J. Cent. South Univ. Technol. (2008) 15(s1): 039-042

DOI: 10.1007/s11771-008-310-6

Plastic variational principle based on the least work consumption principle

TANG Song-hua(唐松花)^{1, 2}, LUO Ying-she(罗迎社)³, ZHOU Zhu-bao(周筑宝)², WANG Zhi-chao(王智超)¹

(1. College of Civil Engineering and Mechanics, Xiangtan University, Xiangtan 411105, China;

2. School of Civil and Architectural Engineering, Central South University, Changsha 410075, China;

3. Institute of Rheological Mechanics & Material Engineering, Central South University of Forestry & Technology, Changsha 410004, China)

Abstract: Plastic variational principles are foundation to solve the boundary-value problems of plastic mechanics with the variational method (or energy method) and finite element method. The most convenient way of establishing different kinds of variational principles is to set up the extreme principle related to the studied problem. Based on a general new extreme principle—the Least work consumption principle, the variational principles of the rigid-plastic and rigid-viscoplastic material were derived. In comparison with existing methods, the method in this paper is more clear and direct, and the physical meaning is clear-cut. This method can offer a new way for establishing other kinds of variational principles.

Key words: rigid-plastic; rigid-viscoplastic; variational principle; the least work consumption principle

1 Introduction

In order to figure out the energetic parameter, deformation parameter and the distribution of the strain and the stress, the related equation group should be solved on the stated original conditions and the boundary conditions, which is called the boundary-value problem in the plastic mechanics. The variational principle of the plastic mechanics is the foundation for solving the boundary-value problem using the variational method (or energy method) and the finite element method^[1-6].

Ref.[7] derived the variational principles of different kinds of plastic material using the imaginary work principle, which needs to make up a potential function F or F′ satisfying  or , so it sets up limitation to the constitutive relation of materials. And for the imaginary work principle is not a extreme principle, Ref.[7] gave the minimum proving process of the following functional expression (i.e. Eqn.(4－39) in Ref.[7]):

              (1)

As is known that the most convenient way to build various variational principles is to build the extremum principle of a certain functional related to the problem, for example, the least potential energy principle, the least remaining energy principle, Hamilton principle in existence and the least work consumption principle brought forward by Ref.[8]. After such extremum principle is founded, even if the basic equations and fixed-solution conditions which should be satisfied can not (or only partly) be obtained, through solving the extreme value of the functional related to the extremum principle, the lack conditions can be introduced as the restriction conditions by means of Lagrange multiplier method. Thus the conditions of solving the extreme value of the new functional turned from the conditional variational problem to non-conditional variational problem by means of Lagrange multiplier method equal to all basic equations and fixed-solution conditions.

Started from the least energy dissipation principle with new intensions, the least work consumption principle applicable to general situations were derived firstly^[8]. Then based on this least work consumption principle, the variational principles of the rigid-plastic and rigid-viscoplstic material were derived, and the two variational principles are identical with the ones in Ref.[7]. This method has no limitation to the constitutive relation of materials. Besides, the new extremum principle not only involved the potential term, but also the energy dissipation term, so it eliminated the unreasonable condition that the dissipation energy was taken as the potential energy in the elastic variational principle nowadays. In comparison with existing methods, the method in this paper is more clear and direct, and the physical meaning is clear-cut. This method can offer a new way for establishing other kinds of variational principles.

According to the first law of thermodynamics, the mechanical work acted to the system W(t) at t moment will produce potential energy U_S(t), kinetic energy T(t) and dissipation energy U_H(t) in general (that is, the mechanical work acted by the extra forces will make the potential energy, kinetic energy and the dissipation energy increase, so U_S(t), T(t) and U_H(t) here are actually the increments caused by the extra force work. When the original value U_S(0)=U_H(0)=T(0)=0, U_S(t), T(t) and U_H(t) can be considered as the increments). For the signs of the produced potential energy and kinetic energy are contrary, so we have

                    (2)

    Besides Halmilton principle (i.e. the least action principle) gives the following equation:

     (3)

    Because t₁ and t₂ in Eqn.(3) are the optional moments in the process, the following equation can be given as

                           (4)

    The energy dissipation term is not included in Eqn.(4), so Eqn.(4) is independent of the experienced route of the process, and it is applicable to the optional moment t expressed by the time parameter t′. Then from Eqn.(4), we have

                         (5)

that is,

                            (6)

Besides from Eqn.(1), we have

                 (7)

where  is the consuming rate of the whole external work,  is the whole energy dissipation rate,  is the variety rate of the whole potential energy, and  is the variety rate of the whole kinetic energy.

According to the least energy dissipation principle with new intensions in Ref.[8] (i.e. “any energy consuming process will proceed with a way of consuming the least energy”). Here so-called “proceed with a way of consuming the least energy” means that at any moment in the energy consuming process its energy consuming rate will be the minimum of all possible energy consuming rate. So we have

    Compared Eqn.(6) and Eqn.(7), the following equation can be obtained:

            (8)

and Eqn.(8) is the expression of the least work consumption principle.

