MESO-SCALE MODELLING OF CONCRETE BEHAVIOUR UNDER TENSILE LOADING

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1 Congresso de Métodos Numéricos em Engenharia 2015 Lisboa, 29 de Junho a 2 de Julho 2015 c APMTAC, Portugal 2015 MESO-SCALE MODELLING OF CONCRETE BEHAVIOUR UNDER TENSILE LOADING C. Albino 1, D. Dias-da-Costa 2, R. Graça-e-Costa 3, J. Alfaiate 1 and E. Júlio 1 1: CEris-ICIST, DECivil Instituto Superior Técnico, Universidade de Lisboa Av. Rovisco Pais, Lisboa, Portugal capa@sapo.pt, jorge.alfaiate,eduardo.julio@tecnico.ulisboa.pt 2: School of Civil Engineering, The University of Sydney, NSW 2006, Australia ISISE, Dept. of Civil Eng., University of Coimbra, Rua Luís Reis Santos, Coimbra daniel.diasdacosta@sydney.edu.au 3: CEPAC, Universidade do Algarve Campus de Gambelas, Faro, Portugal rcosta@ualg.pt Keywords: Concrete, Discrete crack Approach, Embedded discontinuities, Multiple Cracks Abstract. Concrete is a highly heterogeneous material at meso-scale (i.e., at the scale of the aggregates), where it has a complex behaviour that is still not well understood. For this reason, the design of concrete structures often uses simplifications, namely by adopting homogenised material properties. As the design of high-performance concrete mixtures is becoming widely available, it is necessary to thoroughly assess if these underlying simplifications still hold for the new materials. This will require combined computational and experimental strategies using methods that have been introduced in recent years. This manuscript is part of a numerical study that aims at assessing the role of the aggregate (e.g. shape, stiffness and size) at the meso-scale. The discrete crack approach is adopted using finite elements with embedded discontinuities, enabling the simulation of the process of crack localisation and propagation. Preliminary results showed the importance of considering the interaction of multiple cracks. For this reason, a generalisation of the discrete crack approach is herein presented for embedding several cracks within a single finite element. Examples are used at element level to illustrate the kinematics of the finite element in typical situations that arise in the meso-scale modelling of concrete structures. 1 INTRODUCTION AND MOTIVATION Concrete is a highly heterogeneous material with a very complex behaviour that is still not well understood. Traditionally, numerical simulations have been based on homogenised material properties and, although such approach can be quite reasonable in most cases, it may not hold 1

2 for smaller scales. In fact, at the scale of the aggregates (i.e. meso-scale), there are extremely complex crack patterns that require robust numerical models for simulating the discrete nature of fracture. The work herein described is part of a comprehensive study designed to assess the meso-scale behaviour of concrete under tensile loading and to characterise the role of the aggregate (e.g. shape, stiffness and size) at this scale. Fig. 1 shows the crack pattern for an extremely simplified situation, which consists of a single aggregate embedded in cement past with a uniform tensile stress applied. Even in this case, there is already the need for dealing with multiple cracks crossing a single element see representation in Fig. 1c. Figure 1: Simplified meso-scale model: a) geometry; b) boundary conditions; c) cracked pattern. Existing approaches for embedding discontinuities in finite elements have been typically formulated for a single crack within a finite element [1]. This assumption, although reasonable for macro-scale modelling, does preclude the possibility of accurately simulating complex crack patterns found at the meso-scale and, thus, to properly simulate the material behaviour. This paper presents the first part of a numerical study for assessing the role of the aggregate at the meso-scale. Focus is given to the generalisation of the Discrete Strong Discontinuity Approach (DSDA) [1] to include multiple cracks inside a single finite element. In the following sections, the kinematics of the discontinuities, variational formulation and numerical implementation issues are described. Simple examples are used to illustrate the kinematics of the finite element in typical situations. 2 KINEMATICS AND VARIATIONAL FRAMEWORK This section describes the kinematics of a body, Ω, crossed by several embedded strong discontinuities, Γ d. Each discontinuity, d, defines two subregions subregions, Ω d and Ω+ d, as shown in Fig. 2. The unit vector n is orthogonal with respect to the external boundary, whereas n + d is orthogonal with respect to the surface of the discontinuity, pointing inwards Ω + d see Fig. 2a. The external 2

