A comparison between two analytical models that approximate notch-root elastic–plastic strain–stress components in two-phase, particle-reinforced, metal matrix composites under multiaxial cyclic loading: Theory

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In this paper, two analytical models for estimating surface notch-root elastic–plastic, strain–stress histories in particulate metal matrix composites (PMMCs) when subjected to multiaxial cyclic loads are presented and compared. The models are based

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  A comparison between two analytical models that approximate notch-rootelastic–plastic strain–stress components in two-phase, particle-reinforced,metal matrix composites under multiaxial cyclic loading: Theory Gbadebo M. Owolabi, Meera N.K. Singh  *  Department of Mechanical and Manufacturing Engineering, University of Manitoba, Winnipeg, Man., Canada R3T 2N2 Received 27 August 2004; received in revised form 26 June 2005; accepted 26 September 2005Available online 28 November 2005 Abstract In this paper, two analytical models for estimating surface notch-root elastic–plastic, strain–stress histories in particulate metal matrixcomposites (PMMCs) when subjected to multiaxial cyclic loads are presented and compared. The models are based on the concept of incrementalaverage stress, the endochronic theory of plasticity, and modified forms of the incremental equivalent strain energy density method (ESED) andNeuber’s rule. Each model provides a complete set of equations capable of predicting the elastic–plastic strain–stress histories at the notch rootgiven a corresponding elastic solution. The preliminary analytical results obtained by applying the formulations to a circumferentially notchedround bar show that each model can independently be used to predict the nonlinear heterogeneous behavior at the notch root in PMMCcomponents. q 2005 Elsevier Ltd. All rights reserved. Keywords:  Particulate metal matrix composites; Endochronic theory; Notch analysis; Equivalent strain energy density (ESED) method; Neuber’s rule 1. Introduction Particulate metal matrix composites (PMMCs) are increas-ingly finding applications in the aerospace, automotive, sportsequipment, and electronic industries. This is due to their highstiffness and strength, creep and wear resistance, and superiorperformance at elevated temperatures. Moreover, in contrast totheir long fiber counterparts, PMMCs generally have isotropicproperties and are much easier to produce and machine bystandard methods. However, challenges such as the lack of analytical tools available to predict the mechanical behavior of PMMC components when they are subjected to multiaxialcyclic loads, must be overcome before industries can benefitfrom their superior properties.When components are subjected to cyclic loads, cracks andtherefore failures generally initiate at a structural discontinuity(notch). These stress raisers may include bolt holes, keyways,splines, and shoulder fillets. Although it can be assumed thatthe bulk of a PMMC component remains elastic, the stresses inthe metallic matrix in the vicinity of these concentrations mayexceed the yield stress of the material. In other words, localizedmetal matrix plasticity must often be assumed in design.Two associated research papers [1,2] have been published inthe open literature that have presented fatigue data formultiaxially and cyclically loaded PMMCs. These papers areaimed at examining the validity of using multiaxial fatigue lifeprediction relations, developed for metals, to PMMC com-ponents. Although the papers show the applicability of criticalplane and energy-based models to fatigue life prediction of PMMCs, they highlight the necessity for knowing both theelastic–plastic strain and stress histories at a component’scritical location for application.In order to determine the notch-root elastic–plastic strainand stress histories in homogeneous components, experimen-tal, numerical, and approximate analytical techniques havebeen employed. Although