based on finite element analysis and mechanical threshold stress model

Transcription

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2 Residual stress prediction in laser shock peening based on finite element analysis and mechanical threshold stress model A thesis submitted to the Graduate School of the University of Cincinnati in partial fulfillment of the requirements for the degree of Master of Science in the School of Dynamic Systems of the College of Engineering and Applied Science 2012 by Chinmay J Tophkhane Bachelor of Engineering University of Pune, India 2008 Committee Chair: Dr. Dong Qian

3 Abstract This thesis focuses on a physically based strain rate dependent plasticity model known as the Mechanical Threshold Stress (MTS) model proposed by Follansbee and Kocks. The objective is to develop an algorithm based on the tangent modulus method to resolve the constitutive equation represented by the MTS model and use it to analyze the material response under laser shock peening for the Ni alloy INCONEL 718 (IN718). A user defined subroutine has been developed and integrated with commercial software LS-DYNA. A parametric study is carried out to study the influence of various model parameters on the predicted residual stresses. The developed model is then applied in the study of residual stresses imparted on INCONEL 718 induced by laser shock peening (LSP) process. Finite element analysis is performed for the case of plate made of INCONEL 718 and the residual stress predictions are compared with experimental results. The model predictions are found to be in good agreement with the experimental results. To the best of author s knowledge, this is the first time that an MTS model has been developed for IN 718 with an integrated approach. ii

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5 Acknowledgements I would like to thank my advisor Dr. Dong Qian for his support and guidance during the course of this work. I am grateful to him for his constant encouragement and motivation. Also I would like to thank Dr. Vijay Vasudevan and Dr. Janet Dong for taking time to serve on my thesis evaluation committee. I would like to thank my CAE lab colleagues for their help and support. A special note of thanks to my friend and mentor Sagar Bhamare. Without his help and guidance this work would not have been possible. I really appreciate the patience he has shown while answering all my doubts and queries. I would like to thank all my friends for their direct and indirect support during the course of my graduate studies at University of Cincinnati. Last but not the least; I would like to thank my parents, my sister and all my family members for their love, encouragement and support during my good and tough times. iv

6 Contents 1.Introduction to constitutive material models and motivation Linear elastic constitutive model Elasto - plastic constitutive model Rate dependent constitutive models for high strain rate processes Johnson-Cook model Motivation Outline of the Thesis Mechanical threshold stress (MTS) model Background Theory Thermal activation Structure evolution A constitutive solver based on tangent modulus method Background Derivation of tangent modulus for MTS model Application of tangent modulus method to MTS model Algorithm for implementation of tangent modulus method with MTS model v

7 4. Laser shock peening process Introduction Mathematical modeling of shock wave pressure Application to residual stress prediction Residual stress prediction methodology Finite element analysis Problem configuration Results and discussion MTS model - parametric study - athermal stress ( σ a) variation MTS model - parametric study - variation of mechanical threshold stress characterizing dislocation interactions with interstitial atoms ( σ i) MTS model - parametric study - variation of structure evolution parameters Optimized parameters of MTS model for INCONEL Residual stress prediction in overlapping LSP Conclusions and future work Conclusions Future work vi

8 List of figures Figure 1: Stress Vs. Strain at different strain rates Figure 2: Temperature Vs. Strain at different strain rates Figure 3: Laser shock peening process Figure 4: Geometry of the model used for analysis Figure 5: Temporal profile of shock wave pressure Figure 6: Residual stress Vs. Depth for different values of athermal stress ( σ a) Figure 7: Residual stress Vs. Depth for different values of ( σ i) Figure 8: Residual stress Vs. Depth for different values of A Figure 9: Residual stress Vs. Depth for different values of C Figure 10: Residual stress Vs. Depth for different values of D Figure 11: Residual stress Vs. Depth using proposed model parameters vii

9 Chapter 1 Introduction to constitutive material models and motivation A constitutive model is a model that describes the stress response in a material as a function of the strain, strain rate and any other factors that may contribute. It is typically expressed in a set of equations that represent the basic physics of deformation of a body. In computational analysis techniques such as the finite element method, constitutive equation plays an important role. Choice of a constitutive model primarily depends on the material and nature of loading. For example, we can categorize the constitutive models as linear and nonlinear elastic models and rate dependent and rate independent models, which are briefly described below. 1. Linear elastic constitutive model In an elastic material, the material response is path independent. Hence the response of the material to loading is reversible. It is also independent of the rate of loading. The linear model implies that the stress is a linear function of strain. This linear relationship was first stated by Robert Hooke and is popularly known as Hooke s law. Hooke s expression requires both geometrical properties and material characteristics. Thomas Young came up with an expression for this proportional behavior of materials independent of geometry. He introduced a measure popularly known as the Young s Modulus which depends only on the material properties. A simple unidirectional relationship with Young s modulus is given by σ = E ε (1.1) where σ is the uniaxial stress, E is the Young s modulus and ε is the uniaxial strain. For a triaxial state of stress, the linear elastic relationship can be expressed as 1

10 1 υ υ υ υ 1 υ υ σ ε 11 σ 2 υ υ 1 υ ε 22 σ 3 E (1 2 υ) ε = σ 2 23 (1 υ)(1 2 υ) 2ε 23 + (1 2 υ) σ ε 13 2 σ 12 2ε 12 (1 2 υ) (1.2) where [ σ i] and [ i] modulus and Poisson s ratio. ε are respectively stress and strain tensors. E and υ are respectively Young s The linear elastic constitutive model is applicable only when the material is within the small-strain regime and undergoes small rotations. It fails to predict the correct stresses when a material undergoes large rotations and no longer has a linear relationship between the stress and the strain. It is also not applicable when the material is loaded into the plastic deformation range and undergoes permanent deformation. 1.2 Elasto - plastic constitutive model When a material is loaded beyond its elastic limit, it enters plastic regime and undergoes permanent deformation. In most of the engineering applications the components are loaded within their elastic limits as permanent deformations are undesirable. However, in some applications materials undergo permanent deformations and we need to predict their behavior in the plastic regime. For example in manufacturing processes such as metal forming, the plastic behavior of material is of interest. Unlike elastic regime the behavior of material in elastic-plastic regime is path dependent. Stress and strain cannot be related by a simple equation as the loading history has to be taken into consideration. Hence, to establish a stress-strain relationship one has 2

