Elastic plastic Behavior of an Ellipsoidal Inclusion Embedded in an Elastic Matrix Procedia Engineering 173 ( 2017 ) 1116 – 1121 1877 7058 © 2017 The Authors Published by Elsevier Ltd This is an open[.]
Available online at www.sciencedirect.com ScienceDirect Procedia Engineering 173 (2017) 1116 – 1121 11th International Symposium on Plasticity and Impact Mechanics, Implast 2016 Elastic-plastic Behavior of an Ellipsoidal Inclusion Embedded in an Elastic Matrix Prasun Jana∗ Department of Mechanical Engineering, Indian Institute of Technology (Indian School of Mines), Dhanbad-826004, India Abstract In this paper, the stress-strain behavior of an elastic-plastic ellipsoidal inclusion embedded in an infinite elastic matrix is studied The parent elastic material is considered to be subjected to monotonically increasing far-field stresses Stress-strain behaviors of the inclusion are first studied using the commercial finite element package ABAQUS Then a semi-analytical method is developed using suitable extensions of the formulas used in Eshelby’s inclusion problem (1957) An excellent agreement in stress-strain behavior is found between the finite element results and the semi-analytical method The major contribution of this paper is, therefore, the development of the semi-analytical method that has several advantages over the finite element solutions The semi-analytical method is quick and accurate Additionally, when the inclusion geometry is very thin, the semi-analytical method remains viable while the finite element method would require infeasibly finer mesh refinement © Authors Published by Elsevier Ltd This c 2017 2016The The Authors Published by Elsevier B.V.is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the organizing committee of Implast 2016 Peer-review under responsibility of the organizing committee of Implast 2016 Keywords: ellipsoidal inclusion; elasto-plastic analysis; finite element; Eshelby tensor; semi-analytical approach Introduction Theory of ellipsoidal inclusions due to Eshelby [1,2] has been playing a key role in many micromechanics related problems (see [3,4]) This theory has been extensively used for the homogenization schemes used for predicting failures in composite materials [5] There are also other several areas such as damage mechanics, geomechanics, and biomechanics in which this theory has been used See [6] for a detailed review of recent works on this subject Eshelby [1] formulated the elastic fields within an ellipsoidal inclusion in a homogeneous isotropic elastic infinite medium when the inclusion is subjected to some uniform eigenstrain The eigenstrain here means ‘stress-free strain’ which the inclusion would exhibit if not constrained by the surrounding material Non-elastic strains such as due to thermal expansion, misfit strains, phase transformation, and plastic deformation can serve as eigenstrains If the inclusion has different elastic moduli than the outer material, then it is called an ‘inhomogeneous inclusion’ (see Fig 1) ∗ Corresponding author: Assistant Professor, Department of Mechanical Engineering, IIT (ISM) Dhanbad Tel.: +91 326 223 5053 E-mail address: prasunjana@gmail.com 1877-7058 © 2017 The Authors Published by Elsevier Ltd This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the organizing committee of Implast 2016 doi:10.1016/j.proeng.2016.12.075 1117 Prasun Jana / Procedia Engineering 173 (2017) 1116 – 1121 Cijkl Cijkl ε*ij (a) Ω Ω Ω Cijkl , ε*ij (b) Fig (a) Homogeneous inclusion (or simply an inclusion): The ellipsoidal subdomain (Ω) has same elastic moduli as the outer material and contains eigenstrain (ε∗i j ) (b) Inhomogeneous inclusion: The subdomain (Ω) has different elastic moduli than the outer material and contains eigenstrain (ε∗i j ) When the eigenstrain is zero the subdomain is called an inhomogeneity As mentioned above, there are huge number published research works on this topic of ellipsoidal inclusion problems However, a careful literature survey suggests that most of these studies are for inclusions having elastic material behavior whereas the cases