An inelastic constitutive model for concrete

This section describes the smeared crack model provided in Abaqus/Standard for plain concrete.

The following topics are discussed:

Related Topics
In Other Guides
Concrete smeared cracking


The material library in Abaqus also includes a constitutive model for concrete based on theories of scalar plastic damage, described in Damaged plasticity model for concrete and other quasi-brittle materials, which is available both in Abaqus/Standard and Abaqus/Explicit. In Abaqus/Explicit plain concrete can also be analyzed with the cracking model described in A cracking model for concrete and other brittle materials. It is intended that reinforced concrete modeling be accomplished by combining standard elements, using this plain concrete model, with “rebar elements”—rods, defined singly or embedded in oriented surfaces, that use a one-dimensional strain theory and that may be used to model the reinforcing itself.

These elements are superposed on the mesh of plain concrete elements and are used with standard metal plasticity models that describe the behavior of the rebar material. This modeling approach allows the concrete behavior to be considered independently of the rebar, so this section discusses the plain concrete model only. Effects associated with the rebar/concrete interface, such as bond slip and dowel action, cannot be considered in this approach, except by modifying some aspects of the plain concrete behavior to mimic them, such as the use of “tension stiffening” to simulate load transfer across cracks through the rebar.

The theory described in this section is intended as a model of concrete behavior for relatively monotonic loadings under fairly low confining pressures (less than four to five times the largest compressive stress that can be carried by the concrete in uniaxial compression). Cracking is assumed to be the most important aspect of the behavior, and it dominates the modeling. Cracking is assumed to occur when the stresses reach a failure surface, which we call the “crack detection surface.” This failure surface is taken to be a simple Coulomb line written in terms of the first and second stress invariants, p and q, that are defined below. The anisotropy introduced by cracking is assumed to be important in the simulations for which the model is intended, so the model includes consideration of this anisotropy. The model is a smeared crack model, in the sense that it does not track individual “macro” cracks: rather, constitutive calculations are performed independently at each integration point of the finite element model, and the presence of cracks enters into these calculations by the way the cracks affect the stress and material stiffness associated with the integration point. Various objections have been raised against such smeared crack models. The principal concern is that this modeling approach inherently introduces mesh sensitivity in the solutions, in the sense that the finite element results do not converge to a unique result. For example, since cracking is associated with strain softening, mesh refinement will lead to narrower crack bands. Crisfield (1986) discusses this concern in detail and concludes that Hillerborg's (1976) approach, based on brittle fracture concepts, is adequate to deal with this issue for practical purposes. This aspect of the model is discussed below in the section on cracking. For simplicity of discussion in what follows, the term “crack” is used to mean a direction in which cracking has been detected at the single constitutive calculation point in question: the closest physical concept is that there exists a continuum of micro-cracks at the point, oriented as determined by the model.

When the principal stress components are dominantly compressive, the response of the concrete is modeled by an elastic-plastic theory, using a simple form of yield surface written in terms of the first two stress invariants. Associated flow and isotropic hardening are used. This model significantly simplifies the actual behavior: the associated flow assumption generally overpredicts the inelastic volume strain; the simple yield surface used does not match all data very accurately (the third stress invariant would be needed to improve this aspect of the model); and, especially when the concrete is strained beyond the ultimate stress point, the assumption of constant elastic stiffness does not reproduce the observation that the unloading response is significantly weakened (the elastic response of the material appears to be damaged). In addition, when concrete is subjected to very high pressure stress, it exhibits inelastic response: no attempt has been made to build this behavior into the model. In spite of these limitations the model provides useful predictions for a variety of problems involving inelastic loading of concrete. The limitations are introduced for the sake of computational efficiency. In particular, assuming associated flow leads to enough symmetry in the Jacobian matrix of the constitutive model (the “material stiffness matrix”) that the overall equilibrium equation solution usually does not require nonsymmetric equation solution for this reason. All of these limitations could be removed at some sacrifice in computational cost.

The cracking and compression responses of concrete that are incorporated in the model are illustrated by the uniaxial response of a specimen shown in Figure 1.

Figure 1. Uniaxial behavior of plain concrete.

When concrete is loaded in compression, it initially exhibits elastic response. As the stress is increased, some nonrecoverable (inelastic) straining occurs, and the response of the material softens. An ultimate stress is reached, after which the material softens until it can no longer carry any stress. If the load is removed at some point after inelastic straining has occurred, the unloading response is softer than the initial elastic response: this effect is ignored in the model. When a uniaxial specimen is loaded into tension, it responds elastically until, at a stress that is typically 7–10% of the ultimate compressive stress, cracks form so quickly that—even on the stiffest testing machines available—it is very difficult to observe the actual behavior. For the purpose of developing the model, we assume that the material loses strength through a softening mechanism and that this is dominantly a damage effect, in the sense that open cracks can be represented by loss of elastic stiffness (as distinct from the nonrecoverable straining that is associated with classical plasticity effects, such as what we are using for the compressive behavior model). The model neglects any permanent strain associated with cracking; that is, we assume that the cracks can close completely when the stress across them becomes compressive.

In multiaxial stress states these observations can be generalized through the concept of surfaces of failure and of ultimate strength in stress space. These surfaces are defined below and are fitted to experimental data. Typical surfaces are shown in Figure 2 and Figure 3.

Figure 2. Concrete failure surfaces in plane stress.

Figure 3. Concrete failure surfaces in the (pq) plane.

This model makes no attempt to include prediction of cyclic response or of the reduction in the elastic stiffness caused by inelastic straining because the model is intended for application to relatively monotonic loading cases. Nevertheless, it is likely that—even in such cases—the stress trajectories will not be entirely radial and the model must predict the response in such cases in a reasonable way. An isotropically hardening “compressive” yield surface forms the basis of the model for the inelastic response when the principal stresses are dominantly compressive. In tension once cracking is defined to occur (by the “crack detection surface” of the model), the orientation of the cracks is stored and oriented, damaged elasticity is then used to model the existing cracks. Stress components associated with an open crack are not included in the definition of the crack detection surface for detecting additional cracks at the same point, and we only allow cracks to form in orthogonal directions at a point.

Since Abaqus/Standard is an implicit, stiffness method code and the material calculations used to define the behavior of the concrete are carried out independently at each integration point in that part of the model that is made of concrete, the solution is known at the start of the time increment. The constitutive calculations must provide values of stress and material stiffness at the end of the increment, based on the current estimate of the kinematic solution for the response at the spatial integration point during the increment that provides the (logarithmic) strain, ε, at the end of the increment.

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 system is rectangular Cartesian. This use of a local system simplifies the computation of the damaged elasticity used for the components associated with existing cracks.

The model, thus, consists of a “compressive” yield/flow surface to model the concrete response in predominantly compressive states of stress, together with damaged elasticity to represent cracks that have occurred at a material calculation point, the occurrence of cracks being defined by a “crack detection” failure surface that is considered to be part of the elasticity. The details of this model are now presented.