experimental and numericalmethods are the most accurate, they are expensive and timeconsuming, particularly when the components modeled aresubjected to long histories of nonproportional cyclic loads. Tomake fatigue life prediction tools more conducive to typicaldesign environments and to shorten design lead times, there hasbeen a great effort to develop simplified analytical techniquesthat approximate the actual elastic–plastic notch-root materialbehavior. The most commonly used, simplified techniques International Journal of Fatigue 28 (2006) 910–917www.elsevier.com/locate/ijfatigue0142-1123/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijfatigue.2005.09.010 *  Corresponding author. Tel.: C 1 204 474 7333; fax: C 1 204 275 7507. E-mail address:  singhmn@cc.umanitoba.ca (M.N.K. Singh).  applicable to homogeneous materials subjected to uniaxialcyclic loading are Neuber’s rule [3], and the Moski–Glinkaequivalent strain energy density (ESED) method [4]. These methods use the results of an elastic notch-root analysis topredict the elastic–plastic notch-root strains and stresses. Bothof these methods have been developed in incremental form toaddress homogeneous components subjected to nonproportion-ally applied cyclic loads [5,6].Little research has been conducted on the problem of stressconcentrations in PMMCs, perhaps due to the inherentmodeling difficulties when the composites have a finiteconcentration of inclusions, the matrix material in the vicinityof the discontinuity is elastic–plastic, and when multiaxialloads are applied. For uniaxial cyclic loading, it has beenindicated that the uniaxial simplified methods for notch-rootelastic–plastic strain and stress prediction could be used inconjunction with the stabilized hysteresis loop to predict thefatigue life of PMMC components [7]. In [8], Owolabi and Singh developed a Neuber-type, simplified model fordetermining the elastic–plastic strain and stress histories inPMMC components subjected to multiaxial cyclic loads. Themodel proved to provide accurate results when compared tolimited experimental results on circumferentially notched barssubjected to multiaxial cyclic load paths.This paper presents an alternate analytical model thatestimates the notch-root elastic–plastic strain and stresshistories in PMMC components and compares it to thatdeveloped in [8]. The model is comprised of a cyclic plasticitymodel applicable to PMMCs, previously published by theauthors in [9], and a ESED-type, simplified method. In Section2, the problem is illustrated and in Section 3, the constitutivemodels are presented. Section 4 presents the approximaterelations. The summary of the approximate solutions ispresented in Section 5 and in Section 6 numerical resultsobtained using the models are shown for two loading paths.Conclusions are drawn in Section 7. An experimentalvalidation of both models is presented in a companion paperby the authors [10]. 2. Problem background Fig. 1 shows a representative notched PMMC componentsubjected to an external axial load  P  and torsional load  T  . Sincethe lower stressed bulk material away from the higher stressed Nomenclature i ,  j ,  k  ,  l ,  s ,  t   indices,  i ,  j ,  k  ,  l ,  s ,  t  Z 1, 2, 3 (summationconvention) r r  th component C  ijkl (  f  )  reinforcement stiffness tensor C  ijkl ( m )  matrix stiffness tensor S  ij  components of deviatoric stress tensor S  ijkl  components of Eshelby’s tensor V  o  composite volume V   f   reinforcement volume fraction V  m  matrix volume fraction W   strain energy density U  total strain energy density  z  intrinsic time scale a r   material constants a ,  b  indices,  a ,  b Z 1, 2, 3, no summation is implied D  finite difference 3 ij  components of strain tensor r ( ) memory function s ij  components of stress tensord( $ ) differential form of variable or constant( $ ) e elastic components of a variable( $ ) (  f  )  reinforcement component of a variable or constant( $ ) E elastic–plastic components estimated by ESEDmethod( $ ) N elastic–plastic components