3 loads are applied on Γ t, whereas displacements are prescribed on Γ u. The vector of natural forces, t, and the traction at each discontinuity, t + d, are also represented in the same figure. Figure 2: a) Domain Ω crossed by several discontinuities Γ d and 1-D representation of the displacement field; b) Sub-regions defined by each discontinuity. The total displacement within the body is considered to be the sum of a continuous regular part, û, and an enhanced part, ũ, caused by the discontinuities. Across each discontinuity, the jump in the displacement field is represented by [u] d, such as in the 1-D representation shown in Fig. 2a. Following this consideration, the total displacement for any material point is: u(x) = û(x) + N d H Γd ũ d (x) in Ω, (1) where N d is the number of discontinuities, x are the coordinates of the material point and H Γd is the standard Heaviside function centred at each discontinuity Γ d, such that, H Γd = { 1 in Ω + d 0 otherwise. (2) The jump at each discontinuity, [u] d, can be obtained by calculating the difference between total displacements at both sides of the discontinuity: [u] d = u + d u d = ũ d. (3) For small displacements, the strain field is derived from Eq. 1 as follows: ε = s u = s û + N d H Γd s [u] d + 3 N d δ Γd ([u] d n + d )s in Ω. (4)

4 In the latter Equation, denotes the gradient operator, stands for the dyadic product, δ Γd is the Dirac delta-function and ( ) s is the symmetric part of ( ). The principle of virtual work for the body shown in Fig. 2 is given by: ( s N d δu) : σ(ε)dω + δu bdω + δu tdγ Ω\ N d Γ d Ω\ N d Γ d Γ t δ[u] d t + d dγ = 0, Γ d (5) where δ( ) represents the virtual variation of ( ), σ is the stress tensor and b is the vector containing body forces. This variational formulation can be seen as a particular case of thee-field Veubeke-Hu-Washizu principle [2, 3, 4] applied to several discontinuities. The first integral, in Eq. 5, represents the internal work, whereas the second and third integrals are the external work. The last term is the work produced at the discontinuity. 3 NUMERICAL IMPLEMENTATION Consider a partition of the 2-D domain Ω into finite elements. Each element Ω e can be crossed by Nd e discontinuities, Γe d, each one defining two sub-regions, Ωe d and Ω e+ d. See representation in Fig. 3. For the sake of clarity, only two discontinuities are illustrated. Figure 3: a) Domain Ω e crossed by 2 discontinuities Γ e d ; b) Sub-regions defined by each discontinuity. A local frame is defined for each discontinuity (s d, n + d ), where s d(x) is aligned with Γ e d and n+ d is the normal with respect to the discontinuity. 4

5 3.1 ELEMENT INTERPOLATION Having Eq. 1 into account, the displacement field within each finite element, u e, is interpolated by: u e = N e [a e + N e d (H Γd I H e Γ d )ã e d ] in Ω e \ N d Γ e d (6) where N e contains the usual shape functions of the finite element; a e are the nodal degrees of freedom associated with u e ; I is the (2n 2n) identity matrix; H e Γ d is a (2n 2n) diagonal matrix composed by successively evaluating the Heaviside function at each of the 2n degrees of freedom of the finite element; ã e d are the enhanced nodal degrees of freedom related to ũe. The latter degrees are expressed as a function of the enhanced degrees of freedom of the finite element, w e d, which directly measure the opening of each discontinuity. The enhanced nodal displacements for each discontinuity are defined by: M e wd T = ã e d = Mek wd we d, (7) where M ek wd is a matrix transmitting the opening of the discontinuity. The latter matrix is formed by stacking into rows matrix M e wd, which is evaluated at each regular node and is given by 1 (y yi d )sinα e d (x x i d )sinα e ld e d ld e (y y i d )cosα e d (y y i d )sinα e d (y yi d )cosα e d 1 (x xi d )cosα e d (x xi d )sinα e d (x x i d )cosα e d. (8) Following Eq. 3, the opening of each discontinuity is interpolated by: [u] e d = ue+ u e = N e M ek wd we d = Ne wd we d at Γ e d, (9) where N e wd contains the interpolation functions along the s-axis (see Fig. 3a) and for the corresponding pair of enhanced nodes. The strain field is approximated by: ε e = }{{} LN e a e + (H Γd I H e Γ d )ã e d B e where L is the usual differential operator. [ N e d 5 ] in Ω e \ N d Γ e d, (10)

