mesoscopic analysis of crack propagation in concrete by nonlinear finite element method with crack queuing algorithm

Available online at www.sciencedirect.com ScienceDirect Procedia Engineering 172 (2017) 620 – 627 Modern Building Materials, Structures and Techniques, MBMST 2016 Mesoscopic analysis of crack propagation in concrete by nonlinear finite element method with crack queuing algorithm A.K.H Kwana, P.L Nga,b*, Z.M Wangc a Department of Civil Engineering, The University of Hong Kong, Hong Kong, China b Vilnius Gediminas Technical University, Vilnius, Lithuania c Zhengzhou University, Zhengzhou, China Abstract By means of nonlinear finite element method, the initiation and propagation of cracks in concrete are analysed numerically at the mesoscopic level The concrete is treated and modelled as a three-phase system consisting of coarse aggregate, hardened mortar matrix and interfacial transition zones between coarse aggregate and mortar matrix Nonlinear constitutive properties of the materials before and after cracking are incorporated in the modelling An integrated cracking criterion based on both tensile strength and fracture toughness is proposed to cater for stress concentration at crack tips, and a crack queuing algorithm is employed to simulate the stress redistribution and stress relief upon cracking To illustrate the applications of the proposed method, an example of the cracking process of concrete is presented © Published by Elsevier Ltd This © 2017 2016The TheAuthors Authors Published by Elsevier Ltd.is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the organizing committee of MBMST 2016 Peer-review under responsibility of the organizing committee of MBMST 2016 Keywords: Crack propagation; crack queuing algorithm; mesoscopic analysis; nonlinear finite element analysis; strain softening Introduction Concrete is a composite material made up of cement paste, fine aggregate and coarse aggregate There exists a variety of inhomogeneity in concrete At the macroscopic scale, concrete is usually simplified as a homogeneous material whose stress-strain relation may be described by nonlinear macroscopic constitutive models with strain softening [1], whereas the cracks are smeared into the concrete This approach is often adopted in the analysis of * Corresponding author Tel.: +852-28598024 E-mail address: irdngpl@gmail.com 1877-7058 © 2017 The Authors Published by Elsevier Ltd This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the organizing committee of MBMST 2016 doi:10.1016/j.proeng.2017.02.072 A.K.H Kwan et al / Procedia Engineering 172 (2017) 620 – 627 reinforced concrete structures and members However, macroscopic constitutive models are descriptions of averaged behaviour and are often complicated by strain localization and size effects [2] In order to understand strain localization, evaluate size effect and investigate the effects of microstructural details, mesoscopic study of concrete may be performed [3,4] At the mesoscopic scale, the concrete is treated as a three-phase composite material consisting of coarse aggregate, mortar matrix with fine aggregate combined with it, and interfacial transition zones between the coarse aggregate and the mortar matrix It should be noted that apart from the macroscopic and mesoscopic scales, according to Zaitsev and Wittmann [5], there are several hierarchical levels in the modelling of concrete, namely the macro-, meso-, micro- and nano-levels At microscopic scale, the mortar matrix of the mesolevel is subdivided into fine aggregate and hardened cement paste with pore embedded inside At nanoscopic scale, the hardened cement paste is further divided into big pores (air voids) and hardened cement paste with only small pores (capillary pores) in it Nevertheless, in view of the majority of crack trajectories along the boundaries of coarse aggregates, analysis in the mesoscopic scale is investigated in this paper To conduct mesoscopic study, the micro-structure of concrete is modelled by generating a random aggregate structure in which the size, shape and spatial distributions of the coarse aggregate particles resemble those in real concrete statistically For a given domain, the random aggregate structure is discretized into a triangulated mesh to enable finite element analysis [6] Fig displays the discretization of the random aggregate structure There are two major difficulties associated with the nonlinear finite element analysis Firstly, due to the need to use a very fine mesh to properly simulate the geometrical features of the micro-structure, numerical difficulties would be arisen from the stress concentrations near crack tips As the mesh size is very fine, very high stresses near crack tips would be obtained and the cracks formed would be unstable in the analysis Secondly, the nonlinear analysis has to be carried out by an incremental procedure whereby the loads are applied in discrete increments At one increment step, many elements may crack simultaneously This is in contrast with the reality, where a crack once emerges would cause stress redistribution in the vicinity, and there would be stress relief in the nearby elements so that they would remain uncracked To overcome these difficulties, an integrated cracking criterion based on both tensile strength and fracture toughness is adopted, and a crack queuing algorithm is employed, as depicted herein Fig Discretization of random aggregate structure 621 622 A.K.H Kwan et al / Procedia Engineering 172 (2017) 620 – 627 Finite Element Modelling The concrete represented as a three phase system is simulated by finite elements, where the coarse aggregate and the mortar matrix are modelled by 3-noded triangular constant strain elements, and the interfacial transition zones at the aggregate-mortar interfaces are modelled by 4-noded interface elements To enable nonlinear analysis, secant stiffness is used in the element formulation and numerical iterations For