estimated by Neuber’srule( $ ) ( m )  matrix components of a variable or constant( $ ) p plastic components of a variable( $ ) q  q th step component of a variable( $ ) T total components of a variable Fig. 1. Problemdefinition (a) circumferentially notched PMMCbar; (b)notch root stress state at a given  P  and  T  ; (c) increments in stress and strainfor increments d P and d T  . G.M. Owolabi, M.N.K. Singh / International Journal of Fatigue 28 (2006) 910–917   911  notch-root material imposes constraints on the deformation,nonlinear stress gradients are induced in the vicinity of thenotch-root even if elastic material behavior is assumed.Any external notch is a traction free surface, and as such,there are three unknown stress (Fig. 1b) and correspondinglyfour unknown strain components at a notch root. In otherwords, the material at the notch root is in a state of plane stress.Since an elastic–plastic analysis demands an incrementaldevelopment, the solution to the notch-root problem entailsdefining seven unknown increments in stress and strain(Fig. 1c) for given increments in the external axial (d P ) andtorsional (d T  ) loads.In order to quantify these seven increments, a set of sevenrelations is required. These relations must account for theaforementioned structural and material discontinuities.Following the approach of the approximate theories describedin Section 1, here, it is proposed that four of these relations canbe determined from a combination of a cyclic plasticity modelapplicable to PMMCs (Section 3), and the remaining threerelations, from proposed incremental approximate theories(Section 4). 3. Cyclic constitutive relations for PMMCs When a PMMC component is subjected to an externalloading system, it is commonly assumed that the particulatereinforcement remains elastic due to its relatively highstiffness, while the ductile matrix may deform in an elastic–plastic manner. The components of the total incremental matrix( m ) strain tensor, d 3 T ij ð m Þ , can therefore be decomposed into itsincremental elastic, d 3 e ij ð m Þ , and plastic, d 3 p ij ð m Þ , strain com-ponents, ord 3 T ij ð m Þ Z d 3 e ij ð m Þ C d 3 p ij ð m Þ :  (1)The components of the increment in the reinforcement’s (  f  )total strain tensor, d 3 T ij ð  f  Þ , are comprised of only elasticcomponents, d 3 e ij ð  f  Þ  ord 3 T ij ð  f  Þ Z d 3 e ij ð  f  Þ :  (2)All notch-root elastic–plastic strain and stress historyprediction methods, whether experimental, numerical, orapproximate require the adoption of a cyclic constitutivemodel that describes the relations between the above straincomponents and the stress components for a given increment ordecrement in the external load. There has been a very limitedamount of work presented in the literature that describes thecyclic elastic–plastic constitutive behavior of PMMCs sub- jected to multiaxial loads. Two relevant research investigationshave been published by Lease et al. [11] and Owolabi andSingh [9]. Both publications make use of the incremental meantheory developed by Li and Chen [12] (Section 3.1) to describethe elastic strain components in the constituents. The papersdiffer in describing the plastic components (Section 3.2) of thematrix strains. Since the details of the constitutive models havebeen published, only the highlights of the model in [9] arepresented here. 3.1. Elastic analysis In [9,11], relations for estimating the average incremental elastic stress and strain in the composite constituents wereformulated based on Li and Chen models [12] and arepresented briefly here.If an unreinforced matrix is elastic, the resulting incrementsinelastic stress components in the matrix, d s ij ( m ) ,can be relatedto those associated with the nominally applied loads, d s ij , as:d s ij ð m Þ Z d s ij K V   f  C  klst  ð m Þ ð S  klst  K I Þ  L  K 1 ð C  klst  ð  f  Þ K C  klst  ð m Þ Þ C  K 1 klst  ð m Þ  d s ij :  (3)In Eq. (3)  L  Z ½ð V   f  K 1 Þ C  ijkl ð m Þ ð I K S  ijkl Þ C C  ijkl ð  f  Þ ½ V   f   ð S  ijkl K I Þ K S  ijkl  : (4) V   is the volume fraction