Based on a multi-scale approach comprising a multi-scale material model and a respective ﬁnite-element (FE) analysis tool, the indentation response of porous materials is examined in this paper. The considered material is assumed to consist of a homogeneous Drucker-Prager-type matrix-phase and spherical pores. Non-linear homogenization is employed to derive both a strength criterion and a hardening rule at the macroscopic scale without the need of any additional non-physical material parameters. Hereby, the underlying macroscopic hardening is exclusively controlled by the evolution of the porespace during loading. The material model is implemented in a FE program within the framework of elastoplasticity. The so-obtained analysis tool is applied to the analysis of indentation experiments commonly used for characterization and performance-based optimization of materials

Lackner, Roman

Traxl, Roland

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Pore-space controlled hardening model in plasticity of porous materials: application to the analysis of indentation experiments

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1551PORE-SPACE CONTROLLED HARDENING MODEL INPLASTICITY OF POROUS MATERIALS: APPLICATIONTO THE ANALYSIS OF INDENTATION EXPERIMENTSROLAND TRAXL AND ROMAN LACKNERMaterial Technology Innsbruck (MTI)University of InnsbruckTechnikerstraße 13, A-6020 Innsbruck, Austriae-mail: {Roland.Traxl, Roman.Lackner}@uibk.ac.at, www.uibk.ac.at/mtiKey words: Computational Plasticity, Indentation Analysis, Hardness, Porous Materi-als, Strength HomogenizationAbstract. Based on a multi-scale approach comprising a multi-scale material model anda respective finite-element (FE) analysis tool, the indentation response of porous materialsis examined in this paper. The considered material is assumed to consist of a homoge-neous Drucker-Prager-type matrix-phase and spherical pores. Non-linear homogenizationis employed to derive both a strength criterion and a hardening rule at the macroscopicscale without the need of any additional non-physical material parameters. Hereby, theunderlying macroscopic hardening is exclusively controlled by the evolution of the pore-space during loading. The material model is implemented in a FE program within theframework of elastoplasticity. The so-obtained analysis tool is applied to the analysisof indentation experiments commonly used for characterization and performance-basedoptimization of materials.1 INTRODUCTIONIndentation experiments are commonly employed for determination of strength prop-erties (hardness) of materials. Nowadays indentation analysis is applied on a great varietyof materials and at various length scales, ranging from nanoindentation (e.g. Constan-tinides et al. [6]) to classical hardness measurements according to Tabor [15]. This givesscientists and engineers access to material properties even if ordinary test specimens formechanical (compressive/tensile) testing are not available.Meanwhile there are a number of publications treating indentation analysis for mono-lithic solids, focusing on the phenomenological aspect at the macro-scale (see e.g. Cheng etal. [5] for elastoplasticity, and Pichler et al. [14] for viscous material behaviour). In orderto obtain a better understanding of indentation experiments, micro-mechanical changes1XI International Conference on Computational Plasticity. Fundamentals and Applications COMPLAS XI E. Oñate, D.R.J. Owen, D. Peric and B. Suárez (Eds) 1552Roland Traxl and Roman Lacknertaking place in course of the indentation process are considered. E.g., application of limitanalysis to porous material as presented in Cariou et al. [3] gives access to characteristichardness-packing relations. Still, there are some underlying assumptions and restrictions:The change of material properties caused by the load history (hardening or softening) isneglected as well as piling-up or sinking-in effects. Besides, the so-obtained relations areonly valid for virtually rigid materials and associated yielding.Departing from the work presented in [3], a micromechanics-based material model basedon nonlinear homogenization is developed and implemented in a FE program within theframework of non-associated hardening/softening elastoplasticity. Based on numericalsimulations, the significance of the involved physical mechanism is investigated in regardto