11.5.2 Cracking model for concrete

**Product: **ABAQUS/Explicit

The brittle cracking model in ABAQUS/Explicit is defined by using both the *BRITTLE CRACKING option and the *BRITTLE SHEAR option and, optionally, the *BRITTLE FAILURE option. See “Inelastic behavior,” Section 11.1.1, for a discussion of the concrete models available in ABAQUS.

The brittle cracking model in ABAQUS/Explicit:

provides a capability for modeling concrete in all types of structures: beams, trusses, shells and solids;

can also be useful for modeling other materials such as ceramics or brittle rocks;

is designed for applications in which the behavior is dominated by tensile cracking;

assumes that the compressive behavior is always linear elastic;

must be used with the *ELASTIC material option, which also defines the material behavior completely prior to cracking;

is most accurate in applications where the brittle behavior dominates such that the assumption that the material is linear elastic in compression is adequate;

can be used for plain concrete, even though it is intended primarily for the analysis of reinforced concrete structures;

allows removal of elements with the *BRITTLE FAILURE option; and

is defined in detail in “A cracking model for concrete and other brittle materials,” Section 4.5.3 of the ABAQUS Theory Manual.

Reinforcement in concrete structures is typically provided by means of rebars. Rebars are one-dimensional strain theory elements (rods) that can be defined singly or embedded in oriented surfaces. Rebars are defined in “Defining rebar as an element property,” Section 2.2.4. They are typically used with elastic-plastic material behavior and are superposed on a mesh of standard element types used to model the plain concrete. With this modeling approach, the concrete cracking behavior is considered independently of the rebar. Effects associated with the rebar/concrete interface, such as bond slip and dowel action, are modeled approximately by introducing some “tension stiffening” into the concrete cracking model to simulate load transfer across cracks through the rebar.

ABAQUS/Explicit uses a smeared crack model to represent the discontinuous brittle behavior in concrete. It does not track individual “macro” cracks: instead, constitutive calculations are performed independently at each material point of the finite element model. The presence of cracks enters into these calculations by the way in which the cracks affect the stress and material stiffness associated with the material point.

For simplicity of discussion in this section, the term “crack” is used to mean a direction in which cracking has been detected at the single material calculation point in question: the closest physical concept is that there exists a continuum of micro-cracks in the neighborhood of the point, oriented as determined by the model. The anisotropy introduced by cracking is assumed to be important in the simulations for which the model is intended.

The ABAQUS/Explicit cracking model assumes fixed, orthogonal cracks, with the maximum number of cracks at a material point limited by the number of direct stress components present at that material point of the finite element model (a maximum of three cracks in three-dimensional, plane strain, and axisymmetric problems; two cracks in plane stress and shell problems; and one crack in beam or truss problems). Internally, once cracks exist at a point, the component forms of all vector- and tensor-valued quantities are rotated so that they lie in the local system defined by the crack orientation vectors (the normals to the crack faces). The model ensures that these crack face normal vectors will be orthogonal, so that this local crack system is rectangular Cartesian. For output purposes, the user is offered results of stresses and strains in the global and/or local crack systems.

A simple Rankine criterion is used to detect crack initiation. This criterion states that a crack forms when the maximum principal tensile stress exceeds the tensile strength of the brittle material. Although crack detection is based purely on Mode I fracture considerations, ensuing cracked behavior includes both Mode I (tension softening/stiffening) and Mode II (shear softening/retention) behavior, as described later.

As soon as the Rankine criterion for crack formation has been met, we assume that a first crack has formed. The crack surface is taken to be normal to the direction of the maximum tensile principal stress. Subsequent cracks may form with crack surface normals in the direction of maximum principal tensile stress that is orthogonal to the directions of any existing crack surface normals at the same point.