11 to establish a nonlinear constitutive law to reflect the effects of plastic deformation. Various constitutive models of plasticity have been developed for a variety of applications. They can be broadly classified as rate independent and rate dependent models. Rate independent models are used in applications where the material is not sensitive to the loading rate in the application. Rate dependent models are required for applications involving high working temperatures or high strain rates. 1.3 Rate dependent constitutive models for high strain rate processes The deformation behavior of metals, composites and many other materials is strain rate sensitive. The techniques used for prediction of their behavior at low strain rates may not be applicable for processes involving higher strain rates. Some of the common applications of high strain rate include automobile crash testing, forging and other metal forming processes, impact testing, etc. Laser shock peening is also a process where the material undergoes deformation at a very high strain rate. The deformation mechanisms vary with the strain rate. In high strain rate regime (10 3 /sec and higher), one needs to consider the effects of inertia, wave propagation influences and thermal effects [1]. Various empirical and semi-empirical constitutive models have been developed to model the behavior of materials at high strain rates. Johnson and Cook [2] proposed an empirical model based on the experimental data obtained from various tests including Hopkinson bar test, static tensile test and torsion test. The model takes into account strain hardening, strain rate effects and thermal softening. They further expanded their model with inclusion of a fracture based damage model. Steinberg, Cochran and Guinan [3] proposed a semi-empirical model which is applicable at strain rates as high as 10 5 /sec. This model focuses on the effect of pressure and temperature on 3

12 the shear modulus. Zerilli and Armstrong [4] proposed a model based on dislocation mechanics. This model includes the effects of crystal structure and grain size on the responses of different materials on loading. Preston, Tonks and Wallace [5] proposed a model for simulation of explosive loading and high velocity impacts. This model uses Wallace s theory of overdriven shocks in metals for strain rates higher than 10 9 /sec and neglects work hardening effects at these extremely high strain rates. It is thus applicable to a wide range of strain rates, e.g., 10-3 /sec to /sec. Follansbee and Kocks [6] proposed a physically based semi-empirical model, popularly known as the mechanical threshold stress (MTS) model to predict the deformation of copper. This model is explained in the later sections Johnson-Cook model Johnson-Cook model is one of the most widely used material models because of its simple form. The availability of model parameters obtained from experimental data for a wide range of materials has also been a reason for its success. The model is given by following expression * * ( ) ( ) n m σ = A+ Bε 1+ C ln ε 1 T ɺ (1.3) where σ is the von Mises flow stress, ε is equivalent plastic strain, strain rate and * εɺ is normalized plastic * T is the normalized temperature. A, B, C, m and n are material constants. The term in the first set of brackets represents plastic strain effect through power law hardening. Constant A is the yield stress and B and n represent plastic strain effect. Second term represents for the effect of strain rate on the flow stress. Third term represents the effect of thermal softening on the flow stress. It shall be noted that Johnson-Cook model is an empirical 4

13 model and does not directly establish a relation between the functional form of the constitutive equation and the relevant physical mechanism. 1.4 Motivation As described in the next section, MTS model is a robust material model in accounting for the dependence on the high strain rate (> 10 5 /sec). This particular range of strain rate is frequently encountered in the laser shock peening process, which has been commonly applied to aerospace alloys such as Ti-6Al-4V. Significant benefits have been demonstrated in terms of the amount of residual stresses that are incorporated to negate the effects arising from the fatigue loading condition. There is a continuing interest in extending this technology to the hot section of the aerospace structure and apply to high-temperature alloys such as Ni-based alloys, e.g., IN 718. On the other hand, very little modeling work has been published on developing a validated material model for IN718. Motivated by this gap, the main objective of this thesis is to develop a robust material model for IN718 based on MTS model and validate the model through comparison with experiments. 1.5 Outline of the Thesis The thesis consists of following chapters: In Chapter 2 we describe the mechanical threshold stress (MTS) model. The chapter first gives background of the MTS model and the work that has gone into its development. Further we discuss the theoretical details of the model. Chapter 3 describes the tangent modulus method that has been used to resolve the MTS model for analyzing the LSP process. The chapter mainly outlines derivation of the tangent modulus and related mathematical formulation. We have also included the results of parametric studies that 5

14 have been carried out on the MTS model. Chapter 4 provides basic introduction to the laser shock peening process and mathematical modeling of the pressure pulse. In Chapter 5 we have described the application of MTS model for prediction of residual stresses induced by LSP and compare our predictions with the experimental results. Chapter 6 gives the conclusion of the work and talks about scope for future work. 6

15 Chapter 2 Mechanical threshold stress (MTS) model 2.1 Background The MTS model was proposed by Kocks [7] and Mecking and Kocks [8] where they established a relation to determine the effect of thermal activation on the obstacle strength and subsequently on the flow stress. Mecking and Kocks proposed a phenomenological model which was based on the assumption that the kinetics of plastic flow are determined by the current metallurgical structure of the material [8] [9]. They considered dislocation density as the parameter that represents the dislocation structure. They assumed that the flow stress is strain rate dependent but the rate of evolution of dislocation density is rate independent. Follansbee and Kocks [6] used this model to predict deformation behavior of Copper. They presented a relation to account for the rate dependency of the structure evolution process. Follansbee, Huang and Gray [10] applied the MTS model to nickel-carbon system where they considered strengthening contributions from both the interstitial carbon atoms and the stored dislocations. Chen and Gray [11] adopted the MTS model for flow stress prediction in Tantalum and Tantalum-Tungsten alloys which are is highly temperature dependent and sensitive to strain rate. As MTS model gave better fitting results for large strains, they used the predictions of MTS model to generate parameters for other empirical models used for large strain applications. Goto et al [12] employed the model to predict structural responses of ship and submarine hull steels for United States Navy. They considered the strengthening contributions from interactions of dislocations with vacancies, solute atoms and carbides. Puchi-Cabrera et al [13] applied the MTS model to develop constitutive relationships for aluminum alloys and to analyze the effects of various 7