material characterization in indentation tests.2 MATERIAL MODELThe underlying material is assumed to consist of two phases: a solid phase and apore phase. In the sequel, the existence of a representative elementary volume (REV) isassumed, which is equivalent with the requirementd  L  h , (1)where d represents the characteristic size of pores, L the size of the REV, and h theindentation depth.2.1 Elastic and plastic material propertiesThe pore phase of the considered two-phase material is characterized by its volumefraction ϕ and the shape of the pores. In the following, the pores are assumed to bespherical. The solid (matrix) phase is modeled as an elastoplastic material. Hereby, thedomain of strength compatible stress states, EM , is defined by a yield function accordingto the Drucker-Prager criterion, given by the cohesion c and the friction coefficient α ofthe material:σ ∈ EM ⇔ fM(σ) = σd(σ) + ασm(σ)− c ≤ 0 , (2)where σm is the hydrostatic pressure, and σd the equivalent deviatoric stress:σm =13trσ, σd =√12s : s, s = σ − Iσm . (3)Within the elastic domain, the material is supposed to exhibit linear-elastic behavior,represented by the fourth-order tensor CM .The domain EM remains unchanged during loading (ideal plasticity). While an associ-ated flow rule implies dilatation, a non-associated flow rule is adopted in order to describevolume-preserving yielding of the solid (matrix) phase, giving a plastic potential in theform:g(σ) = σd(σ) . (4)21553Roland Traxl and Roman Lackner2.2 Homogenization of elastic propertiesLet V be the domain of the REV and VM the domain of the solid (matrix) phase. Incase of absence of pore pressure as supposed in the following, the homogenization problemof elastic properties is given in the framework of uniform strain boundary conditions bydivσ = 0σ = CM : εξ = E · xin Vin VMin ∂V(5)Therein, x is the position vector, E the macroscopic strain tensor, and ξ the displacementvector at the boundary of the REV. A macroscopic stress tensor Σ is defined as the averageof the local stress σ:Σ =1V∫VσdV = 〈σ〉V . (6)The homogenized stiffness tensor Chom sought-after relates the macroscopic strain to themacroscopic stress:Σ = Chom : E (7)For homogenization, the Mori-Tanaka scheme is applied, which yields the homogenizedbulk and shear modulus as (see, for instance, [7] for details):khom =4kMµM(1− ϕ)3kMϕ+ 4µM, µhom = µM(1− ϕ)(9kM + 8µM)9kM(1 +23ϕ) + 8µM(1 +32ϕ)(8)Accordingly, the macroscopic stiffness tensor is obtained by:Chom = 3khomJ+ 2µhomK, (9)withJijkl =13δijδkl, K = I− J , (10)where δ is the Kronecker delta and I is the fourth-order unit tensor.2.3 Homogenization of strength propertiesWhile application of homogenization schemes to elastic properties is widely spread,most developments concerning non-linear behavior of composite material are relativelyrecent. An early and widely accepted criterion for porous Mises-type material was estab-lished by Gurson [8]. Since then, a number of contributions dealt with purely cohesivematrix phases, describing the material as fictitious non-linear elastic (see [16] for a sur-vey). In [10], this approach was adopted for frictional solid phases with rigid inclusions31554Roland Traxl and Roman Lacknerand in [7] for porous media, respectively. While these contributions derived the macro-scopic yield criterion analytically, Pastor et al. [12] made use of simulation tools basedon limit analysis.An analytical formulation based on non-associated yielding was reported in [11]. There-in, the non-associated plasticity problem is interpreted as a nonlinear-viscoplastic problemwith a pre-stress depending on the strain rate field ε̇ to account for non-associated yielding.By introducing a representative strain rate (see [7]), this problem can be further simplifiedto a problem yielding an analogous structure as the problem of homogenization of elasticproperties with pre-stress. Using the Mori-Tanaka scheme, the macroscopic yield functionreads [11]:Fhom(Σ, ϕ) =(34ϕ− α2)Σ2m + 2αc(1− ϕ)Σm +(23ϕ+ 1)Σ2d − c2(1− ϕ)2 , (11)where ϕ represents the porosity of the material. For volume preserving yielding, thecorresponding plastic potential is given by [11]:Ghom(Σ, ϕ) =34ϕΣ2m +(23ϕ+ 1)Σ2d . (12)In contrast to the model proposed in [11], ϕ is considered as hardening variable in thispaper, depending on the volumetric change of the material. The total change of volume,∆V , reads:∆V = ∆VI +∆VM . (13)∆VI and ∆VM refer to the volume change in the inclusions (pores) and the matrix phase,respectively. ∆VM , resulting only from elastic material response, is neglected (plasticdeformation was assumed to be volume preserving). Thus, the macroscopic volumetricstrain is given by:Evol =∆VV≈∆VIV=VI +∆VIV−VIV= ϕ− ϕ0 . (14)Since the initial ϕ0 is a constant, expression (14) becomes in rate form:ϕ̇ = Ėvol = tr(Ė) . (15)2.4 Finite-element implementationIn the numerical simulation of indentation tests, geometric nonlinearities are takeninto account. In large-strain elastoplasticity, a common concept is the multiplicative de-composition of the deformation gradient into an elastic and plastic part (F = FeFp), asoriginally proposed in [9]. In case of isotropy, this is equivalent to the additive decompo-sition of the logarithmic strain measure (see, for instance, [13]). Thus, the measures usedin the following refer to the logarithmic strain measure.The material model is defined by415551556Roland Traxl and Roman Lackner  Figure 2: Conical indentation (h: indentation depth, hc: penetration depth considering pile-up/sink-in,θ: half-angle of the cone, A: projected area of indent, P : applied force.irreversible part. As a result, permanent deformation remains after unloading, givingaccess to the hardness of the material usually defined by the ratio of the applied force Pand the projected area A of the residual imprint (H = P/A).In this paper, the indentation of the three-sided pyramidal-shaped Berkovic indenteris treated. As this indenter belongs to the class of geometric self-similar indenter, it doesnot posses a characteristic length scale (unlike, e.g., the radius of spherical indenters).The geometry is completely described by the angle of inclination θ of the pyramid. Con-sequentially, the applied force P can be expressed in case of porous materials treated inthe previous section asP = fP (E, ν, c, α, ϕ0, di, h, θ) . (18)Therein, the first four variables are the elastic and plastic properties of the matrix phaseand ϕ0 is the initial void ratio. The set of variables di represents the size of the pores.Applying the Π-theorem according to Cheng [4] yields:PEh2= ΠP(Ec, ν, α, ϕ0,dih, θ). (19)As the sizes of the pores are assumed to be very small in relation to the indentation depth(i.e., di/h → 0), the function ΠP can be regarded as independent of the specific values ofh and di. Thus, the resulting indentation problem is given byPEh2= ΠP(Ec, ν, α, ϕ0, θ). (20)For describing the sink-in and pile-up effect, the variable hc is introduced (see Figure 2).It defines the vertical distance between the tip of the indenter and the highest point of thematerial which is in contact with the indenter. Analogously to P , the following relationscan be established:hc = fh(E, ν, c, α, ϕ0, di, h, θ) , (21)which yieldshch= Πh(Ec, ν, α, ϕ0, θ). (22)61557Roland Traxl and Roman LacknerFigure 3: FE model.3.2 ResultsThe indentation experiment was simulated with the FE program Abaqus using the usermaterial subroutine and considering geometric nonlinearities. At large strain analysis,Abaqus delivers approximations of logarithmic strains to the user subroutine (c.f. [1]).Thus, the model given in Equation (17) can be employed without modifications.Instead of using the real geometry of the Berkovic-Indenter, the indentation was sim-ulated by means of a cone with an accordant A/h2 ratio, giving a half-angle of the coneof 70.32◦. Hence, the problem becomes axisymmetric and is modeled in two dimensions(see FE model in Figure 3). Through varying the parameters E/c, α, and ϕ0, the yetunknown functions ΠP and Πh are numerically evaluated.ΠP and Πh are evaluated for three different friction coefficients α (0.0, 0.25, and 0.5,see Figure 4). Note, that ϕ0 denotes the initial porosity while the current porosity ϕdepends on the loading history and varies in space.Low values for the E/c-ratios represent purely elastic material behavior while increas-ing E/c leads to an increasing influence induced by plasticity. In the elastic range, Πhapparently does not depend on the porosity ϕ0, while this influence increases for largervalues for E/c. For materials with porosities larger than about 0.3, Πh approaches 1.0 forlarge E/c-ratios. As expected, a higher friction coefficient α causes more stiffness (i.e.,higher ΠP ) in the elastoplastic domain.Using ΠP and Πh, the hardness of the material related to the cohesion is obtained byHc=EcΠPΠ2h π tan2θ. (23)In Figure 5, H/c is displayed and compared with results of Cariou et al.