (Figure 1c), and the loading surface shrinks, degenerating in the limit case into a pair of straight lines representing pure friction (Figure 1d). The process is driven by the energy spent in fracture process, Wcr, the increments of which are taken equal to the increments of plastic work, less frictional work in compression. Total exhaustion of tensile strength (χ = 0)
Additional parameters αχ and αc allow for different shapes of the softening laws (linear
decay for αχ = αc = 0). The plastic model is associated in tension (Q = F), but not in
compression, where dilatancy vanishes progressively for σN → σdil. Dilatancy is also
decreased as the fracture process progresses, so that it vanishes for Wcr = GIIaf. The dilatancy
decay functions also include shape parameters αdil
σ and αdilc (also linear decay for zero values
of shape parameters). The elastic stiffness matrix is diagonal with constant KN and KT, which
can be regarded simply as penalty coefficients
Figure 1. Crack laws: (a) hyperbolic cracking surface F and plastic potential Q; (b) fundamental modes of fracture; (c) evolution of cracking surface; (d) softening laws for χ and c.
Three verification examples of the constitutive model are presented. First, a numerical test in pure tension. The σ-u curves obtained for various values of the fracture energy GI
f are
represented in Figure 2 (other relevant parameters are KN = 1,000 MPa/mm, tensile strength
χ0 = 3 MPa and all shape parameters equal to zero). Note that, even with a zero shape
parameter (linear softening function in terms of Wcr), the resulting softening curve in terms of crack opening is of the exponential type, with total area under each curve equal to the prescribed value of GIf.
The second example is a numerical shear test. First, normal compression of a prescribed value is applied. Then, the shear relative displacement is increased progressively with constant normal stress, until a residual state is reached. The parameters used are KN = KT = 25,000
30 MPa, αdil
σ = 2 and all other shape parameters equal zero. The results are depicted in
Figures 3 and 4. In Figure 3, shear stresses are represented against shear relative displacement for various values of normal compression. Note that, after the peak, all curves tend to a residual value of the shear stress, which corresponds to basic friction times the normal stress. In Figure 4, dilatancy is represented by normal against shear relative displacements.
Figure 2. Pure tension: normal stress versus relative displacement for different values of fracture energy.
Figure 4. Shear under constant compression: evolution of dilatancy for various values of σN.
The third constitutive example corresponds to an experimental test by Hassanzadeh12. Normal and shear relative displacements are prescribed to an interface in two steps. First, only normal (opening) is applied till the peak stress is reached. From this point, shear displacement is applied simultaneously to the normal in a fixed proportion characterized by the angle tanθ = uN/uT.
Tests were run for various θ, starting with the limit case θ = 90° which actually corresponds to continuing the test in pure tension. The parameter values used are KN = KT =
200 MPa/m, tanφ = 0.9, χ0 = 2.8 MPa, c0 = 7 MPa, GIf = 0.1 N/mm, GIIaf = 1.0 N/mm, σdil =
56 MPa, αχ = 0, αc = 1.5, αdilσ = 2.7, αdilc = 3. The results obtained in the second part of the
test are represented in Figures 5 and 6.
In Figure 5, normal stress is represented versus normal relative displacements. While for θ = 90° (pure tension), the usual exponential-type of decay is obtained, the imposition of a certain proportion of shear relative displacement causes the stresses to drop faster, change sign into compression, reach a peak and then vanish asymptotically. This is due to development of shear dilatancy that would exceed the prescribed normal opening rate.
In Figure 6, the shear stresses corresponding to the same tests are plotted against shear relative displacements. In both figures, numerical results (continuous lines) are represented together with experimental dots, showing how the proposed model not only gives logical numerical results, but is also capable of fitting experimental data obtained during non-trivial fully coupled normal/shear loading scenarios. Additional details of the constitutive formulation and verification examples can be found in2,11.
Figure 5. Hassanzadeh's tests: normal stress versus relative displacement.
3 NUMERICAL IMPLEMENTATION IN THE FE CODE "DRAC"
The model has been implemented into a set of subroutines that have been added to the constitutive libraries of the FE code DRAC13. This is a research-oriented geotechnical/structural FE program with 2D/3D capabilities, various element types including interfaces, post-processing module DRAC-VIU, etc. which has been in-house developed through the years at the Dept. of Geotechnical Engineering and Geo-Sciences of ETSECCPB-UPC.
