Governing equations

Dierent shape functions can be used for the interpolation than for the approximation of the displacement. Linear shape functions are particularly useful, as the discontinuity is a straight line in the reference element, and also when mapped in the real element.

This simplies the integration signicantly, as an element sub-division algorithm is more easily constructed, than for a curved discontinuity. However, linear shape functions only allows for a piecewise linear representation of discontinuities. Bi-linear shape functions produce curved discontinuities, and an ecient element sub-division algorithm is dicult to construct. For this reason, if bi-linear or higher order shape functions are used, linearization of the discontinuity is often used. The discontinuity is linearized by drawing a straight line between the points on the element edges that are cut by the discontinuity, see gure 4.11.

Figure 4.11. (a) A curved interface in a bi-linear element can be (b) linearized by neglecting the curvature of the interface or (c) the element is decomposed into two triangles and linear interpolation is assumed.

The nodal values of the level set function are interpolated using the same shape functions as in the approximation of the displacement. However, it is unknown whether a linearization of the interface is performed. In either case, when the sub-division has been performed, standard Gauss quadrature is adopted. The element decomposition approach is favorable in computational implementation, because existing nite element integration schemes do not need any modication. In the following section the weak form of the equilibrium equations are presented. Systèmes [2010]

4.9 Governing equations

The weak form of the equilibrium equations, the displacement ¯u being the primary variable, can be stated by the principle of virtual work, see equation 4.21.

Z

Group B122b - Spring 2011 4. The eXtended Finite Element Method

Ω The domain of integration.

Γt The traction surface.

 Strain tensor in vector form.

σ Stress tensor in vector form.

b Prescribed body force vector.

tˆ Prescribed traction vector.

where superscript T refers to the transpose. Prescribed displacements u are imposed on Γ_u, while tractions ˆtare imposed on Γt. The internal boundary of the crack, Γc, is assumed to be traction-free. The domain Ω is bounded by Γ = Γt∪ Γ_c∪ Γ_u, as shown in gure 4.12.

Figure 4.12. Domain Ω supported on Γu and loaded on Γt. An internal crack is dened along the boundary Γc.

σ · n = ˆt on Γt

σ · n = 0 on Γc

u = ˆu on Γu

The two-dimensional nite element discretized form of the weak form is stated in 4.22.

Z and vertical direction, respectively. ft contains the tractions tx and ty in the horizontal 42

4.9. Governing equations Master Thesis

and vertical direction, respectively. D^ep is the elasto-plastic constitutive matrix chosen according to the adopted material model of the concrete, see chapter 5. The strain distribution matrix B is given by equation 4.23.

B =







∂xN_std^T ∂xN_enr^T

∂yN_std^T ∂yN_enr^T

∂_yN_std^T ∂_xN_std^T ∂_yN_enr^T ∂_xN_enr^T





 (4.23)

where N_std^T are standard FEM shape functions, and N_enr^T are the local enrichment functions given by equation 4.4. The strain distribution matrix is of dimension 3×2·(nel+ n^∗_el), and n_el and n^∗_el are the numbers of element nodes and enriched element nodes, respectively.

Discontinuous modeling in

Abaqus 5

This chapter presents a summary of the methods described in chapters 2-4, used in Abaqus 6.10 in relation to discontinuous modeling. With regard to the methods of discontinuous modeling, preliminary choices made in Abaqus 6.10 for this report regarding the element type used for the numerical discretization and the material properties of the examined concrete and steel are presented. The choices are common for the benchmark tests considering concrete and the reinforced concrete beam considered in the problem formulation of this project.

In Abaqus 6.10 cracking in concrete and steel is modeled in a discrete fashion, that is, a strong discontinuity is introduced in the displacement eld by the Phantom-node method, in which the displacement jump is reproduced by introducing the jump function. For the remainder of this report the Phantom-node method will be referred to as the XFEM because the methods are equivalent. The crack is represented as an open interface, but since the crack is always extended to the boundary of the element, in which it is present, only one set function is used for the topological description of a crack. The level-set function is chosen as the signed-distance function and is interpolated using the same interpolation functions as the approximation of the displacement. The crack is modeled by inserting a cohesive segment in the cracked element. The cohesive segments method is based on the ctitious crack model by A. Hillerborg and Peterson [1976], that is, non-linear fracture mechanics is used. The adopted crack initiation criterion is the maximum principal stress criterion, in which a crack is initiated if the maximum principal stress reaches the tensile strength of the concrete. The crack propagates perpendicular to the direction of the maximum principal stress. The evolution of the crack is governed by the fracture energy, which represents the tension-softening behavior of the concrete during cracking. The relationship between the crack opening displacement and the closing stresses acting on the crack is linear, in which case the critical crack opening displacement, wc, is

1/2 · f_t· w_max= G_f ⇔ w_max = 2 · G_f/f_t (5.1)

Group B122b - Spring 2011 5. Discontinuous modeling in Abaqus

Figure 5.1. Linear relationship between the closing stress and crack opening displacement upon cracking.

