The Prandtl-Reuss equations revisited

Transcription

1 ZAMM Z. Angew. Math. Mech. 94, No. 3, (2014) / DOI /zamm The Prandtl-Reuss equations revisited Otto T. Bruhns Institute of Mechanics, Ruhr-University Bochum, Universitätsstr. 150, Bochum, Germany Received 27 October 2013, accepted 11 November 2013 Published online 25 November 2013 Communicated by Holm Altenbach Key words Plasticity, Prandtl-Reuss-theory, deformation theory, hypoelasticity, large deformations, logarithmic rate. At the beginning of the last century two different types of constitutive relations to describe the complex behavior of elastoplastic material were presented. These were the deformation theory originally developed by Hencky and the Prandtl-Reuss theory. Whereas the former provides a direct solid-like relation of stress as function of strain, the latter has been based on an additive composition of elastic and plastic parts of the increments of strains. These in turn were taken as a solid- and fluid-like combination of the de Saint-Venant/Lévy theory with an incremental form of Hookes law. Even nowadays this Prandtl-Reuss theory is still accepted within the restriction of small elastic deformations, i.e. it is generally stated in most textbooks on plasticity that this theory due to a number of defects can not be applied to large deformations. In the present article it is shown that this restrictive statement may be no longer true. Introducing a specific objective time derivative it could be shown that these defects disappear. 1 Introduction The present article is the second in a series of contributions intended to highlight the history of plasticity. The first was devoted to the life and works of Heinrich Hencky and the rapid development of plasticity at the beginning of the last century [3]. This second contribution will discuss somewhat more in detail the significant input of the young Hungarian András or Endre Reuss to this period of progress in plasticity. In addition to and in competition with the deformation theory of Hencky, Reuss in 1930 has introduced the first incremental flow law, where the recoverable elastic behavior of the material has been considered by an incremental elastic-like formulation. He therefore has extended the de Saint-Venant/Lévy approach for rigid plastic behavior simply by adding the elastic and plastic parts of the total strain rate to gain a unified relation for elasto-plastic behavior. These so-called Prandtl-Reuss equations rapidly were accepted in science and application. First problems emerged, when after World War II these relations were transferred to application within large deformations. The objectivity of the incorporated rates was questioned. As a consequence, several possible objective rates were discussed, some of them producing spurious effects in the results of calculation. Moreover, the (alleged) dissipative character of the elastic-like (hypoelastic) part of the Prandtl-Reuss equations was taken as argument to discredit these relations as only applicable for small at least small elastic deformations. This contribution will show that there also exists a hypoelastic formulation for the incremental elastic part of the Prandtl- Reuss equations without any dissipation even if large deformations appear. Thus, this formulation may help to vindicate the Prandtl-Reuss equations as a generally applicable tool to solve problems of finite elasto-plastic behavior. 2 Some historical remarks The beginning of plasticity theory may be identified with the publication of Henri Tresca in 1868 [92] 1. Based on the observations of a series of experiments he published a hypothesis where metals begin to flow when the largest shear stress reaches a critical value 2, or in commonly used today s notations: τ max = 1 2 (σ 1 σ 3 )=k, (1) with σ 1 the largest and σ 3 the smallest value of the principal stresses, and k the experimentally observed shear yield limit. otto.bruhns@rub.de, Phone: Reference is also made to a previous publication in 1864 [91]. 2 He probably may have resorted also to earlier works by Coulomb [9].

2 188 O. T. Bruhns: The Prandtl-Reuss equations revisited Using this condition, Barré de Saint-Venant in 1870 has presented his five equations of hydro-stereodynamics [82] for a problem of plane deformations, emanating from a material behavior of a viscous fluid 3 σ = pi +2μ v ( D 1 3 tr(d)i ), D 1 3 tr(d)i = 1 2μ v (σ + pi), (2) where σ is the stress tensor, p the hydrostatic pressure, and μ v the viscosity. I herein is a unit tensor and D the rate of deformation (stretching) tensor, the symmetrical part of the velocity gradient. In addition to balance equations, the (assumed) incompressibility, as well as Tresca s yield criterion (1), this was a relationship of the following form D xx D yy D xy = σ xx σ yy σ xy, (3) where nothing else but a proportionality between the deviators of stress and strain rate has been expressed. In the same year, Maurice Lévy [41] has transferred this proposition to the general spatial problem: D 1 3 tr(d)i = c τ, τ = σ 1 tr(σ)i, (4) 3 with τ the deviatoric stresses, and c a constant of proportionality. Thus, the basics of a simple flow theory were established. Because of its associated mathematical difficulties 4 this theory, however, did not find any application. It took more than 30 years until these ideas were taken up again. Moreover, from the very beginning the development of plasticity was in this conflict: Does the body under consideration behave like a solid or more fluid-like and does this possibly also depend on the specific problem to be solved? In the former case, the strength properties of the structure might be in major focus and accordingly the plastic behavior is interpreted as the behavior of a solid. 