Cracking is irrecoverable in the sense that, once a crack has occurred at a point, it remains throughout the rest of the calculation. However, crack closing and reopening may take place along the directions of the crack surface normals. The model neglects any permanent strain associated with cracking; that is, it is assumed that the cracks can close completely when the stress across them becomes compressive.

The postfailure behavior for direct straining across cracks is modeled with the *BRITTLE CRACKING option.

In reinforced concrete the specification of postfailure behavior generally means giving the postfailure stress as a function of strain across the crack using the *BRITTLE CRACKING, TYPE=STRAIN option (Figure 11.5.2–1). In cases with little or no reinforcement, this introduces mesh sensitivity in the results, in the sense that the finite element predictions do not converge to a unique solution as the mesh is refined because mesh refinement leads to narrower crack bands.

**Figure 11.5.2–1** *BRITTLE CRACKING, TYPE=STRAIN input.

In practical calculations for reinforced concrete, the mesh is usually such that each element contains rebars. Then, the interaction between the rebars and the concrete tends to mitigate this effect, provided that a reasonable amount of “tension stiffening” is introduced in the cracking model to simulate this interaction. This requires an estimate of the tension stiffening effect, which depends on factors such as the density of reinforcement, the quality of the bond between the rebar and the concrete, the relative size of the concrete aggregate compared to the rebar diameter, and the mesh. A reasonable starting point for relatively heavily reinforced concrete modeled with a fairly detailed mesh is to assume that the strain softening after failure reduces the stress linearly to zero at a total strain about ten times the strain at failure. Since the strain at failure in standard concretes is typically , this suggests that tension stiffening that reduces the stress to zero at a total strain of about is reasonable. This parameter should be calibrated to each particular case. In static applications too little tension stiffening will cause the local cracking failure in the concrete to introduce temporarily unstable behavior in the overall response of the model. Few practical designs exhibit such behavior, so that the presence of this type of response in the analysis model usually indicates that the tension stiffening is unreasonably low.

Usage: *BRITTLE CRACKING, TYPE=STRAIN

When there is no reinforcement in significant regions of the model, the tension stiffening approach described above will introduce unreasonable mesh sensitivity into the results. However, it is generally accepted that Hilleborg's (1976) fracture energy proposal is adequate to allay the concern for many practical purposes. Hilleborg defines the energy required to open a unit area of crack in Mode I () as a material parameter, using brittle fracture concepts. With this approach the concrete's brittle behavior is characterized by a stress-*displacement* response rather than a stress-*strain* response. Under tension a concrete specimen will crack across some section and its length, after it has been pulled apart sufficiently for most of the stress to be removed (so that the elastic strain is small), will be determined primarily by the opening at the crack, which does not depend on the specimen's length.

In ABAQUS/Explicit this fracture energy cracking model can be invoked by using the *BRITTLE CRACKING, TYPE=DISPLACEMENT option and specifying the postfailure stress as a tabular function of displacement across the crack, as illustrated in Figure 11.5.2–2.

**Figure 11.5.2–2** *BRITTLE CRACKING, TYPE=DISPLACEMENT input.

**Figure 11.5.2–3** *BRITTLE CRACKING, TYPE=GFI input.

*BRITTLE CRACKING, TYPE=DISPLACEMENT *BRITTLE CRACKING, TYPE=GFI

The implementation of the stress-displacement concept in a finite element model requires the definition of a characteristic length associated with a material point. The characteristic crack length is based on the element geometry: for beams and trusses we use the element length; for shell and planar elements we use the square root of the element area; for solid elements we use the cube root of the element volume. We use this definition of the characteristic crack length because the direction in which cracks will occur is not known in advance. Therefore, elements with large aspect ratios will have rather different behavior depending on the direction in which they crack: some mesh sensitivity remains because of this effect. Elements that are as close to square as possible are, therefore, recommended unless the user can predict the direction in which cracks will form.