The modeling of cracked concrete is illustrated in gure 5.2.

Figure 5.2. The modeling concept used in Abaqus 6.10 to model cracks in concrete.

The constitutive behavior of concrete is modeled using the so-called Concrete Damaged Plasticity material model, abbreviated CDP. In tension the CDP material model is used in coordination with the XFEM and the cohesive segments method, that is, CDP is used until tensile crack initiation is detected, at which point a cohesive segment is inserted and the XFEM is activated. For compression the CDP material model is used without the XFEM. The CDP material model is a continuum-damage based constitutive relation. The stress-strain relation is given by equation 5.2.

σ = (1 − d)D₀^el: ( − ^pl) = D^el: ( − ^pl) (5.2) 46

Master Thesis

D^el₀ Initial, undamaged elastic stiness matrix of concrete.

D^el Degraded elastic stiness matrix of concrete.

d Scalar degradation variable.

The scalar degradation variable has an initial value of zero for intact material and increases towards one for complete loss of material stiness. Based on the observation that concrete is anisotropic, the scalar degradation variable is dierent in tension and compression. Since the scalar degradation variable is not used for tension, it will not be further discussed in this report. The compressive scalar degradation variable is determined according to equation 5.3.

d_c= 1 − σ_c/f_c (5.3)

Unless stated otherwise the concrete examined in this report has the following material properties:

fc= 30 M P a Characteristic compressive strength.

fcm= 38 M P a Mean compressive strength.

f_t= 1.95 M P a Mean tensile strength.

Ecm= 33 GP a The secant Young's Modulus between σ = 0 MP a and σ = 0.4fcd. ν = 0.2 Poisson's ratio.

ψ = 38^◦ Angle of dilatancy.

G = 0.08 N/mm Fracture energy.

The ultimate tensile strength is determined by ft=√

f_cm· 0.1.

Abaqus 6.10 requires a specication of the uniaxial relationship between the inelastic strain and the stress after yielding has occurred. The uniaxial behavior is then extended to multiaxial directions. The uniaxial tensile and compressive response of the examined concrete is shown in gures 5.3 and 5.4, respectively.

Figure 5.3. Response of concrete due to uniaxial tension. The blue circles represent the values used in Abaqus 6.10.

Group B122b - Spring 2011 5. Discontinuous modeling in Abaqus

The blue circles on the uniaxial tensile response in gure 5.4 are used as input in Abaqus 6.10 to describe the tensile behavior of the concrete. However, Abaqus 6.10 requires the cracking strain versus stress. At the onset of cracking, the cracking strain is equal to zero and the total strain is equal to ft/E_cm= 5.24 · 10⁻⁵. The exact values given as input in Abaqus 6.10 are given in table 5.1. More details regarding the stress-strain relations of the CDP material model are shown in appendix A.

Stress [MPa] Cracking strain [-]

1.732 0

Table 5.1. The relationship between tenstile stress and direct cracking strain used as input for tensile behavior in the CDP model.

Figure 5.4. Reponse of concrete due to uniaxial compression. The blue circles represent the values used in Abaqus 6.10.

The non-linear compressive reponse shown in gure 5.4 is based on an empirical stress-strain relation given in EN 1992-1-1 [2004]. The blue circles on the uniaxial compressive response in gure 5.4 are used as input in Abaqus 6.10 to describe the compressive behavior of the concrete. Note that the points shown in gure 5.4 are not directly used, as Abaqus 6.10 requires the specication of the inelastic strains. For this reason the elastic strain at each point has been subtracted from the total strain. The exact values are given in table 5.2.

The non-linear relationship is shown in equation 5.4.

48

Master Thesis

Table 5.2. Stress-strain values used as input for the compressive behavior in the CDP model.

σ_c= kη − η²

1 + (k − 2)η · f_cm (5.4)

η c/c1

k 1.05 · E^|_f^c1|

c

c1 = 0.22% is the strain corresponding to the ultimate compressive strength for the examined concrete in this report according to EN 1992-1-1 [2004]. As previously

Group B122b - Spring 2011 5. Discontinuous modeling in Abaqus

order to capture its three-dimensional behavior. A main dierence in the description of the constitutive behavior of concrete between one-dimensional and three-dimensional modeling, is the introduction of the yield surface. The yield surface is described in a three-dimensional principal stress space, and is used to describe the stress evolution upon yielding. However, the shape of the yield surface is controlled by so-called hardening variables, also referred to as the equivalent plastic strains, ˜^pl. The stress-strain relations for the general three-dimensional behavior of concrete is given by equation 5.5. Note, that equation 5.5 is the eective stress contrary to the Cauchy stress in 5.2, and is obtained by dividing equation 5.2 by (1 − d). Further information regarding the hardening variables is given in A.