of the subscripted constituent,  C  ijkl represents the components of the stiffness tensor of thesubscripted constituent,  I  is the identity tensor, and  S  ijkl  arethe components of Eshelby’s tensor [13]. Eshelby’s tensor canbe found from the inclusions geometry and the matrixPoisson’s ratio.The components of the incremental matrix elastic straintensor are related to the above increments in the correspondingstress tensor by generalized Hooke’s law as:d 3 e ij ð m Þ Z C  K 1 ijkl ð m Þ  d s kl ð m Þ :  (5)The components in the incremental stress, d s ij (  f  ) , and elasticstrain tensors in the reinforcement can be found, respectively,fromd s ij ð  f  Þ Z d s ij C V  m C  klst  ð m Þ ð S  klst  K  I  Þ  L  K 1 ð C  klst  ð  f  Þ K C  klst  ð m Þ Þ C  K 1 klst  ð m Þ  d s ij ;  (6)andd 3 e ij ð  f  Þ Z C  K 1 ijkl ð  f  Þ  d s kl ð  f  Þ :  (7)The increments in the composite mean strain components,d 3 ij , can be obtained using the weighted sum of the work doneby the stress increments of the constituents or,d s ij d 3 ij Z V  m  d s ij ð m Þ d 3 ij ð m Þ C V   f   d s ij ð  f  Þ d 3 ij ð  f  Þ :  (8)Eq. (8), developed by Li and Chen [12], applies to both elastic and elastic–plastic material behavior.Using Eqs. (3)–(8) that outline the incremental mean fieldtheory, the average increment in stress and strain in an elasticPMMC can be determined knowing the strain or stress fieldthat would arise in the component if it was hypothetically madeof a homogenous matrix material, and the composite materialproperties. 3.2. Elastic–plastic behavior  In [9], the Mro´z [14] multi-surface model and theendochronic theory [15,16], used in conjunction with the G.M. Owolabi, M.N.K. Singh / International Journal of Fatigue 28 (2006) 910–917  912  incremental mean field theory [12] and Eq. (8), wereevaluated for their use in predicting the elastic–plasticconstitutive behavior of PMMCs under multiaxial loading.It was found that both models yield accurate results for thelimited number of load paths tested. However, it has beenshown [16] that for homogeneous materials, the endochronictheory of plasticity can better predict some of the transienteffects associated with nonproportional variable amplitudecyclic loading and thus will be adopted here.Using the endochronic theory of plasticity [15], assuming cyclically stable behavior, in [9] the relationship defining thematrix plastic strain increment is given as D 3 p ij ð m Þ Z a ij ð m Þ  D  zb ;  (9)where a ij ð m Þ Z 12  ð D S  ij ð m Þ Þ q C X nr  Z 1 ð S  r ij ð m Þ Þ q K 1 ð 1 K e K a r   D  z Þ "# ;  (10)and b Z X nr  Z 1 C  r  ð 1 K e K a r   D  z Þ a r  "# :  (11)In Eqs. (9)–(11),  q  is the number of steps into which the loadpath is divided,  S  ij ( m )  are the components of the matrixdeviatoric stress tensor,  S  ij ð m Þ Z s ij ð m Þ K s kk  ð m Þ d ij  =  3,  d ij  is theKro¨neker delta, and  z  is the intrinsic time scale associated withthe deformation history of the material.  C  r   and  a r   are materialconstants determined by fitting a Dirichlet series function,  r (  z ),with  n  terms in the form of  r ð  z Þ Z X nr  Z 1 C  r   e K a r   z ;  (12)to the uniaxial cyclic stress–plastic strain curve of thehomogeneous matrix material.As shown in [9,16] for stress-controlled simulations, for agiven increment in load, the increment in the intrinsic timescale can be found using: ð D S  ij ð m Þ Þ q Z 2 1 D  z X nr  Z 1 C  r  a r  ð S  r ij ð m Þ Þ q K 1 ð 1 K e K a r   D  z Þ "# ð D 3 p ij ð m Þ Þ q C X nr  Z 1 ð S  r ij ð m Þ Þ q K 1 ð 1 K e K a r   D  z Þ "# : (13)Rearranging and taking the inner product of Eq. (13) yield: b 2 K a ij a ij Z 0 Z  R ð D  z Þ :  (14)For a given increment of composite stress tensor, Eq. (14)can be used to solve for the increment in the intrinsic timescale,  D  z , using a numerical computation method to find theroots of the equations.The matrix cyclic constitutive relation can be finalized bycombining Eqs. (1), (5), and (9)–(14) giving: D 3 ij ð m Þ Z 1 C y m E  m D s ij ð m Þ K y m E  m ð D s kk  ð m Þ Þ d ij C a ij ð m Þ  D  zb :  (15)In Eq. (15),  y m  and  E  m  are the matrix Poisson’s ratio andelastic modulus, respectively. 