A R C H I V E S O F M E T A L L U R G Y A N D M A T E R I A L S Volume Issue 2 DOI: /v

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1 A R C H I V E S O F M E T A L L U R G Y A N D M A T E R I A L S Volume Issue DOI: /v R. B. PĘCHERSKI, P. SZEPTYŃSKI, M. NOWAK AN EXTENSION OF BURZYŃSKI HYPOTHESIS OF MATERIAL EFFORT ACCOUNTING FOR THE THIRD INVARIANT OF STRESS TENSOR ROZSZERZENIE HIPOTEZY WYTĘŻENIA BURZYŃSKIEGO UWZGLĘDNIAJĄCE WPŁYW TRZECIEGO NIEZMIENNIKA TENSORA NAPRĘŻENIA The aim o the paper is to propose an extension o the Burzyński hypothesis o material eort to account or the inluence o the third invariant o stress tensor deviator. In the proposed ormulation the contribution o the density o elastic energy o distortion in material eort is controlled by Lode angle. The resulted yield condition is analyzed and possible applications and comparison with the results known in the literature are discussed. Keywords: Burzyński yield condition, strength dierential eect, pressure sensitivity, Lode angle, third invariant o stress tensor deviator Celem pracy jest propozycja rozszerzenia hipotezy wytężenia Burzyńskiego dla uwzględnienia wpływu trzeciego niezmiennika dewiatora tensora naprężenia. W proponowanym sormułowaniu udział gęstości sprężystej energii odkształcenia postaciowego jest kontrolowany przez unkcję zależną od kąta Lodego. Wyprowadzony waruneastyczności porównano ze znanymi wynikami z literatury oraz przedyskutowano jego możliwe zastosowania. 1. Introduction Experimental investigations related with mechanics o soils and rocks, the strength o concrete as well as ailure processes and phase transormations in advanced metallic materials reveal that assumption about isotropic mechanical properties requires also an account or the third invariant o stress tensor deviator in the ormulation o limit state criteria (ailure criteria). The limit state criteria can determine the limit o elasticity, plasticity, phase transormations or rupture. The third invariant o stress tensor deviator corresponds to the Lode angle on the octahedral plane. The experimental observations show that the eect o Lode angle appears particularly visible, while shear processes are taking place. On the atomic level the shear process in a solid body is accompanied with the change o the coniguration o atoms and valence electrons which are responsible or the bonds between the neighboring atoms. This leads to the change o the symmetry o groups o atoms and results in the change, mostly with downward tendency, o their energy o cohesion [3], [6]. Thereore an extent o the domination o shearing in deormed solid body has an essential inluence on the energy-based measure o material eort. This leads to the conclusion that the contribution o shear modes o stress, quantiied by means o the Lode angle, should be accounted or in the energetic criteria o ailure. Due to the universality and multiscale character o energy, among many proposed hypotheses o material eort those with energy basis seem to be the most worthy considering rom the physical point o view. The hypothesis o variable limit energy o volume change and distortion proposed originally by Burzyński [] is the one which seems to be the most general o all many other are just special cases o this one or certain values o its parameters [7], [13]. The aim o the paper is to propose an extension o the Burzyński hypothesis o material eort to account or the aorementioned inluence o the Lode angle. The main idea o the proposed new ormulation is that the contribution o the density o elastic energy o distortion in the measure o material eort is controlled by Lode angle. The resulted yield condition is analyzed rom the point o view o possible applications and the comparison with the results presented in the literature is discussed [4], [5], [8], [9]. INSTITUTE OF FUNDAMENTAL TECHNOLOGICAL RESEARCH, OF THE POLISH ACADEMY OF SCIENCES, WARSZAWA, 5B PAWIŃSKIEGO STR., POLAND

