Mechanical Behavior
The model is a continuum, plasticity-based, damage model for concrete. It assumes that the main two failure mechanisms are tensile cracking and compressive crushing of the concrete material. The evolution of the yield (or failure) surface is controlled by two hardening variables, ˜εplt and ˜εplc, linked to failure mechanisms under tension and compression loading, respectively. We refer to ˜εplt and ˜εplc as tensile and compressive equivalent plastic strains, respectively. The following sections discuss the main assumptions about the mechanical behavior of concrete.
Uniaxial Tension and Compression Stress Behavior
The model assumes that the uniaxial tensile and compressive response of concrete is characterized by damaged plasticity, as shown in Figure 1.

Under uniaxial tension the stress-strain response follows a linear elastic relationship until the value of the failure stress, σt0, is reached. The failure stress corresponds to the onset of micro-cracking in the concrete material. Beyond the failure stress the formation of micro-cracks is represented macroscopically with a softening stress-strain response, which induces strain localization in the concrete structure. Under uniaxial compression the response is linear until the value of initial yield, σc0. In the plastic regime the response is typically characterized by stress hardening followed by strain softening beyond the ultimate stress, σcu. This representation, although somewhat simplified, captures the main features of the response of concrete.
It is assumed that the uniaxial stress-strain curves can be converted into stress versus plastic-strain curves. (This conversion is performed automatically by Abaqus from the user-provided stress versus “inelastic” strain data, as explained below.) Thus,
where the subscripts t and c refer to tension and compression, respectively; ˜εplt and ˜εplc are the equivalent plastic strains, ˙˜εplt and ˙˜εplc are the equivalent plastic strain rates, θ is the temperature, and fi,(i=1,2,…) are other predefined field variables.
As shown in Figure 1, when the concrete specimen is unloaded from any point on the strain softening branch of the stress-strain curves, the unloading response is weakened: the elastic stiffness of the material appears to be damaged (or degraded). The degradation of the elastic stiffness is characterized by two damage variables, dt and dc, which are assumed to be functions of the plastic strains, temperature, and field variables:
The damage variables can take values from zero, representing the undamaged material, to one, which represents total loss of strength.
If E0 is the initial (undamaged) elastic stiffness of the material, the stress-strain relations under uniaxial tension and compression loading are, respectively:
We define the “effective” tensile and compressive cohesion stresses as
The effective cohesion stresses determine the size of the yield (or failure) surface.
Uniaxial Cyclic Behavior
Under uniaxial cyclic loading conditions the degradation mechanisms are quite complex, involving the opening and closing of previously formed micro-cracks, as well as their interaction. Experimentally, it is observed that there is some recovery of the elastic stiffness as the load changes sign during a uniaxial cyclic test. The stiffness recovery effect, also known as the “unilateral effect,” is an important aspect of the concrete behavior under cyclic loading. The effect is usually more pronounced as the load changes from tension to compression, causing tensile cracks to close, which results in the recovery of the compressive stiffness.
The concrete damaged plasticity model assumes that the reduction of the elastic modulus is given in terms of a scalar degradation variable d as
where E0 is the initial (undamaged) modulus of the material.
This expression holds both in the tensile (σ11>0) and the compressive (σ11<0) sides of the cycle. The stiffness degradation variable, d, is a function of the stress state and the uniaxial damage variables, dt and dc. For the uniaxial cyclic conditions Abaqus assumes that
where st and sc are functions of the stress state that are introduced to model stiffness recovery effects associated with stress reversals. They are defined according to
where
The weight factors wt and wc, which are assumed to be material properties, control the recovery of the tensile and compressive stiffness upon load reversal. To illustrate this, consider the example in Figure 2, where the load changes from tension to compression.

Assume that there was no previous compressive damage (crushing) in the material; that is, ˜εplc=0 and dc=0. Then
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In tension (σ11>0), r*=1; therefore, d=dt as expected.
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In compression (σ11<0), r*=0, and d=(1-wc)dt. If wc=1, then d=0; therefore, the material fully recovers the compressive stiffness (which in this case is the initial undamaged stiffness, E=E0). If, on the other hand, wc=0, then d=dt and there is no stiffness recovery. Intermediate values of wc result in partial recovery of the stiffness.
Multiaxial Behavior
The stress-strain relations for the general three-dimensional multiaxial condition are given by the scalar damage elasticity equation:
where Del0 is the initial (undamaged) elasticity matrix.
The previous expression for the scalar stiffness degradation variable, d, is generalized to the multiaxial stress case by replacing the unit step function r*(σ11) with a multiaxial stress weight factor, r(ˆσ), defined as
where ˆσi are the principal stress components. The Macauley bracket is defined by .
See Damaged plasticity model for concrete and other quasi-brittle materials for further details of the constitutive model.