6 The incremental stress is: dσ e = D e B [da e e + N e d (H Γd I H e Γ d )dã e d ] in Ω e \ N d Γ e d, (11) with d( ) representing the incremental variation of ( ). The traction at each discontinuity, also in incremental format, reads: dt e d = Te d d [u]e d = Te d Ne wd dwe d at Γ e d, (12) where T e d is the tangent stiffness matrix. 3.2 DISCRETISED EQUATIONS Eq. 5 is herein discretised using Eqs. 6 to 12: + Γe d Nd e Γ e d [B e ( δda e + Nd e ( δdw e d T N e wd T T e d Ne wd dwe d ( = δda e + Γe d ( N + δda e d e + Γ e d Nd e { } )] T M ek w δdw e d ) dγ e = D e B e (da e + { } ) T M ek w δdw e d N et b e dω e + { } ) T M ek w δdw e d N et t e dγ e Nd e { M ek w dw e d} ) dω e + (13) with M ek w = (H Γd I H e Γ d )M ek wd. It can be shown that B e H Γd IM ek wd = 0, since the discontinuity is assumed to open as if it were a rigid body motion. Then, by progressively taking δa e = 0 and δw e d = 0, the following discretised set of equations is obtained: K e aa K e ad K e aw Nd da e df e K e da K e dd + Ke d K e dn d K e N d a K e N d d K e N d N d + K e N d dw e d. dw e N d = df e d. df e N d, (14) 6

7 where: K e ad = K e da = K e nm = K e aa = Γe d Γe d Γe d K e d = Γ e d Γe d B et D e B e dω e, (15) B et D e B e d dωe d {1,...,Nd e }, (16) B et d De B e dω e d {1,...,Nd e }, (17) B et n D e B e mdω e m,n {1,...,Nd e }, (18) N et wd Te N e wd dγe d {1,...,Nd e }, (19) B e d = Be H e Γ d M ek wd. (20) The external forces are given by: df e = Γe d N et b e dω e + N et t e dγ e. (21) Γ e d If the body forces are neglected, and all remaining forces are applied at the nodes, it can be shown that df e d = 0 [1]. Each discontinuity has 4 degrees of freedom from the two enhanced nodes which are placed at the edges. Since only three degrees of freedom are required to describe a rigid body motion, a penalty matrix is added to K e d. A detailed discussion on this and other implementation issues can be found in [5, 6]. 4 CASE STUDIES This section presents examples selected to illustrate the performance of the embedded approach. A bilinear finite element (2 2 1 mm 3 ) is considered under a plane stress state and crossed by two discontinuities, Γ e d 1 and Γ e d 2. The constitutive laws are linear elastic for both bulk and discontinuity. The Young s modulus is E = 10 8 N/mm 2, whereas the Poisson ratio is ν = MODE-I OPENING This first example concerns a mode-i crack opening, where the shear stiffness is significantly higher than the normal stiffness, such that no relative sliding occurs along each discontinuity. The corresponding values are, respectively, k s = 10 5 N/mm 3 and k n = 1 N/mm 3. The structural scheme is shown in Fig. 4a, being the first discontinuity Γ e d 1 defined by the two enhanced nodes placed at the edges and with coordinates i 1 ( 1.00, 0.35) and j 1 (1.00, 0.75). The second discontinuity, α e d 2, is symmetric relatively to the x-axis. 7