The constitutive subroutines perform relative displacement-to-stress calculations and include a substepping scheme to reduce integration errors. Substep size is determined depending on the type of prescribed relative displacement, and on variation of the flow rule direction14,15.
The interface elements used are “zero-thickness” isoparametric elements that can be inserted in between standard continuum finite elements. The nodes are grouped in pairs, which match on each side those of the adjacent elements. The formulation follows standard application of the Principle of Virtual Work, and the only special consideration refers to the integration rules which correspond to Newton-Cotes/Lobatto schemes (with integration points in between each pair of nodes) in order to avoid spurious oscillations in the resulting stress profiles. See more details in Gens et al.16.
The iterative strategy at the finite element level includes a version of the arc-length standard procedure based on the norm of displacement increments of all nodes15. This strategy works well in the type of microstructural calculations with interfaces along all possible crack paths, in which initially cracks start opening all over the mesh and later most of them close and deformations localize into one main crack.
4 MICROSTRUCTURAL ANALYSIS OF CONCRETE SPECIMENS
Square concrete specimens with 4x4 and 6x6 arrangements of aggregates are discretized as shown in Figure 7 for the 6x6 arrangement. A number of interface elements are inserted along the aggregate-matrix interface, and also across the mortar matrix in order to allow most relevant failure mechanisms. Continuum elements between interfaces are considered linear elastic.
The geometry of the aggregates was taken from previous numerical work by Stankowski4. The mesh was rebuilt completely to provide straighter crack paths, following the ideas proposed by Vonk5.
In Figures 7a and 7b, the discretization used for the matrix and the aggregate parts of the mesh are shown separately. Figure 7c shows the interface lines in the same mesh, and figure 7d depicts a detail of node arrangements in pairs and interface intersections.
Figure 7. FE discretization of the 6x6 arrangement: a) matrix, b) aggregates, c) interfaces inserted, and d) details of discretization.
4.1 Uniaxial tension
In this case, the load is applied in the form of a prescribed displacement of one side of the specimen, leaving lateral displacement free in the transverse direction. Sum of reactions divided by the size of the specimen gives an average stress. The material parameters are, for the continuum elements: E = 70000 MPa (aggregate), E = 25000 MPa (mortar) and ν = 0.18 (both); for the aggregate-mortar interfaces: KN = KT = l09 MPa/m, tanφ = 0.8, tensile strength χ0 = 3 MPa, c0 = 4.5 MPa, GIf = 0.00003 MPa×m, GIIaf = l0GIf, σdil = 7 MPa, and all other
parameters equal to zero; for the mortar-mortar interfaces the same parameters except for χ0 =
6 MPa, c0 = 9 MPa, GIf = 0.00006 MPa×m. The calculations have been repeated with load
applied in the x and y directions for the 4x4 and 6x6 mesh, and the resulting average stress-average strain curves are represented in Figure 8. It can be noted that all them yield similar results in spite of the relatively different geometries involved.
Figure 9 includes results of the 4x4 specimen loaded vertically, at three different stages of the loading sequence indicated on figure 8, as well as the final deformed mesh (magnification factor = 200). The thickness of the lines represents the amount of energy spent in the fracture process (Wcr) at each point of the interface. It is apparent that, initially, many cracks start developing, and at some point deformations localize in one or two cracks that develop while all other cracks unload. Figure 10 shows the final state in terms of energy and also of deformed state (magnification factor = 200) at the end of the calculations for the other three cases analyzed. Well-known phenomena such as crack bridging and branching can be
Figure 8. Average stress-average strain curves.
(a) (b)
(c) (d)
Figure 10. Final state in terms of energy and also of deformed state at the end of the calculations for the other three cases analyzed.
4.2 Uniaxial compression
In this case, load is also applied via uniform prescribed displacement on one side of the specimen, while lateral displacements are allowed. The material parameters are E = 70000 MPa (aggregate), E = 25000 MPa (mortar) and ν = 0.2 (both); for the aggregate-mortar interfaces: KN = KT = l09 MPa/m, tanφ0 = 0.6, χ0 = 2 MPa, c0 = 7 MPa, GIf = 0.00003 MPa×m,
GIIaf = l0GIf, σdil = 40 MPa, tanφr = 0.2, αdilσ = -2, αφ = 1, and all other parameters equal to
zero; for the mortar-mortar interfaces the same parameters except for χ0 = 4 MPa, c0 = 14
MPa, GIf = 0.00006 MPa×m.