of the elasto-plastic inclusions have not received much attention in the literature Study of these elasto-plastic inclusions may have potential applications in modeling internal energy dissipation (i.e material damping) due to elasto-plastic flaws within the material [7] In this paper, we study the stress-strain behavior of a single elasto-plastic ellipsoidal inclusion embedded in an infinite homogeneous isotropic elastic material For the elasto-plastic inclusion, J2 flow theory with linear isotropic hardening material behavior is considered We begin with a finite element simulation in ABAQUS to study the stress state and dissipation in this inclusion under far-field monotonic loading Then a semi-analytical method is developed for solving this elasto-plastic inclusion using suitable extensions of the formulas used in Eshelby’s inclusion problem Results obtained from this semi-analytical method are found to be in excellent agreement with the finite element results Details of these analyses are discussed below Finite Element analysis A schematic of an ellipsoidal inclusion subjected to far-field stresses are shown in Fig (a) The inclusion size will be taken small such that it behaves as if it is in an infinite material body In the finite element simulation in ABAQUS, a cube of 20 mm edge length is considered with a central ellipsoidal flaw of a1 = mm, a2 = 0.75 mm, and a3 = 0.5 mm A quarter of the meshed model is shown in Fig (b) A total of 82440 eight-node linear brick elements (C3D8RH) (see [8] for the element details) have been used to discretize the volume For better accuracy, high mesh refinement near the inclusion were used Table shows the material properties used for the parent and inclusion material The inclusion material is considered to follow J2 plasticity flow theory with linear isotropic hardening behavior as K(α) = σ0yp + Hα Here, σ0yp and H are material constants and α is the hardening parameter (see [9]) In ABAQUS, pseudostatic analysis is carried out with incremental changes in the far-field loads The far-field load (say σ0 ) is increased slowly up to that stress, in small increments from zero, using proportional loading In our analysis, 200 load steps are used The load steps going from to 200 are denoted using an artificial time that goes from to 1, called the normalized load step A normalized load step of 0.3 then corresponds to the 60th step, at which point the far-field load is 0.3 × σ0 This terminology has been used below The far-field stresses used for our simulation is given in Table The results show that the stress state remains essentially uniform within the ellipsoidal inclusion, both before and after yielding For example, the contour for σ x at a normalized load step of 0.6 is shown in Fig (a) 1118 Prasun Jana / Procedia Engineering 173 (2017) 1116 – 1121 z σzz τxz σxx x y τyz τxy σyy Z z x Fig (a) Ellipsoidal X y ( x2 a1 + y2 a22 + Y (a) z2 a23 (b) = 1) elasto-plastic flaw embedded in an elastic material with far-field stresses σ0i j (b) Quarter portion of the finite element mesh of the 3D model Table Material properties (chosen arbitrarily) used in all the analyses in this paper Elastic parent material Young’s modulus Poisson’s ratio 200 GPa 0.3 Elasto-plastic flaw material Young’s modulus Poisson’s ratio Initial yield strength (σ0yp ) Elasto-plastic modulus (H) 120 GPa 0.28 40 MPa 10 GPa Table Far-field stresses considered in all the analyses in this paper σ0xx σ0yy σ0zz τ0yz τ0zx τ0xy 150 MPa 200 MPa −110 MPa 80 MPa −150 MPa −140 MPa We also select an arbitrary element near the centroid of the flaw, and plot all six stress components against normalized load step in Fig (b) As expected, the stress state varies linearly until it reaches the yield envelope Subsequently, the stresses behaves in non-linear fashion It is also seen that the normal stresses increases almost linearly with far-field stress (i.e., the hydrostatic stress increases linearly) We now proceed to the development of our semianalytical approach for the analysis of this elasto-plastic inclusions Semi-analytical method In this section, a