An important feature of the cracking model is that, whereas crack initiation is based on Mode I fracture only, postcracked behavior includes Mode II as well as Mode I. The Mode II shear behavior is based on the common observation that the shear behavior depends on the amount of crack opening. More specifically, the cracked shear modulus is reduced as the crack opens. Therefore, ABAQUS/Explicit offers a shear retention model in which the postcracked shear stiffness is defined as a function of the opening strain across the crack; the shear retention model must be defined in the cracking model, and zero shear retention should not be used.

In these models the dependence is defined by expressing the postcracking shear modulus, , as a fraction of the uncracked shear modulus:

where is the shear modulus of the uncracked material and the shear retention factor, , depends on the crack opening strain, . The user can specify this dependence in piecewise linear form by using the TYPE=RETENTION FACTOR parameter with the *BRITTLE SHEAR option, as shown in Figure 11.5.2–4.

**Figure 11.5.2–4** *BRITTLE SHEAR, TYPE=RETENTION FACTOR input.

where and are material parameters. This form, shown in Figure 11.5.2–5, satisfies the requirements that as (corresponding to the state before crack initiation) and as (corresponding to complete loss of aggregate interlock). See “A cracking model for concrete and other brittle materials,” Section 4.5.3 of the ABAQUS Theory Manual, for a discussion of how shear retention is calculated in the case of two or more cracks.

*BRITTLE SHEAR, TYPE=RETENTION FACTOR *BRITTLE SHEAR, TYPE=POWER LAW

One experiment, a uniaxial tension test, is required to calibrate the simplest version of the brittle cracking model. Other experiments may be required to gain accuracy in postfailure behavior.

**Figure 11.5.2–5** *BRITTLE SHEAR, TYPE=POWER LAW input.

This test is difficult to perform because it is necessary to have a very stiff testing machine to record the postcracking response. Quite often such equipment is not available; in this situation the user must make an assumption about the tensile failure strength of the material and the postcracking response. For concrete the assumption usually made is that the tensile strength is 7–10% of the compressive strength. Uniaxial compression tests can be performed much more easily, so the compressive strength of concrete is usually known.

The values given for tension stiffening are a very important aspect of simulations using the ABAQUS/Explicit brittle cracking model. The postcracking tensile response is highly dependent on the reinforcement present in the concrete. In simulations of unreinforced concrete, the tension stiffening models that are based on fracture energy concepts should be utilized. If reliable experimental data are not available, typical values that can be used were discussed before: common values of range from 40 N/m (0.22 lb/in) for a typical construction concrete (with a compressive strength of approximately 20 MPa, 2850 lb/in^{2}) to 120 N/m (0.67 lb/in) for a high-strength concrete (with a compressive strength of approximately 40 MPa, 5700 lb/in^{2}). In simulations of reinforced concrete the stress-strain tension stiffening model should be used; the amount of tension stiffening depends on the reinforcement present, as discussed before. A reasonable starting point for relatively heavily reinforced concrete modeled with a fairly detailed mesh is to assume that the strain softening after failure reduces the stress linearly to zero at a total strain about ten times the strain at failure. Since the strain at failure in standard concretes is typically , this suggests that tension stiffening that reduces the stress to zero at a total strain of about is reasonable. This parameter should be calibrated to each particular case.

Calibration of the postcracking shear behavior requires combined tension and shear experiments, which are difficult to perform. If such test data are not available, a reasonable starting point is to assume that the shear retention factor, , goes linearly to zero at the same crack opening strain used for the tension stiffening model.

The *BRITTLE FAILURE option can be used in conjunction with the *BRITTLE CRACKING and *BRITTLE SHEAR options to define brittle failure of the material. When one, two, or all three local direct cracking strain (displacement) components at a material point reach the value defined as the failure strain (displacement), the material point fails and all the stress components are set to zero. If all of the material points in an element fail, the element is removed from the mesh. For example, removal of a first-order reduced-integration solid element takes place as soon as its only integration point fails. However, all through-the-thickness integration points must fail before a shell element is removed from the mesh.