16 alloying elements on the material parameters. Banerjee and Bhawalkar [14] proposed an extended MTS model for predicting deformation motion at higher strain rates and temperatures. They incorporated a model for viscous drag controlled dislocation motion and a model for the overdriven shock regime. 2.2 Theory The MTS model is based on the principle that the mechanical properties of the material are dependent on its current metallurgical structure. Changes in this structure affect the behavior of material under loading. In this model, the changes in the metallurgical structure are quantified using a state variable called the mechanical threshold stress. Mechanical threshold stress is the flow stress at zero temperature [6]. The model describes the yield stress as a function of strain rate, temperature and a combination of several mechanical threshold stresses. These threshold stresses represent different obstacles which affect the strain hardening that takes place in response to loading. The process of strain hardening is modeled by the evolution of the threshold stresses. The flow stress at zero temperature [8] is given as ˆ σ = ˆ αµ b ρ (2.1) where µis the shear modulus, b is the magnitude of the Burger s vector, ρ is the dislocation density and ˆ α is a constant which depends on the strength of dislocation/dislocation interaction. At finite temperature, the flow stress σ [8] is given as σ = s( ɺ ε, T ) ˆ αµ b ρ (2.2) 8

17 Here εɺ is the strain rate and T is the temperature. The above equation shows the flow stress as a product of a rate sensitive term and a structure sensitive term. The threshold stress can be divided into two components [6] ˆ σ = ˆ σ a+ ˆ σ t (2.3) where ˆ σ a is the rate independent athermal stress and ˆ σ t is the rate dependent component. The flow stress can now be written as σ = ˆ σ a+ σ t= ˆ σ a+ s( ɺ ε, T) ˆ σ t (2.4) where σ t is the contribution of ˆ σ t to the flow stress at finite temperature Thermal activation The factor s used in the above expression is the ratio between the applied stress and the mechanical threshold stress [6]. For thermally activated glide regime its value is less than 1. In thermally activated glide regime the interaction kinetics of obstacles is given by an Arrhenius expression G ɺ ε = ɺ ε 0exp kt (2.5) where εɺ 0 is a constant which represents the limiting value of strain rate and k is the Boltzmann constant [6]. G is the activation free energy for dislocation motion which is given by 3 σ t G= goµ b 1 ˆ σ t p q (2.6) where g O is the normalized activation energy, and p and q are statistically averaged constants that characterize the shape of the obstacle profile [6]. Hence the factor s can be written as 9

18 kt ɺ ε 0 s= 1 ln µ b 3 g ɺ ε O 1 q 1 p (2.7) Complex alloys derive their mechanical properties from various strengthening mechanisms [10]. The contributions of these mechanisms to the flow stress can be taken into account by applying the power law for superposition. For two coexisting strengthening mechanisms σ t can be given as ( 1) ( 2) n n n t σ = σ + σ (2.8) n n 1 n n 1 n n ( 1) ( 2) ( 1(, ) ˆ 1) ( 2(, ) ˆ 2) σ t= σ + σ = s ɺ ε T σ + s ɺ ε T σ (2.9) where σ 1 and σ 2are individual contributions of two strengthening mechanisms. ˆ σ 1 and ˆ σ 2 are the corresponding mechanical threshold stresses. From Eqs. (2.4) and (2.9) we can write the flow stress as 1 n n n ( I(, ) 1) ( ε(, ) 2) σ = ˆ σ a+ s ɺ ε T ˆ σ + s ɺ ε T ˆ σ (2.10) In the case of Ni-C system, the yield stress at a given strain rate and temperature is expressed as a power law summation of the yield stress contribution for the interaction of dislocations with interstitial carbon atoms σ I, and the yield stress contribution for the interaction of dislocations with stored dislocations σ ε [10]. The corresponding mechanical threshold stresses are ˆ σ I and σ ˆε. and n=1 [10]. Now, the flow stress can be expressed as 1 n n n ( I(, ) I) ( ε(, ) ε) σ = ˆ σ a+ s ɺ ε T ˆ σ + s ɺ ε T ˆ σ (2.11) From previous measurements with Nickel alloys, the constants selected are p=2/3, q=1 10

19 From Eqs. (2.7) and (2.11) kt ɺ ε 0 I kt ɺ ε σ = ˆ σ a+ 1 ln ˆ σ 1 ln ˆ σε µ b 3 + g ε 3 ɺ OI µ b g ɺ ε Oε (2.12) Structure evolution The metallurgical structure of the material evolves with strain. The process of structure evolution consists of dislocation accumulation and dynamic recovery. It is considered as the balance between the two processes. The athermal component ˆ σ a and the threshold stress characterizing contribution for the interaction of dislocations with interstitial carbon atoms ˆ σ I do not show a significant variation with strain and are assumed to be constant. The threshold stress characterizing contribution for the interaction of dislocations with stored dislocations σ ˆε increases with increasing dislocation density. Hence the hardening rate serves as the evolution equation for this component of threshold stress with strain. d ˆ σ dε ε = = 0( ε ) 1 ˆ σε ɺ F ˆ σ (2.13) ε s where 0 is the rate of hardening due to dislocation accumulation, ˆ σ ε s is the steady state or saturation threshold stress and F is a function of ratio of σ ˆε and ˆ σ ε s. From previous measurements [15] with Ni270, the variation of saturation threshold stress with strain rate can be expressed using following equation ln ˆ σε s= lnɺ ε (2.14) The rate of hardening due to dislocation accumulation can be expressed as 0= lnɺ ε ɺ ε (2.15) and 11

20 ( 2 ˆε ) σ σε tanh ˆ s F = (2.16) tanh(2) Therefore we can write the evolution equation in the rate form as ( 2 ˆ σε ) ˆ tanh ɺ dσε ˆ ˆε s σε = = 0 1 σ ɺ ε (2.17) dt tanh(2) 2.3 MTS model summary Flow stress at zero temperature is given by ˆ σ = ˆ αµ b ρ (2.18) Mechanical threshold stress and its yield stress contribution are related by σ = s( ɺ ε, T) ˆ σ (2.19) where kt ɺ ε 0 s= 1 ln µ b 3 g ε O ɺ 1 q 1 p (2.20) Flow stress for a Ni-C alloy is given by 1 n n n ( I(, ) ˆ I) ( ε(, ) ˆε) σ = σ a+ s ɺ ε T σ + s ɺ ε T σ (2.21) Evolution of σ ˆε is given by ( 2 ˆ σε ) ˆ tanh ɺ dσε ˆ ˆε s σε = = 0 1 σ ɺ ε (2.22) dt tanh(2) From experiments with Ni270 and ln ˆ σε s= lnɺ ε (2.23) 0= lnɺ ε ɺ ε (2.24) 12