2 504. The Burzyński energy-based hypothesis o material eort and its extension.1. The hypothesis o variable limit energy o volume change and distortion Burzyński considered as a measure o material eort variable elastic energy density o distortion and a part o energy density o volumetric change controlled by a pressure dependent inluence unction. The limit condition takes in such a case the ollowing orm []: Φ + η (p) Φ v = K where η (p) = ω + δ 3p, (1) where Φ v is volumetric strain energy density, while Φ is distortional strain energy density. It is in general a three-parameter condition. The great convenience in application o it is due to the act that ater proper substitutions the set o parameters (K, ω, δ) can be replaced by (k t, k c, k s ) which are the limit values o stress under tension, compression and pure shear respectively Those three parameters can be measured experimentally. In case o the Burzyński condition (1) those three values give us ull inormation about strength properties o isotropic solid. Sometimes it is useul to express the limit condition (1) in terms o the other two parameters proposed by Burzyński: the strength dierence ratio κ = k c k t > 1 and the so called plasticity coeicient which is a measure o mutual relations between all three limit quantities v = k ck t 1. Eq. (1) can be rewritten in ks the ollowing orm: 1 + v 3 σ + 3 (1 v) p + 3pk t (κ 1) = κk t, () where p = 1 3 (σ 1 + σ + σ 3 ) denotes hydrostatic stress and σ = (σ σ 3 ) + (σ 3 σ 1 ) + (σ 1 σ ) is the stress invariant introduced by Burzyński []. For certain values o κ and v the condition () is describing certain type o a limit surace in the space o principal stresses (σ 1, σ, σ 3 ): v < 1 ellipsoidal surace with one o its axes parallel to p axis. For κ 1 it is translated along p axis. In such a case the strength dierential (SD) eect is involved. In the case o symmetric elastic range (κ = 1) limit surace is the same as in the case o Rychlewski s [10], [11] limit criterion or isotropic materials; v = 1 paraboloidal surace with its axis parallel to p axis and translated along it. It describes the SD eect. For κ = 1 the paraboloid o revolution becomes a cylinder the criterion or pressure insensitive materials not exhibiting SD eect this case corresponds to the Huber criterion [13]; 1 < v < 3 (k c + k t ) 1 hyperboloidal limit surace 8k c k t (two sheets). Only one sheet has a physical meaning. I v reaches its upper limit value the surace becomes a cone the same as in the Drucker-Prager hypothesis [13], which was proposed independently 5 years later; Note that the qualitative change o surace character is continuous however it changes very rapidly when v is close to 1. As it was already mentioned, the SD eect is involved by accounting or pressure inluence on material eort other authors (Raniecki and Mróz [9]) ormulated limit conditions or pressure insensitive-materials describing the SD eect with use o the Lode angle dependence... The proposed extension o the Burzyński hypothesis The inluence o the Lode angle on the Burzyński measure o material eort could be involved by the variable contribution o the energy density o distortion. The contribution is described by certain inluence unction. The possible propositions o the orm o such a unction are given in the next subsection. Finally the extended material eort hypothesis o variable energy o volume change and distortion in the limit state reads: η (θ) Φ + η v (p) Φ v = K, (3) where the symbols η and η v denote Lode angle inluence unction and pressure inluence unction, respectively. The condition (3) can be expressed in a ollowing way: η (θ) q + η v (p) p = K, (4) where q = 1 [ (σ σ 3 ) + (σ 3 σ 1 ) + (σ 1 σ ) ] (5) 3 and θ = 1 3 arccos 3 3 J 3 J 3/ is the Lode angle..3. Propositions o inluence unctions η v and η.3.1. Pressure inluence unction In case o pressure inluence unction the choice o Burzyński s unction seems to be well justiied since hypothesis ormulation given by Eq.(3) is a consistent

