Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 4278082, 6 pages

https://doi.org/10.1155/2017/4278082

## A Consistent Relationship between the Stress and Plastic Strain Components and Its Application in Deep Drawing Process

School of Mechanical Engineering, Tongji University, Shanghai 201804, China

Correspondence should be addressed to Yong Zhang

Received 9 November 2016; Accepted 15 December 2016; Published 11 January 2017

Academic Editor: Hakim Naceur

Copyright © 2017 Yong Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

As von Mises yield criterion and associated flow rule (AFR) are widely applied in metal forming field, a semitotal deformation consistent relationship between the stress and plastic strain components and the rule of dimensional changes of metal forming processes in a plane-stress state are obtained on the basis of them in this paper. The deduced consistent relationship may be easily used in forming interval of the workpiece. And the rule of dimensional changes can be understood through three plastic strain incremental circles on which the critical points can be easily determined on the same basis. Analysis of stress and plastic strain evolution of aluminum warm deep drawing process is conducted, and the advantage of nonisothermal warm forming process is revealed, indicating that this method has the potential in practical large deformation applications.

#### 1. Introduction

The rule of metal forming process has been investigated since several decades ago. Two main theories were proposed, the law of minimum resistance [1] and the consistent relationship [2]. The former pointed out that the particle of metal material will move to the direction which has the minimum resistance during deformation, and this direction is the direction of dimensional increasing. However, this theory lacks experimental support and is hard to obtain its practical application due to its too simple form of expression. As for the latter one, the relationship between order of total principle strain components and the associated principle stress components was deduced on the basis of Levy-Mises equation. This theory reveals the dramatic relationship between dimensional change of metal and the external forces added to the metal during deformation process and received its applications in qualitative and quantitative analysis [3, 4].

However, it is better to develop a rule of metal forming in a form of plastic strain components. On the one hand, as for the nature of deformation of metal materials, plastic strain components should be highlighted because the elastic strain components will change themselves after unloading. On the other hand, in many practical situations, such as forging and forming, the plastic strains may be 1000 times of elastic strains, and the elastic strain components are always neglected. In this paper, a semitotal deformation consistent relationship between stress and plastic strain components is proposed on the basis of the AFR and von Mises yield criterion. With regard to the plane-stress situation, the critical points on the three plastic strain incremental circles can be easily determined due to the physical meaning of the AFR.

In recent years, the studies of aluminum warm forming are being more and more popular due to the problems of energy shortage and environmental damage. However, the rule of the dimensional changes of this sheet metal in the warm forming process is rarely investigated. In this paper, the analysis of stress and plastic strain evolution of the blank in this potential process is fully conducted.

#### 2. Consistent Relationship and Rule of Dimensional Changes of Metal Forming Process

The existing consistent relationship between the stress and total strain components is on the basis of Levy-Mises equation. In this section, a similar consistent relationship between stress and plastic strain components is deduced on the basis of von Mises yield criterion and the AFR, and the rule of dimensional changes of metal forming processes in plane-stress state is also discussed.

##### 2.1. Instantaneous Deformation Tendency

At a moment during forming process, generally can be acquired, where , , and are the principal stress components. So the order of the deviatoric stress components can be obtained:where is the average stress.

As is given by [5], the AFR can be expressed as where denotes the increment in the plastic strain tensor, is plastic multiplier, and is the yield criterion. It is clear that the increment plastic strain tensor is in a direction which is normal to the tangent to the yield surface at the load point. For a von Mises yield criterion, (2) can be rewritten aswhere means the effective stress and means the deviatoric stress tensor. Equation (3) can be rewritten as

Combine (1) and (4), as and are positive, the following expression can be achieved:So the order of the increment principle plastic strain components is the same as the order principal stress components. In other words, at a moment of forming process if , then order of the associated instantaneous plastic strain is .

Asso and .

From (4), the following equation can be obtained:So if , then . Similarly, if , then , and when , . It can be found clearly that the sign of is identical to the sign of ; if , then .

##### 2.2. Semitotal Deformation Consistent Relationship

The above-mentioned derivation is on the basis of instantaneous deformation. And the obtained conclusions can only reflect the instant tendency of deformation. As for a period, if and the principle directions of stress tensor do not change, the same conclusions can be achieved. The derivation using incremental theory is as follows.

A certain time period of an arbitrary deformation can be divided into several small intervals, such as , where is the deformation time. So the following equation for interval can be obtained due to (3) asAs for interval ,During interval ,As is kept unchanged during this period, so (1) can also be achieved in this section, because the increment plastic multiplier and the effective stress are always positive. Then from (8), (9), and (10),

As the principle directions of stress tensor do not change during this period, so the principle directions of and will not change too. The proof is as follows.

It is well known that the principle directions of stress tensor are the same as the eigenvectors of the matrix . The eigenvalues of matrix can be obtained by the following equation:where is the unit matrix and denote the eigenvalues of matrix . The eigenvalues of matrix can also be obtained in the same way:where denote the eigenvalues of matrix . As , so the eigenvalues of matrix will be by comparing (12). The eigenvectors of the matrix can be achieved by solving the following equation:where is the eigenvector of the matrix . The eigenvectors of matrix can also be obtained in the same way:where is the eigenvector of the matrix . As , so the eigenvectors of matrixes and will be the same by comparing (14). It can be shown by (3) that the principle directions of and are the same; that is, the principle directions of , , and are the same, so if the principle directions of stress tensor do not change, the principle directions of and will not change too.

From (11), if the principle direction of (the principle directions of and are the same as that of ) is kept unchanged during the time period , the corresponding increment principle plastic strain components in (11) can be summed up, and then the following inequation can be obtained:So during a time period , if and the principle directions of or are kept unchanged, then can be achieved.

AsThen and .

can be calculated by combining (8), (9), and (10): So during the time period , if , then ; if , , and when , .

The application of the consistent relationship between stress and plastic strain components has been divided into two types. However, only semitotal deformation has the value of practical application. The reason why the instantaneous deformation is also pointed out is for a general understanding of deformation, and semitotal deformation is on the basis of instantaneous deformation. The consistent relationship of the above-mentioned two types of deformation can be concluded in Table 1.