21 Chapter 3 A constitutive solver based on tangent modulus method 3.1 Background In this section, we will outline a constitutive solver based on tangent modulus method[16]. This is a forward gradient method that is used in conjunction with finite element method for analyzing deformation of solids. It is a single step time integration method and does not involve iterations. Based on this method, a stress update algorithm is developed for implementation of the MTS model. A user defined material model code is developed for this algorithm and integrated with the commercial software LS-DYNA to carry out finite element analysis. 3.2 Derivation of tangent modulus for MTS model From Eq.(2.12), the yield surface can be given as I 0 σ σ a kt ɺ ε ˆ σ kt ɺ ε ˆ σε = + 1 ln 1 ln µ µ µ b g ε µ OI ɺ µ b g ε µ O ε ɺ (3.1) where σ is the effective stress, ɺ ε is the effective strain rate and µ 0 is the shear modulus at 0 K. Eq.(3.1) can be equivalently viewed as a yield function given as f kt ɺ ε 0 kt ɺ ε 0 µ = σ σ a 1 ln ˆ σ I 1 ln ˆ σε σ σ ˆ Y σ, ε, 3 3 = 0 µ b g ε OI ɺ µ b g ε µ O ε ɺ ( ε ɺ T) (3.2) where σ Y is the yield stress. The effective stress is defined as 13

22 σ = 3 : 2 σ σ (3.3) in which σ represents the deviatoric component of the stress tensor given by 1 σ = σ ( σ : i) i 3 (3.4) where σ is the stress tensor and i is the second order identity matrix. Assuming the associated flow rule, visco-plastic strain direction is given by f σ = σ σ (3.5) From Eq.(3.3), we have σ = : 2 σ σ 2 3 (3.6) Hence, the direction of visco-plastic strain is σ σ = 2 σ : σ 2σ 2 σ (3.7) From Eq. (3.4) 1 σ = σ (σ : i)i 3 σ = I P σ σ = P σ d s (3.8) (3.9) (3.10) where and I is the fourth order identity matrix. P s and P d are respectively the pressure and deviatoric projections of the stress tensor which are given by following expressions s 1 P = (i i) 3 (3.11) 14

23 d P = I P σ 3 σ = : P d σ 2σ σ 3σ = σ 2σ s (3.12) (3.13) (3.14) d ( σ' : P = σ' ) From Eqs. (3.5) and (3.14) the direction of visco-plastic strain rate is f 3σ = σ 2σ (3.15) Let 3σ p = 2σ (3.16) then the visco-plastic strain rate is vp D ε p = ɺ (3.17) Let P = L : p. The total rate of deformation can be decomposed into three components e vp T D= D + D + D (3.18) where e D represents the elastic component of the rate of deformation, vp D is the visco-plastic component and T D is the thermal component. The elastic deformation rate is given as e -1 D = L : σ (3.19) s d where L represents elastic constitutive matrix and L= 3BP + 2GP with B and G being the bulk and shear modulus, respectively. σ represents certain objective rate of the Cauchy stress. The thermal deformation rate can be given by T D = αt ɺ i (3.20) 15

24 where T ɺ is temperature rate and α is the thermal expansion coefficient ( α = 0.9). From Eqs. (3.17), (3.18), (3.19) and (3.20) -1 D= L : σ + ɺ ε p + αt ɺ i (3.21) Pre-multiplying Eq. (3.21) by the elastic constitutive matrix σ = L : D ɺ ε L : p αt ɺ L : i (3.22) Based on Eq.(3.3) σ = : 2 σ σ 2 3 (3.23) Differentiating Eq. (3.23) with respect to time leads to 2σ ɺ σ = 3 σ : σ (3.24) ɺ 3 σ d : d σ = (σ: P ) 2σ dt (3.25) 3 σ : d d = P : ( σ ) 2σ dt (3.26) ɺ σ 3 σ = : σ 2σ = p : σ (3.27) (3.28) where σɺ is effective stress rate. From Eq.(3.22), ɺ σ = p : L : D ɺ ε ( p : L : p ) αt ɺ ( p : L : i) (3.29) From Eq. (3.16) we have 3σ s d 3σ p : L: p = :( 3BP + 2 GP ) : (3.30) 2σ 2σ 3σ 3σ = ( G) : σ 2σ (3.31) 16

25 s d ( σ' : P = 0 and σ' : P = σ' ) (3 G) 3σ p : L : p = : σ 3G 2 = σ 2 (3.32) In addition 3σ s d p : L : i= :( 3BP + 2 GP ) : i = 0 (3.33) 2σ d ( P : i= 0) From Eqs. (3.29),(3.32) and (3.33) we have ɺ σ = P : D ɺ ε 3G (3.34) To calculate the rate of change of temperature ( T ɺ ) we assume adiabatic heat condition. The plastic dissipation is given by τ : D vp = σε ɺ (3.35) The plastic dissipation can also be written as ρ T = t τ D 0 Cp χ : vp (3.36) where ρ0is density and C p is specific heat capacity, χ is the Taylor-Quinley factor that represents the portion of plastic dissipation that converts into heat. In the current setting, χ = 0.9 From Eqs. (3.35) and (3.36) χσ ɺ ε Tɺ = ρ C 0 p (3.37) Considering linear interpolation within the increment, the strain rate at is given by ɺ ε = (1 ) ɺ ε n + ε ɺ n+ 1 (3.38) Using Taylor series expansion the strain rate for the next step can be given as 17

