fracture modeling using a microstructural mechanics approach––ii. finite element analysis

Fracture modeling using a microstructuralmechanics approach––II. Finite element analysis

C.S. Chang a,*, T.K. Wang b, L.J. Sluys c, J.G.M. van Mier c

a Department of Civil Engineering, University of Massachusetts, 253 Marston Hall, Box 35205, Amherst, MA 01003-5205, USAb Department of Structural and Foundation Engineering, Polytechnic School of the University of Sao Paulo, Sao Paulo, Brazil

c Faculty of Civil Engineering and Geosciences, Delft University of Technology, Delft, The Netherlands

Received 22 March 2002; accepted 16 April 2002

Abstract

In the accompanying paper [C.S. Chang, T.K. Wang, L.J. Sluys, J.G.M. van Mier, J. Eng. Fract. Mech. (this

volume)], the theoretical aspects of a stress–strain model are described based on a microstructural approach. A finite

element formulation was presented that incorporates the developed stress–strain relationship. In this paper the results

of finite element analyses are described. The paper is focused on two main issues: the difference between the micropolar

and the non-polar microstructural model, and the performance of the model respect to mesh size dependence. Speci-

mens in uniaxial tension test and in biaxial tension–shear tests have been analyzed. To evaluate the applicability of this

method, the finite element results are also compared with measured experimental results.

� 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Fracture; Damage mechanics; Concrete; Microstructural model; Stress–strain relationship; Finite element method

1. Introduction

In the accompanying paper [1], we have proposed the theory and formulation of a microstructural modelfor the analysis of fractures in concrete. In this paper, we evaluate the model performance by examining thepredicted finite element results for specimens under various types of loading conditions.In the finite element analysis, we investigate the following aspects:

(1) The effect of the micropolar modeling (i.e., moment transmitting in the material) on the predicted mechan-ical behavior: This can be investigated by comparing finite element results obtained from the non-polarmicrostructural model and the micropolar microstructural model derived in the accompanying paper.

(2) Sensitivity of mesh size: It has been well recognized that mesh size dependency is a serious problem forfinite element analyses of fracture under strain softening conditions. The problem, however, can be

Engineering Fracture Mechanics 69 (2002) 1959–1976

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*Corresponding author. Tel.: +1-413-545-5401; fax: +1-413-545-4525.

E-mail address: [email protected] (C.S. Chang).

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avoided using the models containing an internal length scale, such as strain gradient or non-local mod-els [2–4]. The present microstructural model is expected not to suffer this problem, since it utilizes anunderlying lattice structure. To investigate the mesh sensitivity of the present model, two types of lab-oratory tests were analyzed by the developed finite element method: a uniaxial tension test and a shearbox test, which correspond to a case of mode-I failure and a case of mode-II failure, respectively.

(3) Effect of the underlying structure: In this approach, an underlying structure (i.e., the orientation and sizeof lattice) is specified for each element. For simplicity, we randomly assigned lattice orientations in ele-ments to reflect the randomly arranged aggregates. In this paper, we investigate the effect of lattice ori-entation on the anisotropic strength of the material, and the variation of the strength for samples withrandomly assigned lattice orientations.

To evaluate the performance of this approach, we also compare the model prediction and experimentalresults for specimens with double notches [5], and for specimens under a biaxial tension–shear loadingcondition [6].

2. Finite element analyses for uniaxial tensile tests

2.1. Non-polar microstructural model

Consider a rectangular bar of concrete (width ¼ 2 mm, height ¼ 46:765 mm, and thickness ¼ 1 mm). Anunderlying lattice structure is assumed to be a horizontally aligned hexagonal network (see Fig. 7 in [1]).The elastic properties for the concrete are given as follows: Young’s modulus E ¼ 21220 MPa and Pois-son’s ratio m ¼ 0:15. Based on the elastic properties, we can determine the length-to-depth ratio of latticebeam lint=h ¼ 0:6916 and the lattice beam modulus Eb ¼ 20842 MPa (see Eqs. (40) and (41) and Fig. 3 in[1]). It is noted that, for the non-polar model [1], the stress–strain relationship depends only on the ratiolint=h. Therefore, the continuum can be represented by an underlying lattice structure with any value of lint,as long as the value of h is proportionally changed to keep the ratio lint=h ¼ 0:6916. For the case of lint ¼ 1mm, the value of h is 1.446 mm.For the derived micropolar microstructural model [1], the stress–strain relationship depends on the

length of lattice beam lint, which can be determined from the bending modulus of concrete. For lint ¼ 1 mm,the bending modulus j ¼ 6090 MPamm2 (see Eq. (42) in [1]).This bar is analyzed by the finite element method formulated in the accompanying paper [1] using three

different mesh sizes (l ¼ 1 mm, l ¼ 1:5 mm and l ¼ 2 mm) as shown in Fig. 1, where l is the length of one

