Thermodynamic foundation of generalized variational principle

One long-standing open question remains regarding the theory of the generalized variational principle, that is, why can the stress-strain relation still be derived from the generalized variational principle while the method of Lagrangian multiplier method is applied in vain? This study shows that the generalized variational principle can only be understood and implemented correctly within the framework of thermodynamics. As long as the functional has one of the combination A ( (cid:15) ij ) − σ ij (cid:15) ij or B ( σ ij ) − σ ij (cid:15) ij , its corresponding variational principle will produce the stress-strain relation without the need to introduce extra constraints by the Lagrangian multiplier method. It is proved herein that the Hu-Washizu functional Π HW [ u i , (cid:15) ij , σ ij ] and Hu-Washizu variational principle comprise a real three-ﬁeld functional. In addition, that Chien’s functional Π Q [ u i , (cid:15) ij , σ ij , λ ] is a much more general four-ﬁeld functional and that the Hu-Washizu functional is its special case as λ = 0 are conﬁrmed.


INTRODUCTION
Variational principles have always played an important role in both theoretical and computational mechanics . Generalized variational mechanics began in the 1950s with the breakthrough works of Reissner [2] on two-field variational principles for elasticity problems, in which the displacement u i and stress σ ij are considered independent fields. The previous literature, however, considered only displacement u i as a single independent field. Reissner introduced a functional F that is defined in terms of 12 arguments: six stresses σ ij and six strains ij : where B(σ ij ) is the elastic complementary energy density.
Reissner proved the following theorem: Among all states of stress and displacement that satisfy the boundary conditions of the prescribed surface displacement, the actually occurring state of stress and displacement is determined by the variational equation: where the symbol V indicates the volume of the elastic body and S p indicates that the surface integrals are to be taken over that part of the surface only where the appropriate surface stress is prescribed. In 1954, Hu published a paper [4] (its English version appeared in 1955 [6]) that borrowed the idea from Reissner [2] and successfully extended Reissner's two-field (displacement-stress) theory to a threefield (displacement-stress-strain) theory by introducing a functional H U given by Hu [6]) proved a theorem as follows: In 1955, Washizu [7] independently proposed the same functional and proved the same theorems as Hu [4,6].
Regarding the history of the generalized variational principle, Felippa [32] published a dedicated paper on the original publication of the generalized variational principle and showed that de Veubeke had developed a much more generalized variational principle in a report dated 1951 [3], in which four fields, namely, displacement, stress, strain, and surface force, were included. de Veubeke's four-field (u i , σ ij , ij , t i ) theory can be presented as follows [32]: δΠ = 0, where the functional The three-field standard form is obtained by setting t i = σ ij n j on S u a priori. Hence, Fellippa proposed that the canonical functional in Eq. 5 be called the de Veubeke-Hu-Washizu functional Π V HW . This proposal has been confirmed by The History of the Theory of Structures Searching for Equilibrium [33].
In 1983, Chien [23], who was Hu's supervisor and communicated Hu's paper to both the Chinese Journal of Physics [4] and Science Sinica [6], pointed out that, regarding all publications and reports of Reissner [2], Hu [4,6] and Washizu [7] did not give any information on how to construct the functional. The formulation of the generalized variational seems mystical, and thus Chien indicated that the trial-and-error method was used when Reissner, Hu, and Washizu formulated their functional [23,28].
To derive the generalized functional in a systematic way, Chien proposed to formulate the functional by using the well-known method of Lagrangian multipliers [21]. This method can be described as follows [21,23,28]: Multiply undetermined Lagrange multipliers by various constraints and add these products to the original functional. Considering these undetermined Lagrange multipliers and the original variables in these new functionals as independent variables of variation, it can be seen that the stationary conditions of these functionals give these undetermined Lagrange multipliers in terms of original variables. The substitutions of these results for Lagrange multipliers into the above functional lead to the functional of these non-conditional variational principles.
With the help of the Lagrangian multipliers, in 1983 Chien [23] successfully reformulated the two-field functional, namely, Π[u i , σ ij ] and Π[u i , ij ], which are called the Hellinger-Resissner functional [2] and De Veubeke functional [3], respectively. However, Chien [23] found that the constitutive relation between stress and strain cannot be included to form a three-field functional Π[u i , ij , σ ij ] due to the zero crisis of corresponding Lagrangian multipliers, as it is known to be impossible to incorporate this condition of constraint into a functional whenever the corresponding Lagrange multiplier turns out to be zero. Therefore, Chien claimed that the functional H U is not a three-field, but rather a two-field, functional. To address this point of view, Chien elegantly wrote a monograph on the generalized variational principle [28] and, to overcome the difficulty, he proposed a method of a higher-order Lagrange multiplier, a fourfield functional Π Q that is suggested to be expressed as follows: Chien proved that for no zero λ = 0, the δΠ Q [u i , σ ij , ij , λ] = 0 will produce balance equations, strain-displacement relations, stress-strain relations, and corresponding boundary conditions. Owing to the arbitrary nature of the Lagrangian multiplier λ, there are an infinite number of functionals Regarding Chien's questioning [23,28], no explanation from Hu, to the best of our knowledge, has been found in the literature. Because the formulation of the generalized variational principle has been recognized as a key contribution by a Chinese scholar to mechanics worldwide, and in particular, considering its importance in finite-element formulation, it is vital that Chien's question can be clearly answered. Otherwise, it will continue to cause confusion to both scholars and students. The task of answering this question has become a newcomer's responsibility.