26 ɺ ε ɺ ε ɺ ε ɺ ε ˆ n+ 1= ɺ ε n+ tn ɺ σ + ɺ σε+ Tɺ (3.39) σ ˆ σε T n n n Solving for εɺ yields ( 2 ˆ σε ) tanh ɺ ε ɺ ε ˆε s χσε n tn 0 1 σ ɺ ε ɺ ɺ ε = ɺ ε + ɺ σ + ɺ ε + (3.40) σ ˆ σε tanh(2) T ρ n n n 0C p ( 2 ˆ σε ) tanh ε ˆε s ε χσ ε ε ɺ 1 tn 0 1 σ ɺ ɺ ɺ + = ɺ ε n+ t ɺ n σ (3.41) ˆ σε tanh(2) T ρ n n 0C p σ n Using Eq.(3.34) for σɺ, we have ( 2 ˆ σε ) tanh ε ˆε s ε χσ ε ε ɺ 1 tn 0 1 σ ɺ ɺ ɺ + = ɺ ε n+ tn : ɺ ε 3G ˆ σε tanh(2) T ρ n n 0C p σ n (3.42) ( P D ) Introducing the plastic modulus H H ( 2 ˆ σε ) ɺ ε tanh ɺ ε ˆ σ ˆ σ T χσ ɺ ε tanh(2) ɺ ε ρ0c σ n σ n ε ε s = 3G 0 1 P (3.43) and εɺ = ξ tn H σ n (3.44) ɺ ε ɺ ɺ ε ε 1 + tn H = ɺ ε n+ tn P : D σ σ n n Eq.(3.42) can be written as εɺ ξ [ 1 + ξ] = εɺ n+ P : D (3.45) H 18

27 εɺ = σ ξ tn H n From Eq.(3.45) the effective visco-plastic strain rate is given by ɺ ɺ ε n ξ ε = + P : D (3.46) ( 1+ ξ ) H ( 1+ ξ ) Substituting this back to the constitutive equation yields vp T σ ɺ = L: ( D -D -D ) (3.47) = L: ( D ɺ ε p αt ɺ i) (3.48) αχσ ɺ ε = L: ( D ɺ εp i) (3.49) ρ 0C p αχσ = L:D ɺ εl: ( p+ i) ρ C p ɺ ε nl ξ L αχσ = L:D ( + ( P : D)) : ( p+ i) ( 1+ ξ) H( 1+ ξ) ρc p ɺ ε nl αχσ ξl αχσ = L:D : ( p+ i) ( P : D) : ( p+ i) ( 1+ ξ) ρc p H( 1+ ξ) ρc p ɺ ε n αχσ = L:D P + (3λ + 2 µ ) i ( 1+ ξ) ρc p ξ ( ) ( ) αχσ P P + (3λ + 2 µ ) i P D H 1+ ξ ρc p (3.50) ( L : i= (3λ + 2 µ ), L : p = P ) Based on the expression above, the tangent modulus matrix can be introduced as ξ αχσ H + C p tan ( ) ( ) 1 (3 2 ) L = L P P + λ + µ i P ξ ρ (3.51) and 19

28 ɺ tan ε n αχσ σ = L :D P + (3λ + 2 µ ) i ( 1+ ξ) ρc p (3.52) 3.3 Application of tangent modulus method to MTS model According to Eq.(3.1) kt ɺ ε 0 kt ɺ ε 0 µ σ = σ a+ 1 ln ˆ σ I 1 ln ˆ σε µ b g ε OI ɺ µ b g ε µ O ε ɺ (3.53) According to relation proposed by Varshini [17] the temperature dependence of shear modulus is given by = 0 µ µ e D0 T 0 T 1 (3.54) where D 0 and T0 are constants. We can write the above equation as µ = 1 µ 0 µ 0 D 0 T 0 ( e T 1) (3.55) Differentiating with respect to temperature ' d µ µ D T e = = dt µ 0 µ 0 2 µ 0T T 0 T 0 0 T 0 ( e T 1) 2 (3.56) and 20

29 ' T 0 d T T 1 DT T 0e = = 1 + T 2 0 dt µ µ µ µ T( e T 1 ) (3.57) To calculate the partial derivatives to be used in the Taylor series in Eq. (3.39), we first need to calculate the total derivative of σ in Eq. (3.53). Differentiating Eq. (3.53) leads to dσ = σ + σ µ b g µ b g kt ɺ ε 0 kt ɺ ε 0 µ 1 ln ˆ I 1 ln ˆ 3 3 ε 0 OI ɺ ε O ɺ ε ε µ ' dt 1 2 ' 3 kt 0 kt T 1 T ɺ ε ln d ln dt 2 b 3 + g 3 ɺ ɺ ε ε ˆ σ I OI ɺ ε b goi µ ɺ ε µ ɺ µ ε kt ɺ ε ln b 3 go ɺ µ ε ε 3 2 d ˆ σε 1 2 ' 3 kt ɺ ε 0 kt T 1 T ln d ln dt ˆ ɺ ɺ ε µ ε + σε 0 b g O ɺ b g O ɺ ɺ µ ε ε ε µ µ ε µ ε µ (3.58) We can write the above equation in the form ɺ ˆ (3.59) K 2dσ = K1dε + K 3dσε+ K 4dT in which we have K 2= 1 K ˆ σ IkT kt ɺ ε 0 3 ˆ σ kt ε kt ε 0 µ = 1 ln dɺ ɺ ε 1 ln (3.60) 0 2µ b g ɺ OIε µ b g ε OI ɺ 2µ b goεε µ b g ε µ O ɺ ε ɺ 3kT ˆ σ I kt ɺ ε ˆ σ kt ɺ ε = 1 ln 1 ln 2 3 goi 3 + goε µ 3 0b ɺ ε µ b g ε OI ɺ µ b g ε O ε ɺ ε 0 (3.61) 21

30 K ɺ ε 0 µ kt = 1 ln 3 0 µ b g ε µ Oε ɺ (3.62) K 4 = + µ b g µ b g kt ɺ ε 0 kt ɺ ε 0 µ 1 ln ˆ σ I 1 ln ˆ σε 3 ε 3 ε µ 0 OI ɺ Oε ɺ 1 ' 2 3 ˆ σ Ik T ε 0 kt ε ɺ ɺ0 ln 1 ln 2b 3 g µ ε 3 OI ɺ µ b g ε OI ɺ µ + 1 ' ˆ σεk T ɺ ε 0 µ kt ε ɺ 0 ln 1 2b 3 ln g µ ε Oε ɺ µ b 3 g ε Oε ɺ ' = + µ b g µ b g kt ɺ ε 0 kt ɺ ε 0 µ 1 ln ˆ I 1 ln ˆ 3 σ 3 σε 0 OI ɺ ε O ɺ ε ε µ ' 3k T ɺ ε ˆ σ kt ɺ ε ˆ σ kt ɺ ε ln 1 ln 1 ln goi 3 go b b g 3 ɺ µ ε OI ɺ ε b go ɺ µ ε µ ε ε I 0 ε 0 ' (3.63) Rearranging the terms in Eq. (3.59) ɺ ˆ (3.64) K1dε = K 2dσ K 3dσε K 4dT K 2 K 3 K 4 d ɺ ε = dσ dσε ˆ dt K K K (3.65) Thus ɺ ε K 2 2µ 0b 3ɺ ε = = 1 1 σ K ˆ σ I kt ɺ ε 0 ˆ σ ε kt ɺ ε 0 3kT 1 ln 1 ln g OI 3 + O b g ε 3 OI g ε µ ɺ µ b g ε O ε ɺ (3.66) 22