semi-analytical formulation of this elaso-plastic inclusion problem is developed using classic formulas due to Eshelby (see [1,3]) Let σ0i j be the far-field stress and and ε∗i j be the eigenstrain The far-field strain is ε0kl , satisfying σ0i j = Ci jkl ε0kl , where Ci jkl is the elasticity tensor of the parent material 1119 Prasun Jana / Procedia Engineering 173 (2017) 1116 – 1121 50 Z Stress components (in MPa) S, S11 (Avg: 75%) +2.237e+02 +2.033e+02 +1.829e+02 +1.624e+02 +1.420e+02 +1.216e+02 +1.012e+02 +8.078e+01 +6.036e+01 +3.995e+01 +1.953e+01 −8.845e−01 −2.130e+01 ODB: elliptical_a1_b0p75_c0p5_run9.odb X 30 σx 20 τyz 10 σz −10 τxy −20 −30 Step: Step−1 Increment 120: Step Time = 0.6000 Primary Var: S, S11 σy 40 τzx 0.2 0.4 0.6 0.8 Normalized load step Y (b) (a) Fig (a) Plot of σx in one quarter of the model at a normalized load step of 0.6 The variation of σx within the inclusion is small (b) Stress components inside ellipsoidal elasto-plastic flaws The uniform stress state within the ellipsoidal inhomogeneous inclusion is then given by σi j = CiΩjkl (ε0kl + εkl − ε∗kl ), with εi j = S i jkl ε∗∗ kl , (1) where CiΩjkl is the elasticity tensor of the inclusion material, and S i jkl represents components of the Eshelby tensor The undetermined ε∗∗ kl in Eq (1) is obtained from the equivalence equation ∗ ∗∗ ∗∗ CiΩjkl (ε0kl + S klmn ε∗∗ mn − εkl ) = Ci jkl (εkl + S klmn εmn − εkl ) (2) We note that the above two equations can be directly obtained from [3] Solving Eq (2) for ε∗∗ i j and substituting in Eq (1), we obtain σi j = Ai jkl ε0kl + Bi jkl ε∗kl , (3) where Ai jkl and Bi jkl are tensors given in Appendix A Now, we address the problem of a elasto-plastic inclusion with isotropic hardening material behavior In this case, the plastic strain in the ellipsoidal inclusion is the eigenstrain And this eigenstarin is not known in advance but must be obtained as the solution progresses Since we will work with rate-independent incremental plasticity, we will use the rate form of Eq (3) as σ˙ i j = Ai jkl ε˙ 0kl + Bi jkl ε˙ klp or σ ˙ = A : ε˙ + B : ε˙ p (4) p We have used a different symbol (ε˙ i j ) in Eq (4) for the eigenstrain rate as it corresponds to the plastic strain in our case We note here that Eq (4) can be used for both purely elastic and elasto-plastic deformations The plastic strain rate ε˙ p = as long as the material behaves elastically (initial loading, during unloading, etc.) When the stress state inside the inclusion reaches the yield envelop plastic strain will be developed The yield envelope in the present case is based on the von Mises yield criterion (J2 flow theory), √ f (σ, α) = S : S − K(α) = 0, (5) where S is the deviatoric stress tensor of σ, and K(α) = σ0yp + Hα corresponds to the linear isotropic hardening material behavior (see [9]) 1120 Prasun Jana / Procedia Engineering 173 (2017) 1116 – 1121 0.35 σy 40 30 σx 20 τyz 10 σz −10 τxy −20 −30 Dissipation (in N-mm) Stress components (in MPa) 50 τzx 0.2 0.4 0.6 0.8 Semi-analytical Abaqus 0.3 0.25 0.2 0.15 0.1 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Normalized load steps Normalized load step (a) (b) Fig (a) Comparison of semi-analytical method results against the ABAQUS results (b) Comparison of plastic dissipation within the inclusion The rate of these plastic strain components is normal to the yield envelope Therefore, one obtains ε˙ p = γ ∂f ∂σ (6) for some γ Now from the consistency condition, we get ˙ + ∂α f · α˙ = 0, f˙ = ∂σ f : σ (7) with α˙ = γ 23 (see [9]) Above equations can be used for an analysis of monotonic far-field loading In the beginning of loading the material will behave as elastic The stresses within the inclusion increase linearly as per Eq (4) with ε˙ p = 0, and the Eq (5) has to be used for checking the onset of yielding Once yielding takes place, i.e., in the plastic domain, Eqs (4), (6) and (7) are used simultaneously to solve for σ, ˙ ε˙ p and γ Matlab’s inbuilt integration routine “ode45” has been used to solve these equation to find the stress state Finally, the plastic energy dissipation (Wd ) within the inclusion