In figure 11, the stress-strain curves obtained with the 4x4 and the 6x6 meshes are plotted. The vertical axis represents average uniaxial stress and the right side of the horizontal axis represents the corresponding (prescribed) strain. The left side of the horizontal axis represents average lateral strain. It is apparent that the results obtained with both meshes are quite similar, and they agree well with the typical behavior of concrete in uniaxial compression as observed in experiments17,18.
Figure 11. Average stress-strain curves for the 4x4 and 6x6 meshes under uniaxial compression.
In the left part of the figure, one can see the evolution of the lateral strain. Initially, ε2 evolves
maintaining approximately the Poisson relation. Before reaching the peak load, the lateral strain starts growing faster, and in the softening branch it overcomes the strain prescribed in the loading direction. In figure12, the same behavior is represented in terms of a “volumetric”
-5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 -50 -30 -10 x10-3 1x10-3 2 1 2 3 4 1 2 4x4 mesh 6x6 mesh 1(MPa)
strain defined as (ε1+2ε2), which also matches the typical experimental curves.
Figure 12. Stress vs. “fictitious” volumetric strain curves in uniaxial compression.
In figure13, the evolution of the cracking process of the 6x6 specimen is represented in terms of the modulus of the “plastic” part of the relative displacement vector at each point of the interfaces. The four stages of loading represented in the figure, correspond to the four points marked on the stress-strain curves of figure 11. The state represented in figure 13a corresponds to point 1 with widespread distributed microcracks mainly originated at the concrete-mortar interface. Figure 13b corresponds to point 2, in which the localization process has started. Some of the microcracks are getting connected sideways forming inclined macrocracks, while the rest unload. At point 3 (figure 13c), one can clearly appreciate one or two localization bands well developed which divide the specimen into three main blocks, while the remaining incipient macrocracks have arrested.
In the final stage (figure 13d, point 4 in figure 11) the same scheme is maintained, and the few blocks formed slide with friction with respect to each other. Note that the transition from 13b to 13c represents a significant reduction of stress (points 2 to 3 in figure 11),
-2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 -50 -30 -10 ( + 2 )x101 2 -3 1 2 4x4 mesh 6x6 mesh 1(MPa)
a) b)
c) d)
Figure 13. Localization process in uniaxial compression, represented in terms of the magnitude of plastic strain at the interfaces.
4.3 Brazilian test
The indirect tension or Brazilian test is a frequently used test to obtain tensile strength of concrete. Load is applied on a cylindrical or prismatic specimen, on two narrow bands located at diametrically opposite locations (cylindrical specimen), or at the center of opposite faces (prismatic specimen), as shown in figure 14.
According to linear elasticity, this configuration generates an almost uniform distribution of tensile stress along the line connecting the opposite loads. In practice, although the failure load of the Brazilian test gives a useful estimate of the tensile strength, the numerical values obtained need in general some adjustment due to the fact that the stress state does not correspond to pure tensile loading (significant compressive stress exists in the perpendicular direction to the failure plane).
Figure 14. Geometry and load application in the Brazilian test.
Some experimental results have shown the important influence of the width of the loading platen (b in figure 14), both on the resulting peak load and post-peak behavior19,20. In order to study this effect, calculations with the microstructural model have been carried out for various values of the b/H ratio 0.025, 0.064 y 0.166. For this study, the 6x6 FE mesh has been used, including interfaces also within the aggregates themselves. In this way, aggregate breakage becomes also possible as it is observed in high-strength or light-weight concretes. The material parameters used for the continuum elements are E = 70000 MPa (aggregate), E = 40000 MPa (mortar), ν = 0.2 (both); for the aggregate-mortar interfaces KN = KT = l09 MPa/m, χ0 = 3 MPa, c0 = 10 MPa, tanφ = 0.8, GIf = 0.00003 MPa×m, GIIaf = l0GIf , σdil = 50 MPa, αdilσ
= -1, and all other shape coefficients equal to zero; and for the mortar-mortar and aggregate-aggregate interfaces the same parameters except for χ0 = 6 MPa, c0 = 20 MPa y GIf = 0.00006
MPa×m.
In figure 15, the results are presented in terms of load against crack opening at the center of experiment (both normalized with respect to their peak values).
0 2 4 6 8 10 CMOD/CMOD 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Load/ Load b/H = 0.166 b/H = 0.064 b/H = 0.025 H b 1 2 3 4 peak
From the results of the analysis, CMOD has been taken as the normal relative displacement averaged over the three central interfaces closest to the vertical plane of symmetry of the specimen. In order to facilitate the interpretation of results, in the figure the pre-peak behavior has been simplified as a straight line between the origin and the peak values of stress and CMOD.