Fig. 1. Three finite element mesh sizes (a) l ¼ 1 mm, (b) l ¼ 1:5 mm and (c) l ¼ 2 mm.

1960 C.S. Chang et al. / Engineering Fracture Mechanics 69 (2002) 1959–1976

side of the triangular element. For the purpose of studying mesh size effect, we make the mesh configurationsimilar to the horizontally aligned hexagonal network so that all elements are of same size. In this way, themesh layout in Fig. 1a is identical to its underlying lattice structure. In the other two meshes (Fig. 1b and c),the element sizes are larger than the size of underlying lattice.In this analysis, only the tensile fracture criterion is considered for the lattice beams (see Eq. (31) in [1]).

We assume that the tensile strength of the lattice beam ft ¼ 4:16 MPa. Two weak zones representingmaterial imperfections were assigned to elements in the middle of the bar as marked in Fig. 1. The weakzones have a reduced tensile strength of the lattice beam f 0

t ¼ 3:16 MPa.The computed contours of vertical tensile strain for the bar with mesh size l ¼ 2 mm is shown in Fig. 2

for three stages of deformation, corresponding to the bar stretch u ¼ 0:01930, 0.01933, and 0.01960 mm.The vertical strain contour indicates that fracture starts to develop from the weak zones on the sides (seeFig. 2a) and then extend toward the center of the specimen (see Fig. 2b). Finally the small cracks coalesceinto a major horizontal crack and strain localization occurs in a zone of finite width (see Fig. 2c).The crack patterns are shown in Fig. 3 when the cracks coalesce for the three mesh sizes. It is noted that

at the stages shown in Fig. 2a and b, small cracks in the tensile zone continue to develop. However, atu ¼ 0:0196 mm, a major horizontal crack has been formed such that the bar has no more tensile bearing

Fig. 2. Evolution of vertical strain contours for l ¼ 2 mm at three stages of fracture development (a) from the weak zones on sides,(b) extend toward the center of specimen and (c) coalesce into a major horizontal crack.

Fig. 3. Crack patterns for three mesh sizes at u ¼ 0:0196 mm.

C.S. Chang et al. / Engineering Fracture Mechanics 69 (2002) 1959–1976 1961

capacity, and no additional cracks develop. Thus the width of the crack band will not extend further andthe crack bands shown in Fig. 3 have their maximum widths.The evolution of strain distribution along the longitudinal section of the specimen is shown in Fig. 4 for

a sequence of six loading stages. The displacements at the end of bar corresponding to the six stages are:0.01930, 0.01930, 0.01931, 0.01933, 0.01945, and 0.01960 mm.

2.1.1. Mesh size independencyFor the three different mesh sizes, we compare the computed deformation to show the effect of mesh size.

Fig. 5 is a typical plot of strain distributions along the longitudinal section for three different meshes whenthe bar is stretched to u ¼ 0:01933 mm. The comparison shows that the results are nearly identical for thethree mesh sizes.To further demonstrate mesh size independency, we compare the applied load versus relative displace-

ment curves obtained from analyses with the three different meshes. In each curve, the relative displacementis computed based on the selected two reference points on the specimen, where the distance between the tworeference points is termed as ‘‘reference length’’. To further underline mesh size independency, we compare

Fig. 4. Evolution of strain distribution in longitudinal direction of specimen for l ¼ 2 mm.

