Interpretation of "strain"s in CDP model

Concrete damage plasticity (CDP) model is a very commonly used constitutive model of concrete. Abaqus has provided a basic introduction and operational guidance for this model (linked below).

ConcreteDamaged Plasticity

However, the explanation of the stress-strain curve of concrete in the manual is somewhat lacking, especially for the definition of the four “strain”s, which has caused me some trouble, so I will make further explanation here. And has been animated to show the way.

$\tilde \epsilon_c^{pl}$ and $\epsilon_c^{el}$

Select a point ($\epsilon _c^1$, $\sigma_c^1$ on the strain-softening regime for explanation:

Since this point is already in the stress softening regime, the stiffness should be reduced due to damage to the concrete. If we consider unloading from this point, then the slope of the unloading curve will certainly be less than the initial undamaged elastic modulus ($E_0$. Therefore, the CDP model introduces a damage factor $d$ to represent such damage and reduction in stiffness. Then the equivalent elastic modulus of the unloading curve is calculated as $(1-d)E_0$. The strain change corresponding to the unloading phase is denoted as ‘$\epsilon_c^{el}$’. Such equivalent elastic strain has also included the damage. Accordingly, the remaining portion of the strain is naturally called the equivalent plastic strain $\tilde{\epsilon}_c^{pl}$. (As shown in animation 1)

Different researchers proposed different expressions of damage factor $d$.

Animation1

$\tilde \epsilon_c^{in}$ and $\epsilon_{0c}^{el}$

However, there is no way to define equivalent plastic strain and equivalent elastic strain in Abaqus. This is because Abaqus uses inelastic strain ‘$\tilde\epsilon_c^{in}$’ to replace the equivalent plastic strain ‘$\tilde\epsilon_c^{pl}$’. My explanation for such replacement is that in the context of material science, plastic strain refers specifically to permanent and non-recoverable deformation. Thus, this term does not include some possibly recoverable strains due to other field variables (like temperature). Therefore, it is logical to use the broader term “inelastic strain” to replace it and reorganize the mathematical expression.

So how should we reorganize the term and mathematical relation? If the unloading curve rebounds with the undamaged stiffness, then the modulus during this phase would be $E_0$, and the corresponding strain would be purely elastic strain without any damage ($\epsilon_{0c}^{el}$. Logically, the remaining part would be inelastic strain $\tilde \epsilon_c^{in}$ as shown in animation 2.

Animation2

The the relation is clear, and will be converted by Abaqus automatically:
$$
\tilde \epsilon_c^{pl}+\epsilon_c^{el}=\tilde \epsilon_c^{in}+\epsilon_{0c}^{el}\
\Rightarrow\epsilon_c^{pl}=\tilde\epsilon_c^{in}-\frac{d_c}{1-d_c}\frac{\sigma_c}{E_0}
$$

Some comments

From a physical standpoint, the combination of $\tilde \epsilon_c^{in}$ and $\epsilon_{0c}^{el}$ seems questionable because it does not account for the influence of stress history. Specifically, such combination considers the pure elastic strain based on the current stress state ($\sigma_c^1$. However, before reaching this stress state, the material has already experienced the yield stress which is higher. Therefore, it is debatable whether the pure elastic strain can be simply regarded as $\frac{\sigma_c^1}{E_0}$. Of course, if we do not delve too deeply into the physical nuances and consider it purely in terms of terminology, there is no issue. After all, from a mathematical perspective, it merely represents a transformation of the combination of $\tilde \epsilon_c^{pl}$ and $\epsilon_c^{el}$, without any fundamental change.

Interpretation of "strain"s in CDP model

\(\tilde \epsilon_c^{pl}\) and \(\epsilon_c^{el}\)

\(\tilde \epsilon_c^{in}\) and \(\epsilon_{0c}^{el}\)

Some comments