[3]. In [3], exten-sive parameter studies were performed, examining the material response for a number ofdifferent indenters and material properties. Therein, the material model is based on thesame yield criterion as in the present work but restricted to associated yielding. As theanalysis in [3] is based on limit analysis, neither the loading history and thus sink-in/pile-71558155915601561Roland Traxl and Roman Lacknerpurpose, one application in parameter identification, shown in Figure 6(a), is consideredwhere the properties of the matrix phase are known (E, ν, c, and α) and the initialporosity ϕ0 should be determined. For an experimentally-determined hardness/strength-ratio H/c of 1.95, the scaling relation presented in this paper yields an initial porosity ofϕ0 ≈ 0.4 for α = 0 and ϕ0 ≈ 0.53 for α = 0.25, whereas the scaling relations in [3] un-derestimate the initial porosity with ϕ0 ≈ 0.26 and ϕ0 ≈ 0.45, respectively, representingsomehow the spatial distribution of the porosity below the indenter tip (see Figure 6(b))in an average manner.ACKNOWLEDGMENTThe results reported in this paper were obtained within the research project ”Model-based optimization of recycling building materials for the use as unbounded base-layermaterial”, financially supported by the Austrian Research Promotion Agency (FFG). Theauthors gratefully acknowledge this support!REFERENCES[1] Abaqus Version 6.10 Documentation. Dassault Systèmes (2010).[2] Barthélémy, J.F. and Dormieux, L. Détermination du critère de rupture macro-scopique d’un milieu poreux par homogénéisation non linéaire. Comptes RendesMecanique (2003) 331:271–276.[3] Cariou, S., Ulm, F.J. and Dormieux, L. Hardness-packing density relation forcohesive-frictional porous materials. Journal of the Mechanic and Physics of Solids(2008) 56:924–952.[4] Cheng, Y.-T. and Cheng, C.-M. Scaling, dimensional analysis, and indentation mea-surements. Material, Science and Engineering (2004) R44:91–149.[5] Cheng, Y.-T. and Cheng, C.-M. Scaling relationships in conical indentation of elastic -perfectly plastic solids. Inernational Journal of Solids and Structures (1999) 36:1231–1243.[6] Constantinides, G. , Ravi Chandran, K.S., Ulm F.-J. and Van Vliet K.J. Grid inden-tation analysis of composite microstructure and mechanics: Principles and validation.Materials Science and Engineering A (2006) 430:189–202.[7] Dormieux, L., Kondo, D. and Ulm, F.J.Microporomechanics. Wiley, Chichester, UK,(2006).[8] Gurson, A.L. Continuum theory of ductile rupture by void nucleation and growth:part I - yield criteria and flow rules for porous ductile media. ASME Journal ofEngineering Materials and Technology (1977) 99:2–15.111562Roland Traxl and Roman Lackner[9] Lee, E.H. and Liu, D.T. Finite strain elastic-plastic theory with application to plane-wave analysis. Journal of Applied Physics (1967) 36:1–6.[10] Lemarchand, E., Ulm, F.J. and Dormieux, L. The effect of inclusions on the frictioncoefficient of highly-filled composite materials. Journal of Engineering Mechanics(2002) 128 (8):876–884.[11] Maghous, S., Dormieux, L. and Barthélémy, J.F. Micromechanical approach tothe strength properties of frictional geomaterials. European Journal of MechanicsA/Solids (2009) 28:179–188.[12] Pastor, J., Thoré, Ph. and Pastor, F. Limit analysis and numerical modeling ofspherically porous solids with Coulomb and Drucker-Prager matrices. Journal ofComputational and Applied Mathematics (2010) 234:2162–2174.[13] Perić, D., de Souza Neto, E.A. and Owen, D.R.J. Computational Methodes for Plas-ticity . John Wiley & Sons, Chichester, (2008).[14] Pichler, C., Lackner, R. and Ulm, F.J. Scaling relations for viscoelasticcohesive con-ical indentation. International Journal of Materials Research (2008) 99:836–846.[15] Tabor, D. A simple theory of static and dynamic hardness. Proceedings of the RoyalSociety, Series A (1948) 192:247–274.[16] Zaoui, A., Continuum micromechanics: survey. Journal of Engineering Mechanics(2002) 128 (8):808–816.12

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Pore-space controlled hardening model in plasticity of porous materials: application to the analysis of indentation experiments

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