Fig. 5. Comparison of strain distribution for the three mesh sizes.


the load–displacement curves for four reference lengths: 7, 14, 21 and 46 mm (note: 46 mm is the full length).The four load–displacement curves are converted into the nominal stress–nominal strain curves and plottedin Fig. 6.It is noted that the stress–strain curves of the bar in Fig. 6 are dependent on the selected reference

lengths. When the bar is initially stretched, it behaves elastically and the strain is uniform throughout thebar. Therefore, in the elastic range, the stress–strain curves for all of the four reference lengths are identical.After the peak stress is reached, cracks develop at the middle section of the bar and the strain distributionbecomes non-uniform. Therefore, it is expected that the post-peak portion of the stress–strain curves aredifferent, depending on the selected reference lengths. The two short reference lengths (i.e., z ¼ 7 mm andz ¼ 14 mm) are within the zone of localization (about 14 mm), in which the strain is relatively uniform.Therefore, the post-peak curves are nearly the same for these two cases. As the reference length increases, itcovers portions of undamaged elastic material. Thus the brittleness of the post-peak curve increases rapidlywith reference length. However, for a specific reference length, Fig. 6 indicates that the stress–strain curvesfor three different mesh sizes are identical. These agreements further confirm the character of mesh sizeindependency.

2.1.2. Localization bandAccording to both Figs. 2 and 3, it can be observed that the band of the strain localization is wide

compared to the narrow strain localization band obtained from the analyses using conventional damage orsmeared crack models (see e.g. [7]). In the present finite element analyses, the width of strain localization is�14 mm, which is independent of the element mesh size used. However, in finite element analyses usingconventional damage or smeared crack models, the width of the localized zone is usually equivalent to thesize of a single element. Thus the amount of energy release due to fracture depends on the element size,which does not reflect physical reality.The characteristics of strain localization of the present model is similar to that of strain gradient models

[3,4], non-local models [2,8] or the rate-dependent material [9]. In these so-called higher-order models, an‘‘internal length’’ is involved in the constitutive relationship. Although the stress–strain relationship of thepresent non-polar model does not explicitly contain a length scale, the underlying lattice structure used inthe present model offers the same regularization effect, thus avoiding the problem of mesh size dependency.When a beam in the underlying lattice breaks, the stiffness tensor is degraded and becomes anisotropic withrotated material axes (see Eqs. (38) and (39) in [1]). Due to the redistribution of stresses, beams inneighboring finite elements will break. This causes a smearing effect, which leads to the extension of the

Fig. 6. Stress–strain curves plotted for four reference lengths using the results from three finite element meshes.


crack band over a finite width (and a finite member of elements). The length scale in the continuum for-mulation comes from the underlying lattice structure.

2.1.3. Effect of underlying lattice alignmentAs previously shown in Fig. 7 [1], a hexagonal lattice network has two principal alignments: horizontal

alignment and vertical alignment. For the analyses shown in Section 2.1, we have specified both the un-derlying lattice structure and the mesh layout to be horizontally aligned. It is noted that the computedresults were almost identical when we change the mesh layout to be vertically aligned while the underlyingstructure was remained to be horizontally aligned. The same results are expected because the materialmicrostructures are the same for the two specimens.In this section, we change the microstructure of the specimens by specifying the underlying lattice

structure to be vertically aligned and evaluate its effect on the deformation behavior. It is noted here thatthe meshes used for these analyses are the same as that given in Fig. 1 (l ¼ 1 mm, l ¼ 1:5 mm and l ¼ 2mm), and all material parameters are the same as those used in Section 2.1.The computed contours of vertical strain for mesh size l ¼ 1 mm are shown in Fig. 7 for six stages,

corresponding to a bar stretch u ¼ 0:015, 0.017, 0.019, 0.021, 0.023, and 0.025 mm. At the first three stages,the tensile strain starts to develop from the weak zones at the sides of specimen and gradually form adamaged zone across the mid-section of the bar. Different from the previous case of horizontally alignedlattice structure, the strain localization does not occur at this point because the damaged zone still retainspartial load bearing capacity due to the vertical lattice configuration. At the fourth and fifth stages, thecrack band continues to increase in size and finally, at the six stage, the small cracks coalescence into twomajor curved cracks across the mid-section of the bar, which can be observed from the two curved zones ofstrain localization.The difference in crack patterns in the specimens with vertically and horizontally aligned lattice struc-

tures can be explained by a schematic illustration in Fig. 8. In the case of horizontal alignment, the cracks

Fig. 7. Evolution of vertical strain contours in longitudinal direction for l ¼ 1 mm at six stages of stretch displacement u: (1) 0.015 mm,(2) 0.017 mm, (3) 0.019 mm, (4) 0.021 mm, (5) 0.023 mm, and (6) 0.025 mm.