31 ɺ 3 kt ε 0 2µ b ɺ ɺ ε 1 ln µ b 3 g ɺ ε ε K 3 Oε = = 1 1 ˆ σε K I 0 ε 0 ˆ σ kt ɺ ε ˆ σ kt ɺ ε 3kT 1 ln 1 ln g OI 3 + O b g ε 3 OI g ε µ ɺ µ b g ε O ε ɺ 3 2 (3.67) ' kt ɺ ε 0 kt 0 1 ln ˆ ɺ ε µ I 1 ln ˆε 3 σ + σ 3 0 b g ε OI b g ε µ Oε ε K µ ɺ µ ɺ ɺ 4 = = 1 1 T K I 0 ε 0 3kT ˆ σ kt ɺ ε ˆ σ kt ɺ ε 1 ln 1 ln 2 3 goi 3 + goε µ 3 0b ɺ ε µ b g ε OI ɺ µ b g ε O ε ɺ 1 1 ' 2 2 3k T ɺ ε 0 ˆ σ I kt ɺ ε 0 ˆ σ ε kt ɺ ε 0 ln 1 ln 1 ln 2 3 goi 3 go b µ + ε b g 3 ɺ ε µ ε OI ɺ µ b g ε Oε ɺ kT ˆ σ I kt ε 2 ɺ0 ˆ σ ε kt ɺ ε 0 1 ln 1 ln 2 3 goi 3 + goε µ 3 0b ɺ ε µ b g ε OI ɺ µ b g ε O ε ɺ ' ' kt ɺ ε 0 kt ɺ ε 0 µ ɺ εµ 0 T ɺ ε 0 = 1 ln ˆ I+ 1 ln ˆ ln 3 σ 3 σε 0 b g T OI ɺ b go ɺ ɺ µ ε µ ε µ µ ε ε (3.68) The partial derivatives derived from Eq.(3.66) to (3.68) will be substituted in the expressions involving the evaluation of the tangent modulus, which are part of the procedure to find the stress rate. 3.4 Algorithm for implementation of tangent modulus method with MTS model A stress update algorithm was developed based on the tangent modulus method described above. Initially a stand-alone code was developed using MATLAB. This code was developed to 23

32 understand the behavior of the model under various rates of loading and temperatures. Then a UMAT subroutine was written in FORTRAN and was compiled into a solver which was integrated with LS-DYNA for finite element analysis. The algorithm follows following steps 1. Input strain matrix and calculate strain rates. 2. Effective strain rate, temperature and threshold stress from previous step. 3. Calculate elastic constitutive matrix. 4. Calculate effective test stress using current strain increment and elastic constitutive matrix. 5. Check if the effective test stress is within the yield surface. 6. If condition is elastic, update the stress values using elastic constitutive matrix. 7. If the condition is plastic, calculate approximate effective plastic strain rate using effective test stress and threshold stress from previous step. 8. Update the threshold stress using approximate effective plastic strain rate. 9. Calculate the effective plastic strain rate using updated threshold stress. 10. Calculate partial differentials. 11. Calculate tangent modulus. 12. Update the stress values using tangent modulus. To study the behavior of the MTS model under high rates of loading, we first generated stress-strain curves using the stand-alone MATLAB code. Figure 1 shows the stress-strain curves for various strain rates. We can see the rise in initial yield strength. The increasing stress beyond yield limit shows the strain hardening effect. Figure 2 shows temperature variation with strain at different strain rates. The temperature rise for the imposed strain rate is observed to be moderate. 24

33 1800 Stress Vs Strain Stress in MPa e5/s 5e4/s 1e4/s 5e3/s Strain Figure 1: Stress Vs. Strain at different strain rates. 310 Temperature Vs Strain Temperature in K e5/s 5e4/s 1e4/s 5e3/s Strain Figure 2: Temperature Vs. Strain at different strain rates. 25

34 Chapter 4 Laser shock peening process 4.1 Introduction Peening is a form of surface treatment in which residual stresses are imparted on material surface in order to increase its fatigue life. It falls into the category of cold working processes. The conventional process of peening, popularly known as Shot peening involves bombarding the metal surface being processed with tiny metal balls. A shot when bombarded on the surface forms an indentation or dimple making the material yield under tension. This creates a zone of residual compressive stresses bellow the dimple as the material tries to regain its original shape. Laser shock peening (LSP) is a recently developed method which provides an alternative to conventional shot peening. This process was developed at the Battelle Columbus laboratory [18]. As the name suggests, the process involves use of laser energy. The shots used in conventional peening are replaced by a high energy laser pulse. As the laser beam strikes the surface of the material, a very thin layer of the material is heated which causes it to vaporize. The vapor continues to receive energy and its temperature rises to tens of thousands of degrees. These extremely high temperatures result into ionization and hence transformation of the vapor into plasma.[19] High pressure shockwaves are generated as the heated plasma undergoes rapid expansion. As the shock waves pressure exceeds the Hugoniot elastic limit (HEL), the material undergoes plastic deformation. HEL is a function of dynamic yield strength of the material. For efficient absorption of laser energy incident on the metal surface, it is necessary to minimize the energy losses caused through reflection and absorption in the surrounding environment. This can be achieved by coating the metal surface with an absorbent material and a 26