5 On the contrary, in the latter case a typical forming process might be considered and, thus, the material might be understood as a viscous fluid. To simplify the foregoing theory, in 1913 Richard v. Mises [51] replaced Tresca s yield condition (1) by (τ 1 ) 2 +(τ 2 ) 2 +(τ 3 ) 2 =2k 2, (5) with τ i the principal deviatoric stresses. This can be transformed to give F = tr(τ 2 ) 2 k 2 =0. (6) Together with Eq. (4) 1, the balance equations and the incompressibility condition, this system of partial differential equations at the time still has been considered unsolvable with a few pathological exceptions. In contrast, however, there was the wish to develop practically manageable procedures that could allow to reflect the progress of the material description in the calculation procedures and design rules. Thus, meaningful and yet reasonable simplifications or models were developed. In this sense, also the simplifications introduced by Ludwig Prandtl [71] to consider the material as an (i) ideal-plastic or (ii) elastic-plastic body were to be understood. A further simplification is pursued in 1921 and 1923 by Prandtl [72] and Hencky [21], emerging from the following question: Do there possibly exist special cases such that with the help of balance equations and yield condition (6) alone, solutions to the given problem could be found? For the general spatial problem with its 4 conditions and 6 unknown stress components this obviously would be not the case. If we restrict our considerations, however, e.g. to axially symmetric problems, we note that only 3 unknown stress components are facing 3 conditions. The focus of the following period was therefore on the solution of these statically determinate cases. For problems of plane deformations, moreover, both yield conditions of Tresca (1) and von Mises (6) coincide 6. Following some suggestions by Prandtl [72] and Nádai [53] Hencky in 1923 introduced the slip-line theory [21], which almost three decades later became widely accepted in engineering practice, e.g. of forming processes. Prandtl [73] as well as Carathéodory and Schmidt [7] completed these ideas by graphical solution methods and numerous additional statements. Hilda Geiringer and William Prager [15] contributed to the spreading of this theory. 3 We also refer to the earlier works of Navier [57], Poisson [63], de Saint-Venant [81], and Stokes [87]; see also the report by Truesdell [93]. 4 The interested reader will recognize that, formerly, it was almost impossible to solve systems of partial differential equations. 5 This class of material behavior is typically associated with a relation of stress σ as function of strain ε, i.e.σ = σ(ε), whereas the fluid-like behavior of e.g. Eq. (4) is characterized by a relation σ = σ(d). 6 We note that this is not the case for problems of plane stress.

3 ZAMM Z. Angew. Math. Mech. 94, No. 3 (2014) / Hencky s deformation theory In 1924, Hencky [22] developed a simple constitutive law for elastic as well as elastoplastic behavior, specifying a relationship between stresses and strains. He thereby revisited an energetic approach by Haar and v. Kármán [20]. This so-called deformation theory 7 was the first step to a modern design of structures within the range of plastic deformations. As a result he received ε 1 1+ϕ tr(ε)i = 3 2μ τ, tr(ε) = 1 tr(σ), 3K =3λ +2μ, (7) 3K a model very similar to Hooke s law (refer, e.g. to [2]), where ϕ is a still undetermined Lagrange parameter, λ and μ are the two Lamé s constants of elasticity, and K is the bulk modulus. For ϕ =0, this law changes into that of an linear elastic material. The volumetric deformation is purely elastic and at plastic behavior the shear modulus μ is reduced by (1 + ϕ), the material thus becomes softer. This formulation was rapidly accepted, even if it soon also meets its limits: A neutral change of stresses, as for instance occurs in non-proportional loading, cannot be reflected. For many so-called proportional problems, however, it represents not only the first but also a very simple method. It should be noted that Hencky has assumed that a body under increasing load would be deformed first elastically and then plastically after having reached the yield point. In the interior of the structure, however, still remained a so-called elastic core. This corresponds to the above mentioned concept of plastic behavior as that of a solid material. The above model (7) was assumed to hold during loading of a body; for unloading it must be replaced by a differential form of Hooke s law, i.e. for small deformations with D = ε ε 1 1 tr( ε)i = 3 2μ τ. (8) This amendment was due to Nádai [54], who together with Ilyushin promoted the spreading of this theory. 4 The original work of Prandtl and Reuss Apparently independent 8 from a work by Prandtl [71], where already elastic deformations have been considered in a plane problem, the Hungarian András (Endre) Reuss [76, 77] in 1930 connected the de Saint-Venant/Lévy approach (4) with the description of an (incremental) elastic behavior. 9 For this purpose, like Hencky, he emanated from the v. Mises yield condition (6) and obtained on a comparable way a constitutive law 10 D 1 3 tr(d)i = De + D p = 1 2μ τ +Λτ, ( ) =( ) 1 tr( )I, (9) 3 that with Λ still contains a yet undetermined function, and where the volumetric part tr(d) =tr(d e )= 1 tr( σ) 3K (10) remains purely elastic. Here and in the following a prime will mark a deviator of a second order tensor, and D p stands for the (deviatoric) plastic part of the strain rate and D e for its elastic part. Thus, the basic version of the nowadays commonly used Prandtl-Reuss theory was introduced. It would be instructive to stop here for a while and explore in more detail the question: Who was this A. Reuss, who together with the already generally respected Ludwig Prandtl has introduced the idea that the influence of the recoverable elastic deformations on the description of the complex plastic processes may simply be obtained by adding an incremental form of Hooke s law to the de Saint-Venant/Lévy relation (4) 1? András or Endre Reuss 11 was born on the 1st of July, 1900, in Budapest, where he also died on the 10th of May, Already as high school student of 18 years he was awarded the first Lóránt Eötvös prize of the Hungarian Mathematical 7 According to Prager [66] this terminology was introduced by Ilyushin [34]. Sometimes it is also referred to as a finite stress-strain law (see e.g. [67]) in contrast to the incremental flow law of Prandtl and Reuss. 8 We refer to a footnote of Reuss in [77] where it is stated that while elaborating his 1930 paper, the lecture of Prandtl was not known to him. 9 It should be noted that this incremental form of an elastic relation in general fails to be elastic, e.g. in the sense that a circular process in loading and unloading starting from a virgin state must terminate at a stress- and strain-free situation. This fact, however, was not known at that time. It was Truesdell [93, 95] who more than three decades later introduced the term hypoelasticity to elaborate this difference. 10 In some literature the so-called plastic multiplier is introduced as λ or λ. To avoid any confusion with the first Lamé constant, we here prefer to use Λ. 11 The support of Prof. em. Gyula Béda from Budapest University of Technology (now BME) is most gratefully acknowledged.