extend easier in horizontal direction across the specimen whereas in the case of vertical alignment, theextension of cracks has obstacles in horizontal direction and the curved crack path is easier accessible.The mesh size independency is also observed in this case. It can be seen in Fig. 9 that shows the com-

parison of strain distributions at u ¼ 0:025 mm. According to Fig. 9, the width of the crack band is ap-proximately 18 mm, which is wider than that of the previous case. The distance of the two peaks isapproximately 9 mm that represent the distance between the two curved crack bands.Besides the differences in failure pattern, the computed tensile strengths are also different, �14% higher

for the vertically aligned lattice. It is noted that the two lattice alignments represent two extreme cases. Forother orientations of lattice alignment, the tensile strength is expected to be bounded between the two cases.Until now, the underlying lattice structure in specimens, considered in the analyses, is aligned in the same

direction for all elements, thus representing a simple idealized microstructure. The lattice for each elementmay be randomly oriented to represent randomly arranged aggregates. This case will be considered inSections 2.3 and 3.3 on modeling specimens for laboratory experiments.

2.2. Micropolar microstructural model

The finite element analyses for the uniaxial tension tests discussed above were carried out with the non-polar microstructural model. We have repeated the analyses using the micropolar microstructural model. Itis interesting to note that the computed results obtained from the micropolar model are nearly identical tothe results obtained from the non-polar model. This indicates that the rotation degrees of freedom and thecorresponding moment transmitting in the medium do not play an important role in the analyses under a

Fig. 8. Illustration of crack pattern formation with different lattice alignments.

Fig. 9. Comparison of strain distribution for the three mesh sizes at final stage of loading.


uniaxial tensile load. It is also consistent with the observation that the computed rotations and moments atnodal points are negligible in the analyses using the micropolar model.As indicated for example by Muhlhaus and Vardolakis [10] and de Borst and Sluys [11], the classical

micropolar models can only achieve mesh size independence for mode-II (shear) but not for mode-I (tensile)problems. This is because the rotational degrees of freedom are not activated. The present analyses showthat the microstructural approach can achieve mesh size independence for mode-I problems, and this can bedone with or without a length parameter explicitly revealed in the stiffness tensor.

2.3. Uniaxial tensile test on double notched specimens

Here, we simulate the experiments of uniaxial tensile tests on double-notched specimens carried out atDelft University of Technology [5]. The dimensions of the specimen are 60� 120� 10 mm3 with a notch of10� 2 mm2. Three finite element meshes are made for the double-notched specimen with different meshsizes (872 elements, 1206 elements, and 1664 elements). We will compare the results obtained from the threemeshes to evaluate the influence of mesh size. The fourth mesh with 1608 elements was made for a specimenwith a 15 cm off-centered distance between the two notches (see Fig. 10).The elastic material properties are as follows: Young’s modulus E ¼ 21220 MPa and Poisson’s ratio

m ¼ 0:15. Based on the elastic properties, we can determine the length-to-depth ratio of the lattice beamlint=h ¼ 0:6916 and the lattice beam modulus Eb ¼ 20842 MPa (see Eqs. (40) and (41) and Fig. 3 in [1]). Inthis analysis, only tensile failure mode is considered in the fracture criterion for lattice beams (see Eq. (31)in [1]). The tensile strength of the lattice beam ft ¼ 4:16 MPa.Since the non-polar model and the micropolar model predict nearly identical results for this case, we

present only the finite element results for the non-polar model. The lattice beam length lint ¼ 1 mm, andbeam depth h ¼ 1:446 mm for the underlying lattice structure. In order to better simulate the randommicrostructure of concrete, the lattice orientation is assigned randomly to each element in the finite elementmesh.The crack patterns of the four specimens obtained from the finite element analyses are shown in Fig. 10.

The three meshes are corresponding to (a) 872, (b) 1206 and (c) 1664 elements, in which the three specimenshave different underlying random lattice structure. Due to the differences in lattice structures, analyses ofthe three meshes show three different crack patterns: the up-curved crack, the straight crack and the s-shapecrack. All three patterns resemble observed crack patterns in experiments. In Fig. 10 for the specimen withoff-centered notches, the crack pattern shows a bridge of two cracks developed from both sides. This crackpattern is similar to that observed in the experiment by Shi et al. [5].

Fig. 10. Comparison of crack zones for four finite element meshes. The vertical distance between the notches is 0 mm in (a)–(c) and 15

mm in (d).