3 505 extension o his own idea. Furthermore this unction enables description o various relations between hydrostatic and deviatoric stresses linear, paraboloidal, hyperboloidal and elliptical and it was analyzed precisely by Burzyński in []. It is a two-parameter rational unction which can be written in the orm: η v (p) = ω + δ p.3.. The Lode angle inluence unctions (6) The Lode angle dependence, which is characteristic especially or brittle materials and soils, can be described: two-parameter power unction (Raniecki and Mroz [9]) η (θ) = [1 + α cos (3θ)] β two-parameter exponential unction (Raniecki and Mróz [9]) η (θ) = 1 + α [ 1 e β(1+cos(3θ))] one-parameter trigonometric unction (Lexcellent [5]) [ ] 1 η (θ) = cos arccos [1 α (1 cos (3θ))] 3 two-parameter trigonometric unction (Podgórski [8]) [ ] 1 1 η (θ) = ( π ) cos arccos (α cos (3θ)) β cos 6 β 3 Valuable summary o propositions o Lode angle inluence unctions was made by Bardet in [1]. The earlier attempts to account or the Lode angle eect are discussed by Życzkowski [13] Convexity condition A general discussion on convexity condition was presented in [1] or arbitrary chosen orm o inluence unctions such that K η p (p) > 0: 3 6η η ( + η p η p 3 ( ) η p K η p p + ( η ) + 4 ( η ) + η ( K ηp ) ( ) ηp p η η ) + 4 ( η ) + ( ) ( ηp ) η p + 4η > 0, (7) ( ) η + 4 ( ) η > 0 where:η p (p) = η v (p) p The above conditions can be applied in material identiication procedures basing on numerical itting o the results o simulation using assumed model to the results obtained rom experiment. It is in act an optimization problem in which the given convexity condition can be considered as an inequality constraint on the resultant optimal solution. 3. Yield surace speciication 3.1. Iyer and Lissenden experiments on Inconel 718 An attempt to ind certain orm o a yield condition basing on (4) and experimental data available in literature was made. Inconel 718 alloy was considered. Various tests including tension, compression and pure shear as well as composition o two o those loadings one ollowed by another were perormed or Inconel 718 by Iyer and Lissenden [4]. Values o plasticity limit (considered as an oset limit being a stress causing 0,% permanent plastic strain) and proportionality limit has been estimated basing on uniaxial and pure shear tests. The proportionality limit estimation is especially important since it denotes Hooke s law validity range. However it seems that mathematical ormalism o the considered yield condition could be well used also in case o plasticity limit estimation. Tension: Compression: Pure shear: Plasticity limit t = 779 MPa c = 878 MPa s = 473 MPa Proportionality limit k H t = 500 MPa k H c = 610 MPa s = 33 MPa TABLE 1 One can observe that the considered material exhibit signiicant strength-dierential eect in case o plasticity limit κ = k c k t 1.13 and in case o proportionality limit κ 1.. Mutual relations between those limit stresses can be estimated by Burzyński s plasticity coeicient v = k ck t 1. For the plasticity limit v 0.53, k s or proportionality limit v One can see that this value is close to 0.5, the value distinguished by Burzyńs-

4 506 ki, or which yield condition becomes either a paraboloid o revolution or a cylinder. In act yield condition used in [4] being simple combination o powers o stress tensor invariants, namely: ai1 + b J + c sgn (J 3 ) (J 3 ) /3 1 = 0 a = b = c = MPa MPa MPa (8) is represented in stress space by an ellipsoid with its axis parallel to paxis and with slightly deormed cross-section (see Fig. ). This is due to the assumed elliptical relation between hydrostatic and deviatoric stresses Parameters a, b, c and seven other (viscoplastic) material parameters were ound numerically by itting simulated processes to those observed during experiments The irst attempt was to determine values o parameters so they it the shearing data well, however the obtained result were unrealistic, especially shear threshold stress was much too low. Finally parameters a, b, c o yield condition (8) were ound in [4] by itting numerical simulation to various experimental data, however unequal weights were used or dierent samples in such way so the correlation between model and experiment was optimal. Fig. 1. Ellipsoidal yield surace used in [4] and the paraboloid yield surace according to the proposed criterion given by Eq. (9) Fig.. The comparison o the cross-sections o the two yield suraces, used in [4] and given by Eq. (9), at the deviatoric (octahedral) plane