4 190 O. T. Bruhns: The Prandtl-Reuss equations revisited Fig. 1 Ludwig Prandtl ( ) and András (Endre) Reuss ( ). and Physical Society in a national student competition of mathematics. He was then educated in mechanical engineering and received his degree (Diplom) in 1922 at the Faculty of Mechanical Engineering of Budapest University of Technology (TU). After having completed his studies, in the years he worked as assistant at the Department of Applied Mechanics. In 1924 he left university to join the service of the municipal gas works of Budapest where he remained for 26 years doing valuable engineering as operating engineer. In 1932 he received his doctorate (Dr. techn.) and 1942 he finished his habilitation and became lecturer (Privatdozent) of the TU Budapest. It seems to be worth noting that he wrote his first publication in the field of mechanics at the age of 25 years. Likewise his important 1930 paper appeared before his thesis about the influence of cold working on the yield limit of steel was finished. In 1950, he joined the Hungarian Central Planning for the Chemical Industry. In 1953 he was appointed professor at the chair of Applied Mechanics of the TU Budapest, where he worked until his retirement in His scientific findings have been recognized worldwide. The local scientific community of those days, however, did not appreciate his work. Because of this lack of understanding, the problems of the war years and in addition some family problems, his scientific work was interrupted for several years. Many generalizations of the above theory were given shortly later. So Geiringer and Prager [15] in addition to the yield condition F =0introduced a second condition Φ=0as flow potential 12. The self-evident case F =Φthen is called an associated theory 13 and a normality rule can be derived from the potential property of Φ 14 D p =Λ 1 Φ 2 τ. (11) For the general case of elastoplastic deformations, the yet undetermined parameter Λ can be eliminated with the aid of the so-called condition of consistency, e.g. Φ =0, (12) i.e., during plastic deformation the stress point in stress space must remain on the yield surface. This idea was introduced by Geiringer [14]. Also, the hardening of the material was taken into account by modifying the yield condition. First steps towards a description of an isotropic hardening adapted the parameter k of the uniaxial tension test as function of the history of the corresponding experiments and arrived at, e.g. the following proposals: k 2 = k0 2 + f(k 1 ), k 1 = σ : D p dt, (13) k = k 0 + f(k 2 ), k 2 = tr(d p2 ) dt, (14) where Eq. (13) was introduced by Schmidt [83], and Eq. (14) is due to Odqvist [60]. 15 Here, two different parameters k 1 and k 2 have been defined to describe the progress of the inelastic process under consideration. Whereas in the first case the (accumulated) plastic work w p is used, the second description has defined the so-called arc-length of plastic deformation. 12 This idea of introducing a flow potential in addition to a flow- or yield condition has already been mentioned by v. Mises in [52]. 13 On the contrary, any case F Φis indicated as non-associated. 14 We note that the factor 1 in Eq. (11) has been introduced arbitrarily. As it can be subsumed with Λ, any other factor would not alter the result Similar results could also be found in Fromm [12] and Prager [64].

5 ZAMM Z. Angew. Math. Mech. 94, No. 3 (2014) / 191 σ σ ε ε Fig. 2 Tensile test, cyclic loading with unloading and reloading in different directions: same direction (left) and reverse loading expressing the Bauschinger effect (right). It turned out, however, that these hardening models were not capable to adequately describe a cyclic behavior, in particular the Bauschinger effect (refer to Fig. 2), as they imply a symmetric behavior in tension and compression. Consider, e.g., a cyclic process of loading, unloading and reloading of a simple bar as depicted with Fig. 2. Here, the observed stress-strain behavior is illustrated for reloading in different directions. Whereas in the left case subsequent yielding begins when the stress reaches its former value during plastic flow, this is not the case for reloading in the opposite direction. An isotropic hardening model would not be able to reproduce this behavior. On the contrary, for a continued cyclic process within fixed values of strain it would finally terminate in a purely elastic behavior. To overcome this deficiency, approaches to the description of a kinematic hardening were derived by Reuss and Prager [64, 67] 16 in the form Φ=(τ c ε p ):(τ c ε p ) 2k 2 (15) and allow to take into account the Bauschinger effect. 17 Then, the first evolution equation for this kinematic hardening originated from Melan [48] with Φ=(τ α) :(τ α) 2k 2, α = cd p. (16) A further modification was due to Ziegler [107], where instead of Eq. (16) 2 a slightly different evolution equation for the internal variable 18 α was used. Thus, for small deformations with D = ε, we can conclude that ε = ε e + ε p, (17) and with Eqs. (8) and (11) ε = 1 2μ ( σ ν ) tr( σ)i +Λ 1 Φ 1+ν 2 τ. (18) This is the classical Prandtl-Reuss theory for small deformations. Herein ν is Poisson s ratio, which can be determined from the Lamé constants (refer e.g. to [2]). Another somewhat more general description of this law can often be found in the form ε = C 1 : σ +Λ 1 2 Φ τ, (19) 16 We refer to footnote 4 on page 79 of [64] and page 238 of [67]. Contrary to what is usually stated in most textbooks on plasticity this yield condition was first suggested by Reuss. 17 Here ε p stands for the irreversible plastic part of the strain ε. 18 We should mention here that none of the cited works has used the notion of internal variable. This term has been introduced much later. Instead, kinematical considerations in the stress space, preferably in a two-dimensional space were made. This incidentally has led to the notation of kinematic hardening.