In Fig. 11, the load–displacement curves for the three finite element analyses are plotted and comparedwith the measured curve from the experiment. The displacement represents the averaged relative verticaldisplacement of two points 30 mm apart across the notches at the left edge and at the right edge of thespecimen. In Fig. 11 small unloading regimes are visible in the experimental curve, which are the spotswhere microscopic scans of the specimen surface were made, and the deformation was held constant for awhile, which led to relaxation. According to Fig. 11, the finite element results show that the computedstrength ranges �7% among the three specimens. The randomly aligned lattice structures are intended tosimulate the different microstructures in physical specimens. The range of predicted strength is comparableto that observed in experimental tests.

3. Finite element analyses for shear box tests

Now, we evaluate the model performance for the simulation of specimens in a shear box test where theshear crack is the predominant failure mode. Fig. 12 shows the configuration of a shear box used for thetest. The specimen is placed inside two half-boxes. The lower half-box is fixed. The shear force Ps was

Fig. 11. Load–displacement curves for double-notched specimen obtained from three finite element meshes.

Fig. 12. Boundary and loading conditions for the shear box test.


applied to the upper left frame of the box using a displacement control method, while the top frame of thebox remained horizontal with zero vertical displacement.To evaluate the performance of this model, the specimens (100� 100� 1 mm3) with three different mesh

sizes were considered (l ¼ 6:67 mm, l ¼ 5 mm and l ¼ 3:33 mm), where l is the side length of the triangularelement. The finite element mesh layout is a hexagonal lattice, which is horizontally aligned. No weakelements were assigned in the mesh.In this shear box analysis, only the shear fracture criterion is considered (see Eq. (32) in [1]). The elastic

material properties for the specimens are as follows: Young’s modulus E ¼ 30000 MPa, Poisson’s ratiom ¼ 0:2, and the shear strength of lattice beam fsh is assumed to be 5 MPa.

3.1. Non-polar microstructural model

First, we investigate the crack behavior using the non-polar microstructural model. Based on the elasticproperties, the corresponding lattice properties are lint=h ¼ 0:5774 and the lattice beam modulusEb ¼ 37500 MPa (see Eqs. (40) and (41) in [1]). As discussed previously for the non-polar model, the stress–strain relationship depends only on the ratio lint=h. For the lattice beam length lint ¼ 10 mm, the depth oflattice beam h ¼ 17:32 mm.Fig. 13 shows the computed crack patterns from finite element analyses for the three different mesh sizes.

Fig. 14 shows the computed load–displacement curves for the three different mesh sizes. Since the materialhas the same underlying lattice structure, we expect the crack patterns to be similar despite of the differentmesh sizes, as observed in Fig. 13. The results of load–displacement curves indicate that the non-polarmodel has almost no sensitivity to the mesh size.

3.2. Micropolar microstructural model

We now repeat the same analyses using the micropolar model for the shear box tests shown in Section3.1. Both the material properties and the underlying lattice structure used in the micropolar model areidentical to those used for the non-polar model. Thus the difference between the two analyses can be at-tributed to the extra rotational degrees of freedom and couple stress considered in the micropolar model.Fig. 15 shows the computed crack patterns for the three mesh sizes using the micropolar model. ComparingFigs. 13 and 15, the crack patterns are very different between the micropolar and non-polar models. In themicropolar model, the cracks are extended diagonally and then curved towards horizontal direction due tothe extra rotational degrees of freedom. It is noted that under mode-I loading conditions, the micropolarmodel and the non-polar model predict nearly identical results. However, under mode-II (shear) load-ing conditions, the micropolar model can capture the crack and deformation behavior, which cannot be

Fig. 13. Comparison of crack patterns of shear box tests for three mesh sizes: (a) l ¼ 6:67 mm, (b) l ¼ 5 mm and (c) l ¼ 3:33 mm usingthe non-polar model.


modeled by the non-polar model. Comparing the load–displacement curves in Figs. 14 and 16 for themicropolar and the non-polar case, the elastic behavior seems less stiff and the post peak behavior morebrittle for the micropolar model, due to the effect of rotational degrees of freedom. The load–displacementcurves for three different mesh sizes, as shown in Figs. 14 and 16, demonstrate that the micropolar modelalso reveals almost no sensitivity to mesh size.