¯

σ = D^el₀ : ( − ˜^pl) (5.5)

Tensile cracking is initiated for a principal tensile strain corresponding to a principal tensile stress equal to ft. Similarly, compressive elastic degradation is initiated for a principal compressive stress of fc. At the stage of the loading process where a tensile crack forms a discontinuity in the displacement eld of the cracked element is formed. Upon crack initiation a cohesive segment is inserted and the XFEM is activated in the cracked domain.

In the uncracked domain solely CDP is used. Unless stated otherwise, the steel examined in this report has the following material properties:

f_y = 400 M P a Characteristic yield strength.

E = 210 GP a Modulus of elasticity.

ν = 0.3 Poisson's ratio.

G = 5 N/mm Fracture energy.

The element type used for the numerical discretization in all benchmark tests and in the three-dimensional reinforced concrete beam, investigated in this report, is the C3D8 element from the Abaqus 6.10 library. The C3D8-element is a linear, 8-node, isoparametric three-dimensional hexahedron with full integration. The elements are integrated using eight integration points. The C3D8-element is shown in gure 5.5.

50

Master Thesis

Figure 5.5. 2 × 2 × 2 integration point scheme in a C3D8-element. The node numbering and integration point numbering follows the convention in Abaqus 6.10. The integration points are shown in bold dots. MIT [2011]

The implementation of the XFEM in Abaqus 6.10 will be veried in the following chapter based on the modeling choices made in this chapter.

Verification of the XFEM

and Abaqus 6

This chapter presents four dierent benchmark tests used to evaluate dierent aspects of the eectiveness of the XFEM tool and the CDP material model available in Abaqus 6.10. A description of the individual benchmark tests are given with respect to model setup and the available basis for a comparison, such as existing analytical, experimental or numerical results for the given problem.

Finally the quality of the obtained results is used to assess the quality of the XFEM implementation and the CDP material model in Abaqus 6.10.

In this chapter three benchmark tests are evaluated using the XFEM implementation in Abaqus 6.10, and one benchmark tests is carried out to verify the compressive behavior of the examined concrete using the CDP material model. Benchmark tests are critical to the understanding of the capabilities of the XFEM implementation and the CDP material model in Abaqus 6.10, because they serve as a source for comparison. Moreover, all examined benchmark tests are simple in geometry and loading. This makes the numerical model computationally light, which is convenient in order to verify the XFEM and the CDP material model eciently. Finally the authors believe, that since a learning process is involved, it is advantageous to start with a simple, well-dened benchmark problem before resorting to advanced, three-dimensional modeling of a reinforced concrete beam with more uncertainty in the outcome of the analysis.

The following list of benchmark tests are performed in this chapter.

1. Crack-hole interaction for studying the inuence of a hole on the crack propagation path in a steel plate.

2. Verication of the compressive behavior of concrete using the CDP material model.

3. Crack propagation in a concrete beam in three point bending.

4. Crack formation analysis of a reinforced concrete plate.

The benchmark test introduced in point 1 above deal with analyzing a two-dimensional steel plate with isotropic and linear elastic material behavior. The reason for analyzing

Group B122b - Spring 2011 6. Verication of the XFEM and Abaqus

e.g. concrete, that requires a more sophisticated material model. Moreover, a non-linear material behavior entails that the weak form of the XFEM is described in an incremental form, putting higher demands on the quality of the implementation of the numerical solver in Abaqus 6.10. The material behavior of steel is described using a linear elastic perfectly plastic material model. The material properties of the steel in test 1 and 2 are shown in table 5.

This is followed by the benchmark tests introduced in point 2, 3 and 4, where concrete material properties are implemented. The material properties of the concrete in test 2 and 3 are dierent from the properties of the concrete in test 4, and are thus described in the introduction to the respective benchmarks. The CDP model described in appendix A is used in the context of describing the non-linear behavior of concrete.

All benchmark tests are discretized using the C3D8 element described in section 5.