the sign convention, tension is positive and compression is negative 2.1 Aggregate and mortar matrix elements Homogeneity within an aggregate element and mortar matrix element is assumed To account for the biaxial behaviour, the biaxial stress-strain relation is described via equivalent uniaxial strains [7] with the use of a nonlinear orthotropic material model The equivalent uniaxial strains are defined by: H e1 H1  v2H  v1v2 (1) H e2 H  v1H1  v1v2 (2) where He1 and He2 are the equivalent uniaxial strains, H1 and H2 are the principal strains, v1 and v2 are the Poisson’s ratios, and the subscripts and denote quantities in the major and minor principal directions, respectively The principal stresses V1 and V2 are each assumed to be a single variable function of He1 and He2, respectively, so that the biaxial stress-strain relation is effectively decomposed into two separate uniaxial stress-strain relations Fig Biaxial strength envelope 623 A.K.H Kwan et al / Procedia Engineering 172 (2017) 620 – 627 Under tension, the stress-strain curve is assumed to be linear before cracking and strain-soften linearly after cracking until complete loss of tensile strength, as proposed by Bazant and Oh [8] Under compression, the stressstrain curve proposed by Saenz [9] is adopted For determining the tensile and compressive strengths of the material, the biaxial strength envelope developed by Kupfer and Gerstle [10] as depicted in Fig is used In Fig 2, the principal stresses are normalised by the uniaxial compressive strength fc The biaxial stress state falls into one of the four zones, namely tension-tension (T-T), tension-compression (T-C), compression-tension (C-T), and compressioncompression (C-C) From the stress-strain curves, the principal stresses V1 and V2 are obtained, and the secant stiffness values in the two principal directions E1 and E2 are evaluated as V1/He1 and V2/He2, respectively 2.2 Interfacial transition zone element To simulate the interfacial transition zone between coarse aggregate and mortar matrix, a 4-noded interface element similar to the Goodman type interface element is developed [11] Fig illustrates the topology and connectivity of the interface element The element is of infinitesimal thickness and the nodes are divided into two nodal pairs For each nodal pair, the two nodes share the same coordinates, one node is connected to the aggregate element and the other node is connected to the mortar matrix element Under tension, the interface will crack when the tensile stress exceeds the tensile strength or the shear stress exceeds the shear strength Once cracked, the interface will become separated and the element will have zero stiffness Under compression, the interface will remain intact and is assumed to have a shear capacity given by the Mohr-Coulomb criterion [7] When the shear capacity is exceeded, the interface will fail in shear Nevertheless, it is assumed that after shear failure, the normal stiffness will remain unchanged and the interface can still take up shear in the form of friction Fig Interfacial transition zone element Integrated Cracking Criterion The stress field at a crack tip is singular When a very fine mesh is used, the tensile stresses obtained in the proximity of a crack tip would be extremely high and far exceeding the tensile strength of the constituent material Hence, if only a tensile strength criterion is used, the cracks developed may propagate in an uncontrollable manner To deal with this problem, an integrated cracking criterion combining tensile strength and fracture toughness is proposed As postulated by Erdogan and Sih [12], cracks propagation initiates when σθ(2πr)0.5 ≥ KIC, in which σθ is the circumferential stress (the stress perpendicular to the line joining the point being considered to the crack tip), r is the distance from the point being considered to the crack tip, and KIC is a material property called fracture toughness In graphical representation, the stress field near a crack tip in polar coordinates is illustrated in Fig 4, where r and θ are the polar coordinates with the crack tip as the origin Based on this postulation, the criterion for crack propagation (fracture toughness criterion) is denoted as: Vθ t K IC 2πr (3) 624 A.K.H Kwan et al / Procedia Engineering 172 (2017) 620 – 627 Fig Stress field near crack tip in polar coordinates Combining with the tensile strength criterion, the integrated cracking criterion is set as having developed a tensile stress greater than both the tensile strength and the value given by the above equation When r is small, the fracture toughness criterion controls, but when r is greater than ro (value of r when KIC/(2πr)0.5 is equal to the tensile strength), the tensile strength criterion controls This integrated criterion is implemented by using the following procedures The cracked elements are searched so as to identify and locate the crack tip Consider the cracked element shown in Fig 5, a line A-A is drawn through the centroid of the element perpendicular to the crack The cracking status of adjacent elements (defined as those elements having at least one node in common) is scrutinized If all adjacent elements on the left (right) side of A-A are uncracked, then a crack tip exists at the centroid pointing to the left (right) At this juncture, an element is on the left (right) side of a line if its centroid lies on the left (right) side of the line If there are other cracked elements adjacent to the element being considered on both sides of A-A, then the element is on the crack path and it contains no crack tip Fig Identification of crack tip The cracking criteria of the elements are considered element by element For an element under consideration, the values of tensile strength and ro are first determined It is then searched whether there are any crack tips within a distance of ro from the element centroid If there is no crack tip within the distance ro, the tensile strength