35 transparent overlay. The absorbent coating also protects the material from melting and vaporization at high temperatures. The transparent overlay confines the plasma near the metals surface. This restricts the plasma from expanding away from the surface and laser energy is absorbed more efficiently into the metal surface. The shock wave intensity is found to increase by up to two orders of magnitude with the transparent overlay as compared to peening carried out in vacuum.[20] The expansion of plasma on vacuum is known as direct ablation and the one with the use of transparent overlay is known as confined ablation [21]. Higher shock wave pressures can be achieved with confined ablation as compared to direct ablation mode. Absorbent coating Target metal Transparent overlay Shock wave Laser beam Induced plasma Figure 3: Laser shock peening process 4.2 Mathematical modeling of shock wave pressure As discussed above, a shock wave is generated by the laser beam on striking the metal surface which travels through the material causing a plastic deformation. In order to analyze the mechanics of this process, it is necessary to compute the pressure load induced by the shock 27

36 wave. Various mathematical models have been developed for prediction of the shock wave pressure profile. Fabbro et al [21] proposed a monodimensional analytical model to simulate the pressure pulse. This model assumes the laser irradiation to be uniform. It only considers the temporal behavior of the shock wave. Spatial variation is not considered in this model. Berthe et al [22] studied the inhomogeneities in pressure over a laser spot in their experimental measurements and produced spatial and temporal shock wave profiles. Zhang and Yao [23] modified Fabbro s 1D model for micro scale laser shock processing where spatial non uniformity of shock wave pressure needs to be considered. Francisco [24] proposed a model based on optical emission studies performed on laser ablation of Titanium. This model considers both spatial and temporal behavior of the shock wave. In this work, we have used the 1D model proposed by Fabbro et al. It is assumed that a fraction of the total absorbed laser energy is used to increase the internal energy of the plasma and remaining energy is converted to work. The model is applicable to confined ablation mode of expansion. In this mode, the laser energy deposition takes place at the interface of the materials, i.e., the target metal and the transparent material. The pressure generated as a result of this energy deposition generates shock waves in both the mediums. The thickness of the interface wall created as the interface opens is governed by fluid velocities in both the directions. At a given time t, the interface wall thickness L( t ) can be expressed as t [ 1 2 ] (4.1) 0 L( t) = u ( t) + u ( t) dt 28

37 where u 1 is the fluid velocity behind shock wave in target metal and u 2 is the fluid velocity behind shock wave in transparent material. The fluid velocities can be derived from the shock wave pressure using following relation P= ρ D u = Z u i i i i i (4.2) where ρ is the density of the material and D is the shock velocity. Index i represents the material under consideration. The term Z is known as the shock wave impedance of the material. From Eqs. (4.1) and (4.2) dl( t) 2 = P ( t ) dt Z (4.3) where Z is the effective impedance of coating and is given by = + Z Z Z 1 2 (4.4) where Z 1 and Z 2 are individual impedances of the two materials. As mentioned above, one portion of the deposited laser energy is used to increase the internal energy and remaining energy is converted into work. Hence we can write dl( t) d[ Ei ( t) L] I( t) = P( t) dt + dt (4.5) where I( t ) is the total deposited laser energy per unit surface. d[ E ( t) L ] is the increment in internal energy and P( t) dl is the work of pressure forces. We consider that a fraction α of the internal energy ( E ) is converted into thermal energy ( E ). Pressure P( t ) and thermal energy ( E ) for an ideal gas are related by T T i i 2 P( t) = E ( t) 3 T (4.6) Hence we can write the relation between pressure P( t ) and internal energy E i as 29

38 2 P( t) = αei ( t) (4.7) 3 From Eqs. (4.5) and (4.7) we can write dl( t) 3 d I( t) = P( t) [ P( t) L( t)] dt + 2 α dt (4.8) From Eqs. (4.3) and (4.8) we obtain a second order differential equation of the following form Z 3 Z dl( t) I( t) ( ) + 2 4α dt = 2 dt 3Z L( t) 4α 2 d L t This equation is solved using MATLAB ODE solver to obtain 2 dl( t) dt (4.9). The pressure history P( t ) is then calculated using Eq. (4.3). The pressure history generated from this model depends on the laser intensity input. In some cases, the laser intensity can be fitted to a Gaussian profile, given by I = ae 2 ( t b) 2 2c (4.10) where a represents the peak density, b gives the half time for incident pulse, c is given by FWHM/2.35 and t is the time duration of the pulse. 30

39 Chapter 5 Application to residual stress prediction 5.1 Residual stress prediction methodology FE Model Pressure model Material model Geometrical modeling component Laser parameters - Intensity, pulse width, spot size MTS model - Strain hardening, strain rate effects Meshing FE model Mathematical modeling of shock wave pressure Integration of MTS model with tangent modulus algorithm Temporal pressure distribution Constitutive solver Residual stress prediction by FEM Analysis The methodology used for prediction of residual stresses is shown in the flow chart. The modeling process can be divided into three categories. FE modeling, pressure modeling and material modeling. FE modeling involves creating a geometric model, meshing and setting up boundary conditions. In pressure modeling shock wave pressure history is calculated as shown in Section 4.2. This temporal distribution of pressure when integrated with the FE model provides the nature of load acting on the component. Material modeling involves development of a stress 31

40 update algorithm based on the material model selected. A constitutive solver is developed which is used in conjunction with the FE model for prediction of residual stresses. 5.2 Finite element analysis The first step in FEM analysis is FE modeling. This was carried out using commercial software HYPERMESH. The file is then exported as a LS-DYNA keyword file. The keyword is then further edited in LS-PREPOST. Termination time and time step control parameters are added. Time step is controlled by the Courant number and the loading rate. It was observed that the algorithm required a very small time step when the applied strain rate was high. As the analysis was applied to modeling of high strain rate deformation, we had to select the courant number in such a way that time step was very small. Here we have used a time step of 2x10-11 s. Nature of loading is specified and a temporal profile distribution of pressure load is defined as per the pressure model. A free-free boundary condition was used. The material parameters are also defined at this stage. LSTC provides a FORTRAN file which can be compiled into a solver executable with LS-DYNA. The above mentioned stress update algorithm is incorporated as a user defined material (UMAT 41) model in the FORTRAN file. The solver is executed with the keyword as input file. 5.3 Problem configuration A square plate of size 14 mm x 14 mm x 2 mm was used as the target. Pressure load was applied at the center of the plate over an area of 2mm diameter. The pressure was assumed to be uniform over the area. The analysis was carried out for a three shot LSP process, i.e., the pressure load was applied three times at the same spot. Figure 3 shows the model used for FEM 32