Fig. 14. Comparison of load–displacement curves of shear box tests for three mesh sizes using the non-polar model.

Fig. 15. Comparison of crack patterns of shear box tests for three mesh sizes: (a) l ¼ 6:67 mm, (b) l ¼ 5 mm and (c) l ¼ 3:33 mm(lattice size lint ¼ 10 mm).

Fig. 16. Load–displacement curves of shear box tests for three mesh sizes (lattice size lint ¼ 10 mm).


Under mode-I loading (tension) conditions, such as uniaxial tension, the size of the underlying latticestructure lint does not influence the computed results either in the micropolar model or in non-polar model,as long as the ratio of lint=h is kept constant (see Section 2). However, under mode-II loading conditions,the size of the underlying lattice structure lint is expected to make differences in the computed results fromthe micropolar model, even when the ratio lint=h is kept constant. To investigate this effect, we performanalyses for six specimens with different sizes of underlying lattice structure, i.e., lint ¼ 7:5, 10, 15, 16.5, 18,20 mm. The ratio lint=h for the six lattice structures is fixed at a value 0.5774, such that the Young’smodulus and Poisson’s ratio are identical for the six specimens. The same mesh layout with the length ofeach side of the triangular element l ¼ 10 mm is used for the six specimens.Fig. 17 shows the range of computed crack patterns for the six specimens using the micropolar model.

According to Fig. 17, the computed crack patterns show significant differences. In the case of a specimenwith smaller lattice size, inclined cracks develop and then curve into horizontal direction. For a specimenwith lattice sizes larger than 18 mm, the cracks develop in horizontal direction after the initiation of a shearcrack. These types of curved cracks have also been found by Schlangen [12] and Schlangen and Gaborczi[13] in their analysis for a network of lattice beams.The load–displacement curves of a specimen with lint ¼ 10 mm have already been shown in Fig. 16. In

order to examine the load–displacement behavior for specimens with a large underlying lattice structure,finite element analyses were carried out using lattice beam size lint ¼ 100 mm for three different mesh sizes(l ¼ 6:67 mm, l ¼ 5 mm and l ¼ 3:33 mm). Fig. 18 shows the computed crack patterns and Fig. 19 showsthe load–displacement curves for the three different mesh sizes. From the results shown in Figs. 18 and 19,mesh size independency is again depicted. Comparison of load–displacement curves in Figs. 16 and 19 (forthe cases with lint ¼ 10 mm and lint ¼ 100 mm) shows that, for lattice size lint ¼ 100 mm, the strength isslighter higher and the post-peak behavior is slightly more brittle.

3.3. Comparison with biaxial tension–shear tests

Fig. 20 schematically shows the configuration of the biaxial tension–shear test device used for the ex-periments performed by Nooru-Mohamed [6] and Nooru-Mohamed et al. [14]. The double-edge-notchedspecimen was placed in a special loading frame to allow for the application of various loading paths using

Fig. 17. Crack patterns of shear box tests for lattice sizes ranging from 7.5 to 20 mm.


Fig. 18. Crack patterns of shear box tests for three mesh sizes: (a) l ¼ 6:67 mm, (b) l ¼ 5 mm and (c) l ¼ 3:33 mm (lattice size lint ¼ 100mm).

Fig. 19. Load–displacement curves of shear box tests for three mesh sizes (lattice size lint ¼ 100 mm).

Fig. 20. (a) Configuration of the biaxially loaded double-edge-notched specimen, and (b) the applied global shear force Ps.


either a force or a displacement control method. The behavior was measured for specimens under acombined shear and tensile load.Attention is focused on the largest specimen used in the experiments, for which the box height L ¼ 200

mm, the thickness t ¼ 50 mm and the size of the two symmetrical notches is 25� 5 mm2. The specimen issupported at the bottom and along the right-hand side below the notch. The shear force Ps was first appliedthrough the frame above the notch along the left-hand side of the specimen. Then the tensile force P wasapplied at the top (the frame was glued to the specimen). The relative tensile displacement was measuredbetween the points A and A0 on the left side as well as the points B and B0 on the right side of the shear box(see Fig. 20). The specimen is pre-sheared before failed in tension; therefore, the tensile behavior no longerplays a dominating role.Numerical simulations using the present model were made for two specific experiments conducted by