The C3D8 element is a three-dimensional element and is used for four reasons: rstly, Abaqus 6.10 does not oer the XFEM tool for analyzing non-linear material models with planar elements. Secondly, only linear continuum elements are allowed when the XFEM is activated. Thirdly, in order to use the same element in all benchmark tests, thus eliminating a potential variable between the benchmarks on steel and concrete. Finally, the quality of the C3D8 element is tested as a candidate for modeling of the three-dimensional reinforced concrete beam. Although three-dimensional elements are used, the thickness of the specimens in benchmark test 1, 2 and 3 is irrelevant for the results.

In all numerical simulations a poor element discretization, also referred to as element meshing in most commercial FEM software, can have a signicant eect on the reliability of the results. Therefore the benchmark problems will be modeled with carefully considered element discretization, i.e. structured and symmetrical meshes. In section 6.3 dierent mesh structures will be tested in order to verify the importance hereof.

6.1 Crack-hole interaction

The following benchmark test concern crack-hole interaction for studying the inuence of a hole on the crack propagation path in a steel plate loaded in tension. The purpose of the analysis is to show, that the XFEM in Abaqus 6.10 can simulate crack growth without the need of remeshing while accounting for a complex geometry. Dierent starting locations of the crack were chosen to see how the proximity of the hole aects the stress

eld, and thus the crack propagation path. The dimensions of the steel specimen are 7 mm × 21 mm × 0.1 mm, see gure 6.2. The numerically obtained results on the crack propagation path are compared with numerically obtained results by G. Ventura and Belytschko [2003]. Three models are created with three dierent distances between the crack and the hole. The three models have a distance from the center of the hole to the crack of A: 75 mm, B: 150 mm and C: 225 mm. Figure 6.2 shows setup A with applied boundary conditions and load. The load in the models is increased until Abaqus 6.10 fails to achieve equilibrium in the model. An initial crack of 2 mm is placed in each model.

54

6.1. Crack-hole interaction Master Thesis

Figure 6.1. Load and boundary conditions applied to the steel plate model.

Figure 6.2 shows the numerical results obtained by G. Ventura and Belytschko [2003].

The crack is clearly aected by the presence of the hole. It propagates towards the hole and ends on the periphery of the hole. The numerical results are shown in gure 6.3, and are seen to be similar with the dierence being the smoothness of the crack paths. It is observed, that the crack path becomes less aected by the hole the further the initial crack is placed from the hole.

Figure 6.2. Numerical results of crack paths in a steel plate obtained by G. Ventura and Belytschko [2003].

Group B122b - Spring 2011 6. Verication of the XFEM and Abaqus

Figure 6.3. Crack propagation paths for three dierent initial crack locations: A: 75 mm, B:

150 mmand C: 225 mm are the distances from the center of the hole to the crack.

The results show that Abaqus 6.10 is capable of modeling crack propagation with respect to a change in geometry, such as the hole in the benchmark problem, without the need for remeshing.

6.2 Verication of the CDP material model

In the following benchmark test a concrete cylinder, with the properties shown in table 5 and the dimensions shown in gure 6.4, is examined using the CDP material model available in Abaqus 6.10. Figure 5.4 shows the stress-strain relation given as input to the model. The benchmark test is carried out to verify, that the stress-strain response in compression is identical to the empirical stress-strain relation given as input, based on EN 1992-1-1 [2004], see equation 5.4.

Figure 6.4. Dimensions in mm and boundary conditions for a section of the examined cylinder.

Note that, due to symmetry about the z-axis, only a section of the cylinder is modeled.

The section is supported from moving in the longitudinal and horizontal direction in the way shown in gure 6.4. The loading is displacement based in order to get the post-peak response of the stress-strain relation as output. The stress-strain relation is requested in an arbitrary integration point. The location of the integration point is irrelevant because 56

6.3. Crack propagation in a concrete beam in three point bending Master Thesis

the stress-state is uniform throughout the cylinder. Figure 6.5 shows the stress-strain relation given as output from Abaqus 6.10. The gure also compares the relationship to the expected relationship from EN 1992-1-1 [2004].

Figure 6.5. Stress-strain relation for a cylinder loaded in compression calculated in Abaqus 6.10 and according to EN 1992-1-1 [2004].

The compressive behavior is as expected, and for this reason the implementation of the CDP material model in Abaqus 6.10 is acceptable. The tensile behavior is examined in 7

6.3 Crack propagation in a concrete beam in three point bending

At this point two benchmark problems concerning the compressive response in concrete and crack propagation in steel have been successfully used to verify the CDP material model and the XFEM tool in Abaqus 6.10. In this section a concrete beam in three point

At this point two benchmark problems concerning the compressive response in concrete and crack propagation in steel have been successfully used to verify the CDP material model and the XFEM tool in Abaqus 6.10. In this section a concrete beam in three point

In document XFEM for reinforced concrete (Page 49-118)