6 192 O. T. Bruhns: The Prandtl-Reuss equations revisited where C is the elastic stiffness tensor with components C ijkl = λδ ij δ kl + μ(δ ik δ jl + δ il δ jk ) for an isotropic material. Thus, for a v. Mises material with yield condition (6) and isotropic hardening according to Eq. (13), we determine ε = C 1 : σ +Λτ. (20) Since the expression σ : ε p = ẇ p (21) in Eq. (13) is the plastic (dissipative) part of the power, we may reformulate this relation as k 2 = k f(w p ). (22) Now with the help of the consistency condition (12) Φ =2τ : τ 2 dk2 dw p ẇp =0, (23) and introducing Eq. (21), Λ can be worked out to give Λ= 1 dk2 τ : τ, H =2k2 H dw p, (24) where H = H(w p ) is a hardening function which has to be determined from appropriate experiments. With these results we finally arrive at ε = C 1 : σ + 1 H (τ : τ )τ. (25) For application especially for numerical calculations with nowadays usual computer codes an inversion of Eq. (25) is necessary. We therefore multiply it (double contraction) from the left by C. Due to the specific symmetry properties of C,wehaveC : C 1 = I, wherei is the fourth-order identity tensor, and thus ( σ = C : ε 1 ) H (τ : τ )τ. (26) Now again multiplying this result from the left by τ yields From this we can derive τ : σ = τ : τ = τ : C : ε 1 H τ : τ = τ : C : ε 1+ 1 H τ : C : τ (τ : τ )τ : C : τ. (27) (28) and finally introducing this expression into Eq. (26), the inverted form 19 of relation (25) becomes ( σ = C : ε τ τ : C : ε H + τ : C : τ ). (29) Compared with the finite stress-strain relations of Hencky s deformation theory (7) these Prandtl-Reuss equations correspond to a combination of a fluid- and a solid-like material with the appropriate properties. In passing we should mention that Reuss in [78] also has introduced a less known description of the Tresca yield condition (1) as function of the principal invariants J i of the stress deviator τ, namely with principal invariants F =4J J J 2 2 k2 +96J 2 k 4 64 k 6 =0, (30) J 1 =0, J 2 = 1 2 tr(τ 2 ), J 3 = 1 3 tr(τ 3 ). (31) 19 This calculation was first carried out by Hill [25] for the specific case of an isotropic material.

7 ZAMM Z. Angew. Math. Mech. 94, No. 3 (2014) / 193 The debate as to what might be the better of the above mentioned two theories started of course immediately and dragged on until the fifties and sixties of the last century 20. At the beginning, it seemed as if the deformation theory had some advantages compared to the Prandtl-Reuss theory. Hohenemser [32] and Prager [65] indicated that in comparison with numerous experiments the deformation theory could reflect these results much better. Finally, one also must recognize that in these pre-war years, e.g. in aircraft construction, the deformation theory simply prevailed in practical applications because of its easy handling. This appreciation, however, has changed over the years. Today, the deformation theory seems to be almost unknown, whereas the Prandtl-Reuss theory is generally accepted at least what concerns its application to problems of small deformations. 5 Some amendments Numerous modifications of this classical flow rule have been introduced in the last century in various directions: Firstly, the concept of kinematic hardening (16) has been adopted to account for the Bauschinger effect during cyclic loading. With a slightly modified expression for the plastic part of the power ẇ p, namely s : ε p = ẇ p, s = τ α, (32) this leads to a quite similar Prandtl-Reuss equation ( ) s : C : ε σ = C : ε s, (33) H + s : C : s where compared with Eq. (29) τ has been replaced by the reduced (deviatoric) stresses s and the hardening function H of Eq. (24) 2 by H = H +2k 2 c, (34) combining the isotropic and the kinematic hardening. Moreover, to describe in a more appropriate manner the behavior during cyclic hardening and to model the different ratchetting effects, the linear relation (16) 2 has been modified by introducing a second term, e.g. ( α = c ε p α 2k 2 ẇp). (35) This generalization was first discussed by Armstrong and Frederick [1] and then propagated by Chaboche in several papers (see e.g. [59]) 21. Applying this modification to the so far discussed Prandtl-Reuss equations, we finally arrive at ( ) s : C : ε σ = C : ε s, (36) Ĥ + s : C : s where now Ĥ = H c tr(sα) =H + c(2k 2 tr(sα)). (37) Here, the only difference is in the hardening function which now also changes with a term tr(sα). Thus, this new second term in Eq. (35) allows for different evolutions in the tensile and the compressive parts of the hysteresis loop. In addition, several kinematic variables can be used, with the generalization α = i α i (i =1...n). Secondly, a further modification was carried out with the yet undetermined isotropic hardening. Here, the evolution of k 2 (which is attributed to the diameter of the yield surface) may be described by a relation of the kind (refer to e.g. [59]) 2k ( ) k 2 2 = b k 2 1 ẇ p, (38) where k 2 is a (stabilized) asymptotic value of k2 with initial condition k 2 (0) = k0 2. This equation can be integrated to give k 2 + k 2 ln(k 2 k 2 )+w p = c. (39) 20 We e.g. refer to Prager [66, 67] and the discussion in [68] between Budiansky and Prager. A first critical remark was already made by Hohenemser and Prager [33] in 1932 toward the variational principle by Haar and v. Kármán, that Hencky has used (refer to their footnote 22). 21 It should be mentioned that in most of these works the progress of the plastic processes is described by accumulated plastic strains k2 according to Eq. (14) 2 rather than by the accumulated plastic work w p. This, however, does not alter the given results.