8 In this example, a load of P = 1 N produces a normal jump at each discontinuity of [u] n = 0.192mm. The corresponding deformed mesh is depicted in Fig. 4b, where it can be observed that the top edge only moves vertically due to the symmetry of the problem. Figure 4: Finite element (dimensions in mm): a) structural scheme; b) resulting deformation. 4.2 MODE-II OPENING This section uses the same finite element to illustrate its behaviour in the case of mode-ii opening. This opening mode is induced by selecting a normal stiffness higher than the shear stiffness, being the corresponding values k n = 10 5 N/mm 3 and k s = 1 N/mm 3, respectively. Both geometry and boundary conditions are represented in Fig. 5a. In this case, a load of P = 1.06 N produces a resulting horizontal displacement of u e d 1 (x) = u e d 2 (x) = 0.5mm. Fig. 5b shows the deformed element, with the different subdomains sliding relatively to each other as expected. Due to the symmetry of the problem, the top edge of the element only has horizontal displacements. 4.3 GENERAL OPENING This section presents a general opening mode of the discontinuities. In this case, both normal and shear stiffnesses have the same value, i.e. k n = k s = 1 N/mm 3. The load is defined in Fig. 6a, where P = 0.5 N. The enhanced nodes used to define the first discontinuity Γ e d 1 are placed at i 1 ( 1.00, 0.50) and j 1 (1.00, 0.75). The second discontinuity, Γ e d 2, is defined by i 2 ( 1.00, 0.35) and j 2 (1.00,0.75). The resulting deformation is represented in Fig. 6b, where it can be observed the adequate behaviour of the element. In this case, the top-right node has a displacement of ( 0.16, 0.64) mm, whereas the top-left node has a displacement of ( 0.16,1.23) mm. 8

9 Figure 5: Finite element (dimensions in mm): a) structural scheme; b) resulting deformation. Figure 6: Finite element (dimensions in mm): a) structural scheme; b) resulting deformation. 5 CONCLUSIONS The work herein presented is part of a comprehensive numerical study that aims at characterising the concrete behaviour at meso-scale, i.e. at the scale of the aggregate. At this scale, the simulation of the discrete nature of fracture and the complex crack patterns associated with damage propagation requires the need to handle multiple cracks. This is the case even for very simple situations, such as concrete samples under tensile loading with few aggregates. Most existing embedded formulations have been developed and tested with underlying simplifications, such as embedding a single discontinuity in each element. For this reason, there are 9

10 limitations in what concerns the simulation of the material behaviour. Within this scope, this work presented a generalisation of the Discrete Strong Discontinuity Approach (DSDA) [1] to handle multiple discontinuities. The proposed formulation is general and can be easily implemented into any finite element code. Furthermore, since the opening of the discontinuity is assumed to be transmitted as a rigid body motion, there is no need to perform partial integration of the element stiffness, as with X-FEM based approaches. The selected examples, although still at element level, already showed good performance in what concerns mode-i, mode-ii and general crack openings, with all results in agreement with known solutions. ACKNOWLEDGEMENTS The authors would also like to extend their acknowledgement to the support provided by FEDER funds through the Operational Programme for Competitiveness Factors - COMPETE - by Portuguese funds through FCT - Portuguese Foundation for Science and Technology under Project No. FCOMP FEDER (FCT ref. PTDC/ECM/119214/2010). REFERENCES [1] D. Dias-da-Costa, J. Alfaiate, L. J. Sluys, and E. Júlio. A discrete strong discontinuity approach. Engineering Fracture Mechanics, 76(9): , [2] B. M. F. de Veubeke. Diffusion des inconnues hyperstatiques dans les voilures à longeron couplés. Bulletin du Service Technique de l Aéronautique, Imprimeríe Marcel Hayez, 24:56, [3] Hu H.-C. On some variational principles in the theory of elasticity and the theory of plasticity. Scientia Sinica, 4:33 54, [4] K. Washizu. On the variational principles of elasticity and plasticity. Technical report, Aeroelastic and Structures Research Laboratory, Massachusetts Institute of Technology, Cambridge, March [5] D. Dias-da-Costa, J. Alfaiate, L. J. Sluys, and E. Júlio. Towards the generalization of a discrete strong discontinuity approach. Computer Methods in Applied Mechanics and Engineering, 198(47-48): , [6] D. Dias-da-Costa, J. Alfaiate, L. J. Sluys, P. Areias, and E. Júlio. An embedded formulation with conforming finite elements to capture strong discontinuities. International Journal for Numerical Methods in Engineering, 93(2): ,