41 analysis. Laser intensity of 8 GW/cm 2 and pulse width of 34 ns were selected to calculate the temporal profile of pressure shown in Figure 4. Figure 4: Geometry of the model used for analysis 33

42 4500 Temporal pressure profile Pressure in MPa Time in Seconds x 10-7 Figure 5: Temporal profile of shock wave pressure 5.4 Results and discussion Results were obtained from the FEM analysis in the form of element stresses. To study the residual stress distribution along the depth, the element stresses were averaged over an area of 1mm diameter for each of the element layers in the mesh. The results are then compared with experimental results. The experiments were carried out on 14mm x 14mm x 2mm square plate of INCONEL 718. The plate was peened with three laser shots of 8J energy. INCONEL 718 popularly known as IN718 is a nickel based superalloy. INCONEL exhibits very high yield strength and creep-rupture strengths. INCONEL finds applications where the components are subjected to cyclic loads and high temperatures. It is widely used in spacecrafts, gas turbines, nuclear reactors and high temperature fasteners. In case if Ni based 34

43 superalloys, along with dislocation motion precipitation hardening also contributes towards the strengthening of the material [25]. The expression of MTS model used here has been previously used for modeling behavior of Nickel-Carbon alloys [15]. As mentioned above the term ˆ σ I represents the contribution for the interaction of dislocations with interstitial carbon atoms in Ni- C alloys. It is assumed here that ˆ σ I accounts for both the strengthening contribution of precipitation hardening and contribution for the interaction of dislocations with interstitial carbon atoms. A parametric study was carried out to find out effects of various parameters in the model on the predicted residual stresses. From Eq.(2.12), the yield stress is expressed as kt ɺ ε 0 I kt ɺ ε σ = σ a+ 1 ln ˆ σ 1 ln ˆ σε µ b 3 + g ε 3 OI ɺ µ b g ε O ε ɺ (5.1) From Eq.(2.14) and Eq.(2.15) the structure evolution expressions can be expressed in the form 0= A+ B lnɺ ε+ C ɺ ε (5.2) and ln ˆ σε s= D+ E lnɺ ε (5.3) Residual stresses were calculated for different values of parameters in equations (5.1), (5.2) and (5.3). The results were compared with experimental results. The parametric study was carried out on the square plate model shown in Figure 4 subjected to 8GW/cm 2 laser pulse with pressure profile shown in Figure 5. 35

44 5.4.1 MTS model - parametric study - athermal stress ( σ a) variation This study focuses on the effect of variation in the athermal component of the threshold stress σ a on the predicted residual stress. Remaining parameters in the model are kept constant. As mentioned above The athermal contribution does not change with strain. It is estimated to be the yield stress in undeformed material. 100 Residual stress Vs Depth for different values of Athermal stress Residual stress in MPa Sigma =800MPa a Sigma =1000MPa a Sigma =1200MPa a Experimental Depth in mm Figure 6: Residual stress Vs. Depth for different values of athermal stress ( σ a) Figure 6 shows the predicted residual stresses for different values of athermal stress. It can be noticed that for different values of athermal stress, the distribution of residual stresses along the depth follows a similar pattern. But their magnitude shows significant variation. With increasing athermal stress contribution, the residual stress decreases. Comparing with the 36

45 experimental results, the most agreeable results are obtained at 1200MPa. This value of σ a is close to the yield strength of INCONEL 718 which is 1241MPa [26] MTS model - parametric study - variation of mechanical threshold stress characterizing dislocation interactions with interstitial atoms ( σ i) In this study we have analyzed the effect of variation in mechanical threshold stress contribution due to interaction of dislocations with interstitial atoms ( σ i). This factor does not evolve with strain. But it can be noticed from Eq. (5.1) that the contribution of σ i to the yield stress is a function of strain rate and temperature. 100 Residual stress Vs Depth for different values of Sigma i 0 Residual stress in MPa Sigma =600MPa i Sigma =700MPa i Sigma =800MPa i Sigma =900MPa i Experimental Depth in mm Figure 7: Residual stress Vs. Depth for different values of ( σ i) 37

46 Figure 7 shows the predicted residual stresses for different values of σ i. It can be observed that at lower values of σ i higher residual stresses are obtained. This effect of σ i is more significant near the surface as compared to deeper sections of the material. Comparing with the experimental data σ i of around 600MPa gives the most agreeable results MTS model - parametric study - variation of structure evolution parameters The purpose of this study is to analyze the effects varying structure evolution parameters from Eq.s (5.2) and (5.3) on residual stresses. We have focused only on the parameters having significant contribution in structure evolution. 100 Residual stress Vs Depth for different values of parameter A 0 Residual stress in MPa A=1000MPa A=5000MPa A=10000MPa Experimental Depth in mm Figure 8: Residual stress Vs. Depth for different values of A 38

47 100 Residual stress Vs Depth for different values of parameter C 0 Residual stress in MPa C=5MPa C=7MPa C=9MPa C=11MPa Experimental Depth in mm Figure 9: Residual stress Vs. Depth for different values of C Figure 8 shows predicted residual stress for different values of parameter A keeping all other parameters constant. Figure 9 shows results of a similar study conducted for parameter C. Figure 10 shows results of parametric study conducted for parameter D. It can be seen that both parameters do not show a significant effect on the residual stress values. A small variation can be seen near the surface where higher residual stresses are obtained for lower values of structure evolution parameters. 39

48 100 Residual stress Vs Depth for different values of parameter D 0 Residual stress in MPa D=7 D=8 D=9 D=10 Experimental Depth in mm Figure 10: Residual stress Vs. Depth for different values of D Optimized parameters of MTS model for INCONEL 718 From the parametric study it can be observed that the athermal stress ( σ a) and mechanical threshold stress characterizing dislocation interactions with interstitial atoms ( σ i) have a significant influence on the residual stress predictions. Comparing with the experimental data, the closest agreement was obtained for σ a = 1241 MPa and σ i = 700 MPa. Variation in the structure evolution parameters did not show any significant effect on the residual stress predictions. Hence we have used the same expression as proposed by Follansbee et al [15] for Ni