CHIEN'S QUESTION ON THREE-FIELD VARIATIONAL PRINCIPLE
To propose our understanding of the issue, a brief review of one-, two-, and three-field variational principles, as well as of Chien's question, is presented now.
Let V be the volume of an elastic body, S u the boundary surface where displacement is given, and S σ the boundary surface where external force is given. Letting S be the total boundary surface, then S = S u + S σ .
Assuming the body is subjected to the action of distributed body force f i (i = 1, 2, 3), S p is the portion of the boundary surface subjected to the action of external surface forcep i and S u the portion of the boundary surface where the displacementū i is given. Under statical equilibrium, the stress state in the body is denoted by stress tensor σ ij . Displacement u i , strain ij , and stress σ ij satisfy the following five conditions, that is, which is the balance equation, and σ ij,j = ∂σij ∂xj , where j is a dummy index; which is a strain-displacement relation; which are stress-strain relations; which is the boundary conditions for a given surface displacement; and which is the boundary conditions for a given external surface force; n j is normal unit vector of surface S σ .
For a one-field potential functional, Its extreme condition δΠ[u i ] = 0 leads to the balance equation σ ij,j + f i = 0 with constraints of ij = 1 2 (u i,j + u j,i ) and σ ij = ∂ ij , boundary condition u i =ū i on S u , and σ ij n j =p i on S σ .
If one wishes to eliminate the constraint of straindisplacement relation ij = 1 2 (u i,j + u j,i ), according to Chien [23], a symmetric tensor of Lagrangian multiplier λ ij can be introduced and form a functional as The Lagrangian multiplier λ ij can be determined by ∂ ij ; therefore, the two-field de Veubeke functional is If one wishes to carry on this process and eliminate the stress-strain relations, σ ij = ∂A( ij ) ∂ ij , one can do so by introducing another symmetric tensor of Lagrangian multiplier η ij to form a new functional as The Lagrangian multiplier η ij can be determined by δΠ[u i , ij , σ ij ] = 0, which leads to η ij = 0 and kl ∂ 2 A(eij ) ∂eij ∂e kl = 0, since ∂eij ∂e kl > 0, thus giving an incorrect result ij = 0.
These results reveal that the stress-strain relation cannot be included in the functional, Π[u i , ij , σ ij ], by the Lagrangian multiplier method. In other words, it is impossible to remove all the constraints simply because the related Lagrange multiplier is equal to zero in the stationary condition. This Lagrangian multiplier method crisis was discovered by Chien in 1983 [23], when he published a monograph and provided a comprehensive discussion of the issue [28].