Use the *BRITTLE FAILURE option to define element failure.

Usage: *BRITTLE FAILURE, CRACKS=n

The failure strain must be given on the data lines of this option when it is used with the *BRITTLE CRACKING, TYPE=STRAIN option. The failure displacement must be given on the data lines of this option when it is used with the *BRITTLE CRACKING, TYPE=DISPLACEMENT or GFI option. The failure strain (displacement) can be specified as a function of temperature and/or predefined field variables.

The CRACKS parameter allows the user to control how many cracks at a material point must fail before the material point is considered to have failed; the default is one crack. The value of CRACKS can only be one for beam and truss elements; it cannot be greater than two for plane stress and shell elements; and it cannot be greater than three otherwise.

The brittle failure criterion is a crude way of modeling failure in ABAQUS/Explicit and should be used with care. The main motivation for including this option is to help in computations where not removing an element that can no longer carry stress may lead to excessive distortion of that element and subsequent premature termination of the simulation. For example, in a monotonically loaded structure whose failure mechanism is expected to be dominated by a single tensile macrofracture (Mode I cracking), it may be reasonable to use the brittle failure criterion to remove elements. On the other hand, the fact that the brittle material loses its ability to carry tensile stress does not preclude it from withstanding compressive stress; therefore, it may not be appropriate to remove elements if the material is expected to carry compressive loads after it has failed in tension. An example may be a shear wall subjected to cyclic loading as a result of some earthquake excitation; in this case cracks that develop completely under tensile stress will be able to carry compressive stress when load reversal takes place.

Thus, the effective use of the brittle failure criterion relies on the user having some knowledge of the structural behavior and potential failure mechanism. The use of the brittle failure criterion based on an incorrect user assumption of the failure mechanism will generally result in an incorrect simulation.

The CRACKS parameter allows the user to control how many cracks must open to beyond the failure value before a material point is considered to have failed. The default (one crack) should be used for most structural applications where failure is dominated by Mode I type cracking. However, there are cases in which the CRACKS parameter should be set to a higher number because multiple cracks need to form to develop the eventual failure mechanism. One example may be an unreinforced, deep concrete beam where the failure mechanism is dominated by shear; in this case it is possible that two cracks need to form at each material point for the shear failure mechanism to develop.

Again, the appropriate choice of the value of the CRACKS parameter relies on user knowledge of the structural and failure behaviors.

It is possible to use the brittle failure criterion in brittle cracking elements for which rebar are also defined; the obvious application is the modeling of reinforced concrete. When such elements fail according to the brittle failure criterion, the brittle cracking contribution to the element stress carrying capacity is removed but the rebar contribution to the element stress carrying capacity is not removed. However, if the user also includes the shear failure option in the rebar material definition, the rebar contribution to the element stress carrying capacity will also be removed if the shear failure criterion specified for the rebar is satisfied. This allows the modeling of progressive failure of an under-reinforced concrete structure where the concrete fails first followed by ductile failure of the reinforcement.

ABAQUS/Explicit offers a variety of elements for use with the cracking model: beam, shell, plane stress, plane strain, axisymmetric, and three-dimensional. Plane triangular, triangular prism, and tetrahedral elements are not recommended for use in reinforced concrete analysis since these elements do not support the use of rebar.

In addition to the standard output identifiers available in ABAQUS/Explicit (see “ABAQUS/Explicit output variable identifiers,” Section 4.2.2), the following output variables relate directly to material points that use the brittle cracking model:

CKE | All cracking strain components. |

CKLE | All cracking strain components in local crack axes. |

CKEMAG | Cracking strain magnitude. |

CKLS | All stress components in local crack axes. |

CRACK | Crack orientations. |

CKSTAT | Crack status of each crack. |

STATUS | Status of element (brittle failure model). The status of an element is 1.0 if the element is active and 0.0 if the element is not. |