    Summarized above the least work consumption principle can be obtained as follows: any external work consuming process acting on the system will be proceeding in a way of consuming the least external work on the corresponding restraint conditions. Here “external work consuming” means the external work is completely turned into potential energy, kinetic energy and dissipated energy of the system. Here “restraint conditions” means the control equations and fixed-solution conditions that the physical quantities in the expression of “external work consumption” should satisfy. Here so-called “proceeding in a way of consuming the least external work” means at any moment in the process of external work consuming, the consuming rate of external work takes the minimal value of all possible ones. The variational principle of various kinds of mechanics problems (whether the energy dissipation is needed to be considered) can be expressed as the conditional variational problem solving the minimal value to the functional expression of the consuming rate of external work on the condition of satisfying all basic equations and fixed-solution conditions.

From the expression Eqn.(8) of the least work consumption principle, it is clear that there are three kinds of expressions for the least work consumption principle given as

           (9)

                  (10)

            (11)

where  F_i and  are the body forces and surface forces respectively, and  is the variety rate of the displacement component (i=1, 2, 3).

As the expression of the least work consumption principle(i.e. Eqn.(8))set no limitations to the constitutive relations of materials, and it involves not only potential energy and kinetic energy, but also the energy dissipation, it can be considered as a extreme principle more general than Hamilton principle, the least potential energy principle and the least complementary energy principle. In Refs.[9-11] this principle were applied in the elastic, plastic and creep situations. In this paper this principle will be used to derive the rigid-plastic and rigid-viscoplastic variational principle, for it is also applicable to the dissipation situation. 

3 Variational principles of rigid-plastic material based on the least work consumption principle

3.1 Basic equations and fixed-solution conditions of rigid-plastic mechanics

According to Ref.[7], the basic equations and the boundary conditions which should be satisfied in the rigid-plastic area are as follows.

Equilibrium equations:            (12)

Mises yield conditions:         (13)

Geometry equations:       (14)

Constitutive relations:          (15)

Volume conditions:               (16)

Stress boundary conditions:       (17)

Displacement boundary conditions:    (18)

where  are the stresses,  are the deviatoric stresses,  are the strains,  are the equivalent strain rates, and

 is the velocity components,  is the yield stress,  are the Kronecker symbol,  is the surface forces components,  is the boundary velocity components and  is the extra normal direction cosine of boundary surfaces.

3.2 Variational principles of the rigid-plastic material

According to Eqns.(12)-(18), we have the following conclusions:

                            (19)

                         (20)

and ，, so

                            (21)

Then from the third expression of the least work consumption principle, that is,

The following equation can be obtained as

                 (22)

From Ref.[8], we have

                        (23)

Substituting Eqn.(23) into Eqn.(22), we get

          (24)

After compared, Eqn.(24) is just the first variational principle of the rigid-plastic material, and the term in the big bracket is the same as Eqn.(4－39) in Ref.[7].

4 Variatioanl principle of rigid-viscoplstic material

The constitutive equation of the rigid-viscoplstic material is

                                (25)

Eqn.(25) is the sole-value function, so , and

(26)

where  is the equivalent stress.

And; is the equivalent stress rate, .

From Ref.[7],

                         (27)

where  is the static yield stress,  is the fluid parameter.

Eqn.(27) can be changed to

                           (28)

then Eqn.(26) is integrated into

and , , then we have

                             (29)

Substituting Eqn.(29) into Eqn.(24), the variatioanl principle of the rigid-viscoplstic material is:

               (30)

Eqn.(30) is just the variational principle of the rigid- viscoplstic material, which is identical with Eqn.(4-57) in Ref.[7].

Therefore the complete deriving process of the rigid-plastic variatioanl principle and the  rigid- viscoplstic variational principle are given. The two variational principles were built using the least work consumption principle, i.e., a extreme principle, so the minimum proving are not needed.

5 Conclusions

1) The most convenient way to build various variational principles is to build the extremum principle of a certain functional related to the problem, for example, the least potential energy principle, the least remaining energy principle, Hamilton principle in existence and the least work consumption principle brought forward by Ref.[8]. The variational principles of the rigid-plastic and rigid-viscoplstic material are derived using a general new extreme principle—the Least work consumption principle.

    2) Compared with the method using the imaginary work principle, the method using the least work consumption principle in this paper sets no limitations to the constitutive relations of materials, which is the same as the imaginary work principle. Besides, the new extremum principle involves not only the potential term, but also the energy dissipation term, so it eliminates the unreasonable condition that the dissipation energy is taken as the potential energy in the plastic variational principle nowadays. The method is clearer and direct, the physical meaning is clear-cut, and can offer a new way for establishing other kinds of variational principles.

3) The least work consumption principle is a new general extreme principle involving the potential energy, kinetic energy and dissipation energy, so it has a wider applicable prospect than the extreme principle now available (for example, Hamilton principle, the least potential energy principle and the least complementary energy principle).

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(Edited by YANG You-ping)

Received date: 2008-06-25; Accepted date: 2008-08-05

Corresponding author: TANG Song-hua, Associate professor; Tel: +86-731-5165473; E-mail: tangsh2003@yahoo.com.cn