criterion is used On the other hand, if there are crack tips within the distance ro, the value of r is determined with respect to the closest crack tip and the fracture toughness criterion per Eq (3) is applied using this value of r Crack Queuing Algorithm In nonlinear analysis, the loading is applied in discrete increments At one increment step, multiple elements may develop stresses in excess of the cracking criterion and become cracked If all these elements are allowed to crack simultaneously, this would be against the reality because upon the formation of a crack, there would be stress redistribution and the tensile stress perpendicular to the crack would be relieved thereby eluding the formation of A.K.H Kwan et al / Procedia Engineering 172 (2017) 620 – 627 other cracks in the vicinity The stress redistribution has to be accounted for by re-analysing the element stresses before allowing any other cracks to be formed Otherwise, false cracks caused by numerical instability would appear upon the exertion of a load increment, and crack analysis results would be erroneous [13] To overcome this problem, the crack queuing algorithm is introduced The implementation of the crack queuing algorithm is as follows In a loading step, the concrete element with the highest tensile stress to tensile strength ratio is searched If the ratio is less than 1.0, no new crack is formed and the analysis can proceed to the next load increment step Otherwise, if the ratio is equal to or higher than 1.0, only this concrete element is allowed to crack, and its stiffness matrix is adjusted to reflect the crack formation Then, the whole structure is re-analysed at the same loading level so as to allow the stress redistribution to take place Having finished the re-analysis, the structure is checked again to see whether there are still elements whose stress/strength ratios reach 1.0 or above If there are any such elements, then as before, only the element with the highest stress/strength ratio is allowed to crack This process is repeated until all elements have stress/strength ratio less than 1.0 Only after this process is completed would the analysis proceed to the next load increment step By way of the crack queuing algorithm, the elements whose stresses exceed the cracking limits are queued up and only allowed to crack one after the other The queue changes after each re-analysis because some elements in the queue (elements adjacent to cracks but not in front of crack tips) may become disqualified due to stress relief while other elements may jump into the queue (elements in front of crack tips) due to propagation of crack tips towards them This phenomenon reflects what actually happens and if it is not properly allowed for, the crack pattern produced would not be realistic Numerical Example A plain concrete specimen of size 150 mm by 150 mm subjected to uniaxial tension is analysed numerically The specimen is supported at the bottom edge and loaded by prescribed vertical displacement applied at the top edge In the analysis, the value of KIC is taken from Carpinteri [14] and other material properties are taken from Wang [6] Fig displays the various stages of analysis, which shows the crack propagation process in concrete (the deformations are magnified for ease of visualisation) It is noted that the cracks are initiated at the interfacial transition zone Upon the formation of the first crack, the stress-strain curve starts to descend Therefore, the strainsoftening of concrete is closely related to the crack propagation process Besides, it is observed that the cracks formed are localised within a narrow band, which indicates the strain localisation within a short gauge length This could be more clearly visualised by comparing the axial strains over two different gauge lengths The stress-strain curves obtained using gauge lengths of 50 mm and 150 mm are plotted in Fig These two curves reveal the size dependence in numerical analysis Before cracking, the strains measured by the two gauges are almost identical, but after cracking, the shorter gauge yields a higher strain value while the longer gauge yields a lower strain value 625 626 A.K.H Kwan et al / Procedia Engineering 172 (2017) 620 – 627 (a) 9.63 μm (b) 10.50 μm (c) 11.18 μm (d) 11.73 μm (e) 16.97 μm (f) 22.67 μm Fig Analysis results of crack propagation in concrete Fig Stress-strain curve of concrete specimen under tension A.K.H Kwan et al / Procedia Engineering 172 (2017) 620 – 627 627 Conclusions A nonlinear finite element method for mesoscopic analysis of crack propagation in concrete has been developed The concrete is modelled as a composite material comprising the coarse aggregate, hardened mortar matrix and interfacial transition zones between coarse aggregate and mortar matrix A nonlinear orthotropic material model based on equivalent uniaxial strain is used for simulating the biaxial behaviour The interfacial transition zone is simulated by an interface element Two problems associated with crack analyses have been successfully tackled First, an integrated cracking criterion combining tensile strength and fracture toughness is proposed to overcome the crack tip instability due to the use of very fine meshes Second, the crack queuing algorithm is employed to cater for the stress redistribution and stress relief during crack propagation The method has been applied to analysing concrete subjected to uniaxial tension The results have revealed the following phenomena: (1) the cracks are initiated at interfacial transition zone; (2) the tension capacity of concrete starts to decrease upon the formation of first crack; and (3) due to strain localisation, the descending branch of stress-strain curve is dependent on the gauge length References [1] ASCE, Finite Element Analysis of Reinforced Concrete, ASCE/ACI Committee 447, American Society of Civil Engineers, New York, 1991 [2] Z.P 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