Nooru-Mohamed [6] corresponding to two values of the pre-sheared force: Ps ¼ 5 kN and Ps ¼ 10 kN. Inboth numerical simulations, the shear force Ps is applied using a force control method and then keptconstant while the normal tensile loading is imposed using a displacement control method, which corre-sponds the loading path 4 from the experiments by Nooru-Mohamed [6]. The specimens used in theseexperiments were made from normal weight concrete. The elastic material parameters used in the numericalsimulation are as follow: Young’s modulus E ¼ 15000 MPa, and Poisson’s ratio m ¼ 0:2. Based on theelastic properties, the corresponding lattice properties are: lint=h ¼ 0:5774 and the lattice beam modulusEb ¼ 18750 MPa (see Eqs. (40) and (41) in [1]). In this shear box analysis, the fracture criterion used is givenin Eq. (30) in [1], which considers both failure modes due to tensile and shear loading. The tensile strengthof the lattice beam ft ¼ 4:65 MPa, and the shear strength of the lattice beam fsh ¼ 2:32 MPa.Fig. 21 shows the crack patterns observed from the biaxial tension–shear experiments by Nooru-Mo-

hamed [6] for the two cases of shear loads. According to the experiments, the specimen with Ps ¼ 5 kNreveals a crack pattern similar to that in uniaxial tension tests. The cracks initiate at the notched area on thesides and then develop towards the center of the specimen in horizontal direction. For the specimen with

Fig. 21. Crack patterns observed from experiments: (a) Ps ¼ 5 kN and (b) Ps ¼ 10 kN.


Ps ¼ 10 kN, the crack pattern is different. The cracks initiate at the notched area but develop along aninclined direction. Then the cracks extend and curve towards a horizontal direction.The finite element mesh is shown in Fig. 22. For the nodal points at the top and the bottom, the dis-

placements are fixed in vertical direction and the rotations are fixed. The forces corresponding to Ps areapplied on the nodal points at the lower-right side and at the upper-left side of the box. The orientation ofthe underlying lattice for each element is randomly assigned. For the non-polar model, the computed crackpatterns for both Ps ¼ 5 and 10 kN are horizontal cracks similar to what was observed in uniaxial tensiletests. The non-polar model fails to predict the difference in crack patterns due to the magnitude of the pre-shear load as observed in experiments. This is expected since the non-polar model is not capable of pre-dicting crack patterns for mode-II (shear) conditions as discussed in Section 3.1.On the other hand, the analyses from the micropolar model are expected to depict the different crack

patterns. In the micropolar model, the size of the underlying lattice lint can be determined from the bendingmodulus of the concrete. Unfortunately, the experimentally measured bending modulus was not available.We performed the analyses with various values of lint, the computed crack patterns and load–displacementcurves were found to be very similar for lint < 10 mm. In this range of values for lint, the computed crackpatterns are in reasonable agreement with the experimental results. Therefore the results for lint ¼ 10 mmare presented here.Fig. 23 shows the crack patterns computed from the micropolar model. The predicted horizontal cracks

and curved cracks for the two different pre-sheared loads are similar to the measured crack patterns. Thisdemonstrates that the added micropolar rotations and the ability of transmitting couple stresses are crucialto simulate the correct fracture mechanism.Fig. 24 shows a comparison between the experimentally measured curves and the predicted load–

displacement curves obtained from the micropolar model. According to the experimental results

Fig. 22. Finite element mesh used for the double-edge-notched specimen.

Fig. 23. Predicted crack patterns obtained from the micropolar model: (a) Ps ¼ 5 kN, (b) Ps ¼ 10 kN.


(Nooru-Mohamed, [6]), the peak axial tensile capacity was 15 kN for the case of low pre-sheared load (i.e.,Ps ¼ 5 kN). The tensile capacity is reduced to 10.5 kN (a 30% decrease) for the case of high pre-sheared load(i.e., Ps ¼ 10 kN). According to the finite element analyses with the micropolar model (see Fig. 24), thedifference in axial tensile capacity for the two pre-sheared load cases are predicted. Nooru-Mohamedjustified this result by the fact that the presence of a pre-sheared load causes significant damage, which inturn reduces the axial tensile capacity of the specimen. In the analyses with the micropolar model, thepresence of moments substantially increases the shear load on the lattice beams, thus causing a significantdamage to the material and results in a reduction of the peak load. The values of shear strength for a latticebeam fsh in the fracture criterion also have a significant effect. If the value of fsh is increased from 2.32 to4.65 MPa, the analyses show that the difference in axial tensile capacity between the two cases of pre-sheared load was reduced from 30% to 20%.In Fig. 24, the descending branch of the computed load–displacement curve shows more brittle behavior