5 Why a paraboloid yield surace? Reerring to the Burzyński hypothesis relatively low value o limit shear stress is characteristic or ellipsoidal yield suraces (v < 0.5) so it seems that assuming dierent character o a yield surace (i.e. paraboloidal, what indicates calculated v 0.5) might provide higher precision in model identiication It is also common assumption (based on experimental results) that metals are either pressure-insensitive (cylindrical yield surace as in Huber-Mises hypothesis [13]) or that hydrostatic stress is a sae stress state (paraboloidal yield surace). Thus paraboloidal yield surace seems to be much more natural and reasonable choice than the ellipsoidal one, which is typical or instance in case o oams and cellular materials. Furthermore elliptic relation between deviatoric and hydrostatic stress is symmetric with respect to both o those quantities (sign insensitive) so only the Lode angle might be considered as the cause o SD eect. Assuming that the dierence between values o limit stresses under tension and compression depends only on dierent modes o shearing (Lode angle) omitting the inluence o pressure is unjustiiable. The only objective veriication o a surace character would be perorming experiments under high pressure Yield condition speciication Yield surace given by condition (4) was determined or Inconel 718 assuming that relation between hydrostatic and deviatoric stresses is paraboloidal. To obtain such relation Burzyński s unction was chosen as pressure inluence unction. Analysis o the cross-section o yield surace given by Eq.(8) parallel to octahedral plane indicates that the Lode angle dependence has a hexagonal character. Among all presented Lode angle inluence unctions only the one by Podgórski enables description o such a dependence. However it had to be slightly modiied since its maximums were rotated reerring to the maximums obtained rom Eq.(8). Finally yield condition o a ollowing orm was considered: q cos( π 6 β) cos [ 1 3 arccos [ α cos ( 3 ( θ π ))] β ] + + ( ω + δ p) p K = 0 (9) Unknown parameters can be determined rom a nonlinear system o equations each equation is obtained rom satisying yield condition (9) in simple loading cases like uniaxial tension or compression or pure shear or which values o p, q and θ are known. For higher precision more complex stress states could be analyzed i.e. shearing with tension, shearing with compression or (as Burzyński proposed) biaxial and triaxial uniorm tension and compression. Yet dierent method o yield condition speciication was chosen. Since surace given by Eq.(8) was determined by optimal itting it to various multiaxial tests, it seems that its character is quite well estimated especially or small values o hydrostatic stress - no tests under high pressure were perormed and hydrostatic component o a stress tensor in each test was also not signiicantly high due to lack o stress concentration in the samples. That s why an attempt o itting paraboloidal yield surace given by Eq.(4) to the surace (not only single points in stress space corresponding with basic strength tests) given by Eq.(8) was made. To achieve proper itting Levenberg-Marquardt algorithm was implemented replacing yield surace used rom [4] by a large set o points. For reasons presented below, the itting tooace only or positive values o hydrostatic stress. As it was mentioned in subsection 3., the aim o the authors was to determine a paraboloidal yield surace dierent character o paraboloidal and ellipsoidal relation between q and p or negative values o pressure would cause too big error or any eicient use o Levenberg-Marquardt algorithm. Another reason is that results presented in [4], are much more reliable or positive values o pressure the material was characterized in [4] basing on ive tests (tension, compression and three shearing tests) yet each o them had a dierent inluence on the inal result due to use o inequal weights. The only source o hydrostatic component in those tests are uniaxial tests. Tension test was taken with the weight 40% while compression test with the weight 10%. As a measure o an error o itting a distance between both suraces along q direction was considered. It appeared that the resultant values o unknown parameters o unction given by Eq.