4. Approximate relations Virtually all of the relations proposed to date that are used toapproximate the elastic–plastic notch-root strains and stressesin homogeneous components are based on an equation of notch-root parameters in two geometrically identical notchedbodies. One of the bodies is assumed to exhibit elasticbehavior, and the other, elastic–plastic behavior with the samematerial constants as the elastic body. Consequently, implicitto these simplified models is the assumption that the plasticzone at the notch-root is small relative to the surroundingelastic field.Following the same methodology, in this section, two sets of relations are defined to address notch-root elastic–plasticbehavior in heterogeneous PMMC components. Since thesemethods have their roots in the ESED method and Neuber’srule, both srcinally proposed for homogeneous bodies in planestress, they are categorized as such below. 4.1. Incremental equivalent strain energy density method (ESED) for PMMCs The ESED method [4] was originally developed for abody in plane stress, where there is a uniaxial stresscondition at the notch root. In developing the method,Molski and Glinka [4] proposed that the actual strain energydensity at an elastic–plastically deforming notch-root isexactly the same as if the material was to hypotheticallyremain elastic. Following the notation given in Fig. 1, thismodel can be written as: ð  3 e22 0 s 22  d 3 e22 Z ð  3 E22 0 s 22  d 3 E22 ;  (16)where e stands for the elastic notch root parameters and Estands for the notch root elastic–plastic parameters asestimated by ESED method.In [5], a generalized incremental strain energy densitymethod was developed to address the path dependent nature of the elastic–plastic strain–stress history at the notch root in ahomogeneous body subjected to nonproportional multiaxialcyclic loads. The relations state that for a given increment inthe applied load, the increment in the strain energy density atthe notch root in an elastic-plastic body is the same as thatwhich would be obtained at the notch root if the body were tohypothetically remain elastic as the loads were applied. Thehypothesis can be represented mathematically asd W  e ij Z d W  E ij  ;  (17) G.M. Owolabi, M.N.K. Singh / International Journal of Fatigue 28 (2006) 910–917   913  or, s e ij  D 3 e ij Z s E ij  D 3 E ij :  (18)In this paper, a modified form of the above incrementalESED method is proposed, based on a definition of macrostrain energy density,  W  , proposed by Duva and Hutchinson[17] for composite systems. For a two-phase composite systemwith a material volume  V  o , Duva and Hutchinson’s definitionof macro strain energy density reduces to: W  Z 1 V  o ð  V  m W  m  d V  C ð  V   f  W   f   d V  0B@1CA ;  (19)where  W   f   and  W  m  are the reinforcement and matrix strainenergy densities, respectively, and  V   f   and  V  m  are thereinforcement and matrix volume fractions, respectively.For a composite system under nonproportional loading,here, it is proposed that for a given increment in external load,the corresponding increment in the constituents weighted strainenergy density at the notch root in an elastic–plastic body canbe approximated by that which would have been obtained if thecomposite system were to remain elastic through out theloading history. This hypothesis can be expressed as D W  e ij Z ð 1 K V   f   Þ D W  E ij ð m Þ C V   f   D W  e ij ð  f  Þ ;  (20)or, s e ij  D 3 e ij Z ð 1 K V   f   Þ s E ij ð m Þ  D 3 E ij ð m Þ C V   f  s e ij ð  f  Þ  D 3 e ij ð  f  Þ :  (21)If   V   f  Z 0, thus indicating a homogeneous material, Eq. (21),reduces to Eq. (18), and forplane stress, the srcinal form of theESED method rule given in [4]. Using Eq. (21) along with the elastic–plastic constitutive relations (Eq. (15)) will only givefive out of seven equations needed to completely define thenotch-root elastic strains and stress for the composite matrix.Consequently, two