The influence of the relative platen size can be clearly seen in the figure. For small b/H there is a peak and a clear softening descending branch. As b/H increases, the softening is reduced and, for the largest value used of b/H=0.166, after a brief descending part, the curve increases again and a second round-shaped peak is reached, which is higher than the first one. This change in response is consistent with experimental observations21. Additional insight of the material behavior in this test is obtained from the cracking patterns observed at various stages of the loading process. In figure 16, the cracking sequence is represented in terms of the magnitude of the plastic strain vector at each point of the interfaces, which correspond to the points 1 to 4 marked on the load-displacement curve for b/H=0.166 in figure 15.
(1) (2)
(3) (4)
Figure 16. Cracking states at loading stages marked 1 to 4 in fig. 15, for b/H=0.166 (magnitude of plastic relative displacements at the interfaces).
At the time of the first peak (point 1) in figure 15, a first vertical crack forms in the center of the specimen. At point 2, already in the descending branch, the crack propagates towards
on friction and shear is observed near the lower loading platen: two inclined cracks develop creating a triangle of material that moves with it like a wedge22,23,24.
In figure 17, the crack patterns for two different platen sizes (b/H = 0.025 and 0.064) are represented at a similar stage of loading.
Figure 17. Cracking state at similar stage of loading, for b/H=0.025 (left) and 0.064 (right).
5 CONCLUDING REMARKS
A microstructural model for concrete fracture has been proposed, based on “zero-thickness” interface elements equipped with a fracture-based constitutive law. These interface elements are located in between standard continuum elements with elastic behavior, and represent all potential crack lines embedded in the mesh. The constitutive law has the structure of work-softening elasto-plasticity, with two parameters, which represent fracture energies in models I and IIa. Interface elements require special integration techniques well documented in the literature, and the overall FE calculation requires an arc-length scheme to overcome sharp peaks and widespread changes in interfaces from closed to opening, and then back to closed state.
The microstructural model proposed can represent some of the most salient features of experimental concrete fracture, such as microcracking, localization and macroscopic crack formation. The qualitative agreement is obtained on the average stress-strain curves as well as in the microcrack patterns and their coalescence and evolution. This type of model also opens the possibility to study the influence of aggregate shape and size, relative strengths of aggregate-matrix, aggregate-aggregate and matrix-matrix interfaces, and a number of other effects impossible to account for using macroscopic models.
A similar approach is being applied to cancellous (porous) bone and other materials25,26, with very promising results as described in another paper in the same conference proceedings.
So far, analyses have been confined to 2D, which in spite of the promising results obtained, represents a drastic limitation. Work is under way to extend the same approach to 3D. Mesh generation and 3D constitutive models for interfaces are some of the efforts currently pursued.
ACKNOWLEDGEMENTS
The authors wish to thank DGICYT-MEC (Madrid, Spain) for the support received through research projects PB96-0500 and AMB96-0953. The first author wants to acknowledge the support received from Generalitat de Catalunya (Barcelona, Spain), through Conselleria d’Obres Públiques and Direcció General de Recerca (projects GRQ 93-3012, GRQ 99-00135), and from CSIC (Madrid, Spain) through a post-doctoral fellowship.
REFERENCES
[1] A. Hillerborg, M. Modéer, P.E. Petersson, “Analysis of crack formation and crack growth in concrete by means of Fracture Mechanics and Finite Elements”. Cement and Concrete
Research, 6(6):773-781, (1976).
[2] C.M. López, Microstructural analysis of concrete fracture using interface elements.
Application to various concretes (In Spanish). Doctoral Thesis. Universitat Politecnica de
Catalunya. ETSECCCP-UPC, E-08034 Barcelona, Spain (1999).
[3] J.G. Rots, Computational modeling of concrete fracture. PhD thesis, Delft University of Technology, The Netherlands (1988).
[4] T. Stankowski, T. Numerical simulation of progressive failure in particle composites. PhD thesis, Dept. CEAE, University of Colorado, Boulder, CO 80309-0428, USA (1990). [5] R. Vonk, Softening of concrete loaded in compression. PhD thesis, Technische
Universiteit Eindhoven, Postbus 513, 5600 MB Eindhoven, Netherlands (1992).
[6] I. Carol, y P.C. Prat, “A statically constrained microplane for the smeared analysis of concrete cracking”. In Bicanic and Mang, editors, Computer aided analysis and design of
concrete structures, 919-930 (1990).