(9) obtained rom the perormed computation depend strongly on their initial values used in iteration i the number o estimated parameters was greater than two. However the number o parameters could be reduced by assuming some o their values i.e.: ω = 0 due to act that relation between p and q was assumed paraboloidal. β = π 6 due to act that Lode s angle dependence has a hexagonal character. α = 0.9 smoothing o corners o an octahedral cross-section depend on the value o this parameter (or α = 0 no Lode s angle dependence; α = 1, β = π 6 strictly hexagonal dependence like i.e. in Coulomb-Tresca-Guest hypothesis). This value was estimated so that octahedral cross-sections o both suraces correspond suiciently well. Parameters δ 174, 85 MPa and K MPa were ound using the mentioned Levenberg-Marquardt

6 508 algorithm. Comparison o both suraces given by Eq.(8) and Eq.(9) and their octahedral cross-sections comparison are shown in Fig.(1) and Fig.(). Surace given by Eq. (9) was presented in orm o points only to make the igure o two intersecting suraces clear. 4. Summary A proposition o a yield criterion o clear physical (energetic) interpretation or isotropic and homogeneous materials involving pressure sensitivity with the resulting SD eect and the Lode angle dependence and was introduced. Many commonly used yield criteria are involved as special cases or certain orm o inluence unctions and certain values o their parameters. It is an extension o the Burzyński yield condition accounting or inluence o the third invariant o stress tensor deviator. General convexity condition or obtained yield surace or arbitrary chosen orms o inluence unctions was derived. Certain orm o yield condition or Inconel 718 alloy was ormulated reerring to experiments by Iyer and Lissenden [4] by yield surace identiication. Acknowledgements The paper has been prepared within the ramework o the research project N N o the Ministry o Science and Higher Education o Poland. REFERENCES [1] J. P. B a r d e t, Lode dependences or isotropic pressure sensitive materials, J Appl Mech 57, (1990). [] W. B u r z y ń s k i, Studium nad hipotezami wytężenia, Akademia Nauk Technicznych, Lwów, (198); see also: Selected passages rom Włodzimierz Burzyński s doctoral dissertation Study on material eort hypotheses, Eng Trans 57, (009). [3] J. G i l l m a n, Electronic basis o the strength o materials, Cambridge University Press (003). [4] S. K. I y e r, C. J. L i s s e n d e n, Multiaxial constitutive model accounting or the strength-dierential in Inconel 718, Int J Palst 19, (003). [5] C. L e x c e l l e n t, A. V i v e t, C. B o u v e t, S. C a l l o c h, P. B l a n, Experimental and numerical determinations o the initial surace o phase transormation under biaxial loading in some polycrystalline shape-memory alloys, J MechPhys Sol 50, (00). [6] K. N a l e p k a, R. B. P ę c h e r s k i, Modelling o the interactions In the copper crystal applied in the structure (1 1 1) Cu / ( ) Al O 3, Arch Metall Mat 54, (009). [7] R. B. P ę c h e r s k i, Burzyński yield condition vis-à-vis the related studies reported in the literature, Eng Trans 56, (009). [8] J. P o d g ó r s k i, Limit state condition and the dissipation unction or isotropic materials, Arch Mech 36, (1984). [9] B. R a n i e c k i, Z. M r ó z, Yield or martensitic phase transormation conditions and dissipation unctions or isotropic, pressure-insensitive alloys exhibiting SD eect, Acta Mech 195, (008). [10] J. R y c h l e w s k i, CEIIINOSSSTTUV, Mathematical structure o elastic bodies [in Russian], Technical Report 17, Inst. Mech. Probl. USSR Acad. Sci., Moskva (1983). [11] J. R y c h l e w s k i, Elastic energy decomposition and limit criteria, Advances in Mechanics 7, (1984) (in Russian). [1] P. S z e p t y ń s k i, Convexity condition or generalized yield surace o isotropic homogeneous bodies, Opuscu Engng Trans, 31, submitted or publication (011). [13] M. Ż y c z k o w s k i, Combined loadings in the theory o plasticity, PWN, Warszawa (1981). Received: 10 January 011.