8 194 O. T. Bruhns: The Prandtl-Reuss equations revisited Applying, e.g., the above initial condition, the constant c is determined and we arrive at an implicit description for k 2 = k 2 (w p ) w p = 2 ( k 2 k0 2 b + k2 ln k2 k 2 ) k0 2. (40) k2 This is a logarithmic function that starts out from a virgin state w p =0with the initial value k0 2 and approaches asymptotically the final value of k. 2 Finally, numerous applications of plasticity theory e.g. in metal forming processes made it necessary to extend these relations to larger deformations. Whereas Hencky s theory right from the beginning and even from its setting was restricted to applications within infinitesimal small deformations, this was not the case for the Prandtl-Reuss theory. Having in mind its setting as a combination of fluid-like and solid-like materials, its description of the fluid-like part originally was introduced as a relation of the stress σ as function of the rate of deformation (stretching) tensor D. Thus, it should be straightforward to replace the rates of the strains in Eq. (17) by the respective parts of the rate of deformation, viz. D = D e + D p. (41) This, however, would cause two new problems related with the elastic part of the aforementioned decomposition: (i) First, if the rates of the strain in Eq. (17) are replaced by the stretching, a relation between strain and stretching would become necessary 22, e.g. to determine the strains in the plastically deformed structure. (ii) Then the question arises, what kind of rate should be used for the herein incorporated stresses and stress-like quantities? These questions will be discussed more detailed in the next sections. 6 The step toward finite deformations It is noticeable that until the 1930 s in the mathematical representation of the so far considered contributions, we find Cartesian coordinates, small deformations, and all relations written in detailed component notation. Hencky [23] in 1925 was one of the first to introduce tensor analysis into mechanics, to avoid the maze of formulas, that so far has prevented from calculating finite deformations... he said 23. To this end in [24] he introduced a logarithmic strain measure 24, with the advantage to allow a correct superposition of strains and, moreover, to describe a total compression in a physically reasonable way 25 26, h = 1 2 ln B, B = FFT, (42) h = e (0) = 1 2 ln B = 1 n (ln χ σ )B σ, (43) 2 σ=1 where F is the deformation gradient and χ σ and B σ are n distinct eigenvalues and eigenprojections, respectively, of the left Cauchy-Green tensor B. 27 Moreover, e (0) is an Eulerian strain of the family of the Seth-Hill-Doyle-Ericksen strains (refer e.g. to [100]) e (m) = 1 2m (Bm I), (44) for m =0. The corresponding Lagrangean counterparts are E (m) = 1 2m (Cm I), (45) 22 Provided such a relation exists. We will see that it took several decades to properly answer this first question. 23 We note that in this work he also for the first time introduced convective coordinates. 24 This strain is sometimes called natural strain or simply logarithmic strain. Here, we will use the term Hencky strain h. 25 Imagine an elastic body subject to an axial force. Any compression to zero length then must be related with an infinitely large force. 26 In most of his papers Hencky introduced a description in principal axes. Thus, in numerous discussions he was confronted with related problems if non-principal axes were used. He then usually referred to a possible transformation into principal axes. He also noted that this approach was quite common among technologists (refer e.g. to Ludwik, [45]). Nevertheless, this measure was considered as essentially intractable and of particular usefulness (refer e.g. to Fitzgerald [13] and Gurtin and Spear [19] and the references therein). Knowing the today s computational possibilities, however, we should overcome this position and use Hencky strains in descriptions of finite deformations. 27 In today s notation.

9 ZAMM Z. Angew. Math. Mech. 94, No. 3 (2014) / 195 and in particular for m =0, H = 1 2 ln C, C = F T F (46) are the Lagrangean Hencky strains with the property H = R T hr. (47) Herein C is the right Cauchy-Green tensor and according to the polar decomposition of the deformation gradient F = RU = VR (48) U and V are right and left stretch tensors, respectively, and R is the rotation tensor. Thus, due to the above correspondence (47) the Lagrangean Hencky strain H may be interpreted as the back-rotated Eulerian Hencky strain h. Hencky introduced Lagrangean and Eulerian descriptions and discussed in this context the importance of time derivatives occurring in the relevant constitutive laws. For a Lagrangean analysis these were the material time derivatives. In an Eulerian description, which he preferred for physical reasons, he noted that the time derivative must be independent of the respective rigid-body rotation or simply objective in today s notation. In [24] he therefore replaced the specified time derivative of the stress tensor by σ σ = σ + σw Wσ. (49) Herein, the spin tensor (vorticity) W has been introducedas skew-symmetricalpart of the velocity gradientl W = 1 ( L L T ), D = 1 ( L + L T ), L = FF (50) Unfortunately, the original work by Hencky contained a small error. Instead of the above spin tensor he used an alternative definition, which differs by a minus sign. Due to this deviation, however, his derivative (49) loses its objectivity. 28 We immediately recognize the so-called Jaumann derivative, previously already discussed in this context by Zaremba [105] and Jaumann [35]. But even this from today s perspective important statement was ignored. Until much later after the end of World War II these ideas were taken up again, e.g. by Oldroyd [62], Rivlin and Ericksen [80], Noll [58], Thomas [89, 90], Truesdell [94, 95], Hill [26], Prager [70], and others. Prager also in a work of 1960 [69] mentioned that Jaumann s work does not seem to be well known: the definition... is frequently used in the recent literature without reference to Jaumann. 7 Different objective rates and hypoelasticity The first who rediscovered the above mentioned problem of finite deformations was Oldroyd [62]. Like Hencky he introduced convected coordinates and a convected differentiation with respect to time which must replace the material derivative when equations of state are transformed from a convected to a fixed system of reference. The physical significance of this operation is explained then as a total differentiation following a material element, independent of a fixed frame of reference or the way the material is moving in space. As convected coordinates were used, four different relations 29 may be given in a spatial description depending on the different representations of the (second rank) tensor components as covariant, contravariant or mixed quantities. For the stress tensor a contravariant description was preferred and thus the upper convected derivative becomes 30 (δ cc /δt)σ = σ u = σ σl T Lσ (51) for the Cauchy stresses. In addition, the remaining three forms are: 31 (δ c c /δt)σ = σ m = σ σl T + L T σ, (δ c c /δt)σ = σ m = σ + σl Lσ, (52) (δ cc /δt)σ = σ l = σ + σl + L T σ, 28 This may explain why Hencky s original proposition (49) later was criticized by Truesdell in [93]. 29 It can be shown that these different forms are particular cases of the Lie derivative (refer to [47, 85]). 30 We note that here the weight of the rate of stress tensor is zero for an incompressible material (refer also to [16]). 31 The last of these forms is sometimes also called Cotter-Rivlin or lower convected rate according to [10]. Moreover, due to the symmetry of the Cauchy stress tensor, the two mixed descriptions coincide.