than that of the measured curve. The same trend is also observed between the computed and measuredload–displacement curves for uniaxial tensile tests (see Fig. 11). Thus the loss of load bearing capacity withrespect to deformation after material damage should be a more gradual process than that predicted by themodel. It is noted that the fracture behavior of a lattice beam in the present model is assumed to be purelybrittle; the lattice beam is either undamaged or completely fractured with zero bearing capacity. If thelattice beam were allowed to damage gradually, the predicted after-peak curve would fit better to themeasured curve.The underlying lattice structure used in the model is only a simple regular hexagonal lattice structure,

which is regarded as the first approximation of the real complex microstructure. The purpose of this study israther to evaluate the approach of modeling a continuum with an underlying structure, and to examine thekey features that can be captured by such a model with the simplest underlying structure. It is interesting tonote that even with the simplest underlying structure, the model shows element size independency. Fur-thermore, realistic deformation behavior and crack patterns can be simulated.

4. Summary and conclusion

The results of finite element analyses for concrete specimens under different loading conditions have beenpresented. The finite element analyses are based on the derived stress–strain relationships for a continuumwith underlying lattice structure [1]. In the finite element analyses, we investigate the following aspects:

Fig. 24. Comparison of load–displacement curves between experimental data and numerical simulation using the micropolar model

(lint ¼ 10 mm).


(1) The effect of the couple stresses in the micropolar model: For uniaxial tension tests, the non-polarmodel and the micropolar model give nearly identical results because under mode-I conditions, the rota-tional degrees of freedom are not activated. However, when loading paths involving shear (mode-IIconditions) are considered, such as the specimen in a shear box test, the crack pattern obtained from thenon-polar model becomes very different from that of the micropolar model. In the case of a micropolarmodel, the crack pattern of the specimen is significantly influenced by the size of the underlying latticestructure. Furthermore, the analysis of the biaxial tension–shear experiments show that the micropolarmodel is capable of capturing the crack patterns and load–displacement behavior as observed in the test.The non-polar model only gives reliable results when shear is not a dominating factor.(2) Sensitivity of mesh size: The microstructural consideration can be regarded as a useful regularization

technique leading to mesh size independency. Based on the proposed microstructural model, the characterof mesh size independency has been observed in the finite element analyses on mode-I loading condition(uniaxial tension tests) and mode-II loading condition (shear box tests). The present microstructural modelpredicts an interesting crack behavior––a wide band of strain localization as opposed to the narrow, meshdependent band predicted in the classical damage models. The characteristics with respect to strain lo-calization of the present model are similar to those of strain gradient models [3,4], non-local models [2,8]and rate-dependent models [9]. Due to the rotation of material axes after damage, cracks can extend toneighboring finite elements and form a crack band over a finite width, which is a function of the size of theunderlying lattice structure.(3) Effect of the underlying structure: One of the interesting results is that the predicted crack pattern and

tensile capacity of a specimen are influenced by the orientation of its underlying lattice structure. It may beviewed that the randomly aligned lattice structure indirectly represents the arrangement of aggregates inconcrete. For the analyses of three specimens with random lattice structures in uniaxial tensile tests, weobserved that the range of strengths varies by approximately 7%, which is consistent with the range ofmeasured strength in experiments as a result of the microstructure variation in physical specimens.Crack patterns can also be affected by the size of the underlying lattice structure for specimens in shear

box tests. The predicted crack patterns vary from straight horizontal cracks to diagonally curved cracks fordifferent sizes of underlying lattice. These types of curved cracks have also been found by Schlangen [12]and Schlangen and Gaborczi [13] in their numerical simulations and experimentally by Nooru-Mohamed[6].Although the underlying lattice structure used in the model can only be regarded as the first approxi-

mation of a real complex microstructure, the microstructural model shows potential advantages in theanalyses of deformation and fracture. The most notable advantage is that the model results in mesh sizeindependency. And despite the simple microstructure assumed in the model, the finite element analyses cancapture the experimentally measured crack patterns and deformation behavior of specimens under varioustypes of loading.

References

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fracture modeling using a microstructural mechanics approach––ii. finite element analysis

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