is computed, using ˙ d = (σ : ε˙ p )V, W (8) where V is the volume of the ellipsoid Results and discussions Here, we compare the results from the above semi-analytical method with the finite element results presented earlier Figure (a) shows the comparison The stress plots shown in this figure actually contain two sets of visually indistinguishable superposed graphs, one from the semi-analytical calculation and another from the ABAQUS simulation We also compared the plastic dissipation from this two approaches in Fig (b) Figures (a) and (b) establish that the results from the semi-analytical method are in excellent agreement with the finite element results However, it can be noted that the semi-analytical method has advantages over the finite element solutions The semi-analytical method is quick and accurate Additionally, when the inclusion geometry is very thin, the semi-analytical method remains viable while the finite element method would require infeasibly finer mesh refinement Prasun Jana / Procedia Engineering 173 (2017) 1116 – 1121 1121 Conclusions In this work, the stress-strain behaviour of an elasto-plastic ellipsoidal inclusion embedded within an infinite elastic parent material has been studied The study began with finite element simulation in ABAQUS In the second part, a semi-analytical method was developed using Eshelby’s formula for ellipsoidal inclusion We showed that the results from this semi-analytical method are in excellent agreement with the ABAQUS simulation results We also pointed out that the semi-analytical method has advantages over the finite element solutions For very thin inclusion geometry, the semi-analytical method will remain viable while the finite element analysis would require infeasibly finer mesh refinement Therefore, the major contribution of this paper is the development of the semi-analytical method In future, this work will be extended for the combined hardening material model and for cyclically applied far-field stresses Acknowledgements The author thanks Sukanta Chakraborty, ISM Dhanbad, for useful discussions on this research topic Appendix A Details of tensors Ai jkl and Bi jkl The tensors Ai jkl and Bi jkl (written here as A and B for simplicity) of Eq (3) are given by −1 −1 A = C Ω − C Ω S C + (C Ω − C)S (C − C Ω ), and B = C ∗ S C + (C Ω − C)S C Ω − C Ω , where S is the Eshelby tensor, and C and C Ω are the stiffness tensors References [1] [2] [3] [4] [5] [6] J.D Eshelby, The determination of the elastic field of an ellipsoidal inclusion, and related problems, Proc R Soc Lond A 241 (1957) 376–396 J.D Eshelby, The elastic field outside an ellipsoidal inclusion Proc R Soc Lond A 252 (1959) 561–569 T Mura, Micromechanics of Defects in Solids, second ed., Martinus Nijhoff, Dordrecht, Netherlands, 1987 J Qu, M Cherkaoui, Fundamentals of Micromechanics of Solids Wiley, Hoboken, New Jersey, 2006 T Mori, K Tanaka, Average stress in matrix and average elastic energy of materials with misfitting inclusions Acta Metall 21 (1973) 571–574 K Zhou, J.J Hoh, X Wang, L.M Keer, J.H Pang, B Song, Q.J Wang, A review of recent works on inclusions Mech Mater 60 (2013) 144158 [7] P Jana, A Chatterjee, Power-law damping from dispersed elasto-plastic flaws with Weibull-distributed strengths, Int J Mech Sci 87 (2014) 137–149 [8] ABAQUS Inc., Analysis User’s Manual, Release 6.9 Documentation for ABAQUS, 2009 [9] S.C Simo, T.J.R Hughes, Computational Inelasticity, Springer-Verlag, New York, 1998 ... stress-strain behavior of a single elasto -plastic ellipsoidal inclusion embedded in an in? ??nite homogeneous isotropic elastic material For the elasto -plastic inclusion, J2 flow theory with linear isotropic... problem of a elasto -plastic inclusion with isotropic hardening material behavior In this case, the plastic strain in the ellipsoidal inclusion is the eigenstrain And this eigenstarin is not known in. .. Prasun Jana / Procedia Engineering 173 (2017) 1116 – 1121 1121 Conclusions In this work, the stress-strain behaviour of an elasto -plastic ellipsoidal inclusion embedded within an in? ??nite elastic