10 196 O. T. Bruhns: The Prandtl-Reuss equations revisited where here the superscript or subscript index c is used as a mnemonic notation to distinguish contavariant or covariant, respectively (refer to Hill [29]). The perception that even in this accordingly corrected Prandtl-Reuss law still errors can occur, is generally attributed to Truesdell. For purely elastic processes, e.g. during unloading of a plastically deformed material, the constitutive law formulated in rates of stresses and deformation (which not until later is denoted hypoelastic by Truesdell [95]) does not have the properties of an elastic body. This can be shown easily if a (so-called non-radial) circular process is calculated and thereby at the end an accumulated dissipation is observed. 32 In a comprehensive work on the foundations of elasticity and fluid dynamics [93] Truesdell discussed the general form of a material where the rate of stress is related to the rate of deformation, rate of stress = f(rate of deformation) (53) for a material without a natural time. Applying the principle of invariance against a rigid rotation, he arrived at the following expression for the rate of stress σ = σ σl T Lσ + σ tr(d) (54) in a spatial description, where σ is named a relative time flux. This result is expressed as Cauchy s flux principle: if the defining equations for an ideal material involve the stress rate, they shall do so only through the relative time flux. 33 Thus, the defining relation for the simplest law satisfying this principle is σ = A : D, (55) where the material tensor A maybeafunctionofσ. This class of ideal materials is named hypo-elastic bodies with the properties that in general they neither have a preferred state nor a preferred stress (refer to [95]). 34 As special case an isotropic hypo-elastic body of grade zero is considered, where the right-hand side of (55) appears to be independent of σ (refer to [95]). In the simplest case this expression will depend upon the rate of deformation in the same way as the stress depends upon the strain in the classical elasticity theory, viz. σ =2μD + λtr(d)i. (56) It was shown that these equations reduce to those of classical linear elasticity under the usual assumptions of infinitesimal deformations. Under the assumption of invariance against a rigid body motion Noll 35 [58] and Thomas [89,90] arrived at a description for the rate of stress 36 on the left-hand side of Eq. (53) in the form of the Jaumann rate Eq. (49). Hill in [28] finally combined the convected derivatives Eq. (52) and the Jaumann rate by introducing a relation 37 σ= σ + σw Wσ m(σd + Dσ), (57) which turns over to the Jaumann rate (49), the Oldroyd rate (51) and the Cotter-Rivlin rate (52) 3, respectively, for m = 0, 1, 1. During ensuing years possible relations between hypo-elasticity and elasticity were discussed extensively. From this discussion it turned out that most hypo-elastic materials fail to have the properties of an elastic let alone a hyperelastic (Green elastic) material. It was Bernstein [5] who showed that hypo-elastic rate constitutive relations have to meet specific integrability conditions to gain the properties of an elastic material. Nevertheless these rates preferably the Jaumann rate were widely accepted and used in the computer codes of the Prandtl-Reuss equations. The latter (the preference of the Jaumann rate) may be due to Prager [69] who in an elementary discussion of different at the time existing stress rates and with the objective to avoid the non-uniqueness in the definition of stress rates has introduced an additional restriction which has to be satisfied by an appropriate stress rate. This additional condition was derived from the plastic behavior of the second constituent of the composite material. During plastic deformation, a state of stress is said to be at the yield limit when the 32 But even this can already be read in the work of Jaumann, who then denotes his rate material a material with fluxes (Fluxionen). He states that the behavior of this rate material is indeed not elastic and only in the limit of infinitesimal deformations changes to that of an elastic body. 33 It is evident that this is the same result as Oldroyd s upper convected rate with a weight of the rate of stress tensor equal to 1, orinotherwords, where the differential operation is applied to the weighted Cauchy stress Jσ, wherej is the Jacobian of deformation and J = Jtr(D). 34 We note that Truesdell in [94] emphasized that this new theory does not employ any concept of strain. 35 Noll called his general invariance requirement principle of isotropy of space. 36 Thomas called this an absolute time derivative. 37 Hill in his work indeed preferred the Kirchoff stress T, which is the weighted Cauchy stress Jσ, rather than the Cauchy stress σ itself.