[7] I. Carol y P. Prat, “Smeared analysis of concrete fracture using a microplane based multicrack model with static constraint”. In J.G.M. van Mier, J. G. Rots, and A. Bakker, editors, Fracture processes in concrete, rock and ceramics, pages 619-628, Noordwijk, The Netherlands. E & FN SPON (1991).
[8] P.C. Prat, I. Carol, R. Gettu, “Numerical analysis of mixed-mode fracture processes”.
Anales de Mecánica de la Fractura, 9:75-80, (1992).
[9] I. Carol, P.C. Prat, R. Gettu, “Numerical analysis of mixed-mode fracture of quasi-brittle materials using a multicrack constitutive model”. In H. P. Rossmanith and K. J. Miller, editors, Mixed-mode fatigue and fracture, pp. 319-332. Mechanical Engineering Publications Ltd., London. ESIS Publication 14, (1993).
[10]I. Carol, P.C. Prat, “A multicrack model based on the theory of multisurface plasticity and two fracture energies”. In Owen, D.R.J., Oñate, E., and Hinton, E., editors,
Computational plasticity (COMPLAS IV), Vol. 2 pp.1583-1594, Barcelona, Pineridge
Press, (1995).
[11]I. Carol, P.C. Prat y C.M. López, “A normal/shear cracking model. Application to discrete crack analysis”. J. of Engineering Mechanics. Vol. 123,No 8 (1997).
[13]P.C. Prat, A. Gens, I. Carol, A. Ledesma, y J.A. Gili, DRAC: “A computer software for the analysis of rock mechanics problems”. In H. Liu, editor, Application of computer
methods in rock mechanics, 2: 1361 - 1368, Xian, China. Shaanxi Science and
Technology Press (1993).
[14]J. Marques, “Stress computation in elastoplasticity”. Engng Comput., 1: 42-51 (1984). [15]M. Crisfield, Non-linear finite element analysis of solids and structures, Volume 1,
(1991).
[16]A. Gens, I. Carol, E. Alonso, “A constitutive model for rock joints. Formulation and numerical implementation”. Computers and Geotechnics, 9:3-20, (1990).
[17]J. G. M. Van Mier, Strain-softening of Concrete under Multiaxial Loading Conditions. PhD thesis, Eindhoven University of Technology, Eindhoven, The Netherlands. (1984). [18]J. G. M. Van Mier, Fracture Processes of Concrete. CRC Press (1997).
[19]T. Tang, S.P. Shah, y C. Ouyang, “Fracture Mechanics and Size Effect of Concrete in Tension”. Journal of Structural Engineering, ASCE, 118, pp. 3169-3185 (1992).
[20]C. Rocco, G.V. Guinea, J. Planas, J. y M. Elices, “Variación de la resistencia a la tracción en el ensayo brasileño”. Anales de Mecánica de la Fractura, 12, pp. 227-232 (1995). [21]C. Rocco, G.V. Guinea, J. Planas y M. Elices, “Estudio experimental sobre los
mecanismos de rotura del ensayo de compresión diametral”. Anales de Mecánica de la
Fractura, 14, pp. 170-175 (1997).
[22]Z.P. Bazant, M. Kazemi; T. Hasegawa and J. Mazars, “Size effect in Brazilian split cylinder test”. ACI Material Journal, 88 (3). pp. 325-332 (1991).
[23]A. Castro-Montero, Z. Jia and S. Shah, “Evaluation of damage in Brazilian test using holographic interferometry”. ACI Materials Journal, 2, pp. 268-275 (1995).
[24]S. Carmona, Caracterización de la fractura del hormigón y de vigas de hormigón
armado. Tesis Doctoral, Escuela Técnica Superior de Ingenieros de Caminos, Canales y
Puertos, Universitat Politècnica de Catalunya, Barcelona, España (1997).
[25]M. Pini, C.M. López, I. Carol and R. Contro, “Microstructural analysis of cancellous bone using interface elements”. In Rossmanith, H.P.(Ed.), Damage and Failure of
Interfaces-(DFI-1), Viena, Austria (1997).
[26]I. Carol and C:M. López, “Failure Analysis of Quasi-Brittle Materials Using Interface Elements”. In Mechanics of Quasi-Brittle Materials and Structures. Gilles Pijaudier-Cabot, Zdenek Bittnar y Bruno Gérard (Eds.), pp. 289-305 (1999).
