On Plastic Dislocation Density Tensor

This article attempts to clarify an issue regarding the proper definition of plastic dislocation density tensor. This study shows that the Ortiz's and Berdichevsky's plastic dislocation density tensors are equivalent with each other, but not with Kondo's one. To fix the problem, we propose a modified version of Kondo's plastic dislocation density tensor.

Plastic deformation is everywhere, from bending a fork to panel beating a car body. It is easy to be intrigued by a subject that pervades so many aspects of peoples' daily lives.
It is clear that a unification of the definition for plastic dislocation density tensor is still an issue. which the above-mentioned definition is the proper one? What is the relationship between those definitions? If the definition is not well defined, how to fix it?
Phenomenologically the total elasto-plastic deforma- can be decomposed into the multiplication of elastic gradient namely F = F e F p , is due to Bilby et. al. [3], Kröner [6], Lee and Liu [20], and Lee [21]. The elastic deformation gradient F e = g i ⊗ e i = F e ij g i ⊗ e j = F e ij g i e j and plastic gradient F p = e i ⊗G i = F p ij e i ⊗G j = F p ij e i G j , G i , e i , g i are the base vectors corresponding to the reference, intermediate and current configuration, respectively. The deformation decomposition is shown in Figure 1.
It should be noted that the elastic deformation F e and plastic deformation F p cannot be gradients of global maps, they are therefore called incompatible, namely F e × ∇ = 0 and F p × ∇ = 0 as well, where the operator ∇ = G k ∇ k is gradient operator, and ∇ k is covariant derivative in reference configuration. Nevertheless, both F e and F p are orientation preserving so that J p = det(F p ) > 0 and J e = det(F e ) > 0. This means, F p and F e have inverse deformations, denoted correspondingly by (F p ) −1 and (F e ) −1 .
In this short article, we will show that the Ortiz's and Berdichevsky's plastic dislocation density tensor are equivalent, while not equivalent with Kondo's one. To fix Kondo's problem, we can change Kondo's definition to following form Thus, we have modified Kondo's plastic dislocation density tensor as follows  Figure 1: Elasto-plastic deformation configuration: elastic deformation F e , plastic deformation F p and total deformation F both Ortiz's and Berdichevsky's plastic dislocation density tensor. To verify these, we need to prove a tensor identity at first.

Lemma 1 Giving two 2nd order tensors
where where the unit tensor I = G m G m = δ ij G i G j in reference configuration. Therefore, we have proven the tensor identity, which has never been seen in literature.
Despite the incompatibility of elastic and plastic deformation, namely, F e and F p , the total deformation F is compatible, it means that the total deformation must be gradient of global maps, thus it must satisfy compatible condition [23], namely, the incompatible tensor Inc(F ) = F × ∇ = 0, which leads to Inc(F ) = (F e · F p ) × ∇ = 0.
Applying the identity of tensor proved in the Lemma, we have Using the previous definitions of plastic dislocation density tensor, the above expression can be rewritten as Therefore, we have their relationships: F · T Berdichevsky + T Modified−Kondo · F p = 0, F · T Berdichevsky − F e · T Oritz = 0.
Clearly the relations 7,8 and 9 reveal that three definition of the plastic dislocation tensity tensor are equivalent.
In summary, this study shows that both Ortiz's and Berdichevsky's plastic dislocation density tensors are equivalent, and are proper definition. Although Kondo's definition is not proper one, it can be fixed by the modified version in Eq. 2.