11 ZAMM Z. Angew. Math. Mech. 94, No. 3 (2014) / 197 yield function is zero. To avoid contradictions, the yield function should be stationary when the stress rate vanishes. This implies that the invariants of the stress tensor should be stationary, too. Applying this stationary condition 38 to the above introduced stress rates, Prager argued that the convected (non-corotational) rates presented with the Oldroyd rates (51) and (52) and the Truesdell rate (54) could not be recommended as these rates do not imply stationarity of the stress invariants. Thus, with the Jaumann rate the above Prandtl-Reuss theory, in particular, the J 2 -flow theory with a v. Mises type yield function (6), was developed by many researchers. However, the foundation of this classical theory was shaken by an unexpected discovery of spurious phenomena known as shear oscillations. It seems that Lehmann [38] 39 was the first to reveal that a rigid plastic J 2 -flow theory with kinematic hardening would predict an oscillating shear stress response to monotonically progressing simple shearing (refer to Fig. 3). Fig. 3 Shear stress τ and normal stresses σ 2 = σ 3 vs. shearing ϑ =tanγ during simple shear for a rigid plastic J 2-flow theory with kinematic hardening (refer to Lehmann [38]). Ten years later, this phenomenon was rediscovered by Nagtegaal and de Jong [56]. On the other hand, Dienes [11] demonstrated that a similar phenomenon would emerge also for the hypoelastic rate equation (56) which was assumed to describe purely elastic behavior. The question how to avoid these unexpected results, however, is then often reduced to a search for a proper definition of the objective rate. Dienes with his current theory used the Green-Naghdi rate or polar rate (refer to [17, 18]) where the material time derivative is applied to a rotated stress tensor, thus replacing the vorticity W of the Jaumann rate by a skew-symmetric rate of rotation Ω R with σ R = σ + σω R Ω R σ, Ω R = ṘRT, (58) where R is the rotation tensor obtained from the deformation gradient through the polar decomposition F = RU = VR. In an entirely comparable way, e.g. the spin Ω E and the rotation R E of the Eulerian triad may be used to define another objective rate 40 σ E = σ + σω E Ω E σ, Ω E = ṘE R ET. (59) This rate is sometimes called Sowerby-Chu rate (refer to [86], see also Szabó and Balla [88]). The hypoelastic rate equation (56) is intended for finite elastic deformation behavior. Although it may be clear that the small deformation case yields the conventional elastic relation, i.e., Hooke s law, essential difference emerges for the direct extension of Eq. (9) or the first part of Eq. (18) to finite deformation. Typical examples were presented with Kleiber [37] and Szabó and Balla [88]. Kleiber also discussed non-radial cyclic processes of a hypoelastic material with a Jaumann derivative and observed a significant linearly increasing accuracy loss with the number of cycles. It was stated that within several cycles (with not uncommon rotations) the solution could lead to an energy error comparable with the maximum energy attained during the cycle thus rendering the results totally useless 41. In view of this result he concluded that it 38 We emphasize, however, that this is only a sufficient condition for stationarity of the invariants. 39 Unfortunately, this paper was written in German and submitted for publication to a Romanian journal in Due to severe production problems in those days (lack of paper) the article appeared not until These circumstances may explain why this paper was widely ignored. 40 It may be clear that numerous additional objective rates could be defined, e.g. simply by combining the hitherto mentioned rules. 41 The accumulated energy should vanish at the end of a closed cycle for an elastic material.

12 198 O. T. Bruhns: The Prandtl-Reuss equations revisited might be doubtful that the use of an objective rate on the left-hand side of Eq. (55) (with a constant material tensor A) would yield more accurate results than those corresponding to a simple (non-objective) material time derivative. Another unexpected finding was made by Simo and Pister [85], who in 1984 with reference to the works of Truesdell [95], Bernstein [5], and Green and McInnis [18] demonstrated that for each of the at that time well-known objective rates, except for certain unrealistic particular cases for elastic constants, the widely-used hypoelastic rate equation (56) fails to be exactly integrable to define an elastic, in particular, hyperelastic relation. Since this finding, a general tendency is to believe (see, e.g., [36, 44, 84]) that the just-mentioned non-integrability property would likely be true for all possible rates. This would imply that the classical Eulerian elastoplasticity and the Prandtl-Reuss theory might be self-inconsistent in the sense of formulating elastic behavior via the hypoelastic equation. In fact, a non-integrable hypoelastic formulation is path-dependent and dissipative, and thus would deviate essentially from the recoverable elastic-like behavior. 8 The logarithmic rate and related properties The development of the logarithmic rate as a remedy out of the above mentioned dilemma has begun with the wish to resolve the following question: Can the stretching D be represented as a direct flux of an Eulerian strain measure, say e? This question is closely related to a second one, namely, whether there exists a relation between the stretching and the logarithmic strain? Although the stretching is frequently referred to as the rate of deformation tensor or the Eulerian strain rate, it was believed for a long time that the former would not be the case (see e.g. Hill [29], see also Ogden [61]). With respect to the latter there have been different attempts to relate the logarithmic Hencky strain h = 1 ln B =lnv 2 (60) to the stretching. Hill [27] showed that the material time derivative of the Lagrangean counterpart Ḣ equals R T DR within a second order term (see also Rice [79] and Hill [29]). The problem of specifying the Lagrangean stress tensor conjugate to the Lagrangean logarithmic strain H was partly solved by Hill [29] and finally completed by Hoger [31], namely with the following relation ˆT = R T TR, (61) where T = Jσ is the so-called Kirchhoff stress, a weighted Cauchy stress. With respect to the Eulerian logarithmic strain Hoger in [31] like Macvean [46] and Ogden [61] earlier stated that h as a direct consequence from the above mentioned non-existence of any relation with the stretching does not have a conjugate stress, in general. Her proof, however, was somewhat flawed as the material time derivative of ln V introduced in [31] failed to be objective (see also [40]). 42 Gurtin and Spear [19] and Hoger [30] derived conditions when under very specific circumstances the Jaumann rate of the logarithmic strain (ln V ) equals the stretching. It was shown, e.g., that a condition DV = VD (62) would be necessary and sufficient for this case. Moreover, it was shown that (ln V ) would be a very good approximation to D for sufficiently small deformations 43. Inspired by their works and using an explicit formula for the gradient of the general strain measure e with respect to the stretch tensor V (see [8] and [96]) Xiao et al. [98] could prove that an objective corotational rate of the logarithmic strain measure ln V could be identical with the stretching tensor D, and furthermore that in all possible strain measures only ln V would enjoy this property, i.e. any corotational rate of any other strain measure could not be identical with D. This result was gained by broadening the work-conjugacy relation introduced by Hill [27], ẇ = tr(sė) =tr(td). (63) AstressS and a strain E are said to be a conjugate pair if tr(sė) represents the stress power ẇ. Relation (63) now has been generalized by introducing a pair of objective Eulerian stress and strain measures, say (s, e), both symmetric, and an Eulerian spin Ω = Q T Q = Q T Q, (64) 42 Instead, she should have introduced an objective time derivative and thus broaden the definition of conjugacy introduced by Hill [27]. We will come back to that point shortly. 43 A result which already has been demonstrated by Jaumann in 1911.