THERMODYNAMIC FOUNDATION OF GENERALIZED VARIATIONAL PRINCIPLE
For the sake of brainstorming on the stress-strain relation, a quick brief of constitutive theory from a thermodynamics perspective is presented here.
Following the above thinking, it is easy to know that a functional has included a stress-strain relation if it contains either the terms In other words, if the structure of the functional was in the following form, it implies that the stress-strain relation and due to the arbitrary variation δ ij , one therefore has the following stress-strain relation: Similarly, if the structure of the functional was in the following form, it implies that the stress-strain relation ij = ∂B(σij ) ∂σij is included.
With this understanding, an examination of the following Hu-Washizu functional is necessary: The combination of the underlined terms in the above functional is exactly the term of A( ij ) − σ ij ij . Therefore, the Hu-Washizu functional Π HW [u i , ij , σ ij ] has already included the stress-strain relation σ ij = ∂A( ij ) ∂ ij . The Hu-Washizu functional Π HW [u i , ij , σ ij ] is a real three-field functional. This key point was not understood by Hu [4,6], Reissner [2], and de Veubeke [3] when they constructed their own generalized functional by the trial-and-error method. This situation is very similar to the formulation of the Schrödinger wave equation in quantum mechanics. The Schrödinger equation provides a way to calculate the wave function of a system and how it changes dynamically in time. However, the Schrödinger equation does not directly say exactly what the wave function is. For instance, Schrödinger originally viewed the electron's wave function as its charge density was smeared across space, but Born reinterpreted the absolute square value of the wave function as the electron's probability density distributed across space.
Of course, Chien's functional Π Q [u i , ij , σ ij ] in Eq. 6 is not only a three-field functional, but also much more general one, since it contains all elements, such as A( ij), B(σ ij ), and σ ij ij . The arbitrary nature of λ provides some kind flexibility in constructing a generalized functional.

CONCLUSIONS
It has been shown in this study that the generalized variational principle can only be correctly understood and implemented within the framework of thermodynamics. As long as the functional has any one of the combination A( ij ) − σ ij ij or B(σ ij ) − σ ij ij , its corresponding variational principle can produce the stress-strain relation without the need to introduce extra constraints by the Lagrangian multiplier method.
It has been proved that the Hu-Washizu functional Π HW [u i , ij , σ ij ] is a real three-field functional and therefore that the Hu-Washizu variational principle is a threefield variational principle. In addition, that Chien's functional Π Q [u i , ij , σ ij , λ] is a much more general four-field functional has been confirmed.
Owing to Chien's academic acumen, he discovered the problems and carried out meticulous research. His research inspired the author to think further on this issue. Although it was finally proved that the result of the Hu-Washizu functional was only correct in form, the current understanding has risen to a new level, leading to the resolution of the historic academic controversy on the issue of constructing a three-field functional.

ACKNOWLEDGMENTS
The author is honored to have benefited from personal connections with both Prof. Chien and Prof. Hu. Professor Chien supervised Prof. Kai-yuan Ye, the author's Ph.D. supervisor. Professor Hu was the committee chairman for the author's post-doctoral final progress report when he completed his post-doctoral research at Tsinghua University in 1991. In the great discovery of the generalized variational principle, both Prof. Qian and Prof. Hu made original contributions. Their academic thoughts are very important to our understanding of the generalized variational principle. Now that both Prof. Chien and Prof. Hu have passed away, if some valuable answer to the Qian question can be provided it can serve as the best tribute to both. Therefore, it is my privilege to dedicate this paper to the memories of Prof. Chien and Prof. Hu for their great contribution to the theory of the generalized variational principle.
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