additional independent equations arerequired.To obtain the additional equations, it is proposed here thatfor the composite system, the weighted contribution of each of the elastic–plastic strain–stress components of the constituentsto the increment in strain energy density at the notch root is thesame as the contribution of each stress–strain components tothe increment in composite strain energy density at the notchroot when obtained from an elastic analysis. This proposedhypothesis can be expressed as: s e ab  D 3 e ab s e ij  D 3 e ij Z ð 1 K V   f   Þ s E ab ð m Þ  D 3 E ab ð m Þ C V   f  s e ab ð  f  Þ  D 3 e ab ð  f  Þ ð 1 K V   f   Þ s E ij ð m Þ  D 3 E ij ð m Þ C V   f  s e ij ð  f  Þ  D 3 e ij ð  f  Þ :  (22)In Eq. (22), the indices,  i ,  j ,  a ,  b Z 1, 2, 3 and summation isassumed over  i ,  j , while for  a ,  b , summation is not implied.Substituting Eq. (21) into Eq. (22) results in: s e ab  D 3 e ab Z ð 1 K V   f   Þ s E ab ð m Þ  D 3 E ab ð m Þ C V   f  s e ab ð  f  Þ  D 3 e ab ð  f  Þ :  (23)The developed Eq. (23) provides three equations. Conse-quently, a combination of four equations from the elastic–plastic constitutive equation, Eq. (15), with Eq. (23) yieldsthe required set of seven equations necessary to completelydefine the composite system matrix notch-root elastic–plasticstrain–stress increments.It should be emphasized here that the elastic components ein Eq. (23) are not nominal values, but values obtained at thenotch root from an elastic analysis. These values can be foundusing Eqs. (3)–(8), the external loads and the stressconcentration factors of the notched geometry modeled. 4.2. Incremental Neuber’s rule for PMMCs [8] The original Neuber’s rule [3] was proposed for ahomogeneous notched body under pure shear stress, but it isoften used for notched bodies under tension and bending loads.Owolabi and Singh [8] laid out a detailed methodology forapproximating the notch-root elastic–plastic strains in PMMCcomponents subjected to multiaxial loading using a modifiedform of the Neuber’s rule. As shown in [8], for an increment inload on the composite system, the corresponding increment inthe constituents-weighted total strain energy density at thenotch root in an elastic–plastic body,  U N , can be approximatedby which that would have been obtained if the compositesystem were to remain elastic through out the loading history, U e . This proposed hypothesis can be expressed using: D U e Z ð 1 K V   f   Þ D U N ð m Þ C V   f   D U e ð  f  Þ :  (24)The formulation can be re-written as: s e ij  D 3 e ij C 3 e ij  D s e ij Z ð 1 K V   f   Þð s N ij ð m Þ  D 3 N ij ð m Þ C 3 N ij ð m Þ  D s N ij ð m Þ Þ C V   f   ð s e ij ð  f  Þ  D 3 e ij ð  f  Þ C 3 e ij ð  f  Þ  D s e ij ð  f  Þ Þ :  (25)In Eq. (25), N stands for the notch-root elastic–plasticparameters as estimated by Neuber’s method. If   V   f  Z 0, thusindicating a homogeneous material, Eq. (25), reduces to theincremental homogeneous form of Neuber’s given in [5], and for plane stress, the srcinal form of the Neuber’s rule given in[3].Two additional equations are required to completely definethe notch-root elastic–plastic strain and stress. In [8], Owolabiand Singh proposed that for the composite system, theweighted contribution of each of the elastic–plastic strain–stress component to the increment in total energy density at thenotch root is the same as the contribution of each stress–straincomponent to the increment in composite total strain energydensity at the notch root when obtained from elastic analysis.Combining this hypothesis with Eq. (25) yields: s e ab  D 3 e ab C 3 e ab  D s e ab Z ð 1 K V   f   Þð s N ab ð m Þ  D 3 N ab ð m Þ C 3 N ab ð m Þ  D s N ab ð m Þ Þ C V   f   ð s e ab ð  f  Þ  D 3 e ab ð  f  Þ C 3 e ab ð  f  Þ  D s e ab ð  f  Þ Þ :  (26)Eq. (26) provides three additional Neuber-type relationshipsnecessary to use in conjunction with the cyclic constitutive G.M. Owolabi, M.N.K. Singh / International Journal of Fatigue 28 (2006) 910–917  914
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