13 ZAMM Z. Angew. Math. Mech. 94, No. 3 (2014) / 199 where Q is a proper orthogonal tensor. In an Ω -frame relative to a fixed background frame, this pair becomes (Q sq T, Q eq T ). Then the inner product (Q sq T ):( Q eq T ) (65) is formed by the observer in the Ω -frame just as an observer in the fixed background frame does for a Lagrangean stress and strain pair. Following the idea of Hill, which is concerned with a fixed background frame, the observer in the Ω -frame judges that the pair (s, e) is an Ω -work-conjugate pair if the inner product (65) furnishes the stress power, i.e. (Q sq T ):( Q eq T )=s : e = ẇ, (66) where e is the corotational rate of the strain defined by the spin Ω, e = ė + eω Ω e. (67) A corotational rate of an Eulerian tensor need not be objective. The spin tensors defining objective corotational rates are called material spins (refer to Xiao et al. [100]). We now want to know whether or not a strain measure e and a material spin Ω can be found such that the objective corotational rate of e defined by Ω is identical with the stretching D,i.e. e = ė + eω Ω e = D, e = h and Ω = Ω log. (68) It turned out that this expression 44, where both the strain and the spin can be chosen arbitrarily, holds if and only if e is the logarithmic strain h =lnv.whene =lnv, Eq. (68) has a unique continuous solution for the spin Ω, the logarithmic spin (refer to ( [97, 98, 100, 102]) Ω log = W + m σ τ ( 1+(b σ/b τ ) 1 (b σ /b τ ) + 2 ln(b σ /b τ ) )B σdb τ. (69) In the above, the b 1,, b m and B 1,, B m are the m distinct eigenvalues and the corresponding eigenprojections 45 of the left Cauchy-Green deformation tensor B. We note that explicit basis-free and unified expressions for Ω log have been presented with [98, 104]. The essence behind the above perhaps complicated result is a unique, intrinsic relationship between Hencky s natural strain h =lnv and the natural deformation rate D. The spin Ω log given by Eq. (69) is referred to as the logarithmic spin because of its unique relationship with the logarithmic strain. For an objective Eulerian tensor Λ, the corotational rate defined by the logarithmic spin, i.e., Λ log Λ + ΛΩ log Ω log Λ (70) is called the logarithmic rate of Λ. Then, Eq. (70) together with the unique relationship (68) yields the following exact kinematical relationship 46 between the Hencky strain h and the stretching D: D = h log = ḣ + hωlog Ω log h. (71) 44 It seems that a direct precursor of Eq. (68) is relation (2.13) with Eqs. (2.11) and (2.15) in Bruhns and Lehmann [4], which says that the Jaumann rate of h should exactly give D in some cases; see also: Gurtin and Spear [19] and Hoger [30]. The main idea in finding Eq. (68) was inspired by the wish to express in an Eulerian description of the energy balance the stress power as function of Cauchy stress and objective rate of a conjugate strain. It was the replacement of the spin W and Hencky strain h in the foregoing relation by an arbitrary strain e and spin Ω leading to Eq. (68), namely, to the finding of the unique, intrinsic relationship between h and D. This direct relationship, i.e., Eq. (71) below, was already found in 1991 by Lehmann, et al. [39]; see footnote 46 for details. 45 These quantities are sometimes also called eigenprojectors (refer to Bertram [6]). 46 It seems that Lehmann et al. [39] were the first to establish the relationship (71) in discussing the work-conjugacy between Cauchy stress σ and Hencky strain h, with a seminal, but sometimes perhaps misleading, idea that any given pair of Eulerian stress s = s(σ, V ) and Eulerian strain e = e(v ) could be rendered work-conjugate by defining a corresponding time derivative. With s = σ this would imply that D could be given by an objective rate of any given strain e(v ). It seems that almost at the same time several groups were seeking a solution for this problem, namely to express the stretching D as an objective rate of an Eulerian strain. It is reported that P.A. Zhilin 1995 starting from a quite different idea came to a conclusion comparable to Eq. (71). Unfortunately, however, his result remained unpublished then (refer to [106] and the respective remarks therein). Later, Eq. (71) was also rediscovered by Reinhardt and Dubey [74, 75]. This relationship was derived in the general sense of studying Eq. (68) independently and its intrinsic uniqueness property was thus revealed for the first time in Xiao et al. [97, 98, 100, 102] from different contexts.